Integrand size = 25, antiderivative size = 529 \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^3} \, dx=\frac {b^{5/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e \cot (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{4 a^{7/2} \left (a^2+b^2\right )^3 d e^{3/2}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d e^{3/2}}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d e^{3/2}}+\frac {8 a^4+31 a^2 b^2+15 b^4}{4 a^3 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)}}-\frac {b^2}{2 a \left (a^2+b^2\right ) d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}-\frac {b^2 \left (13 a^2+5 b^2\right )}{4 a^2 \left (a^2+b^2\right )^2 d e \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d e^{3/2}}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d e^{3/2}} \]
1/4*b^(5/2)*(63*a^4+46*a^2*b^2+15*b^4)*arctan(b^(1/2)*(e*cot(d*x+c))^(1/2) /a^(1/2)/e^(1/2))/a^(7/2)/(a^2+b^2)^3/d/e^(3/2)-1/2*(a-b)*(a^2+4*a*b+b^2)* arctan(1-2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/(a^2+b^2)^3/d/e^(3/2)*2^(1/ 2)+1/2*(a-b)*(a^2+4*a*b+b^2)*arctan(1+2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2) )/(a^2+b^2)^3/d/e^(3/2)*2^(1/2)+1/4*(a+b)*(a^2-4*a*b+b^2)*ln(e^(1/2)+cot(d *x+c)*e^(1/2)-2^(1/2)*(e*cot(d*x+c))^(1/2))/(a^2+b^2)^3/d/e^(3/2)*2^(1/2)- 1/4*(a+b)*(a^2-4*a*b+b^2)*ln(e^(1/2)+cot(d*x+c)*e^(1/2)+2^(1/2)*(e*cot(d*x +c))^(1/2))/(a^2+b^2)^3/d/e^(3/2)*2^(1/2)+1/4*(8*a^4+31*a^2*b^2+15*b^4)/a^ 3/(a^2+b^2)^2/d/e/(e*cot(d*x+c))^(1/2)-1/2*b^2/a/(a^2+b^2)/d/e/(a+b*cot(d* x+c))^2/(e*cot(d*x+c))^(1/2)-1/4*b^2*(13*a^2+5*b^2)/a^2/(a^2+b^2)^2/d/e/(a +b*cot(d*x+c))/(e*cot(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.75 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.57 \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^3} \, dx=-\frac {-8 a^2 b^2 \left (3 a^2-b^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\frac {b \cot (c+d x)}{a}\right )-16 a^2 b^2 \left (a^2+b^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},2,\frac {1}{2},-\frac {b \cot (c+d x)}{a}\right )-8 b^2 \left (a^2+b^2\right )^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},3,\frac {1}{2},-\frac {b \cot (c+d x)}{a}\right )-8 a^4 \left (a^2-3 b^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\cot ^2(c+d x)\right )+\sqrt {2} a^3 b \left (3 a^2-b^2\right ) \sqrt {\cot (c+d x)} \left (2 \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+\log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{4 a^3 \left (a^2+b^2\right )^3 d e \sqrt {e \cot (c+d x)}} \]
-1/4*(-8*a^2*b^2*(3*a^2 - b^2)*Hypergeometric2F1[-1/2, 1, 1/2, -((b*Cot[c + d*x])/a)] - 16*a^2*b^2*(a^2 + b^2)*Hypergeometric2F1[-1/2, 2, 1/2, -((b* Cot[c + d*x])/a)] - 8*b^2*(a^2 + b^2)^2*Hypergeometric2F1[-1/2, 3, 1/2, -( (b*Cot[c + d*x])/a)] - 8*a^4*(a^2 - 3*b^2)*Hypergeometric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2] + Sqrt[2]*a^3*b*(3*a^2 - b^2)*Sqrt[Cot[c + d*x]]*(2*ArcT an[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x ]]] + Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[1 + Sqrt[2] *Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(a^3*(a^2 + b^2)^3*d*e*Sqrt[e*Cot[c + d*x]])
Time = 2.81 (sec) , antiderivative size = 499, normalized size of antiderivative = 0.94, number of steps used = 28, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.080, Rules used = {3042, 4052, 27, 3042, 4132, 27, 3042, 4132, 27, 3042, 4136, 27, 3042, 4017, 25, 27, 1482, 1476, 1082, 217, 1479, 25, 27, 1103, 4117, 73, 218}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
\(\Big \downarrow \) 4052 |
\(\displaystyle -\frac {\int -\frac {5 b^2 e \cot ^2(c+d x)-4 a b e \cot (c+d x)+\left (4 a^2+5 b^2\right ) e}{2 (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^2}dx}{2 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {5 b^2 e \cot ^2(c+d x)-4 a b e \cot (c+d x)+\left (4 a^2+5 b^2\right ) e}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^2}dx}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {5 b^2 e \tan \left (c+d x+\frac {\pi }{2}\right )^2+4 a b e \tan \left (c+d x+\frac {\pi }{2}\right )+\left (4 a^2+5 b^2\right ) e}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 4132 |
\(\displaystyle \frac {-\frac {\int -\frac {-16 b e^2 \cot (c+d x) a^3+\left (8 a^4+31 b^2 a^2+15 b^4\right ) e^2+3 b^2 \left (13 a^2+5 b^2\right ) e^2 \cot ^2(c+d x)}{2 (e \cot (c+d x))^{3/2} (a+b \cot (c+d x))}dx}{a e \left (a^2+b^2\right )}-\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\int \frac {-16 b e^2 \cot (c+d x) a^3+\left (8 a^4+31 b^2 a^2+15 b^4\right ) e^2+3 b^2 \left (13 a^2+5 b^2\right ) e^2 \cot ^2(c+d x)}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))}dx}{2 a e \left (a^2+b^2\right )}-\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\int \frac {16 b e^2 \tan \left (c+d x+\frac {\pi }{2}\right ) a^3+\left (8 a^4+31 b^2 a^2+15 b^4\right ) e^2+3 b^2 \left (13 a^2+5 b^2\right ) e^2 \tan \left (c+d x+\frac {\pi }{2}\right )^2}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 a e \left (a^2+b^2\right )}-\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 4132 |
\(\displaystyle \frac {\frac {\frac {2 \int -\frac {b \left (8 a^4+31 b^2 a^2+15 b^4\right ) \cot ^2(c+d x) e^4+b \left (24 a^4+31 b^2 a^2+15 b^4\right ) e^4+8 a^3 \left (a^2-b^2\right ) \cot (c+d x) e^4}{2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{a e^3}+\frac {2 e \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {e \cot (c+d x)}}}{2 a e \left (a^2+b^2\right )}-\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\frac {2 e \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {e \cot (c+d x)}}-\frac {\int \frac {b \left (8 a^4+31 b^2 a^2+15 b^4\right ) \cot ^2(c+d x) e^4+b \left (24 a^4+31 b^2 a^2+15 b^4\right ) e^4+8 a^3 \left (a^2-b^2\right ) \cot (c+d x) e^4}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{a e^3}}{2 a e \left (a^2+b^2\right )}-\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {2 e \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {e \cot (c+d x)}}-\frac {\int \frac {b \left (8 a^4+31 b^2 a^2+15 b^4\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2 e^4+b \left (24 a^4+31 b^2 a^2+15 b^4\right ) e^4-8 a^3 \left (a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) e^4}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a e^3}}{2 a e \left (a^2+b^2\right )}-\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 4136 |
\(\displaystyle \frac {\frac {\frac {2 e \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^4 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{a^2+b^2}+\frac {\int \frac {8 \left (a^3 b \left (3 a^2-b^2\right ) e^4+a^4 \left (a^2-3 b^2\right ) \cot (c+d x) e^4\right )}{\sqrt {e \cot (c+d x)}}dx}{a^2+b^2}}{a e^3}}{2 a e \left (a^2+b^2\right )}-\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\frac {2 e \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^4 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}dx}{a^2+b^2}+\frac {8 \int \frac {a^3 b \left (3 a^2-b^2\right ) e^4+a^4 \left (a^2-3 b^2\right ) \cot (c+d x) e^4}{\sqrt {e \cot (c+d x)}}dx}{a^2+b^2}}{a e^3}}{2 a e \left (a^2+b^2\right )}-\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {2 e \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^4 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}+\frac {8 \int \frac {a^3 b \left (3 a^2-b^2\right ) e^4-a^4 \left (a^2-3 b^2\right ) e^4 \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}}{a e^3}}{2 a e \left (a^2+b^2\right )}-\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 4017 |
\(\displaystyle \frac {\frac {\frac {2 e \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^4 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}+\frac {16 \int -\frac {a^3 e^4 \left (b \left (3 a^2-b^2\right ) e+a \left (a^2-3 b^2\right ) \cot (c+d x) e\right )}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}}{a e^3}}{2 a e \left (a^2+b^2\right )}-\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {\frac {2 e \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^4 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 \int \frac {a^3 e^4 \left (b \left (3 a^2-b^2\right ) e+a \left (a^2-3 b^2\right ) \cot (c+d x) e\right )}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}}{a e^3}}{2 a e \left (a^2+b^2\right )}-\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\frac {2 e \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^4 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 a^3 e^4 \int \frac {b \left (3 a^2-b^2\right ) e+a \left (a^2-3 b^2\right ) \cot (c+d x) e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}}{a e^3}}{2 a e \left (a^2+b^2\right )}-\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 1482 |
\(\displaystyle \frac {\frac {\frac {2 e \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^4 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 a^3 e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {\cot (c+d x) e+e}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a e^3}}{2 a e \left (a^2+b^2\right )}-\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {\frac {\frac {2 e \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^4 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 a^3 e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a e^3}}{2 a e \left (a^2+b^2\right )}-\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {\frac {\frac {2 e \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^4 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 a^3 e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e \cot (c+d x)-1}d\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a e^3}}{2 a e \left (a^2+b^2\right )}-\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\frac {\frac {2 e \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^4 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 a^3 e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {e-e \cot (c+d x)}{\cot ^2(c+d x) e^2+e^2}d\sqrt {e \cot (c+d x)}\right )}{d \left (a^2+b^2\right )}}{a e^3}}{2 a e \left (a^2+b^2\right )}-\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {\frac {\frac {2 e \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^4 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 a^3 e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{a e^3}}{2 a e \left (a^2+b^2\right )}-\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {\frac {2 e \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^4 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 a^3 e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{a e^3}}{2 a e \left (a^2+b^2\right )}-\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\frac {2 e \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^4 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 a^3 e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e-\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}}{\cot (c+d x) e+e+\sqrt {2} \sqrt {e \cot (c+d x)} \sqrt {e}}d\sqrt {e \cot (c+d x)}}{2 \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{a e^3}}{2 a e \left (a^2+b^2\right )}-\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {\frac {\frac {2 e \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^4 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-b \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 a^3 e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{a e^3}}{2 a e \left (a^2+b^2\right )}-\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 4117 |
\(\displaystyle \frac {\frac {\frac {2 e \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {e \cot (c+d x)}}-\frac {\frac {b^3 e^4 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {1}{\sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}d(-\cot (c+d x))}{d \left (a^2+b^2\right )}-\frac {16 a^3 e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{a e^3}}{2 a e \left (a^2+b^2\right )}-\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {\frac {2 e \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {e \cot (c+d x)}}-\frac {-\frac {2 b^3 e^3 \left (63 a^4+46 a^2 b^2+15 b^4\right ) \int \frac {1}{\frac {b \cot ^2(c+d x)}{e}+a}d\sqrt {e \cot (c+d x)}}{d \left (a^2+b^2\right )}-\frac {16 a^3 e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{a e^3}}{2 a e \left (a^2+b^2\right )}-\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\frac {\frac {2 e \left (8 a^4+31 a^2 b^2+15 b^4\right )}{a d \sqrt {e \cot (c+d x)}}-\frac {\frac {2 b^{5/2} e^{7/2} \left (63 a^4+46 a^2 b^2+15 b^4\right ) \arctan \left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a} \sqrt {e}}\right )}{\sqrt {a} d \left (a^2+b^2\right )}-\frac {16 a^3 e^4 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (e \cot (c+d x)+\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (e \cot (c+d x)-\sqrt {2} \sqrt {e} \sqrt {e \cot (c+d x)}+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d \left (a^2+b^2\right )}}{a e^3}}{2 a e \left (a^2+b^2\right )}-\frac {b^2 \left (13 a^2+5 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}}{4 a e \left (a^2+b^2\right )}-\frac {b^2}{2 a d e \left (a^2+b^2\right ) \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))^2}\) |
-1/2*b^2/(a*(a^2 + b^2)*d*e*Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x])^2) + (-((b^2*(13*a^2 + 5*b^2))/(a*(a^2 + b^2)*d*Sqrt[e*Cot[c + d*x]]*(a + b*Co t[c + d*x]))) + ((2*(8*a^4 + 31*a^2*b^2 + 15*b^4)*e)/(a*d*Sqrt[e*Cot[c + d *x]]) - ((2*b^(5/2)*(63*a^4 + 46*a^2*b^2 + 15*b^4)*e^(7/2)*ArcTan[(Sqrt[b] *Cot[c + d*x])/(Sqrt[a]*Sqrt[e])])/(Sqrt[a]*(a^2 + b^2)*d) - (16*a^3*e^4*( ((a - b)*(a^2 + 4*a*b + b^2)*(-(ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/ Sqrt[e]]/(Sqrt[2]*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sq rt[e]]/(Sqrt[2]*Sqrt[e])))/2 - ((a + b)*(a^2 - 4*a*b + b^2)*(-1/2*Log[e + e*Cot[c + d*x] - Sqrt[2]*Sqrt[e]*Sqrt[e*Cot[c + d*x]]]/(Sqrt[2]*Sqrt[e]) + Log[e + e*Cot[c + d*x] + Sqrt[2]*Sqrt[e]*Sqrt[e*Cot[c + d*x]]]/(2*Sqrt[2] *Sqrt[e])))/2))/((a^2 + b^2)*d))/(a*e^3))/(2*a*(a^2 + b^2)*e))/(4*a*(a^2 + b^2)*e)
3.1.87.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2/f Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & & NeQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + Simp[1 /((m + 1)*(a^2 + b^2)*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x], x], x] / ; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || Integ erQ[m]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/f Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ [b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^ n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Simp[ (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^n*((1 + Tan[ e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] & & !GtQ[n, 0] && !LeQ[n, -1]
Time = 0.05 (sec) , antiderivative size = 480, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(-\frac {2 e^{4} \left (-\frac {b^{3} \left (\frac {\left (\frac {15}{8} a^{4} b +\frac {11}{4} a^{2} b^{3}+\frac {7}{8} b^{5}\right ) \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}+\frac {a e \left (17 a^{4}+26 a^{2} b^{2}+9 b^{4}\right ) \sqrt {e \cot \left (d x +c \right )}}{8}}{\left (e \cot \left (d x +c \right ) b +a e \right )^{2}}+\frac {\left (63 a^{4}+46 a^{2} b^{2}+15 b^{4}\right ) \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}\, b}{\sqrt {a e b}}\right )}{8 \sqrt {a e b}}\right )}{a^{3} e^{5} \left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (-3 a^{2} b e +b^{3} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{\left (a^{2}+b^{2}\right )^{3} e^{5}}-\frac {1}{a^{3} e^{5} \sqrt {e \cot \left (d x +c \right )}}\right )}{d}\) | \(480\) |
default | \(-\frac {2 e^{4} \left (-\frac {b^{3} \left (\frac {\left (\frac {15}{8} a^{4} b +\frac {11}{4} a^{2} b^{3}+\frac {7}{8} b^{5}\right ) \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}+\frac {a e \left (17 a^{4}+26 a^{2} b^{2}+9 b^{4}\right ) \sqrt {e \cot \left (d x +c \right )}}{8}}{\left (e \cot \left (d x +c \right ) b +a e \right )^{2}}+\frac {\left (63 a^{4}+46 a^{2} b^{2}+15 b^{4}\right ) \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}\, b}{\sqrt {a e b}}\right )}{8 \sqrt {a e b}}\right )}{a^{3} e^{5} \left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (-3 a^{2} b e +b^{3} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (-a^{3}+3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{\left (a^{2}+b^{2}\right )^{3} e^{5}}-\frac {1}{a^{3} e^{5} \sqrt {e \cot \left (d x +c \right )}}\right )}{d}\) | \(480\) |
-2/d*e^4*(-b^3/a^3/e^5/(a^2+b^2)^3*(((15/8*a^4*b+11/4*a^2*b^3+7/8*b^5)*(e* cot(d*x+c))^(3/2)+1/8*a*e*(17*a^4+26*a^2*b^2+9*b^4)*(e*cot(d*x+c))^(1/2))/ (e*cot(d*x+c)*b+a*e)^2+1/8*(63*a^4+46*a^2*b^2+15*b^4)/(a*e*b)^(1/2)*arctan ((e*cot(d*x+c))^(1/2)*b/(a*e*b)^(1/2)))+1/(a^2+b^2)^3/e^5*(1/8*(-3*a^2*b*e +b^3*e)*(e^2)^(1/4)/e^2*2^(1/2)*(ln((e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c ))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/ 2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2) +1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1))+1/8*(-a^3+3*a*b ^2)/(e^2)^(1/4)*2^(1/2)*(ln((e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2) *2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/ 2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-2*ar ctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)))-1/a^3/e^5/(e*cot(d*x+c ))^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 5133 vs. \(2 (454) = 908\).
Time = 2.31 (sec) , antiderivative size = 10308, normalized size of antiderivative = 19.49 \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^3} \, dx=\text {Too large to display} \]
\[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^3} \, dx=\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \left (a + b \cot {\left (c + d x \right )}\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^3} \, dx=\int { \frac {1}{{\left (b \cot \left (d x + c\right ) + a\right )}^{3} \left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
Time = 22.78 (sec) , antiderivative size = 21158, normalized size of antiderivative = 40.00 \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+b \cot (c+d x))^3} \, dx=\text {Too large to display} \]
((2*e)/a + (e*cot(c + d*x)*(16*a^4*b + 25*b^5 + 49*a^2*b^3))/(4*a^2*(a^4 + b^4 + 2*a^2*b^2)) + (b^2*e^2*cot(c + d*x)^2*(8*a^4 + 15*b^4 + 31*a^2*b^2) )/(4*a^3*(a^4*e + b^4*e + 2*a^2*b^2*e)))/(b^2*d*(e*cot(c + d*x))^(5/2) + a ^2*d*e^2*(e*cot(c + d*x))^(1/2) + 2*a*b*d*e*(e*cot(c + d*x))^(3/2)) + atan ((((-1i/(4*(b^6*d^2*e^3 - a^6*d^2*e^3 + a*b^5*d^2*e^3*6i + a^5*b*d^2*e^3*6 i - 15*a^2*b^4*d^2*e^3 - a^3*b^3*d^2*e^3*20i + 15*a^4*b^2*d^2*e^3)))^(1/2) *(((e*cot(c + d*x))^(1/2)*(471859200*a^22*b^44*d^7*e^16 + 9500098560*a^24* b^42*d^7*e^16 + 91857354752*a^26*b^40*d^7*e^16 + 564502986752*a^28*b^38*d^ 7*e^16 + 2464648527872*a^30*b^36*d^7*e^16 + 8104469069824*a^32*b^34*d^7*e^ 16 + 20769933361152*a^34*b^32*d^7*e^16 + 42351565209600*a^36*b^30*d^7*e^16 + 69534945902592*a^38*b^28*d^7*e^16 + 92434029608960*a^40*b^26*d^7*e^16 + 99508717355008*a^42*b^24*d^7*e^16 + 86342935511040*a^44*b^22*d^7*e^16 + 5 9767095558144*a^46*b^20*d^7*e^16 + 32432589897728*a^48*b^18*d^7*e^16 + 134 11815522304*a^50*b^16*d^7*e^16 + 4030457708544*a^52*b^14*d^7*e^16 + 805425 905664*a^54*b^12*d^7*e^16 + 86608183296*a^56*b^10*d^7*e^16 + 1612709888*a^ 58*b^8*d^7*e^16 + 16777216*a^60*b^6*d^7*e^16 + 167772160*a^62*b^4*d^7*e^16 + 16777216*a^64*b^2*d^7*e^16) + (-1i/(4*(b^6*d^2*e^3 - a^6*d^2*e^3 + a*b^ 5*d^2*e^3*6i + a^5*b*d^2*e^3*6i - 15*a^2*b^4*d^2*e^3 - a^3*b^3*d^2*e^3*20i + 15*a^4*b^2*d^2*e^3)))^(1/2)*(251658240*a^24*b^45*d^8*e^18 - (e*cot(c + d*x))^(1/2)*(-1i/(4*(b^6*d^2*e^3 - a^6*d^2*e^3 + a*b^5*d^2*e^3*6i + a^5...