Integrand size = 23, antiderivative size = 396 \[ \int \frac {\sqrt {\cot (c+d x)}}{(a+b \tan (c+d x))^3} \, dx=\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {b^{3/2} \left (35 a^4+6 a^2 b^2+3 b^4\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 a^{5/2} \left (a^2+b^2\right )^3 d}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{4 a^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {(a-b) \left (a^2+4 a b+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right )^3 d} \]
-1/4*b^(3/2)*(35*a^4+6*a^2*b^2+3*b^4)*arctan(a^(1/2)*cot(d*x+c)^(1/2)/b^(1 /2))/a^(5/2)/(a^2+b^2)^3/d+1/2*b^2*cot(d*x+c)^(3/2)/a/(a^2+b^2)/d/(b+a*cot (d*x+c))^2-1/2*(a+b)*(a^2-4*a*b+b^2)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))/( a^2+b^2)^3/d*2^(1/2)-1/2*(a+b)*(a^2-4*a*b+b^2)*arctan(1+2^(1/2)*cot(d*x+c) ^(1/2))/(a^2+b^2)^3/d*2^(1/2)-1/4*(a-b)*(a^2+4*a*b+b^2)*ln(1+cot(d*x+c)-2^ (1/2)*cot(d*x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)+1/4*(a-b)*(a^2+4*a*b+b^2)*ln (1+cot(d*x+c)+2^(1/2)*cot(d*x+c)^(1/2))/(a^2+b^2)^3/d*2^(1/2)+1/4*b^2*(11* a^2+3*b^2)*cot(d*x+c)^(1/2)/a^2/(a^2+b^2)^2/d/(b+a*cot(d*x+c))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 6.19 (sec) , antiderivative size = 542, normalized size of antiderivative = 1.37 \[ \int \frac {\sqrt {\cot (c+d x)}}{(a+b \tan (c+d x))^3} \, dx=-\frac {-\frac {2 b^{7/2} \left (a^2-3 b^2\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{5/2} \left (a^2+b^2\right )^3}+\frac {2 b^3 \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{a^2 \left (a^2+b^2\right )^3}-\frac {2 b^2 \left (a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 a \left (a^2+b^2\right )^3}+\frac {2 b \left (a^2-3 b^2\right ) \cot ^{\frac {5}{2}}(c+d x)}{5 \left (a^2+b^2\right )^3}-\frac {2 a \left (a^2-3 b^2\right ) \cot ^{\frac {7}{2}}(c+d x)}{7 \left (a^2+b^2\right )^3}-\frac {2 a \left (a^2-3 b^2\right ) \left (7 \cot ^{\frac {3}{2}}(c+d x)-3 \cot ^{\frac {7}{2}}(c+d x)-7 \cot ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )\right )}{21 \left (a^2+b^2\right )^3}+\frac {4 a^2 \cot ^{\frac {9}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (2,\frac {9}{2},\frac {11}{2},-\frac {a \cot (c+d x)}{b}\right )}{9 b \left (a^2+b^2\right )^2}+\frac {2 a^2 \cot ^{\frac {9}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (3,\frac {9}{2},\frac {11}{2},-\frac {a \cot (c+d x)}{b}\right )}{9 b^3 \left (a^2+b^2\right )}+\frac {b \left (3 a^2-b^2\right ) \left (10 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-10 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+40 \sqrt {\cot (c+d x)}-8 \cot ^{\frac {5}{2}}(c+d x)+5 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-5 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{20 \left (a^2+b^2\right )^3}}{d} \]
-(((-2*b^(7/2)*(a^2 - 3*b^2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]]) /(a^(5/2)*(a^2 + b^2)^3) + (2*b^3*(a^2 - 3*b^2)*Sqrt[Cot[c + d*x]])/(a^2*( a^2 + b^2)^3) - (2*b^2*(a^2 - 3*b^2)*Cot[c + d*x]^(3/2))/(3*a*(a^2 + b^2)^ 3) + (2*b*(a^2 - 3*b^2)*Cot[c + d*x]^(5/2))/(5*(a^2 + b^2)^3) - (2*a*(a^2 - 3*b^2)*Cot[c + d*x]^(7/2))/(7*(a^2 + b^2)^3) - (2*a*(a^2 - 3*b^2)*(7*Cot [c + d*x]^(3/2) - 3*Cot[c + d*x]^(7/2) - 7*Cot[c + d*x]^(3/2)*Hypergeometr ic2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/(21*(a^2 + b^2)^3) + (4*a^2*Cot[c + d*x]^(9/2)*Hypergeometric2F1[2, 9/2, 11/2, -((a*Cot[c + d*x])/b)])/(9*b*(a ^2 + b^2)^2) + (2*a^2*Cot[c + d*x]^(9/2)*Hypergeometric2F1[3, 9/2, 11/2, - ((a*Cot[c + d*x])/b)])/(9*b^3*(a^2 + b^2)) + (b*(3*a^2 - b^2)*(10*Sqrt[2]* ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 10*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqr t[Cot[c + d*x]]] + 40*Sqrt[Cot[c + d*x]] - 8*Cot[c + d*x]^(5/2) + 5*Sqrt[2 ]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 5*Sqrt[2]*Log[1 + S qrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(20*(a^2 + b^2)^3))/d)
Time = 1.98 (sec) , antiderivative size = 371, normalized size of antiderivative = 0.94, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.130, Rules used = {3042, 4156, 3042, 4048, 27, 3042, 4128, 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.
\(\displaystyle \int \frac {\sqrt {\cot (c+d x)}}{(a+b \tan (c+d x))^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {\cot (c+d x)}}{(a+b \tan (c+d x))^3}dx\) |
\(\Big \downarrow \) 4156 |
\(\displaystyle \int \frac {\cot ^{\frac {7}{2}}(c+d x)}{(a \cot (c+d x)+b)^3}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}{\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
\(\Big \downarrow \) 4048 |
\(\displaystyle \frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}-\frac {\int -\frac {\sqrt {\cot (c+d x)} \left (3 b^2-4 a \cot (c+d x) b+\left (4 a^2+3 b^2\right ) \cot ^2(c+d x)\right )}{2 (b+a \cot (c+d x))^2}dx}{2 a \left (a^2+b^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\sqrt {\cot (c+d x)} \left (3 b^2-4 a \cot (c+d x) b+\left (4 a^2+3 b^2\right ) \cot ^2(c+d x)\right )}{(b+a \cot (c+d x))^2}dx}{4 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (3 b^2+4 a \tan \left (c+d x+\frac {\pi }{2}\right ) b+\left (4 a^2+3 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
\(\Big \downarrow \) 4128 |
\(\displaystyle \frac {\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}-\frac {\int -\frac {-16 b \cot (c+d x) a^3+\left (8 a^4+3 b^2 a^2+3 b^4\right ) \cot ^2(c+d x)+b^2 \left (11 a^2+3 b^2\right )}{2 \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a \left (a^2+b^2\right )}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\int \frac {-16 b \cot (c+d x) a^3+\left (8 a^4+3 b^2 a^2+3 b^4\right ) \cot ^2(c+d x)+b^2 \left (11 a^2+3 b^2\right )}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\int \frac {16 b \tan \left (c+d x+\frac {\pi }{2}\right ) a^3+\left (8 a^4+3 b^2 a^2+3 b^4\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2+b^2 \left (11 a^2+3 b^2\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
\(\Big \downarrow \) 4136 |
\(\displaystyle \frac {\frac {\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a^2+b^2}+\frac {\int -\frac {8 \left (a^2 b \left (3 a^2-b^2\right )-a^3 \left (a^2-3 b^2\right ) \cot (c+d x)\right )}{\sqrt {\cot (c+d x)}}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a^2+b^2}-\frac {8 \int \frac {a^2 b \left (3 a^2-b^2\right )-a^3 \left (a^2-3 b^2\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {8 \int \frac {\left (a^2-3 b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) a^3+b \left (3 a^2-b^2\right ) a^2}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
\(\Big \downarrow \) 4017 |
\(\displaystyle \frac {\frac {\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 \int -\frac {a^2 \left (b \left (3 a^2-b^2\right )-a \left (a^2-3 b^2\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {\frac {16 \int \frac {a^2 \left (b \left (3 a^2-b^2\right )-a \left (a^2-3 b^2\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}+\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\frac {16 a^2 \int \frac {b \left (3 a^2-b^2\right )-a \left (a^2-3 b^2\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}+\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
\(\Big \downarrow \) 1482 |
\(\displaystyle \frac {\frac {\frac {16 a^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {\frac {\frac {16 a^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {\frac {\frac {16 a^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\frac {\frac {16 a^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {\frac {\frac {16 a^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {\frac {16 a^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\frac {16 a^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {\frac {\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}+\frac {16 a^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
\(\Big \downarrow \) 4117 |
\(\displaystyle \frac {\frac {\frac {b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}d(-\cot (c+d x))}{d \left (a^2+b^2\right )}+\frac {16 a^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {\frac {16 a^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {2 b^2 \left (35 a^4+6 a^2 b^2+3 b^4\right ) \int \frac {1}{a \cot ^2(c+d x)+b}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}+\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {b^2 \cot ^{\frac {3}{2}}(c+d x)}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}+\frac {\frac {b^2 \left (11 a^2+3 b^2\right ) \sqrt {\cot (c+d x)}}{a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}+\frac {\frac {16 a^2 \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )-\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {2 b^{3/2} \left (35 a^4+6 a^2 b^2+3 b^4\right ) \arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} d \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}}{4 a \left (a^2+b^2\right )}\) |
(b^2*Cot[c + d*x]^(3/2))/(2*a*(a^2 + b^2)*d*(b + a*Cot[c + d*x])^2) + ((b^ 2*(11*a^2 + 3*b^2)*Sqrt[Cot[c + d*x]])/(a*(a^2 + b^2)*d*(b + a*Cot[c + d*x ])) + ((2*b^(3/2)*(35*a^4 + 6*a^2*b^2 + 3*b^4)*ArcTan[(Sqrt[a]*Cot[c + d*x ])/Sqrt[b]])/(Sqrt[a]*(a^2 + b^2)*d) + (16*a^2*(-1/2*((a + b)*(a^2 - 4*a*b + b^2)*(-(ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sq rt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2])) + ((a - b)*(a^2 + 4*a*b + b^2)*(-1/2*L og[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2 ]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2])))/2))/((a^2 + b^2)*d))/(2 *a*(a^2 + b^2)))/(4*a*(a^2 + b^2))
3.9.35.3.1 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*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Simp[1 /(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c *(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d) *Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*( n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 2] && LtQ [n, -1] && IntegerQ[2*m]
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*d^2 + c*(c*C - B*d))*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Sim p[1/(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c*m + a*d* (n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b *(d*(B*c - A*d)*(m + n + 1) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ [a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
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]
Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x _)]^(n_.))^(p_.), x_Symbol] :> Simp[d^(n*p) Int[(d*Cot[e + f*x])^(m - n*p )*(b + a*Cot[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, d, e, f, m, n, p}, x] && !IntegerQ[m] && IntegersQ[n, p]
Leaf count of result is larger than twice the leaf count of optimal. \(2070\) vs. \(2(348)=696\).
Time = 1.40 (sec) , antiderivative size = 2071, normalized size of antiderivative = 5.23
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(2071\) |
default | \(\text {Expression too large to display}\) | \(2071\) |
1/4/d*(1/tan(d*x+c))^(1/2)*tan(d*x+c)^(1/2)*(2*2^(1/2)*(a*b)^(1/2)*arctan( -1+2^(1/2)*tan(d*x+c)^(1/2))*a^4*b^3-3*2^(1/2)*(a*b)^(1/2)*ln(-(2^(1/2)*ta n(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*a^6* b+2^(1/2)*(a*b)^(1/2)*ln(-(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/ 2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*a^4*b^3-3*2^(1/2)*(a*b)^(1/2)*ln(-(1+2^(1 /2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(2^(1/2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1))* a^5*b^2-6*2^(1/2)*(a*b)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^6*b-6*2 ^(1/2)*(a*b)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^5*b^2+2*2^(1/2)*(a *b)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^4*b^3-6*2^(1/2)*(a*b)^(1/2) *arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a^6*b+2*2^(1/2)*(a*b)^(1/2)*ln(-(2^(1 /2)*tan(d*x+c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)) )*a^3*b^4*tan(d*x+c)-12*2^(1/2)*(a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^( 1/2))*a^5*b^2*tan(d*x+c)-12*2^(1/2)*(a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d*x+ c)^(1/2))*a^4*b^3*tan(d*x+c)+4*2^(1/2)*(a*b)^(1/2)*arctan(-1+2^(1/2)*tan(d *x+c)^(1/2))*a^3*b^4*tan(d*x+c)-6*2^(1/2)*(a*b)^(1/2)*ln(-(2^(1/2)*tan(d*x +c)^(1/2)-tan(d*x+c)-1)/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*a^5*b^2*t an(d*x+c)+4*2^(1/2)*(a*b)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^6*b*t an(d*x+c)-12*2^(1/2)*(a*b)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^5*b^ 2*tan(d*x+c)-12*2^(1/2)*(a*b)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a^4 *b^3*tan(d*x+c)+4*2^(1/2)*(a*b)^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2)...
Leaf count of result is larger than twice the leaf count of optimal. 4131 vs. \(2 (348) = 696\).
Time = 1.33 (sec) , antiderivative size = 8294, normalized size of antiderivative = 20.94 \[ \int \frac {\sqrt {\cot (c+d x)}}{(a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \]
\[ \int \frac {\sqrt {\cot (c+d x)}}{(a+b \tan (c+d x))^3} \, dx=\int \frac {\sqrt {\cot {\left (c + d x \right )}}}{\left (a + b \tan {\left (c + d x \right )}\right )^{3}}\, dx \]
Time = 0.31 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {\cot (c+d x)}}{(a+b \tan (c+d x))^3} \, dx=-\frac {\frac {{\left (35 \, a^{4} b^{2} + 6 \, a^{2} b^{4} + 3 \, b^{6}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} \sqrt {a b}} + \frac {2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {\frac {11 \, a^{2} b^{3} + 3 \, b^{5}}{\sqrt {\tan \left (d x + c\right )}} + \frac {13 \, a^{3} b^{2} + 5 \, a b^{4}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6} + \frac {2 \, {\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )}}{\tan \left (d x + c\right )} + \frac {a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4}}{\tan \left (d x + c\right )^{2}}}}{4 \, d} \]
-1/4*((35*a^4*b^2 + 6*a^2*b^4 + 3*b^6)*arctan(a/(sqrt(a*b)*sqrt(tan(d*x + c))))/((a^8 + 3*a^6*b^2 + 3*a^4*b^4 + a^2*b^6)*sqrt(a*b)) + (2*sqrt(2)*(a^ 3 - 3*a^2*b - 3*a*b^2 + b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 2*sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*arctan(-1/2*sqrt(2)*(s qrt(2) - 2/sqrt(tan(d*x + c)))) - sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)* log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) + sqrt(2)*(a^3 + 3*a^ 2*b - 3*a*b^2 - b^3)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) )/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - ((11*a^2*b^3 + 3*b^5)/sqrt(tan(d*x + c)) + (13*a^3*b^2 + 5*a*b^4)/tan(d*x + c)^(3/2))/(a^6*b^2 + 2*a^4*b^4 + a^2*b^6 + 2*(a^7*b + 2*a^5*b^3 + a^3*b^5)/tan(d*x + c) + (a^8 + 2*a^6*b^2 + a^4*b^4)/tan(d*x + c)^2))/d
\[ \int \frac {\sqrt {\cot (c+d x)}}{(a+b \tan (c+d x))^3} \, dx=\int { \frac {\sqrt {\cot \left (d x + c\right )}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {\cot (c+d x)}}{(a+b \tan (c+d x))^3} \, dx=\int \frac {\sqrt {\mathrm {cot}\left (c+d\,x\right )}}{{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3} \,d x \]