Integrand size = 23, antiderivative size = 331 \[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3} \, dx=\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {\sqrt {b} \left (15 a^4-18 a^2 b^2-b^4\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{4 a^{3/2} \left (a^2+b^2\right )^3 d}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\cot (c+d x)}}{1+\cot (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right )^3 d}+\frac {b^2 \sqrt {\cot (c+d x)}}{2 a \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac {b \left (9 a^2+b^2\right ) \sqrt {\cot (c+d x)}}{4 a \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))} \] Output:
-1/2*(a-b)*(a^2+4*a*b+b^2)*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2))*2^(1/2)/(a^ 2+b^2)^3/d-1/2*(a-b)*(a^2+4*a*b+b^2)*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))*2^ (1/2)/(a^2+b^2)^3/d+1/4*b^(1/2)*(15*a^4-18*a^2*b^2-b^4)*arctan(a^(1/2)*cot (d*x+c)^(1/2)/b^(1/2))/a^(3/2)/(a^2+b^2)^3/d-1/2*(a+b)*(a^2-4*a*b+b^2)*arc tanh(2^(1/2)*cot(d*x+c)^(1/2)/(1+cot(d*x+c)))*2^(1/2)/(a^2+b^2)^3/d+1/2*b^ 2*cot(d*x+c)^(1/2)/a/(a^2+b^2)/d/(b+a*cot(d*x+c))^2-1/4*b*(9*a^2+b^2)*cot( d*x+c)^(1/2)/a/(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 = 6.11 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.48 \[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3} \, dx=-\frac {\frac {2 b^{5/2} \left (a^2-3 b^2\right ) \arctan \left (\frac {\sqrt {a} \sqrt {\cot (c+d x)}}{\sqrt {b}}\right )}{a^{3/2} \left (a^2+b^2\right )^3}-\frac {2 b^2 \left (a^2-3 b^2\right ) \sqrt {\cot (c+d x)}}{a \left (a^2+b^2\right )^3}+\frac {2 b \left (a^2-3 b^2\right ) \cot ^{\frac {3}{2}}(c+d x)}{3 \left (a^2+b^2\right )^3}-\frac {2 a \left (a^2-3 b^2\right ) \cot ^{\frac {5}{2}}(c+d x)}{5 \left (a^2+b^2\right )^3}-\frac {2 b \left (3 a^2-b^2\right ) \left (\cot ^{\frac {3}{2}}(c+d x)-\cot ^{\frac {3}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )\right )}{3 \left (a^2+b^2\right )^3}+\frac {4 a^2 \cot ^{\frac {7}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (2,\frac {7}{2},\frac {9}{2},-\frac {a \cot (c+d x)}{b}\right )}{7 b \left (a^2+b^2\right )^2}+\frac {2 a^2 \cot ^{\frac {7}{2}}(c+d x) \operatorname {Hypergeometric2F1}\left (3,\frac {7}{2},\frac {9}{2},-\frac {a \cot (c+d x)}{b}\right )}{7 b^3 \left (a^2+b^2\right )}-\frac {a \left (a^2-3 b^2\right ) \left (10 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-10 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+40 \sqrt {\cot (c+d x)}-8 \cot ^{\frac {5}{2}}(c+d x)+5 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-5 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{20 \left (a^2+b^2\right )^3}}{d} \] Input:
Integrate[1/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^3),x]
Output:
-(((2*b^(5/2)*(a^2 - 3*b^2)*ArcTan[(Sqrt[a]*Sqrt[Cot[c + d*x]])/Sqrt[b]])/ (a^(3/2)*(a^2 + b^2)^3) - (2*b^2*(a^2 - 3*b^2)*Sqrt[Cot[c + d*x]])/(a*(a^2 + b^2)^3) + (2*b*(a^2 - 3*b^2)*Cot[c + d*x]^(3/2))/(3*(a^2 + b^2)^3) - (2 *a*(a^2 - 3*b^2)*Cot[c + d*x]^(5/2))/(5*(a^2 + b^2)^3) - (2*b*(3*a^2 - b^2 )*(Cot[c + d*x]^(3/2) - Cot[c + d*x]^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2]))/(3*(a^2 + b^2)^3) + (4*a^2*Cot[c + d*x]^(7/2)*Hypergeom etric2F1[2, 7/2, 9/2, -((a*Cot[c + d*x])/b)])/(7*b*(a^2 + b^2)^2) + (2*a^2 *Cot[c + d*x]^(7/2)*Hypergeometric2F1[3, 7/2, 9/2, -((a*Cot[c + d*x])/b)]) /(7*b^3*(a^2 + b^2)) - (a*(a^2 - 3*b^2)*(10*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqr t[Cot[c + d*x]]] - 10*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[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 + Sqrt[2]*Sqrt[Cot[c + d* x]] + Cot[c + d*x]]))/(20*(a^2 + b^2)^3))/d)
Time = 1.93 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.10, 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, 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.
| \(\displaystyle \int \frac {1}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4156 |
| \(\displaystyle \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a \cot (c+d x)+b)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (-\tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}{\left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4048 |
| \(\displaystyle \frac {b^2 \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}-\frac {\int -\frac {b^2-4 a \cot (c+d x) b+\left (4 a^2+b^2\right ) \cot ^2(c+d x)}{2 \sqrt {\cot (c+d x)} (b+a \cot (c+d x))^2}dx}{2 a \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {b^2-4 a \cot (c+d x) b+\left (4 a^2+b^2\right ) \cot ^2(c+d x)}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))^2}dx}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {b^2+4 a \tan \left (c+d x+\frac {\pi }{2}\right ) b+\left (4 a^2+b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {-\frac {\int \frac {-\left (\left (9 a^2+b^2\right ) \cot ^2(c+d x) b^2\right )+\left (7 a^2-b^2\right ) b^2-8 a \left (a^2-b^2\right ) \cot (c+d x) b}{2 \sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{b \left (a^2+b^2\right )}-\frac {b \left (9 a^2+b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {-\left (\left (9 a^2+b^2\right ) \cot ^2(c+d x) b^2\right )+\left (7 a^2-b^2\right ) b^2-8 a \left (a^2-b^2\right ) \cot (c+d x) b}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{2 b \left (a^2+b^2\right )}-\frac {b \left (9 a^2+b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {-\left (\left (9 a^2+b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^2 b^2\right )+\left (7 a^2-b^2\right ) b^2+8 a \left (a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right ) b}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 b \left (a^2+b^2\right )}-\frac {b \left (9 a^2+b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {8 \left (b \left (a^2-3 b^2\right ) a^2+b^2 \left (3 a^2-b^2\right ) \cot (c+d x) a\right )}{\sqrt {\cot (c+d x)}}dx}{a^2+b^2}+\frac {b^2 \left (15 a^4-18 a^2 b^2-b^4\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {b \left (9 a^2+b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {b^2 \left (15 a^4-18 a^2 b^2-b^4\right ) \int \frac {\cot ^2(c+d x)+1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}dx}{a^2+b^2}-\frac {8 \int \frac {b \left (a^2-3 b^2\right ) a^2+b^2 \left (3 a^2-b^2\right ) \cot (c+d x) a}{\sqrt {\cot (c+d x)}}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {b \left (9 a^2+b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {b^2 \left (15 a^4-18 a^2 b^2-b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {8 \int \frac {a^2 b \left (a^2-3 b^2\right )-a b^2 \left (3 a^2-b^2\right ) \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {b \left (9 a^2+b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {-\frac {\frac {b^2 \left (15 a^4-18 a^2 b^2-b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}-\frac {16 \int -\frac {a b \left (a \left (a^2-3 b^2\right )+b \left (3 a^2-b^2\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {b \left (9 a^2+b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {16 \int \frac {a b \left (a \left (a^2-3 b^2\right )+b \left (3 a^2-b^2\right ) \cot (c+d x)\right )}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^4-18 a^2 b^2-b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {b \left (9 a^2+b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {16 a b \int \frac {a \left (a^2-3 b^2\right )+b \left (3 a^2-b^2\right ) \cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^4-18 a^2 b^2-b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {b \left (9 a^2+b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {-\frac {\frac {16 a b \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \int \frac {\cot (c+d x)+1}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^4-18 a^2 b^2-b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {b \left (9 a^2+b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {-\frac {\frac {16 a b \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {1}{2} \int \frac {1}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} \int \frac {1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^4-18 a^2 b^2-b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {b \left (9 a^2+b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {-\frac {\frac {16 a b \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\cot (c+d x)-1}d\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^4-18 a^2 b^2-b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {b \left (9 a^2+b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {\frac {16 a b \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \int \frac {1-\cot (c+d x)}{\cot ^2(c+d x)+1}d\sqrt {\cot (c+d x)}+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^4-18 a^2 b^2-b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {b \left (9 a^2+b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {-\frac {\frac {16 a b \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^4-18 a^2 b^2-b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {b \left (9 a^2+b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {16 a b \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^4-18 a^2 b^2-b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {b \left (9 a^2+b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {16 a b \left (\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\cot (c+d x)}}{\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\cot (c+d x)}+1}{\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1}d\sqrt {\cot (c+d x)}\right )+\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {b^2 \left (15 a^4-18 a^2 b^2-b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}}{2 b \left (a^2+b^2\right )}-\frac {b \left (9 a^2+b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-\frac {\frac {b^2 \left (15 a^4-18 a^2 b^2-b^4\right ) \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-\tan \left (c+d x+\frac {\pi }{2}\right )} \left (b-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2+b^2}+\frac {16 a b \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {b \left (9 a^2+b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {-\frac {\frac {b^2 \left (15 a^4-18 a^2 b^2-b^4\right ) \int \frac {1}{\sqrt {\cot (c+d x)} (b+a \cot (c+d x))}d(-\cot (c+d x))}{d \left (a^2+b^2\right )}+\frac {16 a b \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {b \left (9 a^2+b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {-\frac {\frac {16 a b \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}-\frac {2 b^2 \left (15 a^4-18 a^2 b^2-b^4\right ) \int \frac {1}{a \cot ^2(c+d x)+b}d\sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}-\frac {b \left (9 a^2+b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}}{4 a \left (a^2+b^2\right )}+\frac {b^2 \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {b^2 \sqrt {\cot (c+d x)}}{2 a d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}+\frac {-\frac {b \left (9 a^2+b^2\right ) \sqrt {\cot (c+d x)}}{d \left (a^2+b^2\right ) (a \cot (c+d x)+b)}-\frac {\frac {16 a b \left (\frac {1}{2} (a-b) \left (a^2+4 a b+b^2\right ) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} (a+b) \left (a^2-4 a b+b^2\right ) \left (\frac {\log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d \left (a^2+b^2\right )}+\frac {2 b^{3/2} \left (15 a^4-18 a^2 b^2-b^4\right ) \arctan \left (\frac {\sqrt {a} \cot (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} d \left (a^2+b^2\right )}}{2 b \left (a^2+b^2\right )}}{4 a \left (a^2+b^2\right )}\) |
Input:
Int[1/(Sqrt[Cot[c + d*x]]*(a + b*Tan[c + d*x])^3),x]
Output:
(b^2*Sqrt[Cot[c + d*x]])/(2*a*(a^2 + b^2)*d*(b + a*Cot[c + d*x])^2) + (-(( b*(9*a^2 + b^2)*Sqrt[Cot[c + d*x]])/((a^2 + b^2)*d*(b + a*Cot[c + d*x]))) - ((2*b^(3/2)*(15*a^4 - 18*a^2*b^2 - b^4)*ArcTan[(Sqrt[a]*Cot[c + d*x])/Sq rt[b]])/(Sqrt[a]*(a^2 + b^2)*d) + (16*a*b*(((a - b)*(a^2 + 4*a*b + b^2)*(- (ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt [Cot[c + d*x]]]/Sqrt[2]))/2 + ((a + b)*(a^2 - 4*a*b + b^2)*(-1/2*Log[1 - S qrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[C ot[c + d*x]] + Cot[c + d*x]]/(2*Sqrt[2])))/2))/((a^2 + b^2)*d))/(2*b*(a^2 + b^2)))/(4*a*(a^2 + b^2))
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*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]
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]
Time = 0.31 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.04
| method | result | size |
| derivativedivides | \(-\frac {-\frac {2 b \left (\frac {\left (-\frac {9}{8} a^{4}-\frac {5}{4} a^{2} b^{2}-\frac {1}{8} b^{4}\right ) \cot \left (d x +c \right )^{\frac {3}{2}}-\frac {b \left (7 a^{4}+6 a^{2} b^{2}-b^{4}\right ) \sqrt {\cot \left (d x +c \right )}}{8 a}}{\left (b +a \cot \left (d x +c \right )\right )^{2}}+\frac {\left (15 a^{4}-18 a^{2} b^{2}-b^{4}\right ) \arctan \left (\frac {a \sqrt {\cot \left (d x +c \right )}}{\sqrt {a b}}\right )}{8 a \sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (a^{3}-3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(344\) |
| default | \(-\frac {-\frac {2 b \left (\frac {\left (-\frac {9}{8} a^{4}-\frac {5}{4} a^{2} b^{2}-\frac {1}{8} b^{4}\right ) \cot \left (d x +c \right )^{\frac {3}{2}}-\frac {b \left (7 a^{4}+6 a^{2} b^{2}-b^{4}\right ) \sqrt {\cot \left (d x +c \right )}}{8 a}}{\left (b +a \cot \left (d x +c \right )\right )^{2}}+\frac {\left (15 a^{4}-18 a^{2} b^{2}-b^{4}\right ) \arctan \left (\frac {a \sqrt {\cot \left (d x +c \right )}}{\sqrt {a b}}\right )}{8 a \sqrt {a b}}\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {\left (a^{3}-3 a \,b^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {\cot \left (d x +c \right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}{\cot \left (d x +c \right )+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\right )\right )}{4}}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(344\) |
Input:
int(1/cot(d*x+c)^(1/2)/(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-1/d*(-2*b/(a^2+b^2)^3*(((-9/8*a^4-5/4*a^2*b^2-1/8*b^4)*cot(d*x+c)^(3/2)-1 /8*b*(7*a^4+6*a^2*b^2-b^4)/a*cot(d*x+c)^(1/2))/(b+a*cot(d*x+c))^2+1/8*(15* a^4-18*a^2*b^2-b^4)/a/(a*b)^(1/2)*arctan(a*cot(d*x+c)^(1/2)/(a*b)^(1/2)))+ 2/(a^2+b^2)^3*(1/8*(a^3-3*a*b^2)*2^(1/2)*(ln((cot(d*x+c)+2^(1/2)*cot(d*x+c )^(1/2)+1)/(cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2)+1))+2*arctan(1+2^(1/2)*cot (d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*cot(d*x+c)^(1/2)))+1/8*(3*a^2*b-b^3)*2^ (1/2)*(ln((cot(d*x+c)-2^(1/2)*cot(d*x+c)^(1/2)+1)/(cot(d*x+c)+2^(1/2)*cot( d*x+c)^(1/2)+1))+2*arctan(1+2^(1/2)*cot(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)* cot(d*x+c)^(1/2)))))
Leaf count of result is larger than twice the leaf count of optimal. 2370 vs. \(2 (293) = 586\).
Time = 0.67 (sec) , antiderivative size = 4771, normalized size of antiderivative = 14.41 \[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate(1/cot(d*x+c)^(1/2)/(a+b*tan(d*x+c))^3,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3} \, dx=\int \frac {1}{\left (a + b \tan {\left (c + d x \right )}\right )^{3} \sqrt {\cot {\left (c + d x \right )}}}\, dx \] Input:
integrate(1/cot(d*x+c)**(1/2)/(a+b*tan(d*x+c))**3,x)
Output:
Integral(1/((a + b*tan(c + d*x))**3*sqrt(cot(c + d*x))), x)
Time = 0.51 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3} \, dx=\frac {\frac {{\left (15 \, a^{4} b - 18 \, a^{2} b^{3} - b^{5}\right )} \arctan \left (\frac {a}{\sqrt {a b} \sqrt {\tan \left (d x + c\right )}}\right )}{{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 2 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {\frac {7 \, a^{2} b^{2} - b^{4}}{\sqrt {\tan \left (d x + c\right )}} + \frac {9 \, a^{3} b + a b^{3}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6} + \frac {2 \, {\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )}}{\tan \left (d x + c\right )} + \frac {a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}}{\tan \left (d x + c\right )^{2}}}}{4 \, d} \] Input:
integrate(1/cot(d*x+c)^(1/2)/(a+b*tan(d*x+c))^3,x, algorithm="maxima")
Output:
1/4*((15*a^4*b - 18*a^2*b^3 - b^5)*arctan(a/(sqrt(a*b)*sqrt(tan(d*x + c))) )/((a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*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)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) + sqrt(2)*(a^3 - 3*a^2*b - 3*a*b^2 + b^3)*log(sq rt(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) - ((7*a^2*b^2 - b^4)/sqrt(tan(d*x + c)) + (9*a^3*b + a*b^3)/tan(d*x + c)^(3/2))/(a^5*b^2 + 2*a^3*b^4 + a*b^6 + 2*(a^ 6*b + 2*a^4*b^3 + a^2*b^5)/tan(d*x + c) + (a^7 + 2*a^5*b^2 + a^3*b^4)/tan( d*x + c)^2))/d
\[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3} \, dx=\int { \frac {1}{{\left (b \tan \left (d x + c\right ) + a\right )}^{3} \sqrt {\cot \left (d x + c\right )}} \,d x } \] Input:
integrate(1/cot(d*x+c)^(1/2)/(a+b*tan(d*x+c))^3,x, algorithm="giac")
Output:
integrate(1/((b*tan(d*x + c) + a)^3*sqrt(cot(d*x + c))), x)
Timed out. \[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3} \, dx=\int \frac {1}{\sqrt {\mathrm {cot}\left (c+d\,x\right )}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3} \,d x \] Input:
int(1/(cot(c + d*x)^(1/2)*(a + b*tan(c + d*x))^3),x)
Output:
int(1/(cot(c + d*x)^(1/2)*(a + b*tan(c + d*x))^3), x)
\[ \int \frac {1}{\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^3} \, dx=\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right ) \tan \left (d x +c \right )^{3} b^{3}+3 \cot \left (d x +c \right ) \tan \left (d x +c \right )^{2} a \,b^{2}+3 \cot \left (d x +c \right ) \tan \left (d x +c \right ) a^{2} b +\cot \left (d x +c \right ) a^{3}}d x \] Input:
int(1/cot(d*x+c)^(1/2)/(a+b*tan(d*x+c))^3,x)
Output:
int(sqrt(cot(c + d*x))/(cot(c + d*x)*tan(c + d*x)**3*b**3 + 3*cot(c + d*x) *tan(c + d*x)**2*a*b**2 + 3*cot(c + d*x)*tan(c + d*x)*a**2*b + cot(c + d*x )*a**3),x)