Integrand size = 23, antiderivative size = 241 \[ \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\frac {\left (8 a^2-15 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{4 a^{7/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{3/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{3/2} d}+\frac {b^2 \left (7 a^2+15 b^2\right )}{4 a^3 \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {5 b \cot (c+d x)}{4 a^2 d \sqrt {a+b \tan (c+d x)}}-\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}} \]
1/4*(8*a^2-15*b^2)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/a^(7/2)/d-arcta nh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/(a-I*b)^(3/2)/d-arctanh((a+b*tan( d*x+c))^(1/2)/(a+I*b)^(1/2))/(a+I*b)^(3/2)/d+1/4*b^2*(7*a^2+15*b^2)/a^3/(a ^2+b^2)/d/(a+b*tan(d*x+c))^(1/2)+5/4*b*cot(d*x+c)/a^2/d/(a+b*tan(d*x+c))^( 1/2)-1/2*cot(d*x+c)^2/a/d/(a+b*tan(d*x+c))^(1/2)
Time = 5.85 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.07 \[ \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\frac {\frac {\left (8 a^2-15 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {-\frac {4 a^3 \left (a+\sqrt {-b^2}\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {-b^2}}}\right )}{\sqrt {a-\sqrt {-b^2}}}+\frac {4 a^3 \left (-a+\sqrt {-b^2}\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+\sqrt {-b^2}}}\right )}{\sqrt {a+\sqrt {-b^2}}}+\frac {7 a^2 b^2+15 b^4+5 a b \left (a^2+b^2\right ) \cot (c+d x)-2 a^2 \left (a^2+b^2\right ) \cot ^2(c+d x)}{\sqrt {a+b \tan (c+d x)}}}{a \left (a^2+b^2\right )}}{4 a^2 d} \]
(((8*a^2 - 15*b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/a^(3/2) + (( -4*a^3*(a + Sqrt[-b^2])*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^ 2]]])/Sqrt[a - Sqrt[-b^2]] + (4*a^3*(-a + Sqrt[-b^2])*ArcTanh[Sqrt[a + b*T an[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/Sqrt[a + Sqrt[-b^2]] + (7*a^2*b^2 + 15 *b^4 + 5*a*b*(a^2 + b^2)*Cot[c + d*x] - 2*a^2*(a^2 + b^2)*Cot[c + d*x]^2)/ Sqrt[a + b*Tan[c + d*x]])/(a*(a^2 + b^2)))/(4*a^2*d)
Time = 2.01 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.16, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.957, Rules used = {3042, 4052, 27, 3042, 4132, 27, 3042, 4133, 27, 3042, 4136, 27, 3042, 4022, 3042, 4020, 25, 73, 221, 4117, 73, 221}
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.
| \(\displaystyle \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (c+d x)^3 (a+b \tan (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 4052 |
| \(\displaystyle -\frac {\int \frac {\cot ^2(c+d x) \left (5 b \tan ^2(c+d x)+4 a \tan (c+d x)+5 b\right )}{2 (a+b \tan (c+d x))^{3/2}}dx}{2 a}-\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\cot ^2(c+d x) \left (5 b \tan ^2(c+d x)+4 a \tan (c+d x)+5 b\right )}{(a+b \tan (c+d x))^{3/2}}dx}{4 a}-\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {5 b \tan (c+d x)^2+4 a \tan (c+d x)+5 b}{\tan (c+d x)^2 (a+b \tan (c+d x))^{3/2}}dx}{4 a}-\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle -\frac {-\frac {\int -\frac {\cot (c+d x) \left (8 a^2-15 b^2-15 b^2 \tan ^2(c+d x)\right )}{2 (a+b \tan (c+d x))^{3/2}}dx}{a}-\frac {5 b \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}}{4 a}-\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {\cot (c+d x) \left (8 a^2-15 b^2-15 b^2 \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^{3/2}}dx}{2 a}-\frac {5 b \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}}{4 a}-\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {8 a^2-15 b^2-15 b^2 \tan (c+d x)^2}{\tan (c+d x) (a+b \tan (c+d x))^{3/2}}dx}{2 a}-\frac {5 b \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}}{4 a}-\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}\) |
| \(\Big \downarrow \) 4133 |
| \(\displaystyle -\frac {\frac {\frac {2 \int \frac {\cot (c+d x) \left (-8 b \tan (c+d x) a^3-b^2 \left (7 a^2+15 b^2\right ) \tan ^2(c+d x)+\left (8 a^2-15 b^2\right ) \left (a^2+b^2\right )\right )}{2 \sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}-\frac {2 b^2 \left (7 a^2+15 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{2 a}-\frac {5 b \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}}{4 a}-\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {\cot (c+d x) \left (-8 b \tan (c+d x) a^3-b^2 \left (7 a^2+15 b^2\right ) \tan ^2(c+d x)+\left (8 a^2-15 b^2\right ) \left (a^2+b^2\right )\right )}{\sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}-\frac {2 b^2 \left (7 a^2+15 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{2 a}-\frac {5 b \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}}{4 a}-\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {-8 b \tan (c+d x) a^3-b^2 \left (7 a^2+15 b^2\right ) \tan (c+d x)^2+\left (8 a^2-15 b^2\right ) \left (a^2+b^2\right )}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}-\frac {2 b^2 \left (7 a^2+15 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{2 a}-\frac {5 b \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}}{4 a}-\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle -\frac {\frac {\frac {\left (8 a^2-15 b^2\right ) \left (a^2+b^2\right ) \int \frac {\cot (c+d x) \left (\tan ^2(c+d x)+1\right )}{\sqrt {a+b \tan (c+d x)}}dx+\int -\frac {8 \left (\tan (c+d x) a^4+b a^3\right )}{\sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}-\frac {2 b^2 \left (7 a^2+15 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{2 a}-\frac {5 b \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}}{4 a}-\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\left (8 a^2-15 b^2\right ) \left (a^2+b^2\right ) \int \frac {\cot (c+d x) \left (\tan ^2(c+d x)+1\right )}{\sqrt {a+b \tan (c+d x)}}dx-8 \int \frac {\tan (c+d x) a^4+b a^3}{\sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}-\frac {2 b^2 \left (7 a^2+15 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{2 a}-\frac {5 b \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}}{4 a}-\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\frac {\left (8 a^2-15 b^2\right ) \left (a^2+b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-8 \int \frac {\tan (c+d x) a^4+b a^3}{\sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}-\frac {2 b^2 \left (7 a^2+15 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{2 a}-\frac {5 b \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}}{4 a}-\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle -\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}-\frac {-\frac {5 b \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}+\frac {-\frac {2 b^2 \left (7 a^2+15 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (8 a^2-15 b^2\right ) \left (a^2+b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-8 \left (\frac {1}{2} a^3 (b+i a) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} a^3 (-b+i a) \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx\right )}{a \left (a^2+b^2\right )}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}-\frac {-\frac {5 b \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}+\frac {-\frac {2 b^2 \left (7 a^2+15 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (8 a^2-15 b^2\right ) \left (a^2+b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-8 \left (\frac {1}{2} a^3 (b+i a) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx-\frac {1}{2} a^3 (-b+i a) \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx\right )}{a \left (a^2+b^2\right )}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle -\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}-\frac {-\frac {5 b \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}+\frac {-\frac {2 b^2 \left (7 a^2+15 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (8 a^2-15 b^2\right ) \left (a^2+b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-8 \left (-\frac {i a^3 (-b+i a) \int -\frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}-\frac {i a^3 (b+i a) \int -\frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}\right )}{a \left (a^2+b^2\right )}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}-\frac {-\frac {5 b \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}+\frac {-\frac {2 b^2 \left (7 a^2+15 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (8 a^2-15 b^2\right ) \left (a^2+b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-8 \left (\frac {i a^3 (-b+i a) \int \frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}+\frac {i a^3 (b+i a) \int \frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}\right )}{a \left (a^2+b^2\right )}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}-\frac {-\frac {5 b \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}+\frac {-\frac {2 b^2 \left (7 a^2+15 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (8 a^2-15 b^2\right ) \left (a^2+b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-8 \left (\frac {a^3 (b+i a) \int \frac {1}{-\frac {i \tan ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}-\frac {a^3 (-b+i a) \int \frac {1}{\frac {i \tan ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}\right )}{a \left (a^2+b^2\right )}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}-\frac {-\frac {5 b \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}+\frac {-\frac {2 b^2 \left (7 a^2+15 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\left (8 a^2-15 b^2\right ) \left (a^2+b^2\right ) \int \frac {\tan (c+d x)^2+1}{\tan (c+d x) \sqrt {a+b \tan (c+d x)}}dx-8 \left (\frac {a^3 (b+i a) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}-\frac {a^3 (-b+i a) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}\right )}{a \left (a^2+b^2\right )}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle -\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}-\frac {-\frac {5 b \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}+\frac {-\frac {2 b^2 \left (7 a^2+15 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {\left (8 a^2-15 b^2\right ) \left (a^2+b^2\right ) \int \frac {\cot (c+d x)}{\sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{d}-8 \left (\frac {a^3 (b+i a) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}-\frac {a^3 (-b+i a) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}\right )}{a \left (a^2+b^2\right )}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}-\frac {-\frac {5 b \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}+\frac {-\frac {2 b^2 \left (7 a^2+15 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {2 \left (8 a^2-15 b^2\right ) \left (a^2+b^2\right ) \int \frac {1}{\frac {a+b \tan (c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \tan (c+d x)}}{b d}-8 \left (\frac {a^3 (b+i a) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}-\frac {a^3 (-b+i a) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}\right )}{a \left (a^2+b^2\right )}}{2 a}}{4 a}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\cot ^2(c+d x)}{2 a d \sqrt {a+b \tan (c+d x)}}-\frac {-\frac {5 b \cot (c+d x)}{a d \sqrt {a+b \tan (c+d x)}}+\frac {-\frac {2 b^2 \left (7 a^2+15 b^2\right )}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {-\frac {2 \left (8 a^2-15 b^2\right ) \left (a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}-8 \left (\frac {a^3 (b+i a) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}-\frac {a^3 (-b+i a) \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}\right )}{a \left (a^2+b^2\right )}}{2 a}}{4 a}\) |
-1/2*Cot[c + d*x]^2/(a*d*Sqrt[a + b*Tan[c + d*x]]) - ((-5*b*Cot[c + d*x])/ (a*d*Sqrt[a + b*Tan[c + d*x]]) + ((-8*(-((a^3*(I*a - b)*ArcTan[Tan[c + d*x ]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d)) + (a^3*(I*a + b)*ArcTan[Tan[c + d*x]/ Sqrt[a + I*b]])/(Sqrt[a + I*b]*d)) - (2*(8*a^2 - 15*b^2)*(a^2 + b^2)*ArcTa nh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d))/(a*(a^2 + b^2)) - (2*b^ 2*(7*a^2 + 15*b^2))/(a*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]]))/(2*a))/(4* a)
3.6.45.3.1 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/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f) Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2 Int[(a + b*Tan[e + f*x])^m*( 1 - I*Tan[e + f*x]), x], x] + Simp[(c - I*d)/2 Int[(a + b*Tan[e + f*x])^m *(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !IntegerQ[m]
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[((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*b^2 + a^2*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*Sim p[A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2)) - a*C*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - b*C)*Tan[e + f*x] - d*(A*b^2 + a^2*C)*( m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, 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]
Timed out.
hanged
Leaf count of result is larger than twice the leaf count of optimal. 2182 vs. \(2 (201) = 402\).
Time = 0.55 (sec) , antiderivative size = 4381, normalized size of antiderivative = 18.18 \[ \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \]
[-1/8*(4*((a^6*b + a^4*b^3)*d*tan(d*x + c)^3 + (a^7 + a^5*b^2)*d*tan(d*x + c)^2)*sqrt(((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2 - 6* a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + a^3 - 3*a*b^2)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))*log(-(3*a^2 - b^2)*sqrt(b*tan(d*x + c) + a) + (2*(a^7 + 3*a^5 *b^2 + 3*a^3*b^4 + a*b^6)*d^3*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^ 4)) - (3*a^4 - 4*a^2*b^2 + b^4)*d)*sqrt(((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^ 6)*d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^ 4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + a^3 - 3*a*b^2)/(( a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))) - 4*((a^6*b + a^4*b^3)*d*tan(d*x + c)^3 + (a^7 + a^5*b^2)*d*tan(d*x + c)^2)*sqrt(((a^6 + 3*a^4*b^2 + 3*a^2 *b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) + a^3 - 3* a*b^2)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2))*log(-(3*a^2 - b^2)*sqrt( b*tan(d*x + c) + a) - (2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d^3*sqrt(-( 9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - (3*a^4 - 4*a^2*b^2 + b^4)*d)*sq rt(((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^2*sqrt(-(9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^...
\[ \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\int \frac {\cot ^{3}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Timed out. \[ \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Timed out} \]
Time = 5.90 (sec) , antiderivative size = 6216, normalized size of antiderivative = 25.79 \[ \int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \]
((2*b^4)/(a*(a^2 + b^2)) - ((25*b^4 + 9*a^2*b^2)*(a + b*tan(c + d*x)))/(4* a^2*(a^2 + b^2)) + (b^2*(7*a^2 + 15*b^2)*(a + b*tan(c + d*x))^2)/(4*a^3*(a ^2 + b^2)))/(d*(a + b*tan(c + d*x))^(5/2) - 2*a*d*(a + b*tan(c + d*x))^(3/ 2) + a^2*d*(a + b*tan(c + d*x))^(1/2)) + log((((((8*a^3*d^2 - 24*a*b^2*d^2 )^2/64 - b^6*d^4 - a^6*d^4 - 3*a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) + a^3*d^ 2 - 3*a*b^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^ (1/2)*((((((8*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^4 - a^6*d^4 - 3*a^2*b^4 *d^4 - 3*a^4*b^2*d^4)^(1/2) + a^3*d^2 - 3*a*b^2*d^2)/(4*(a^6*d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2)*(251658240*a^24*b^30*d^8 - (a + b*tan(c + d*x))^(1/2)*((((8*a^3*d^2 - 24*a*b^2*d^2)^2/64 - b^6*d^4 - a^6*d ^4 - 3*a^2*b^4*d^4 - 3*a^4*b^2*d^4)^(1/2) + a^3*d^2 - 3*a*b^2*d^2)/(4*(a^6 *d^4 + b^6*d^4 + 3*a^2*b^4*d^4 + 3*a^4*b^2*d^4)))^(1/2)*(134217728*a^27*b^ 28*d^9 + 1409286144*a^29*b^26*d^9 + 6643777536*a^31*b^24*d^9 + 18522046464 *a^33*b^22*d^9 + 33822867456*a^35*b^20*d^9 + 42278584320*a^37*b^18*d^9 + 3 6641439744*a^39*b^16*d^9 + 21743271936*a^41*b^14*d^9 + 8455716864*a^43*b^1 2*d^9 + 1946157056*a^45*b^10*d^9 + 201326592*a^47*b^8*d^9) + 2382364672*a^ 26*b^28*d^8 + 9948889088*a^28*b^26*d^8 + 23924310016*a^30*b^24*d^8 + 36071 014400*a^32*b^22*d^8 + 34292629504*a^34*b^20*d^8 + 18555600896*a^36*b^18*d ^8 + 2483027968*a^38*b^16*d^8 - 3841982464*a^40*b^14*d^8 - 2852126720*a^42 *b^12*d^8 - 855638016*a^44*b^10*d^8 - 100663296*a^46*b^8*d^8) + (a + b*...