Integrand size = 25, antiderivative size = 338 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\frac {\arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{(i a-b)^{5/2} d}+\frac {\text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{(i a+b)^{5/2} d}+\frac {2 b^2 \left (7 a^2+8 b^2\right )}{3 a^3 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac {4 b \sqrt {\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac {4 b^2 \left (4 a^4+15 a^2 b^2+8 b^4\right )}{3 a^4 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}} \]
arctan((I*a-b)^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1/2))*cot(d*x+c)^( 1/2)*tan(d*x+c)^(1/2)/(I*a-b)^(5/2)/d+arctanh((I*a+b)^(1/2)*tan(d*x+c)^(1/ 2)/(a+b*tan(d*x+c))^(1/2))*cot(d*x+c)^(1/2)*tan(d*x+c)^(1/2)/(I*a+b)^(5/2) /d+4/3*b^2*(4*a^4+15*a^2*b^2+8*b^4)/a^4/(a^2+b^2)^2/d/cot(d*x+c)^(1/2)/(a+ b*tan(d*x+c))^(1/2)-2/3*cot(d*x+c)^(3/2)/a/d/(a+b*tan(d*x+c))^(3/2)+2/3*b^ 2*(7*a^2+8*b^2)/a^3/(a^2+b^2)/d/cot(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(3/2)+4* b*cot(d*x+c)^(1/2)/a^2/d/(a+b*tan(d*x+c))^(3/2)
Time = 6.38 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.49 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}-\frac {2 \left (-\frac {6 b}{a d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac {3 \left (\frac {a b \sqrt {\tan (c+d x)}}{3 (i a-b) (a+b \tan (c+d x))^{3/2}}+\frac {16 b^2 \sqrt {\tan (c+d x)}}{3 a (a+b \tan (c+d x))^{3/2}}-\frac {a b \sqrt {\tan (c+d x)}}{3 (i a+b) (a+b \tan (c+d x))^{3/2}}+\frac {32 b^2 \sqrt {\tan (c+d x)}}{3 a^2 \sqrt {a+b \tan (c+d x)}}-\frac {-\frac {3 \sqrt [4]{-1} a^2 \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(-a+i b)^{3/2}}+\frac {(5 a-2 i b) b \sqrt {\tan (c+d x)}}{(a-i b) \sqrt {a+b \tan (c+d x)}}}{3 (i a+b)}+\frac {-\frac {3 \sqrt [4]{-1} a^2 \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(a+i b)^{3/2}}+\frac {(5 a+2 i b) b \sqrt {\tan (c+d x)}}{(a+i b) \sqrt {a+b \tan (c+d x)}}}{3 (i a-b)}\right )}{2 a d}\right )}{3 a}\right ) \]
Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-2/(3*a*d*Tan[c + d*x]^(3/2)*(a + b *Tan[c + d*x])^(3/2)) - (2*((-6*b)/(a*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)) - (3*((a*b*Sqrt[Tan[c + d*x]])/(3*(I*a - b)*(a + b*Tan[c + d* x])^(3/2)) + (16*b^2*Sqrt[Tan[c + d*x]])/(3*a*(a + b*Tan[c + d*x])^(3/2)) - (a*b*Sqrt[Tan[c + d*x]])/(3*(I*a + b)*(a + b*Tan[c + d*x])^(3/2)) + (32* b^2*Sqrt[Tan[c + d*x]])/(3*a^2*Sqrt[a + b*Tan[c + d*x]]) - ((-3*(-1)^(1/4) *a^2*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[ c + d*x]]])/(-a + I*b)^(3/2) + ((5*a - (2*I)*b)*b*Sqrt[Tan[c + d*x]])/((a - I*b)*Sqrt[a + b*Tan[c + d*x]]))/(3*(I*a + b)) + ((-3*(-1)^(1/4)*a^2*ArcT an[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] )/(a + I*b)^(3/2) + ((5*a + (2*I)*b)*b*Sqrt[Tan[c + d*x]])/((a + I*b)*Sqrt [a + b*Tan[c + d*x]]))/(3*(I*a - b))))/(2*a*d)))/(3*a))
Time = 2.20 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.14, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.840, Rules used = {3042, 4729, 3042, 4052, 27, 3042, 4132, 27, 3042, 4133, 27, 3042, 4132, 27, 3042, 4099, 3042, 4098, 104, 216, 219}
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.
| \(\displaystyle \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cot (c+d x)^{5/2}}{(a+b \tan (c+d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 4729 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {1}{\tan (c+d x)^{5/2} (a+b \tan (c+d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 4052 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2 \int \frac {3 \left (2 b \tan ^2(c+d x)+a \tan (c+d x)+2 b\right )}{2 \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2}}dx}{3 a}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {\int \frac {2 b \tan ^2(c+d x)+a \tan (c+d x)+2 b}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2}}dx}{a}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {\int \frac {2 b \tan (c+d x)^2+a \tan (c+d x)+2 b}{\tan (c+d x)^{3/2} (a+b \tan (c+d x))^{5/2}}dx}{a}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {-\frac {2 \int -\frac {a^2-8 b^2-8 b^2 \tan ^2(c+d x)}{2 \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}dx}{a}-\frac {4 b}{a d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}}{a}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {\frac {\int \frac {a^2-8 b^2-8 b^2 \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}dx}{a}-\frac {4 b}{a d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}}{a}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {\frac {\int \frac {a^2-8 b^2-8 b^2 \tan (c+d x)^2}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}dx}{a}-\frac {4 b}{a d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}}{a}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 4133 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {\frac {\frac {2 \int \frac {3 a^4-3 b \tan (c+d x) a^3-14 b^2 a^2-16 b^4-2 b^2 \left (7 a^2+8 b^2\right ) \tan ^2(c+d x)}{2 \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}dx}{3 a \left (a^2+b^2\right )}-\frac {2 b^2 \left (7 a^2+8 b^2\right ) \sqrt {\tan (c+d x)}}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}}{a}-\frac {4 b}{a d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}}{a}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {\frac {\frac {\int \frac {3 a^4-3 b \tan (c+d x) a^3-14 b^2 a^2-16 b^4-2 b^2 \left (7 a^2+8 b^2\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}dx}{3 a \left (a^2+b^2\right )}-\frac {2 b^2 \left (7 a^2+8 b^2\right ) \sqrt {\tan (c+d x)}}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}}{a}-\frac {4 b}{a d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}}{a}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {\frac {\frac {\int \frac {3 a^4-3 b \tan (c+d x) a^3-14 b^2 a^2-16 b^4-2 b^2 \left (7 a^2+8 b^2\right ) \tan (c+d x)^2}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}dx}{3 a \left (a^2+b^2\right )}-\frac {2 b^2 \left (7 a^2+8 b^2\right ) \sqrt {\tan (c+d x)}}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}}{a}-\frac {4 b}{a d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}}{a}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {\frac {\frac {\frac {2 \int \frac {3 \left (a^4 \left (a^2-b^2\right )-2 a^5 b \tan (c+d x)\right )}{2 \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}-\frac {4 b^2 \left (4 a^4+15 a^2 b^2+8 b^4\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{3 a \left (a^2+b^2\right )}-\frac {2 b^2 \left (7 a^2+8 b^2\right ) \sqrt {\tan (c+d x)}}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}}{a}-\frac {4 b}{a d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}}{a}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {\frac {\frac {\frac {3 \int \frac {a^4 \left (a^2-b^2\right )-2 a^5 b \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}-\frac {4 b^2 \left (4 a^4+15 a^2 b^2+8 b^4\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{3 a \left (a^2+b^2\right )}-\frac {2 b^2 \left (7 a^2+8 b^2\right ) \sqrt {\tan (c+d x)}}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}}{a}-\frac {4 b}{a d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}}{a}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {\frac {\frac {\frac {3 \int \frac {a^4 \left (a^2-b^2\right )-2 a^5 b \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx}{a \left (a^2+b^2\right )}-\frac {4 b^2 \left (4 a^4+15 a^2 b^2+8 b^4\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{3 a \left (a^2+b^2\right )}-\frac {2 b^2 \left (7 a^2+8 b^2\right ) \sqrt {\tan (c+d x)}}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}}{a}-\frac {4 b}{a d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}}{a}-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}\right )\) |
| \(\Big \downarrow \) 4099 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {4 b}{a d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b^2 \left (7 a^2+8 b^2\right ) \sqrt {\tan (c+d x)}}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {4 b^2 \left (4 a^4+15 a^2 b^2+8 b^4\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {3 \left (\frac {1}{2} a^4 (a-i b)^2 \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} a^4 (a+i b)^2 \int \frac {i \tan (c+d x)+1}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx\right )}{a \left (a^2+b^2\right )}}{3 a \left (a^2+b^2\right )}}{a}}{a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {4 b}{a d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b^2 \left (7 a^2+8 b^2\right ) \sqrt {\tan (c+d x)}}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {4 b^2 \left (4 a^4+15 a^2 b^2+8 b^4\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {3 \left (\frac {1}{2} a^4 (a-i b)^2 \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} a^4 (a+i b)^2 \int \frac {i \tan (c+d x)+1}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx\right )}{a \left (a^2+b^2\right )}}{3 a \left (a^2+b^2\right )}}{a}}{a}\right )\) |
| \(\Big \downarrow \) 4098 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {4 b}{a d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b^2 \left (7 a^2+8 b^2\right ) \sqrt {\tan (c+d x)}}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {4 b^2 \left (4 a^4+15 a^2 b^2+8 b^4\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {3 \left (\frac {a^4 (a+i b)^2 \int \frac {1}{(1-i \tan (c+d x)) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{2 d}+\frac {a^4 (a-i b)^2 \int \frac {1}{(i \tan (c+d x)+1) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}d\tan (c+d x)}{2 d}\right )}{a \left (a^2+b^2\right )}}{3 a \left (a^2+b^2\right )}}{a}}{a}\right )\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {4 b}{a d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b^2 \left (7 a^2+8 b^2\right ) \sqrt {\tan (c+d x)}}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {4 b^2 \left (4 a^4+15 a^2 b^2+8 b^4\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {3 \left (\frac {a^4 (a-i b)^2 \int \frac {1}{\frac {(i a-b) \tan (c+d x)}{a+b \tan (c+d x)}+1}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}}{d}+\frac {a^4 (a+i b)^2 \int \frac {1}{1-\frac {(i a+b) \tan (c+d x)}{a+b \tan (c+d x)}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}}{d}\right )}{a \left (a^2+b^2\right )}}{3 a \left (a^2+b^2\right )}}{a}}{a}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {4 b}{a d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b^2 \left (7 a^2+8 b^2\right ) \sqrt {\tan (c+d x)}}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {4 b^2 \left (4 a^4+15 a^2 b^2+8 b^4\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {3 \left (\frac {a^4 (a+i b)^2 \int \frac {1}{1-\frac {(i a+b) \tan (c+d x)}{a+b \tan (c+d x)}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}}{d}+\frac {a^4 (a-i b)^2 \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {-b+i a}}\right )}{a \left (a^2+b^2\right )}}{3 a \left (a^2+b^2\right )}}{a}}{a}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {4 b}{a d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {2 b^2 \left (7 a^2+8 b^2\right ) \sqrt {\tan (c+d x)}}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {4 b^2 \left (4 a^4+15 a^2 b^2+8 b^4\right ) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {3 \left (\frac {a^4 (a-i b)^2 \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {-b+i a}}+\frac {a^4 (a+i b)^2 \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {b+i a}}\right )}{a \left (a^2+b^2\right )}}{3 a \left (a^2+b^2\right )}}{a}}{a}\right )\) |
Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-2/(3*a*d*Tan[c + d*x]^(3/2)*(a + b *Tan[c + d*x])^(3/2)) - ((-4*b)/(a*d*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x ])^(3/2)) + ((-2*b^2*(7*a^2 + 8*b^2)*Sqrt[Tan[c + d*x]])/(3*a*(a^2 + b^2)* d*(a + b*Tan[c + d*x])^(3/2)) + ((3*((a^4*(a - I*b)^2*ArcTan[(Sqrt[I*a - b ]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a - b]*d) + (a^4* (a + I*b)^2*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a + b]*d)))/(a*(a^2 + b^2)) - (4*b^2*(4*a^4 + 15*a^2*b^2 + 8*b^4)*Sqrt[Tan[c + d*x]])/(a*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]]))/(3 *a*(a^2 + b^2)))/a)/a)
3.9.74.3.1 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_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + Simp[1 /((m + 1)*(a^2 + b^2)*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x], x], x] / ; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || Integ erQ[m]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[A^2/f Subst[Int[(a + b*x)^m*((c + d*x)^n/(A - B*x)), x], x, Tan[e + f* x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[A^2 + B^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(A + I*B)/2 Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 - I*T an[e + f*x]), x], x] + Simp[(A - I*B)/2 Int[(a + b*Tan[e + f*x])^m*(c + d *Tan[e + f*x])^n*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A , B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[A^2 + B^2, 0]
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[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m Int[ActivateTrig[u]/(c*Tan[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownTangentIntegrandQ[u, x]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 18.10 (sec) , antiderivative size = 19915, normalized size of antiderivative = 58.92
Leaf count of result is larger than twice the leaf count of optimal. 11729 vs. \(2 (284) = 568\).
Time = 2.65 (sec) , antiderivative size = 11729, normalized size of antiderivative = 34.70 \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\int { \frac {\cot \left (d x + c\right )^{\frac {5}{2}}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Exception generated. \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(con st gen &
Timed out. \[ \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^{5/2}}{{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2}} \,d x \]