Integrand size = 23, antiderivative size = 291 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{5/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{5/2} d}-\frac {2 a^2 \tan ^3(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}-\frac {4 a \left (8 a^4+15 a^2 b^2+4 b^4\right ) \sqrt {a+b \tan (c+d x)}}{3 b^4 \left (a^2+b^2\right )^2 d}+\frac {2 \left (8 a^4+15 a^2 b^2+b^4\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b^3 \left (a^2+b^2\right )^2 d} \]
-arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/(a-I*b)^(5/2)/d-arctanh((a+ b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/(a+I*b)^(5/2)/d-4/3*a*(8*a^4+15*a^2*b^2 +4*b^4)*(a+b*tan(d*x+c))^(1/2)/b^4/(a^2+b^2)^2/d+2/3*(8*a^4+15*a^2*b^2+b^4 )*(a+b*tan(d*x+c))^(1/2)*tan(d*x+c)/b^3/(a^2+b^2)^2/d-4*a^2*(a^2+2*b^2)*ta n(d*x+c)^2/b^2/(a^2+b^2)^2/d/(a+b*tan(d*x+c))^(1/2)-2/3*a^2*tan(d*x+c)^3/b /(a^2+b^2)/d/(a+b*tan(d*x+c))^(3/2)
Time = 5.97 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.21 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\frac {2 \left (-\frac {3 b \left (-2 a b^2+a^2 \sqrt {-b^2}+\left (-b^2\right )^{3/2}\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {-b^2}}}\right )}{2 \sqrt {-b^2} \left (a^2+b^2\right )^2 \sqrt {a-\sqrt {-b^2}}}-\frac {3 b \left (2 a b^2+a^2 \sqrt {-b^2}+\left (-b^2\right )^{3/2}\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+\sqrt {-b^2}}}\right )}{2 \sqrt {-b^2} \left (a^2+b^2\right )^2 \sqrt {a+\sqrt {-b^2}}}+\frac {a^3 \left (8 a^2+7 b^2\right )}{b^3 \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {6 a \tan ^2(c+d x)}{b (a+b \tan (c+d x))^{3/2}}+\frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^{3/2}}-\frac {3 a^2 \left (8 a^4+15 a^2 b^2+5 b^4\right )}{b^3 \left (a^2+b^2\right )^2 \sqrt {a+b \tan (c+d x)}}\right )}{3 b d} \]
(2*((-3*b*(-2*a*b^2 + a^2*Sqrt[-b^2] + (-b^2)^(3/2))*ArcTanh[Sqrt[a + b*Ta n[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/(2*Sqrt[-b^2]*(a^2 + b^2)^2*Sqrt[a - Sq rt[-b^2]]) - (3*b*(2*a*b^2 + a^2*Sqrt[-b^2] + (-b^2)^(3/2))*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/(2*Sqrt[-b^2]*(a^2 + b^2)^2*Sqrt [a + Sqrt[-b^2]]) + (a^3*(8*a^2 + 7*b^2))/(b^3*(a^2 + b^2)*(a + b*Tan[c + d*x])^(3/2)) - (6*a*Tan[c + d*x]^2)/(b*(a + b*Tan[c + d*x])^(3/2)) + Tan[c + d*x]^3/(a + b*Tan[c + d*x])^(3/2) - (3*a^2*(8*a^4 + 15*a^2*b^2 + 5*b^4) )/(b^3*(a^2 + b^2)^2*Sqrt[a + b*Tan[c + d*x]])))/(3*b*d)
Time = 1.77 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.10, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {3042, 4048, 27, 3042, 4128, 27, 3042, 4130, 27, 3042, 4113, 3042, 4022, 3042, 4020, 25, 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.
\(\displaystyle \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\tan (c+d x)^5}{(a+b \tan (c+d x))^{5/2}}dx\) |
\(\Big \downarrow \) 4048 |
\(\displaystyle \frac {2 \int \frac {3 \tan ^2(c+d x) \left (2 a^2-b \tan (c+d x) a+\left (2 a^2+b^2\right ) \tan ^2(c+d x)\right )}{2 (a+b \tan (c+d x))^{3/2}}dx}{3 b \left (a^2+b^2\right )}-\frac {2 a^2 \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\tan ^2(c+d x) \left (2 a^2-b \tan (c+d x) a+\left (2 a^2+b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^{3/2}}dx}{b \left (a^2+b^2\right )}-\frac {2 a^2 \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\tan (c+d x)^2 \left (2 a^2-b \tan (c+d x) a+\left (2 a^2+b^2\right ) \tan (c+d x)^2\right )}{(a+b \tan (c+d x))^{3/2}}dx}{b \left (a^2+b^2\right )}-\frac {2 a^2 \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 4128 |
\(\displaystyle \frac {\frac {2 \int \frac {\tan (c+d x) \left (-2 a \tan (c+d x) b^3+\left (8 a^4+15 b^2 a^2+b^4\right ) \tan ^2(c+d x)+8 a^2 \left (a^2+2 b^2\right )\right )}{2 \sqrt {a+b \tan (c+d x)}}dx}{b \left (a^2+b^2\right )}-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{b \left (a^2+b^2\right )}-\frac {2 a^2 \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\int \frac {\tan (c+d x) \left (-2 a \tan (c+d x) b^3+\left (8 a^4+15 b^2 a^2+b^4\right ) \tan ^2(c+d x)+8 a^2 \left (a^2+2 b^2\right )\right )}{\sqrt {a+b \tan (c+d x)}}dx}{b \left (a^2+b^2\right )}-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{b \left (a^2+b^2\right )}-\frac {2 a^2 \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\int \frac {\tan (c+d x) \left (-2 a \tan (c+d x) b^3+\left (8 a^4+15 b^2 a^2+b^4\right ) \tan (c+d x)^2+8 a^2 \left (a^2+2 b^2\right )\right )}{\sqrt {a+b \tan (c+d x)}}dx}{b \left (a^2+b^2\right )}-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{b \left (a^2+b^2\right )}-\frac {2 a^2 \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 4130 |
\(\displaystyle \frac {\frac {\frac {2 \int -\frac {-3 \left (a^2-b^2\right ) \tan (c+d x) b^3+2 a \left (8 a^4+15 b^2 a^2+4 b^4\right ) \tan ^2(c+d x)+2 a \left (8 a^4+15 b^2 a^2+b^4\right )}{2 \sqrt {a+b \tan (c+d x)}}dx}{3 b}+\frac {2 \left (8 a^4+15 a^2 b^2+b^4\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}}{b \left (a^2+b^2\right )}-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{b \left (a^2+b^2\right )}-\frac {2 a^2 \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\frac {2 \left (8 a^4+15 a^2 b^2+b^4\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}-\frac {\int \frac {-3 \left (a^2-b^2\right ) \tan (c+d x) b^3+2 a \left (8 a^4+15 b^2 a^2+4 b^4\right ) \tan ^2(c+d x)+2 a \left (8 a^4+15 b^2 a^2+b^4\right )}{\sqrt {a+b \tan (c+d x)}}dx}{3 b}}{b \left (a^2+b^2\right )}-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{b \left (a^2+b^2\right )}-\frac {2 a^2 \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {2 \left (8 a^4+15 a^2 b^2+b^4\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}-\frac {\int \frac {-3 \left (a^2-b^2\right ) \tan (c+d x) b^3+2 a \left (8 a^4+15 b^2 a^2+4 b^4\right ) \tan (c+d x)^2+2 a \left (8 a^4+15 b^2 a^2+b^4\right )}{\sqrt {a+b \tan (c+d x)}}dx}{3 b}}{b \left (a^2+b^2\right )}-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{b \left (a^2+b^2\right )}-\frac {2 a^2 \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 4113 |
\(\displaystyle \frac {\frac {\frac {2 \left (8 a^4+15 a^2 b^2+b^4\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}-\frac {\int \frac {-6 a b^4-3 \left (a^2-b^2\right ) \tan (c+d x) b^3}{\sqrt {a+b \tan (c+d x)}}dx+\frac {4 a \left (8 a^4+15 a^2 b^2+4 b^4\right ) \sqrt {a+b \tan (c+d x)}}{b d}}{3 b}}{b \left (a^2+b^2\right )}-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{b \left (a^2+b^2\right )}-\frac {2 a^2 \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\frac {2 \left (8 a^4+15 a^2 b^2+b^4\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}-\frac {\int \frac {-6 a b^4-3 \left (a^2-b^2\right ) \tan (c+d x) b^3}{\sqrt {a+b \tan (c+d x)}}dx+\frac {4 a \left (8 a^4+15 a^2 b^2+4 b^4\right ) \sqrt {a+b \tan (c+d x)}}{b d}}{3 b}}{b \left (a^2+b^2\right )}-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{b \left (a^2+b^2\right )}-\frac {2 a^2 \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 4022 |
\(\displaystyle -\frac {2 a^2 \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {2 \left (8 a^4+15 a^2 b^2+b^4\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}-\frac {-\frac {3}{2} i b^3 (a-i b)^2 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {3}{2} i b^3 (a+i b)^2 \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx+\frac {4 a \left (8 a^4+15 a^2 b^2+4 b^4\right ) \sqrt {a+b \tan (c+d x)}}{b d}}{3 b}}{b \left (a^2+b^2\right )}}{b \left (a^2+b^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 a^2 \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {2 \left (8 a^4+15 a^2 b^2+b^4\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}-\frac {-\frac {3}{2} i b^3 (a-i b)^2 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {3}{2} i b^3 (a+i b)^2 \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx+\frac {4 a \left (8 a^4+15 a^2 b^2+4 b^4\right ) \sqrt {a+b \tan (c+d x)}}{b d}}{3 b}}{b \left (a^2+b^2\right )}}{b \left (a^2+b^2\right )}\) |
\(\Big \downarrow \) 4020 |
\(\displaystyle -\frac {2 a^2 \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {2 \left (8 a^4+15 a^2 b^2+b^4\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}-\frac {-\frac {3 b^3 (a+i b)^2 \int -\frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}-\frac {3 b^3 (a-i b)^2 \int -\frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}+\frac {4 a \left (8 a^4+15 a^2 b^2+4 b^4\right ) \sqrt {a+b \tan (c+d x)}}{b d}}{3 b}}{b \left (a^2+b^2\right )}}{b \left (a^2+b^2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {2 a^2 \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {2 \left (8 a^4+15 a^2 b^2+b^4\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}-\frac {\frac {3 b^3 (a+i b)^2 \int \frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}+\frac {3 b^3 (a-i b)^2 \int \frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}+\frac {4 a \left (8 a^4+15 a^2 b^2+4 b^4\right ) \sqrt {a+b \tan (c+d x)}}{b d}}{3 b}}{b \left (a^2+b^2\right )}}{b \left (a^2+b^2\right )}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {2 a^2 \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {2 \left (8 a^4+15 a^2 b^2+b^4\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}-\frac {-\frac {3 i b^2 (a-i b)^2 \int \frac {1}{-\frac {i \tan ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{d}+\frac {3 i b^2 (a+i b)^2 \int \frac {1}{\frac {i \tan ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{d}+\frac {4 a \left (8 a^4+15 a^2 b^2+4 b^4\right ) \sqrt {a+b \tan (c+d x)}}{b d}}{3 b}}{b \left (a^2+b^2\right )}}{b \left (a^2+b^2\right )}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {2 a^2 \tan ^3(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {-\frac {4 a^2 \left (a^2+2 b^2\right ) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {2 \left (8 a^4+15 a^2 b^2+b^4\right ) \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}-\frac {\frac {4 a \left (8 a^4+15 a^2 b^2+4 b^4\right ) \sqrt {a+b \tan (c+d x)}}{b d}+\frac {3 i b^3 (a+i b)^2 \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}-\frac {3 i b^3 (a-i b)^2 \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}}{3 b}}{b \left (a^2+b^2\right )}}{b \left (a^2+b^2\right )}\) |
(-2*a^2*Tan[c + d*x]^3)/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) + ( (-4*a^2*(a^2 + 2*b^2)*Tan[c + d*x]^2)/(b*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]]) + ((2*(8*a^4 + 15*a^2*b^2 + b^4)*Tan[c + d*x]*Sqrt[a + b*Tan[c + d* x]])/(3*b*d) - (((3*I)*(a + I*b)^2*b^3*ArcTan[Tan[c + d*x]/Sqrt[a - I*b]]) /(Sqrt[a - I*b]*d) - ((3*I)*(a - I*b)^2*b^3*ArcTan[Tan[c + d*x]/Sqrt[a + I *b]])/(Sqrt[a + I*b]*d) + (4*a*(8*a^4 + 15*a^2*b^2 + 4*b^4)*Sqrt[a + b*Tan [c + d*x]])/(b*d))/(3*b))/(b*(a^2 + b^2)))/(b*(a^2 + b^2))
3.6.46.3.1 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*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_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && !LeQ[m, -1]
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[((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[C*(a + b*Tan[e + f*x])^m*((c + d*Tan[ e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C *(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f*x] - (C* m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[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] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[ c, 0] && NeQ[a, 0])))
Leaf count of result is larger than twice the leaf count of optimal. \(2208\) vs. \(2(261)=522\).
Time = 0.09 (sec) , antiderivative size = 2209, normalized size of antiderivative = 7.59
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(2209\) |
default | \(\text {Expression too large to display}\) | \(2209\) |
-6/d/b^4*a*(a+b*tan(d*x+c))^(1/2)+2/d*b^2/(a^2+b^2)^(5/2)/(2*(a^2+b^2)^(1/ 2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/ 2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a-1/2/d*b^2/(a^2+b^2)^(7/2)*ln(b*tan(d* x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2 ))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^2-1/2/d*b^2/(a^2+b^2)^3*ln((a+b*tan(d*x +c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*( 2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a+1/2/d*b^2/(a^2+b^2)^3*ln(b*tan(d*x+c)+a+(a+ b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2 +b^2)^(1/2)+2*a)^(1/2)*a+1/2/d*b^2/(a^2+b^2)^(7/2)*ln((a+b*tan(d*x+c))^(1/ 2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*(2*(a^2+b ^2)^(1/2)+2*a)^(1/2)*a^2-6/d*b^2/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^(1/2)-2*a)^( 1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a ^2+b^2)^(1/2)-2*a)^(1/2))*a^3-5/d*b^4/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^(1/2)-2 *a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/ (2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a-2/d*b^2/(a^2+b^2)^(5/2)/(2*(a^2+b^2)^(1/2 )-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2 ))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a+6/d*b^2/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^( 1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^( 1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a^3+5/d*b^4/(a^2+b^2)^(7/2)/(2*(a^2+b ^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2...
Leaf count of result is larger than twice the leaf count of optimal. 3401 vs. \(2 (257) = 514\).
Time = 0.33 (sec) , antiderivative size = 3401, normalized size of antiderivative = 11.69 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
1/6*(3*((a^4*b^6 + 2*a^2*b^8 + b^10)*d*tan(d*x + c)^2 + 2*(a^5*b^5 + 2*a^3 *b^7 + a*b^9)*d*tan(d*x + c) + (a^6*b^4 + 2*a^4*b^6 + a^2*b^8)*d)*sqrt((a^ 5 - 10*a^3*b^2 + 5*a*b^4 + (a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5 *a^2*b^8 + b^10)*d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^ 2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12 *b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2* b^18 + b^20)*d^4)))/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b ^8 + b^10)*d^2))*log((5*a^4 - 10*a^2*b^2 + b^4)*sqrt(b*tan(d*x + c) + a) + ((3*a^12 + 14*a^10*b^2 + 25*a^8*b^4 + 20*a^6*b^6 + 5*a^4*b^8 - 2*a^2*b^10 - b^12)*d^3*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 2 52*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b ^20)*d^4)) - (5*a^7 - 25*a^5*b^2 + 31*a^3*b^4 - 3*a*b^6)*d)*sqrt((a^5 - 10 *a^3*b^2 + 5*a*b^4 + (a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b ^8 + b^10)*d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)))/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b ^10)*d^2))) - 3*((a^4*b^6 + 2*a^2*b^8 + b^10)*d*tan(d*x + c)^2 + 2*(a^5*b^ 5 + 2*a^3*b^7 + a*b^9)*d*tan(d*x + c) + (a^6*b^4 + 2*a^4*b^6 + a^2*b^8)...
\[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\int \frac {\tan ^{5}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
Timed out. \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Timed out} \]
Time = 16.33 (sec) , antiderivative size = 4681, normalized size of antiderivative = 16.09 \[ \int \frac {\tan ^5(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
atan(((((1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^ 2*10i - 10*a^3*b^2*d^2))^(1/2)*(48*a*b^20*d^4 - ((1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^18 *d^5 + 7680*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 1344 0*a^13*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b^2*d^5))/4 + 368*a^3*b^18*d^4 + 1216*a^5*b^16*d^4 + 2240*a^7*b ^14*d^4 + 2464*a^9*b^12*d^4 + 1568*a^11*b^10*d^4 + 448*a^13*b^8*d^4 - 64*a ^15*b^6*d^4 - 80*a^17*b^4*d^4 - 16*a^19*b^2*d^4))/2 - ((a + b*tan(c + d*x) )^(1/2)*(320*a^4*b^14*d^3 - 16*b^18*d^3 + 1024*a^6*b^12*d^3 + 1440*a^8*b^1 0*d^3 + 1024*a^10*b^8*d^3 + 320*a^12*b^6*d^3 - 16*a^16*b^2*d^3))/2)*(1/(a^ 5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2*10i - 10*a^3 *b^2*d^2))^(1/2)*1i - (((1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^2 *5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d^2))^(1/2)*(48*a*b^20*d^4 + ((1/(a^5*d ^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2*10i - 10*a^3*b^ 2*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 16128*a^11 *b^12*d^5 + 13440*a^13*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b^2*d^5))/4 + 368*a^3*b^18*d^4 + 1216*a^5*b^16* d^4 + 2240*a^7*b^14*d^4 + 2464*a^9*b^12*d^4 + 1568*a^11*b^10*d^4 + 448*...