Integrand size = 21, antiderivative size = 108 \[ \int \frac {\tan ^5(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {\log (\cos (c+d x))}{a d}-\frac {\left (a^2-b^2\right )^2 \log (a+b \sec (c+d x))}{a b^4 d}+\frac {\left (a^2-2 b^2\right ) \sec (c+d x)}{b^3 d}-\frac {a \sec ^2(c+d x)}{2 b^2 d}+\frac {\sec ^3(c+d x)}{3 b d} \]
[Out]
Time = 0.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3970, 908} \[ \int \frac {\tan ^5(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {\left (a^2-b^2\right )^2 \log (a+b \sec (c+d x))}{a b^4 d}+\frac {\left (a^2-2 b^2\right ) \sec (c+d x)}{b^3 d}-\frac {a \sec ^2(c+d x)}{2 b^2 d}-\frac {\log (\cos (c+d x))}{a d}+\frac {\sec ^3(c+d x)}{3 b d} \]
[In]
[Out]
Rule 908
Rule 3970
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (b^2-x^2\right )^2}{x (a+x)} \, dx,x,b \sec (c+d x)\right )}{b^4 d} \\ & = \frac {\text {Subst}\left (\int \left (a^2 \left (1-\frac {2 b^2}{a^2}\right )+\frac {b^4}{a x}-a x+x^2-\frac {\left (a^2-b^2\right )^2}{a (a+x)}\right ) \, dx,x,b \sec (c+d x)\right )}{b^4 d} \\ & = -\frac {\log (\cos (c+d x))}{a d}-\frac {\left (a^2-b^2\right )^2 \log (a+b \sec (c+d x))}{a b^4 d}+\frac {\left (a^2-2 b^2\right ) \sec (c+d x)}{b^3 d}-\frac {a \sec ^2(c+d x)}{2 b^2 d}+\frac {\sec ^3(c+d x)}{3 b d} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.91 \[ \int \frac {\tan ^5(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {-\frac {b^4 \log (\cos (c+d x))}{a}-\frac {\left (a^2-b^2\right )^2 \log (a+b \sec (c+d x))}{a}+b \left (a^2-2 b^2\right ) \sec (c+d x)-\frac {1}{2} a b^2 \sec ^2(c+d x)+\frac {1}{3} b^3 \sec ^3(c+d x)}{b^4 d} \]
[In]
[Out]
Time = 1.24 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(\frac {\frac {\left (-a^{4}+2 a^{2} b^{2}-b^{4}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{b^{4} a}-\frac {a}{2 b^{2} \cos \left (d x +c \right )^{2}}-\frac {-a^{2}+2 b^{2}}{b^{3} \cos \left (d x +c \right )}+\frac {\left (a^{2}-2 b^{2}\right ) a \ln \left (\cos \left (d x +c \right )\right )}{b^{4}}+\frac {1}{3 b \cos \left (d x +c \right )^{3}}}{d}\) | \(115\) |
default | \(\frac {\frac {\left (-a^{4}+2 a^{2} b^{2}-b^{4}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{b^{4} a}-\frac {a}{2 b^{2} \cos \left (d x +c \right )^{2}}-\frac {-a^{2}+2 b^{2}}{b^{3} \cos \left (d x +c \right )}+\frac {\left (a^{2}-2 b^{2}\right ) a \ln \left (\cos \left (d x +c \right )\right )}{b^{4}}+\frac {1}{3 b \cos \left (d x +c \right )^{3}}}{d}\) | \(115\) |
risch | \(\frac {i x}{a}+\frac {2 i c}{a d}+\frac {2 a^{2} {\mathrm e}^{5 i \left (d x +c \right )}-4 b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-2 a b \,{\mathrm e}^{4 i \left (d x +c \right )}+4 a^{2} {\mathrm e}^{3 i \left (d x +c \right )}-\frac {16 b^{2} {\mathrm e}^{3 i \left (d x +c \right )}}{3}-2 a b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 a^{2} {\mathrm e}^{i \left (d x +c \right )}-4 b^{2} {\mathrm e}^{i \left (d x +c \right )}}{d \,b^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{b^{4} d}+\frac {2 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{b^{2} d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a d}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{4} d}-\frac {2 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{2} d}\) | \(303\) |
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.19 \[ \int \frac {\tan ^5(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {3 \, a^{2} b^{2} \cos \left (d x + c\right ) + 6 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{3} \log \left (a \cos \left (d x + c\right ) + b\right ) - 6 \, {\left (a^{4} - 2 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} \log \left (-\cos \left (d x + c\right )\right ) - 2 \, a b^{3} - 6 \, {\left (a^{3} b - 2 \, a b^{3}\right )} \cos \left (d x + c\right )^{2}}{6 \, a b^{4} d \cos \left (d x + c\right )^{3}} \]
[In]
[Out]
\[ \int \frac {\tan ^5(c+d x)}{a+b \sec (c+d x)} \, dx=\int \frac {\tan ^{5}{\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.02 \[ \int \frac {\tan ^5(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\frac {6 \, {\left (a^{3} - 2 \, a b^{2}\right )} \log \left (\cos \left (d x + c\right )\right )}{b^{4}} - \frac {6 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a b^{4}} - \frac {3 \, a b \cos \left (d x + c\right ) - 6 \, {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, b^{2}}{b^{3} \cos \left (d x + c\right )^{3}}}{6 \, d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 560 vs. \(2 (104) = 208\).
Time = 1.53 (sec) , antiderivative size = 560, normalized size of antiderivative = 5.19 \[ \int \frac {\tan ^5(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {\frac {3 \, {\left (a^{3} - 2 \, a b^{2}\right )} \log \left ({\left | a + b - \frac {2 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} \right |}\right )}{b^{4}} - \frac {6 \, {\left (a^{3} - 2 \, a b^{2}\right )} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{b^{4}} - \frac {3 \, {\left (a^{4} - 2 \, a^{2} b^{2} + 2 \, b^{4}\right )} \log \left (\frac {{\left | 2 \, b + \frac {2 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {2 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - 2 \, {\left | a \right |} \right |}}{{\left | 2 \, b + \frac {2 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {2 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + 2 \, {\left | a \right |} \right |}}\right )}{b^{4} {\left | a \right |}} + \frac {11 \, a^{3} - 12 \, a^{2} b - 22 \, a b^{2} + 20 \, b^{3} + \frac {33 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {24 \, a^{2} b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {78 \, a b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {48 \, b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {33 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {12 \, a^{2} b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {78 \, a b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {12 \, b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {11 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {22 \, a b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{b^{4} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{3}}}{6 \, d} \]
[In]
[Out]
Time = 15.08 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.10 \[ \int \frac {\tan ^5(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d}-\frac {\frac {2\,\left (3\,a^2-5\,b^2\right )}{3\,b^3}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (2\,a^2+a\,b-4\,b^2\right )}{b^3}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (a^2+a\,b-b^2\right )}{b^3}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}+\frac {a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )\,\left (a^2-2\,b^2\right )}{b^4\,d}-\frac {\ln \left (a+b-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )\,{\left (a^2-b^2\right )}^2}{a\,b^4\,d} \]
[In]
[Out]