Integrand size = 21, antiderivative size = 121 \[ \int \frac {\tan ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=-\frac {\log (\cos (c+d x))}{a^2 d}+\frac {\left (a^2-b^2\right ) \left (3 a^2+b^2\right ) \log (a+b \sec (c+d x))}{a^2 b^4 d}-\frac {2 a \sec (c+d x)}{b^3 d}+\frac {\sec ^2(c+d x)}{2 b^2 d}+\frac {\left (a^2-b^2\right )^2}{a b^4 d (a+b \sec (c+d x))} \]
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Time = 0.13 (sec) , antiderivative size = 121, 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} \[ \int \frac {\tan ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\left (a^2-b^2\right )^2}{a b^4 d (a+b \sec (c+d x))}+\frac {\left (3 a^2+b^2\right ) \left (a^2-b^2\right ) \log (a+b \sec (c+d x))}{a^2 b^4 d}-\frac {\log (\cos (c+d x))}{a^2 d}-\frac {2 a \sec (c+d x)}{b^3 d}+\frac {\sec ^2(c+d x)}{2 b^2 d} \]
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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)^2} \, dx,x,b \sec (c+d x)\right )}{b^4 d} \\ & = \frac {\text {Subst}\left (\int \left (-2 a+\frac {b^4}{a^2 x}+x-\frac {\left (a^2-b^2\right )^2}{a (a+x)^2}+\frac {\left (a^2-b^2\right ) \left (3 a^2+b^2\right )}{a^2 (a+x)}\right ) \, dx,x,b \sec (c+d x)\right )}{b^4 d} \\ & = -\frac {\log (\cos (c+d x))}{a^2 d}+\frac {\left (a^2-b^2\right ) \left (3 a^2+b^2\right ) \log (a+b \sec (c+d x))}{a^2 b^4 d}-\frac {2 a \sec (c+d x)}{b^3 d}+\frac {\sec ^2(c+d x)}{2 b^2 d}+\frac {\left (a^2-b^2\right )^2}{a b^4 d (a+b \sec (c+d x))} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.88 \[ \int \frac {\tan ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {-\frac {b^4 \log (\cos (c+d x))}{a^2}+\frac {(a-b) (a+b) \left (3 a^2+b^2\right ) \log (a+b \sec (c+d x))}{a^2}-2 a b \sec (c+d x)+\frac {1}{2} b^2 \sec ^2(c+d x)+\frac {\left (a^2-b^2\right )^2}{a (a+b \sec (c+d x))}}{b^4 d} \]
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Time = 1.27 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.05
method | result | size |
derivativedivides | \(\frac {\frac {\left (-3 a^{2}+2 b^{2}\right ) \ln \left (\cos \left (d x +c \right )\right )}{b^{4}}+\frac {1}{2 b^{2} \cos \left (d x +c \right )^{2}}-\frac {2 a}{b^{3} \cos \left (d x +c \right )}-\frac {a^{4}-2 a^{2} b^{2}+b^{4}}{a^{2} b^{3} \left (b +a \cos \left (d x +c \right )\right )}+\frac {\left (3 a^{4}-2 a^{2} b^{2}-b^{4}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{b^{4} a^{2}}}{d}\) | \(127\) |
default | \(\frac {\frac {\left (-3 a^{2}+2 b^{2}\right ) \ln \left (\cos \left (d x +c \right )\right )}{b^{4}}+\frac {1}{2 b^{2} \cos \left (d x +c \right )^{2}}-\frac {2 a}{b^{3} \cos \left (d x +c \right )}-\frac {a^{4}-2 a^{2} b^{2}+b^{4}}{a^{2} b^{3} \left (b +a \cos \left (d x +c \right )\right )}+\frac {\left (3 a^{4}-2 a^{2} b^{2}-b^{4}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{b^{4} a^{2}}}{d}\) | \(127\) |
risch | \(\frac {i x}{a^{2}}+\frac {2 i c}{a^{2} d}-\frac {2 \left (3 a^{4} {\mathrm e}^{5 i \left (d x +c \right )}-2 a^{2} b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+b^{4} {\mathrm e}^{5 i \left (d x +c \right )}+3 a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}+6 a^{4} {\mathrm e}^{3 i \left (d x +c \right )}-6 a^{2} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+2 b^{4} {\mathrm e}^{3 i \left (d x +c \right )}+3 a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}+3 a^{4} {\mathrm e}^{i \left (d x +c \right )}-2 a^{2} b^{2} {\mathrm e}^{i \left (d x +c \right )}+b^{4} {\mathrm e}^{i \left (d x +c \right )}\right )}{d \,b^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} a^{2} \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{2}}{b^{4} d}+\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{2} d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{b^{4} d}-\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{b^{2} d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{2} d}\) | \(385\) |
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Time = 0.32 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.81 \[ \int \frac {\tan ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=-\frac {3 \, a^{3} b^{2} \cos \left (d x + c\right ) - a^{2} b^{3} + 2 \, {\left (3 \, a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left ({\left (3 \, a^{5} - 2 \, a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )^{3} + {\left (3 \, a^{4} b - 2 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) + 2 \, {\left ({\left (3 \, a^{5} - 2 \, a^{3} b^{2}\right )} \cos \left (d x + c\right )^{3} + {\left (3 \, a^{4} b - 2 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\cos \left (d x + c\right )\right )}{2 \, {\left (a^{3} b^{4} d \cos \left (d x + c\right )^{3} + a^{2} b^{5} d \cos \left (d x + c\right )^{2}\right )}} \]
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\[ \int \frac {\tan ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\int \frac {\tan ^{5}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.23 \[ \int \frac {\tan ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=-\frac {\frac {3 \, a^{3} b \cos \left (d x + c\right ) - a^{2} b^{2} + 2 \, {\left (3 \, a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}}{a^{3} b^{3} \cos \left (d x + c\right )^{3} + a^{2} b^{4} \cos \left (d x + c\right )^{2}} + \frac {2 \, {\left (3 \, a^{2} - 2 \, b^{2}\right )} \log \left (\cos \left (d x + c\right )\right )}{b^{4}} - \frac {2 \, {\left (3 \, a^{4} - 2 \, a^{2} b^{2} - b^{4}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{2} b^{4}}}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 568 vs. \(2 (119) = 238\).
Time = 1.56 (sec) , antiderivative size = 568, normalized size of antiderivative = 4.69 \[ \int \frac {\tan ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\frac {2 \, {\left (3 \, a^{5} - 3 \, a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - a b^{4} + b^{5}\right )} \log \left ({\left | a + b + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{3} b^{4} - a^{2} b^{5}} + \frac {2 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{2}} - \frac {2 \, {\left (3 \, a^{2} - 2 \, 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 {9 \, a^{2} - 8 \, a b - 6 \, b^{2} + \frac {18 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {8 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {16 \, b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {9 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {6 \, b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{b^{4} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}} - \frac {2 \, {\left (3 \, a^{5} + 5 \, a^{4} b - 4 \, a^{2} b^{3} - 3 \, a b^{4} - b^{5} + \frac {3 \, a^{5} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {3 \, a^{4} b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {2 \, a^{3} b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {2 \, a^{2} b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {a b^{4} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {b^{5} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{{\left (a + b + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )} a^{2} b^{4}}}{2 \, d} \]
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Time = 14.90 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.36 \[ \int \frac {\tan ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a^2\,d}-\frac {\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-6\,a^3-3\,a^2\,b+a\,b^2+2\,b^3\right )}{a\,b^3}-\frac {2\,\left (-3\,a^3-3\,a^2\,b+a\,b^2+b^3\right )}{a\,b^3}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (a-b\right )\,\left (3\,a^2+3\,a\,b+b^2\right )}{a\,b^3}}{d\,\left (\left (b-a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (3\,a-b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (-3\,a-b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a+b\right )}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )\,\left (3\,a^2-2\,b^2\right )}{b^4\,d}-\frac {\ln \left (a+b-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )\,\left (-3\,a^4+2\,a^2\,b^2+b^4\right )}{a^2\,b^4\,d} \]
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