Integrand size = 21, antiderivative size = 74 \[ \int \frac {\tan ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\log (\cos (c+d x))}{a^2 d}+\frac {\left (a^2+b^2\right ) \log (a+b \sec (c+d x))}{a^2 b^2 d}+\frac {a^2-b^2}{a b^2 d (a+b \sec (c+d x))} \]
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Time = 0.09 (sec) , antiderivative size = 74, 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} \[ \int \frac {\tan ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {a^2-b^2}{a b^2 d (a+b \sec (c+d x))}+\frac {\left (a^2+b^2\right ) \log (a+b \sec (c+d x))}{a^2 b^2 d}+\frac {\log (\cos (c+d x))}{a^2 d} \]
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Rule 908
Rule 3970
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {b^2-x^2}{x (a+x)^2} \, dx,x,b \sec (c+d x)\right )}{b^2 d} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {b^2}{a^2 x}+\frac {a^2-b^2}{a (a+x)^2}+\frac {-a^2-b^2}{a^2 (a+x)}\right ) \, dx,x,b \sec (c+d x)\right )}{b^2 d} \\ & = \frac {\log (\cos (c+d x))}{a^2 d}+\frac {\left (a^2+b^2\right ) \log (a+b \sec (c+d x))}{a^2 b^2 d}+\frac {a^2-b^2}{a b^2 d (a+b \sec (c+d x))} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.84 \[ \int \frac {\tan ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx=-\frac {\frac {b-\frac {b^3}{a^2}}{b+a \cos (c+d x)}+\log (\cos (c+d x))-\frac {\left (a^2+b^2\right ) \log (b+a \cos (c+d x))}{a^2}}{b^2 d} \]
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Time = 0.84 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {-\frac {a^{2}-b^{2}}{a^{2} b \left (b +a \cos \left (d x +c \right )\right )}+\frac {\left (a^{2}+b^{2}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{b^{2} a^{2}}-\frac {\ln \left (\cos \left (d x +c \right )\right )}{b^{2}}}{d}\) | \(72\) |
default | \(\frac {-\frac {a^{2}-b^{2}}{a^{2} b \left (b +a \cos \left (d x +c \right )\right )}+\frac {\left (a^{2}+b^{2}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{b^{2} a^{2}}-\frac {\ln \left (\cos \left (d x +c \right )\right )}{b^{2}}}{d}\) | \(72\) |
risch | \(-\frac {i x}{a^{2}}-\frac {2 i c}{a^{2} d}-\frac {2 \left (a^{2}-b^{2}\right ) {\mathrm e}^{i \left (d x +c \right )}}{a^{2} b d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{b^{2} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{2} d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{2} d}\) | \(163\) |
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Time = 0.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.38 \[ \int \frac {\tan ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx=-\frac {a^{2} b - b^{3} - {\left (a^{2} b + b^{3} + {\left (a^{3} + a b^{2}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) + {\left (a^{3} \cos \left (d x + c\right ) + a^{2} b\right )} \log \left (-\cos \left (d x + c\right )\right )}{a^{3} b^{2} d \cos \left (d x + c\right ) + a^{2} b^{3} d} \]
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\[ \int \frac {\tan ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\int \frac {\tan ^{3}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \frac {\tan ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx=-\frac {\frac {a^{2} - b^{2}}{a^{3} b \cos \left (d x + c\right ) + a^{2} b^{2}} + \frac {\log \left (\cos \left (d x + c\right )\right )}{b^{2}} - \frac {{\left (a^{2} + b^{2}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{2} b^{2}}}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (74) = 148\).
Time = 0.65 (sec) , antiderivative size = 313, normalized size of antiderivative = 4.23 \[ \int \frac {\tan ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\frac {{\left (a^{3} - a^{2} b + a b^{2} - b^{3}\right )} \log \left ({\left | a + b + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{3} b^{2} - a^{2} b^{3}} - \frac {\log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{2}} - \frac {\log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{b^{2}} - \frac {a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3} + \frac {a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {a^{2} b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {a b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{{\left (a + b + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )} a^{2} b^{2}}}{d} \]
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Time = 14.31 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.68 \[ \int \frac {\tan ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\ln \left (a+b-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )\,\left (\frac {1}{a^2}+\frac {1}{b^2}\right )}{d}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a^2\,d}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}{b^2\,d}-\frac {2\,\left (a+b\right )}{a\,b\,d\,\left (\left (b-a\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a+b\right )} \]
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