Integrand size = 21, antiderivative size = 64 \[ \int \csc ^2(c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {a^3 \cot (c+d x)}{d}+\frac {3 a^2 b \log (\tan (c+d x))}{d}+\frac {3 a b^2 \tan (c+d x)}{d}+\frac {b^3 \tan ^2(c+d x)}{2 d} \]
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Time = 0.07 (sec) , antiderivative size = 64, 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 = {3597, 45} \[ \int \csc ^2(c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {a^3 \cot (c+d x)}{d}+\frac {3 a^2 b \log (\tan (c+d x))}{d}+\frac {3 a b^2 \tan (c+d x)}{d}+\frac {b^3 \tan ^2(c+d x)}{2 d} \]
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Rule 45
Rule 3597
Rubi steps \begin{align*} \text {integral}& = \frac {b \text {Subst}\left (\int \frac {(a+x)^3}{x^2} \, dx,x,b \tan (c+d x)\right )}{d} \\ & = \frac {b \text {Subst}\left (\int \left (3 a+\frac {a^3}{x^2}+\frac {3 a^2}{x}+x\right ) \, dx,x,b \tan (c+d x)\right )}{d} \\ & = -\frac {a^3 \cot (c+d x)}{d}+\frac {3 a^2 b \log (\tan (c+d x))}{d}+\frac {3 a b^2 \tan (c+d x)}{d}+\frac {b^3 \tan ^2(c+d x)}{2 d} \\ \end{align*}
Time = 2.52 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.97 \[ \int \csc ^2(c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {\csc (c+d x) \sec ^2(c+d x) \left (3 a \left (a^2-b^2\right ) \cos (c+d x)+\left (a^3+3 a b^2\right ) \cos (3 (c+d x))-2 b \left (b^2-3 a^2 \log (\cos (c+d x))-3 a^2 \cos (2 (c+d x)) (\log (\cos (c+d x))-\log (\sin (c+d x)))+3 a^2 \log (\sin (c+d x))\right ) \sin (c+d x)\right )}{4 d} \]
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Time = 2.52 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {\frac {b^{3}}{2 \cos \left (d x +c \right )^{2}}+3 a \,b^{2} \tan \left (d x +c \right )+3 a^{2} b \ln \left (\tan \left (d x +c \right )\right )-a^{3} \cot \left (d x +c \right )}{d}\) | \(55\) |
default | \(\frac {\frac {b^{3}}{2 \cos \left (d x +c \right )^{2}}+3 a \,b^{2} \tan \left (d x +c \right )+3 a^{2} b \ln \left (\tan \left (d x +c \right )\right )-a^{3} \cot \left (d x +c \right )}{d}\) | \(55\) |
risch | \(\frac {-2 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+6 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+2 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-4 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-2 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-2 i a^{3}-6 i a \,b^{2}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}-\frac {3 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{2}}{d}\) | \(162\) |
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Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (62) = 124\).
Time = 0.28 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.98 \[ \int \csc ^2(c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {3 \, a^{2} b \cos \left (d x + c\right )^{2} \log \left (\cos \left (d x + c\right )^{2}\right ) \sin \left (d x + c\right ) - 3 \, a^{2} b \cos \left (d x + c\right )^{2} \log \left (-\frac {1}{4} \, \cos \left (d x + c\right )^{2} + \frac {1}{4}\right ) \sin \left (d x + c\right ) - 6 \, a b^{2} \cos \left (d x + c\right ) + 2 \, {\left (a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - b^{3} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right )} \]
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\[ \int \csc ^2(c+d x) (a+b \tan (c+d x))^3 \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{3} \csc ^{2}{\left (c + d x \right )}\, dx \]
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none
Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.88 \[ \int \csc ^2(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {b^{3} \tan \left (d x + c\right )^{2} + 6 \, a^{2} b \log \left (\tan \left (d x + c\right )\right ) + 6 \, a b^{2} \tan \left (d x + c\right ) - \frac {2 \, a^{3}}{\tan \left (d x + c\right )}}{2 \, d} \]
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Time = 0.71 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.09 \[ \int \csc ^2(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {b^{3} \tan \left (d x + c\right )^{2} + 6 \, a^{2} b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) + 6 \, a b^{2} \tan \left (d x + c\right ) - \frac {2 \, {\left (3 \, a^{2} b \tan \left (d x + c\right ) + a^{3}\right )}}{\tan \left (d x + c\right )}}{2 \, d} \]
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Time = 4.17 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.97 \[ \int \csc ^2(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {b^3\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2\,d}-\frac {a^3\,\mathrm {cot}\left (c+d\,x\right )}{d}+\frac {3\,a^2\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}+\frac {3\,a\,b^2\,\mathrm {tan}\left (c+d\,x\right )}{d} \]
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