Integrand size = 21, antiderivative size = 130 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=b \left (3 a^2-b^2\right ) x+\frac {b \left (3 a^2-b^2\right ) \cot (c+d x)}{d}+\frac {a \left (a^2-3 b^2\right ) \cot ^2(c+d x)}{2 d}-\frac {3 a^2 b \cot ^3(c+d x)}{4 d}+\frac {a \left (a^2-3 b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d} \]
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Time = 0.30 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3646, 3709, 3610, 3612, 3556} \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {a \left (a^2-3 b^2\right ) \cot ^2(c+d x)}{2 d}+\frac {b \left (3 a^2-b^2\right ) \cot (c+d x)}{d}+\frac {a \left (a^2-3 b^2\right ) \log (\sin (c+d x))}{d}+b x \left (3 a^2-b^2\right )-\frac {3 a^2 b \cot ^3(c+d x)}{4 d}-\frac {a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d} \]
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Rule 3556
Rule 3610
Rule 3612
Rule 3646
Rule 3709
Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d}+\frac {1}{4} \int \cot ^4(c+d x) \left (9 a^2 b-4 a \left (a^2-3 b^2\right ) \tan (c+d x)-b \left (3 a^2-4 b^2\right ) \tan ^2(c+d x)\right ) \, dx \\ & = -\frac {3 a^2 b \cot ^3(c+d x)}{4 d}-\frac {a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d}+\frac {1}{4} \int \cot ^3(c+d x) \left (-4 a \left (a^2-3 b^2\right )-4 b \left (3 a^2-b^2\right ) \tan (c+d x)\right ) \, dx \\ & = \frac {a \left (a^2-3 b^2\right ) \cot ^2(c+d x)}{2 d}-\frac {3 a^2 b \cot ^3(c+d x)}{4 d}-\frac {a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d}+\frac {1}{4} \int \cot ^2(c+d x) \left (-4 b \left (3 a^2-b^2\right )+4 a \left (a^2-3 b^2\right ) \tan (c+d x)\right ) \, dx \\ & = \frac {b \left (3 a^2-b^2\right ) \cot (c+d x)}{d}+\frac {a \left (a^2-3 b^2\right ) \cot ^2(c+d x)}{2 d}-\frac {3 a^2 b \cot ^3(c+d x)}{4 d}-\frac {a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d}+\frac {1}{4} \int \cot (c+d x) \left (4 a \left (a^2-3 b^2\right )+4 b \left (3 a^2-b^2\right ) \tan (c+d x)\right ) \, dx \\ & = b \left (3 a^2-b^2\right ) x+\frac {b \left (3 a^2-b^2\right ) \cot (c+d x)}{d}+\frac {a \left (a^2-3 b^2\right ) \cot ^2(c+d x)}{2 d}-\frac {3 a^2 b \cot ^3(c+d x)}{4 d}-\frac {a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d}+\left (a \left (a^2-3 b^2\right )\right ) \int \cot (c+d x) \, dx \\ & = b \left (3 a^2-b^2\right ) x+\frac {b \left (3 a^2-b^2\right ) \cot (c+d x)}{d}+\frac {a \left (a^2-3 b^2\right ) \cot ^2(c+d x)}{2 d}-\frac {3 a^2 b \cot ^3(c+d x)}{4 d}+\frac {a \left (a^2-3 b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.71 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.91 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {4 b \left (-3 a^2+b^2\right ) \cot (c+d x)-2 a \left (a^2-3 b^2\right ) \cot ^2(c+d x)+4 a^2 b \cot ^3(c+d x)+a^3 \cot ^4(c+d x)+2 (a-i b)^3 \log (i-\cot (c+d x))+2 (a+i b)^3 \log (i+\cot (c+d x))}{4 d} \]
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Time = 0.57 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.05
method | result | size |
derivativedivides | \(\frac {\frac {\left (-a^{3}+3 a \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (3 a^{2} b -b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )-\frac {a^{3}}{4 \tan \left (d x +c \right )^{4}}+a \left (a^{2}-3 b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )+\frac {a \left (a^{2}-3 b^{2}\right )}{2 \tan \left (d x +c \right )^{2}}-\frac {a^{2} b}{\tan \left (d x +c \right )^{3}}+\frac {b \left (3 a^{2}-b^{2}\right )}{\tan \left (d x +c \right )}}{d}\) | \(137\) |
default | \(\frac {\frac {\left (-a^{3}+3 a \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (3 a^{2} b -b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )-\frac {a^{3}}{4 \tan \left (d x +c \right )^{4}}+a \left (a^{2}-3 b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )+\frac {a \left (a^{2}-3 b^{2}\right )}{2 \tan \left (d x +c \right )^{2}}-\frac {a^{2} b}{\tan \left (d x +c \right )^{3}}+\frac {b \left (3 a^{2}-b^{2}\right )}{\tan \left (d x +c \right )}}{d}\) | \(137\) |
norman | \(\frac {\frac {b \left (3 a^{2}-b^{2}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{d}+b \left (3 a^{2}-b^{2}\right ) x \left (\tan ^{4}\left (d x +c \right )\right )-\frac {a^{3}}{4 d}+\frac {a \left (a^{2}-3 b^{2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a^{2} b \tan \left (d x +c \right )}{d}}{\tan \left (d x +c \right )^{4}}+\frac {a \left (a^{2}-3 b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {a \left (a^{2}-3 b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(151\) |
parallelrisch | \(\frac {-\left (\cot ^{4}\left (d x +c \right )\right ) a^{3}-4 \left (\cot ^{3}\left (d x +c \right )\right ) a^{2} b +2 \left (\cot ^{2}\left (d x +c \right )\right ) a^{3}-6 \left (\cot ^{2}\left (d x +c \right )\right ) a \,b^{2}+12 a^{2} b d x -4 b^{3} d x +4 a^{3} \ln \left (\tan \left (d x +c \right )\right )-12 \ln \left (\tan \left (d x +c \right )\right ) a \,b^{2}-2 \ln \left (\sec ^{2}\left (d x +c \right )\right ) a^{3}+6 \ln \left (\sec ^{2}\left (d x +c \right )\right ) a \,b^{2}+12 \cot \left (d x +c \right ) a^{2} b -4 \cot \left (d x +c \right ) b^{3}}{4 d}\) | \(153\) |
risch | \(3 a^{2} b x -b^{3} x -i a^{3} x +3 i a \,b^{2} x -\frac {2 i a^{3} c}{d}+\frac {6 i a \,b^{2} c}{d}-\frac {2 i \left (-2 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+3 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-6 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+2 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-6 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+12 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-3 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-2 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+3 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-10 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+3 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+4 a^{2} b -b^{3}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}-\frac {3 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{2}}{d}\) | \(301\) |
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Time = 0.25 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.17 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{4} + {\left (3 \, a^{3} - 6 \, a b^{2} + 4 \, {\left (3 \, a^{2} b - b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{4} - 4 \, a^{2} b \tan \left (d x + c\right ) + 4 \, {\left (3 \, a^{2} b - b^{3}\right )} \tan \left (d x + c\right )^{3} - a^{3} + 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{2}}{4 \, d \tan \left (d x + c\right )^{4}} \]
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Time = 1.76 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.58 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=\begin {cases} \tilde {\infty } a^{3} x & \text {for}\: c = 0 \wedge d = 0 \\x \left (a + b \tan {\left (c \right )}\right )^{3} \cot ^{5}{\left (c \right )} & \text {for}\: d = 0 \\\tilde {\infty } a^{3} x & \text {for}\: c = - d x \\- \frac {a^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {a^{3}}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {a^{3}}{4 d \tan ^{4}{\left (c + d x \right )}} + 3 a^{2} b x + \frac {3 a^{2} b}{d \tan {\left (c + d x \right )}} - \frac {a^{2} b}{d \tan ^{3}{\left (c + d x \right )}} + \frac {3 a b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {3 a b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {3 a b^{2}}{2 d \tan ^{2}{\left (c + d x \right )}} - b^{3} x - \frac {b^{3}}{d \tan {\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.04 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {4 \, {\left (3 \, a^{2} b - b^{3}\right )} {\left (d x + c\right )} - 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 4 \, {\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac {4 \, a^{2} b \tan \left (d x + c\right ) - 4 \, {\left (3 \, a^{2} b - b^{3}\right )} \tan \left (d x + c\right )^{3} + a^{3} - 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{4}}}{4 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (124) = 248\).
Time = 1.61 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.32 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 72 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 360 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 192 \, {\left (3 \, a^{2} b - b^{3}\right )} {\left (d x + c\right )} + 192 \, {\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 192 \, {\left (a^{3} - 3 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {400 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1200 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 360 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 96 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 72 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
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Time = 4.94 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.12 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^3 \, dx=-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (3\,a\,b^2-a^3\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,{\left (a+b\,1{}\mathrm {i}\right )}^3}{2\,d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^4\,\left ({\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {3\,a\,b^2}{2}-\frac {a^3}{2}\right )-{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (3\,a^2\,b-b^3\right )+\frac {a^3}{4}+a^2\,b\,\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{2\,d} \]
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