Optimal. Leaf size=92 \[ 4 a b \left (a^2-b^2\right ) x-\frac {b^2 \left (6 a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac {a^4 \log (\sin (c+d x))}{d}+\frac {3 a b^3 \tan (c+d x)}{d}+\frac {b^2 (a+b \tan (c+d x))^2}{2 d} \]
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Rubi [A]
time = 0.13, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3647, 3718,
3705, 3556} \begin {gather*} \frac {a^4 \log (\sin (c+d x))}{d}-\frac {b^2 \left (6 a^2-b^2\right ) \log (\cos (c+d x))}{d}+4 a b x \left (a^2-b^2\right )+\frac {3 a b^3 \tan (c+d x)}{d}+\frac {b^2 (a+b \tan (c+d x))^2}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3647
Rule 3705
Rule 3718
Rubi steps
\begin {align*} \int \cot (c+d x) (a+b \tan (c+d x))^4 \, dx &=\frac {b^2 (a+b \tan (c+d x))^2}{2 d}+\frac {1}{2} \int \cot (c+d x) (a+b \tan (c+d x)) \left (2 a^3+2 b \left (3 a^2-b^2\right ) \tan (c+d x)+6 a b^2 \tan ^2(c+d x)\right ) \, dx\\ &=\frac {3 a b^3 \tan (c+d x)}{d}+\frac {b^2 (a+b \tan (c+d x))^2}{2 d}-\frac {1}{2} \int \cot (c+d x) \left (-2 a^4-8 a b \left (a^2-b^2\right ) \tan (c+d x)-2 b^2 \left (6 a^2-b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=4 a b \left (a^2-b^2\right ) x+\frac {3 a b^3 \tan (c+d x)}{d}+\frac {b^2 (a+b \tan (c+d x))^2}{2 d}+a^4 \int \cot (c+d x) \, dx+\left (b^2 \left (6 a^2-b^2\right )\right ) \int \tan (c+d x) \, dx\\ &=4 a b \left (a^2-b^2\right ) x-\frac {b^2 \left (6 a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac {a^4 \log (\sin (c+d x))}{d}+\frac {3 a b^3 \tan (c+d x)}{d}+\frac {b^2 (a+b \tan (c+d x))^2}{2 d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.42, size = 94, normalized size = 1.02 \begin {gather*} \frac {-(a+i b)^4 \log (i-\tan (c+d x))+2 a^4 \log (\tan (c+d x))-(a-i b)^4 \log (i+\tan (c+d x))+6 a b^3 \tan (c+d x)+b^2 (a+b \tan (c+d x))^2}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 85, normalized size = 0.92
method | result | size |
derivativedivides | \(\frac {a^{4} \ln \left (\sin \left (d x +c \right )\right )+4 a^{3} b \left (d x +c \right )-6 a^{2} b^{2} \ln \left (\cos \left (d x +c \right )\right )+4 a \,b^{3} \left (\tan \left (d x +c \right )-d x -c \right )+b^{4} \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(85\) |
default | \(\frac {a^{4} \ln \left (\sin \left (d x +c \right )\right )+4 a^{3} b \left (d x +c \right )-6 a^{2} b^{2} \ln \left (\cos \left (d x +c \right )\right )+4 a \,b^{3} \left (\tan \left (d x +c \right )-d x -c \right )+b^{4} \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(85\) |
norman | \(\left (4 a^{3} b -4 a \,b^{3}\right ) x +\frac {b^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {4 a \,b^{3} \tan \left (d x +c \right )}{d}+\frac {a^{4} \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(93\) |
risch | \(4 a^{3} b x -4 a \,b^{3} x -i a^{4} x +6 i a^{2} b^{2} x -i b^{4} x +\frac {12 i a^{2} b^{2} c}{d}-\frac {2 i b^{4} c}{d}-\frac {2 i a^{4} c}{d}+\frac {2 i b^{3} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+4 a \,{\mathrm e}^{2 i \left (d x +c \right )}+4 a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {6 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{2} b^{2}}{d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) b^{4}}{d}+\frac {a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(185\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 89, normalized size = 0.97 \begin {gather*} \frac {b^{4} \tan \left (d x + c\right )^{2} + 2 \, a^{4} \log \left (\tan \left (d x + c\right )\right ) + 8 \, a b^{3} \tan \left (d x + c\right ) + 8 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )} - {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.01, size = 101, normalized size = 1.10 \begin {gather*} \frac {b^{4} \tan \left (d x + c\right )^{2} + a^{4} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + 8 \, a b^{3} \tan \left (d x + c\right ) + 8 \, {\left (a^{3} b - a b^{3}\right )} d x - {\left (6 \, a^{2} b^{2} - b^{4}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.53, size = 133, normalized size = 1.45 \begin {gather*} \begin {cases} - \frac {a^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a^{4} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + 4 a^{3} b x + \frac {3 a^{2} b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - 4 a b^{3} x + \frac {4 a b^{3} \tan {\left (c + d x \right )}}{d} - \frac {b^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {b^{4} \tan ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\left (c \right )}\right )^{4} \cot {\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.43, size = 90, normalized size = 0.98 \begin {gather*} \frac {b^{4} \tan \left (d x + c\right )^{2} + 2 \, a^{4} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) + 8 \, a b^{3} \tan \left (d x + c\right ) + 8 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )} - {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.97, size = 92, normalized size = 1.00 \begin {gather*} \frac {b^4\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^4}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,{\left (a+b\,1{}\mathrm {i}\right )}^4}{2\,d}+\frac {a^4\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}+\frac {4\,a\,b^3\,\mathrm {tan}\left (c+d\,x\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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