Optimal. Leaf size=97 \[ -\left (\left (a^4-6 a^2 b^2+b^4\right ) x\right )-\frac {4 a b^3 \log (\cos (c+d x))}{d}+\frac {4 a^3 b \log (\sin (c+d x))}{d}+\frac {b^2 \left (a^2+b^2\right ) \tan (c+d x)}{d}-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))^2}{d} \]
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Rubi [A]
time = 0.13, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3646, 3718,
3705, 3556} \begin {gather*} \frac {4 a^3 b \log (\sin (c+d x))}{d}+\frac {b^2 \left (a^2+b^2\right ) \tan (c+d x)}{d}-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))^2}{d}-x \left (a^4-6 a^2 b^2+b^4\right )-\frac {4 a b^3 \log (\cos (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3646
Rule 3705
Rule 3718
Rubi steps
\begin {align*} \int \cot ^2(c+d x) (a+b \tan (c+d x))^4 \, dx &=-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))^2}{d}+\int \cot (c+d x) (a+b \tan (c+d x)) \left (4 a^2 b-a \left (a^2-3 b^2\right ) \tan (c+d x)+b \left (a^2+b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=\frac {b^2 \left (a^2+b^2\right ) \tan (c+d x)}{d}-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))^2}{d}-\int \cot (c+d x) \left (-4 a^3 b+\left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)-4 a b^3 \tan ^2(c+d x)\right ) \, dx\\ &=-\left (a^4-6 a^2 b^2+b^4\right ) x+\frac {b^2 \left (a^2+b^2\right ) \tan (c+d x)}{d}-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))^2}{d}+\left (4 a^3 b\right ) \int \cot (c+d x) \, dx+\left (4 a b^3\right ) \int \tan (c+d x) \, dx\\ &=-\left (a^4-6 a^2 b^2+b^4\right ) x-\frac {4 a b^3 \log (\cos (c+d x))}{d}+\frac {4 a^3 b \log (\sin (c+d x))}{d}+\frac {b^2 \left (a^2+b^2\right ) \tan (c+d x)}{d}-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))^2}{d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.28, size = 94, normalized size = 0.97 \begin {gather*} -\frac {a^4 \cot (c+d x)+\frac {1}{2} i (a-i b)^4 \log (i-\cot (c+d x))-\frac {1}{2} i (a+i b)^4 \log (i+\cot (c+d x))-4 a b^3 \log (\tan (c+d x))-b^4 \tan (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 83, normalized size = 0.86
method | result | size |
derivativedivides | \(\frac {a^{4} \left (-\cot \left (d x +c \right )-d x -c \right )+4 a^{3} b \ln \left (\sin \left (d x +c \right )\right )+6 a^{2} b^{2} \left (d x +c \right )-4 a \,b^{3} \ln \left (\cos \left (d x +c \right )\right )+b^{4} \left (\tan \left (d x +c \right )-d x -c \right )}{d}\) | \(83\) |
default | \(\frac {a^{4} \left (-\cot \left (d x +c \right )-d x -c \right )+4 a^{3} b \ln \left (\sin \left (d x +c \right )\right )+6 a^{2} b^{2} \left (d x +c \right )-4 a \,b^{3} \ln \left (\cos \left (d x +c \right )\right )+b^{4} \left (\tan \left (d x +c \right )-d x -c \right )}{d}\) | \(83\) |
norman | \(\frac {\frac {b^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{d}+\left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) x \tan \left (d x +c \right )-\frac {a^{4}}{d}}{\tan \left (d x +c \right )}+\frac {4 a^{3} b \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {2 a b \left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\) | \(105\) |
risch | \(-4 i a^{3} b x +4 i a \,b^{3} x -a^{4} x +6 a^{2} b^{2} x -b^{4} x +\frac {8 i a \,b^{3} c}{d}-\frac {8 i a^{3} b c}{d}-\frac {2 i \left (a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+a^{4}+b^{4}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {4 b^{3} a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}+\frac {4 a^{3} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(169\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 88, normalized size = 0.91 \begin {gather*} \frac {4 \, a^{3} b \log \left (\tan \left (d x + c\right )\right ) + b^{4} \tan \left (d x + c\right ) - \frac {a^{4}}{\tan \left (d x + c\right )} - {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )} - 2 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.41, size = 114, normalized size = 1.18 \begin {gather*} \frac {2 \, a^{3} b \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 ) - 2 \, a b^{3} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right ) + b^{4} \tan \left (d x + c\right )^{2} - a^{4} - {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} d x \tan \left (d x + c\right )}{d \tan \left (d x + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.97, size = 133, normalized size = 1.37 \begin {gather*} \begin {cases} \tilde {\infty } a^{4} x & \text {for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\x \left (a + b \tan {\left (c \right )}\right )^{4} \cot ^{2}{\left (c \right )} & \text {for}\: d = 0 \\- a^{4} x - \frac {a^{4}}{d \tan {\left (c + d x \right )}} - \frac {2 a^{3} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {4 a^{3} b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + 6 a^{2} b^{2} x + \frac {2 a b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - b^{4} x + \frac {b^{4} \tan {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.77, size = 102, normalized size = 1.05 \begin {gather*} \frac {4 \, a^{3} b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) + b^{4} \tan \left (d x + c\right ) - {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )} - 2 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - \frac {4 \, a^{3} b \tan \left (d x + c\right ) + a^{4}}{\tan \left (d x + c\right )}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.96, size = 94, normalized size = 0.97 \begin {gather*} \frac {b^4\,\mathrm {tan}\left (c+d\,x\right )}{d}-\frac {a^4\,\mathrm {cot}\left (c+d\,x\right )}{d}+\frac {4\,a^3\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (a-b\,1{}\mathrm {i}\right )}^4\,1{}\mathrm {i}}{2\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^4\,1{}\mathrm {i}}{2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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