Optimal. Leaf size=69 \[ -a \left (a^2-3 b^2\right ) x-\frac {b^3 \log (\cos (c+d x))}{d}+\frac {3 a^2 b \log (\sin (c+d x))}{d}-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))}{d} \]
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Rubi [A]
time = 0.07, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3646, 3705,
3556} \begin {gather*} -a x \left (a^2-3 b^2\right )+\frac {3 a^2 b \log (\sin (c+d x))}{d}-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))}{d}-\frac {b^3 \log (\cos (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3646
Rule 3705
Rubi steps
\begin {align*} \int \cot ^2(c+d x) (a+b \tan (c+d x))^3 \, dx &=-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))}{d}+\int \cot (c+d x) \left (3 a^2 b-a \left (a^2-3 b^2\right ) \tan (c+d x)+b^3 \tan ^2(c+d x)\right ) \, dx\\ &=-a \left (a^2-3 b^2\right ) x-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))}{d}+\left (3 a^2 b\right ) \int \cot (c+d x) \, dx+b^3 \int \tan (c+d x) \, dx\\ &=-a \left (a^2-3 b^2\right ) x-\frac {b^3 \log (\cos (c+d x))}{d}+\frac {3 a^2 b \log (\sin (c+d x))}{d}-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))}{d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.15, size = 80, normalized size = 1.16 \begin {gather*} -\frac {a^3 \cot (c+d x)-\frac {1}{2} (i a+b)^3 \log (i-\cot (c+d x))+\frac {1}{2} (i a-b)^3 \log (i+\cot (c+d x))-b^3 \log (\tan (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 62, normalized size = 0.90
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\cot \left (d x +c \right )-d x -c \right )+3 a^{2} b \ln \left (\sin \left (d x +c \right )\right )+3 b^{2} a \left (d x +c \right )-b^{3} \ln \left (\cos \left (d x +c \right )\right )}{d}\) | \(62\) |
default | \(\frac {a^{3} \left (-\cot \left (d x +c \right )-d x -c \right )+3 a^{2} b \ln \left (\sin \left (d x +c \right )\right )+3 b^{2} a \left (d x +c \right )-b^{3} \ln \left (\cos \left (d x +c \right )\right )}{d}\) | \(62\) |
norman | \(\frac {\left (-a^{3}+3 b^{2} a \right ) x \tan \left (d x +c \right )-\frac {a^{3}}{d}}{\tan \left (d x +c \right )}+\frac {3 a^{2} b \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(84\) |
risch | \(-3 i x b \,a^{2}+i b^{3} x -a^{3} x +3 a \,b^{2} x -\frac {6 i b \,a^{2} c}{d}+\frac {2 i b^{3} c}{d}-\frac {2 i a^{3}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}-\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(114\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 74, normalized size = 1.07 \begin {gather*} \frac {6 \, a^{2} b \log \left (\tan \left (d x + c\right )\right ) - 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )} - {\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - \frac {2 \, a^{3}}{\tan \left (d x + c\right )}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.04, size = 97, normalized size = 1.41 \begin {gather*} \frac {3 \, a^{2} 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 ) - b^{3} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right ) - 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} d x \tan \left (d x + c\right ) - 2 \, a^{3}}{2 \, 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 = 0.66, size = 114, normalized size = 1.65 \begin {gather*} \begin {cases} \tilde {\infty } a^{3} 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 )^{3} \cot ^{2}{\left (c \right )} & \text {for}\: d = 0 \\- a^{3} x - \frac {a^{3}}{d \tan {\left (c + d x \right )}} - \frac {3 a^{2} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {3 a^{2} b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + 3 a b^{2} x + \frac {b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.14, size = 88, normalized size = 1.28 \begin {gather*} \frac {6 \, a^{2} b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) - 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )} - {\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.91, size = 78, normalized size = 1.13 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^3}{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 {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,{\left (a+b\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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