Optimal. Leaf size=46 \[ -\frac {a \cot ^2(c+d x)}{2 d}-\frac {a \log (\sin (c+d x))}{d}-\frac {b \cot (c+d x)}{d}-b x \]
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Rubi [A] time = 0.06, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3529, 3531, 3475} \[ -\frac {a \cot ^2(c+d x)}{2 d}-\frac {a \log (\sin (c+d x))}{d}-\frac {b \cot (c+d x)}{d}-b x \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3529
Rule 3531
Rubi steps
\begin {align*} \int \cot ^3(c+d x) (a+b \tan (c+d x)) \, dx &=-\frac {a \cot ^2(c+d x)}{2 d}+\int \cot ^2(c+d x) (b-a \tan (c+d x)) \, dx\\ &=-\frac {b \cot (c+d x)}{d}-\frac {a \cot ^2(c+d x)}{2 d}+\int \cot (c+d x) (-a-b \tan (c+d x)) \, dx\\ &=-b x-\frac {b \cot (c+d x)}{d}-\frac {a \cot ^2(c+d x)}{2 d}-a \int \cot (c+d x) \, dx\\ &=-b x-\frac {b \cot (c+d x)}{d}-\frac {a \cot ^2(c+d x)}{2 d}-\frac {a \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [C] time = 0.29, size = 66, normalized size = 1.43 \[ -\frac {a \left (\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right )}{2 d}-\frac {b \cot (c+d x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2(c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 72, normalized size = 1.57 \[ -\frac {a \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{2} + {\left (2 \, b d x + a\right )} \tan \left (d x + c\right )^{2} + 2 \, b \tan \left (d x + c\right ) + a}{2 \, d \tan \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.87, size = 113, normalized size = 2.46 \[ -\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, {\left (d x + c\right )} b - 8 \, a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) + 8 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 4 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {12 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 52, normalized size = 1.13 \[ -\frac {a \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a \ln \left (\sin \left (d x +c \right )\right )}{d}-b x -\frac {b \cot \left (d x +c \right )}{d}-\frac {b c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 58, normalized size = 1.26 \[ -\frac {2 \, {\left (d x + c\right )} b - a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \, a \log \left (\tan \left (d x + c\right )\right ) + \frac {2 \, b \tan \left (d x + c\right ) + a}{\tan \left (d x + c\right )^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.07, size = 83, normalized size = 1.80 \[ -\frac {{\mathrm {cot}\left (c+d\,x\right )}^2\,\left (\frac {a}{2}+b\,\mathrm {tan}\left (c+d\,x\right )\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (\frac {a}{2}+\frac {b\,1{}\mathrm {i}}{2}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (\frac {a}{2}-\frac {b\,1{}\mathrm {i}}{2}\right )}{d}-\frac {a\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.00, size = 83, normalized size = 1.80 \[ \begin {cases} \tilde {\infty } a x & \text {for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\x \left (a + b \tan {\relax (c )}\right ) \cot ^{3}{\relax (c )} & \text {for}\: d = 0 \\\frac {a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {a \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {a}{2 d \tan ^{2}{\left (c + d x \right )}} - b x - \frac {b}{d \tan {\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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