Optimal. Leaf size=83 \[ -\frac{a \left (a^2-3 b^2\right ) \log (\sin (c+d x))}{d}-b x \left (3 a^2-b^2\right )-\frac{5 a^2 b \cot (c+d x)}{2 d}-\frac{a^2 \cot ^2(c+d x) (a+b \tan (c+d x))}{2 d} \]
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Rubi [A] time = 0.141154, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3565, 3628, 3531, 3475} \[ -\frac{a \left (a^2-3 b^2\right ) \log (\sin (c+d x))}{d}-b x \left (3 a^2-b^2\right )-\frac{5 a^2 b \cot (c+d x)}{2 d}-\frac{a^2 \cot ^2(c+d x) (a+b \tan (c+d x))}{2 d} \]
Antiderivative was successfully verified.
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Rule 3565
Rule 3628
Rule 3531
Rule 3475
Rubi steps
\begin{align*} \int \cot ^3(c+d x) (a+b \tan (c+d x))^3 \, dx &=-\frac{a^2 \cot ^2(c+d x) (a+b \tan (c+d x))}{2 d}+\frac{1}{2} \int \cot ^2(c+d x) \left (5 a^2 b-2 a \left (a^2-3 b^2\right ) \tan (c+d x)-b \left (a^2-2 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac{5 a^2 b \cot (c+d x)}{2 d}-\frac{a^2 \cot ^2(c+d x) (a+b \tan (c+d x))}{2 d}+\frac{1}{2} \int \cot (c+d x) \left (-2 a \left (a^2-3 b^2\right )-2 b \left (3 a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=-b \left (3 a^2-b^2\right ) x-\frac{5 a^2 b \cot (c+d x)}{2 d}-\frac{a^2 \cot ^2(c+d x) (a+b \tan (c+d x))}{2 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{5 a^2 b \cot (c+d x)}{2 d}-\frac{a \left (a^2-3 b^2\right ) \log (\sin (c+d x))}{d}-\frac{a^2 \cot ^2(c+d x) (a+b \tan (c+d x))}{2 d}\\ \end{align*}
Mathematica [C] time = 0.329958, size = 96, normalized size = 1.16 \[ \frac{-2 a \left (a^2-3 b^2\right ) \log (\tan (c+d x))-6 a^2 b \cot (c+d x)+a^3 \left (-\cot ^2(c+d x)\right )+(a+i b)^3 \log (-\tan (c+d x)+i)+(a-i b)^3 \log (\tan (c+d x)+i)}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 94, normalized size = 1.1 \begin{align*}{b}^{3}x+{\frac{{b}^{3}c}{d}}+3\,{\frac{a{b}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-3\,x{a}^{2}b-3\,{\frac{b{a}^{2}\cot \left ( dx+c \right ) }{d}}-3\,{\frac{{a}^{2}bc}{d}}-{\frac{{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{{a}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.66, size = 124, normalized size = 1.49 \begin{align*} -\frac{2 \,{\left (3 \, a^{2} b - b^{3}\right )}{\left (d x + c\right )} -{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \,{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac{6 \, a^{2} b \tan \left (d x + c\right ) + a^{3}}{\tan \left (d x + c\right )^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77601, size = 236, normalized size = 2.84 \begin{align*} -\frac{6 \, a^{2} b \tan \left (d x + c\right ) +{\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 )^{2} + a^{3} +{\left (a^{3} + 2 \,{\left (3 \, a^{2} b - b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{2}}{2 \, d \tan \left (d x + c\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.7936, size = 146, normalized size = 1.76 \begin{align*} \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 ^{3}{\left (c \right )} & \text{for}\: d = 0 \\\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 )}} - 3 a^{2} b x - \frac{3 a^{2} b}{d \tan{\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} + b^{3} x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.99117, size = 231, normalized size = 2.78 \begin{align*} -\frac{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 12 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 8 \,{\left (3 \, a^{2} b - b^{3}\right )}{\left (d x + c\right )} - 8 \,{\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 ) + 8 \,{\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{12 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 36 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 12 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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