Optimal. Leaf size=130 \[ \frac{a \left (a^2-3 b^2\right ) \cot ^2(c+d x)}{2 d}+\frac{b \left (3 a^2-b^2\right ) \cot (c+d x)}{d}+\frac{a \left (a^2-3 b^2\right ) \log (\sin (c+d x))}{d}+b x \left (3 a^2-b^2\right )-\frac{3 a^2 b \cot ^3(c+d x)}{4 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d} \]
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Rubi [A] time = 0.231662, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3565, 3628, 3529, 3531, 3475} \[ \frac{a \left (a^2-3 b^2\right ) \cot ^2(c+d x)}{2 d}+\frac{b \left (3 a^2-b^2\right ) \cot (c+d x)}{d}+\frac{a \left (a^2-3 b^2\right ) \log (\sin (c+d x))}{d}+b x \left (3 a^2-b^2\right )-\frac{3 a^2 b \cot ^3(c+d x)}{4 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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Rule 3565
Rule 3628
Rule 3529
Rule 3531
Rule 3475
Rubi steps
\begin{align*} \int \cot ^5(c+d x) (a+b \tan (c+d x))^3 \, dx &=-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d}+\frac{1}{4} \int \cot ^4(c+d x) \left (9 a^2 b-4 a \left (a^2-3 b^2\right ) \tan (c+d x)-b \left (3 a^2-4 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac{3 a^2 b \cot ^3(c+d x)}{4 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d}+\frac{1}{4} \int \cot ^3(c+d x) \left (-4 a \left (a^2-3 b^2\right )-4 b \left (3 a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=\frac{a \left (a^2-3 b^2\right ) \cot ^2(c+d x)}{2 d}-\frac{3 a^2 b \cot ^3(c+d x)}{4 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d}+\frac{1}{4} \int \cot ^2(c+d x) \left (-4 b \left (3 a^2-b^2\right )+4 a \left (a^2-3 b^2\right ) \tan (c+d x)\right ) \, dx\\ &=\frac{b \left (3 a^2-b^2\right ) \cot (c+d x)}{d}+\frac{a \left (a^2-3 b^2\right ) \cot ^2(c+d x)}{2 d}-\frac{3 a^2 b \cot ^3(c+d x)}{4 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d}+\frac{1}{4} \int \cot (c+d x) \left (4 a \left (a^2-3 b^2\right )+4 b \left (3 a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=b \left (3 a^2-b^2\right ) x+\frac{b \left (3 a^2-b^2\right ) \cot (c+d x)}{d}+\frac{a \left (a^2-3 b^2\right ) \cot ^2(c+d x)}{2 d}-\frac{3 a^2 b \cot ^3(c+d x)}{4 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 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{b \left (3 a^2-b^2\right ) \cot (c+d x)}{d}+\frac{a \left (a^2-3 b^2\right ) \cot ^2(c+d x)}{2 d}-\frac{3 a^2 b \cot ^3(c+d x)}{4 d}+\frac{a \left (a^2-3 b^2\right ) \log (\sin (c+d x))}{d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))}{4 d}\\ \end{align*}
Mathematica [C] time = 1.68483, size = 118, normalized size = 0.91 \[ -\frac{-2 a \left (a^2-3 b^2\right ) \cot ^2(c+d x)+4 b \left (b^2-3 a^2\right ) \cot (c+d x)+4 a^2 b \cot ^3(c+d x)+a^3 \cot ^4(c+d x)+2 (a-i b)^3 \log (-\cot (c+d x)+i)+2 (a+i b)^3 \log (\cot (c+d x)+i)}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 159, normalized size = 1.2 \begin{align*} -{b}^{3}x-{\frac{\cot \left ( dx+c \right ){b}^{3}}{d}}-{\frac{{b}^{3}c}{d}}-{\frac{3\,a{b}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-3\,{\frac{a{b}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{b{a}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{d}}+3\,x{a}^{2}b+3\,{\frac{b{a}^{2}\cot \left ( dx+c \right ) }{d}}+3\,{\frac{b{a}^{2}c}{d}}-{\frac{{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,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 = 1.59505, size = 182, normalized size = 1.4 \begin{align*} \frac{4 \,{\left (3 \, a^{2} b - b^{3}\right )}{\left (d x + c\right )} - 2 \,{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 4 \,{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac{4 \, a^{2} b \tan \left (d x + c\right ) - 4 \,{\left (3 \, a^{2} b - b^{3}\right )} \tan \left (d x + c\right )^{3} + a^{3} - 2 \,{\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{4}}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61506, size = 348, normalized size = 2.68 \begin{align*} \frac{2 \,{\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 )^{4} +{\left (3 \, a^{3} - 6 \, a b^{2} + 4 \,{\left (3 \, a^{2} b - b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{4} - 4 \, a^{2} b \tan \left (d x + c\right ) + 4 \,{\left (3 \, a^{2} b - b^{3}\right )} \tan \left (d x + c\right )^{3} - a^{3} + 2 \,{\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (d x + c\right )^{2}}{4 \, d \tan \left (d x + c\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.99658, size = 406, normalized size = 3.12 \begin{align*} -\frac{3 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 24 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 36 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 72 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 360 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 96 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 192 \,{\left (3 \, a^{2} b - b^{3}\right )}{\left (d x + c\right )} + 192 \,{\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 ) - 192 \,{\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{400 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 1200 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 360 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 96 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 36 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 72 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 24 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4}}}{192 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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