Optimal. Leaf size=141 \[ \frac{a^2 \left (2 a^2-11 b^2\right ) \cot ^2(c+d x)}{4 d}+\frac{4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}+\frac{\left (-6 a^2 b^2+a^4+b^4\right ) \log (\sin (c+d x))}{d}+4 a b x \left (a^2-b^2\right )-\frac{5 a^3 b \cot ^3(c+d x)}{6 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d} \]
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Rubi [A] time = 0.341677, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3565, 3635, 3628, 3529, 3531, 3475} \[ \frac{a^2 \left (2 a^2-11 b^2\right ) \cot ^2(c+d x)}{4 d}+\frac{4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}+\frac{\left (-6 a^2 b^2+a^4+b^4\right ) \log (\sin (c+d x))}{d}+4 a b x \left (a^2-b^2\right )-\frac{5 a^3 b \cot ^3(c+d x)}{6 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d} \]
Antiderivative was successfully verified.
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Rule 3565
Rule 3635
Rule 3628
Rule 3529
Rule 3531
Rule 3475
Rubi steps
\begin{align*} \int \cot ^5(c+d x) (a+b \tan (c+d x))^4 \, dx &=-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac{1}{4} \int \cot ^4(c+d x) (a+b \tan (c+d x)) \left (10 a^2 b-4 a \left (a^2-3 b^2\right ) \tan (c+d x)-2 b \left (a^2-2 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac{5 a^3 b \cot ^3(c+d x)}{6 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac{1}{4} \int \cot ^3(c+d x) \left (-2 a^2 \left (2 a^2-11 b^2\right )-16 a b \left (a^2-b^2\right ) \tan (c+d x)-2 b^2 \left (a^2-2 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=\frac{a^2 \left (2 a^2-11 b^2\right ) \cot ^2(c+d x)}{4 d}-\frac{5 a^3 b \cot ^3(c+d x)}{6 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac{1}{4} \int \cot ^2(c+d x) \left (-16 a b \left (a^2-b^2\right )+4 \left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right ) \, dx\\ &=\frac{4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}+\frac{a^2 \left (2 a^2-11 b^2\right ) \cot ^2(c+d x)}{4 d}-\frac{5 a^3 b \cot ^3(c+d x)}{6 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac{1}{4} \int \cot (c+d x) \left (4 \left (a^4-6 a^2 b^2+b^4\right )+16 a b \left (a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=4 a b \left (a^2-b^2\right ) x+\frac{4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}+\frac{a^2 \left (2 a^2-11 b^2\right ) \cot ^2(c+d x)}{4 d}-\frac{5 a^3 b \cot ^3(c+d x)}{6 d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}+\left (a^4-6 a^2 b^2+b^4\right ) \int \cot (c+d x) \, dx\\ &=4 a b \left (a^2-b^2\right ) x+\frac{4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}+\frac{a^2 \left (2 a^2-11 b^2\right ) \cot ^2(c+d x)}{4 d}-\frac{5 a^3 b \cot ^3(c+d x)}{6 d}+\frac{\left (a^4-6 a^2 b^2+b^4\right ) \log (\sin (c+d x))}{d}-\frac{a^2 \cot ^4(c+d x) (a+b \tan (c+d x))^2}{4 d}\\ \end{align*}
Mathematica [C] time = 2.25636, size = 147, normalized size = 1.04 \[ \frac{6 a^2 \left (a^2-6 b^2\right ) \cot ^2(c+d x)+48 a b \left (a^2-b^2\right ) \cot (c+d x)-6 \left (-2 \left (-6 a^2 b^2+a^4+b^4\right ) \log (\tan (c+d x))+(a-i b)^4 \log (\tan (c+d x)+i)+(a+i b)^4 \log (-\tan (c+d x)+i)\right )-16 a^3 b \cot ^3(c+d x)-3 a^4 \cot ^4(c+d x)}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 180, normalized size = 1.3 \begin{align*}{\frac{{b}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-4\,{b}^{3}ax-4\,{\frac{\cot \left ( dx+c \right ) a{b}^{3}}{d}}-4\,{\frac{a{b}^{3}c}{d}}-3\,{\frac{{a}^{2}{b}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}-6\,{\frac{{a}^{2}{b}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{4\,b{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+4\,{\frac{b{a}^{3}\cot \left ( dx+c \right ) }{d}}+4\,x{a}^{3}b+4\,{\frac{b{a}^{3}c}{d}}-{\frac{{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{{a}^{4}\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 = 1.53747, size = 201, normalized size = 1.43 \begin{align*} \frac{48 \,{\left (a^{3} b - a b^{3}\right )}{\left (d x + c\right )} - 6 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 12 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac{16 \, a^{3} b \tan \left (d x + c\right ) + 3 \, a^{4} - 48 \,{\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{3} - 6 \,{\left (a^{4} - 6 \, a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{4}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05052, size = 377, normalized size = 2.67 \begin{align*} \frac{6 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\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} - 16 \, a^{3} b \tan \left (d x + c\right ) + 3 \,{\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 16 \,{\left (a^{3} b - a b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{4} - 3 \, a^{4} + 48 \,{\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{3} + 6 \,{\left (a^{4} - 6 \, a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{12 \, 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 = 2.84577, size = 452, normalized size = 3.21 \begin{align*} -\frac{3 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 32 \, a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 36 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 144 \, a^{2} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 480 \, a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 384 \, a b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 768 \,{\left (a^{3} b - a b^{3}\right )}{\left (d x + c\right )} + 192 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) - 192 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + \frac{400 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 2400 \, a^{2} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 400 \, b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 480 \, a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 384 \, a b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 36 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 144 \, a^{2} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 32 \, a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, a^{4}}{\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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