Optimal. Leaf size=170 \[ \frac{a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{15 d}+\frac{2 a b \left (a^2-b^2\right ) \cot ^2(c+d x)}{d}-\frac{\left (-6 a^2 b^2+a^4+b^4\right ) \cot (c+d x)}{d}+\frac{4 a b \left (a^2-b^2\right ) \log (\sin (c+d x))}{d}-x \left (-6 a^2 b^2+a^4+b^4\right )-\frac{3 a^3 b \cot ^4(c+d x)}{5 d}-\frac{a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d} \]
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Rubi [A] time = 0.382967, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{15 d}+\frac{2 a b \left (a^2-b^2\right ) \cot ^2(c+d x)}{d}-\frac{\left (-6 a^2 b^2+a^4+b^4\right ) \cot (c+d x)}{d}+\frac{4 a b \left (a^2-b^2\right ) \log (\sin (c+d x))}{d}-x \left (-6 a^2 b^2+a^4+b^4\right )-\frac{3 a^3 b \cot ^4(c+d x)}{5 d}-\frac{a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d} \]
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 ^6(c+d x) (a+b \tan (c+d x))^4 \, dx &=-\frac{a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac{1}{5} \int \cot ^5(c+d x) (a+b \tan (c+d x)) \left (12 a^2 b-5 a \left (a^2-3 b^2\right ) \tan (c+d x)-b \left (3 a^2-5 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac{3 a^3 b \cot ^4(c+d x)}{5 d}-\frac{a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac{1}{5} \int \cot ^4(c+d x) \left (-a^2 \left (5 a^2-27 b^2\right )-20 a b \left (a^2-b^2\right ) \tan (c+d x)-b^2 \left (3 a^2-5 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=\frac{a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{15 d}-\frac{3 a^3 b \cot ^4(c+d x)}{5 d}-\frac{a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac{1}{5} \int \cot ^3(c+d x) \left (-20 a b \left (a^2-b^2\right )+5 \left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right ) \, dx\\ &=\frac{2 a b \left (a^2-b^2\right ) \cot ^2(c+d x)}{d}+\frac{a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{15 d}-\frac{3 a^3 b \cot ^4(c+d x)}{5 d}-\frac{a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac{1}{5} \int \cot ^2(c+d x) \left (5 \left (a^4-6 a^2 b^2+b^4\right )+20 a b \left (a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=-\frac{\left (a^4-6 a^2 b^2+b^4\right ) \cot (c+d x)}{d}+\frac{2 a b \left (a^2-b^2\right ) \cot ^2(c+d x)}{d}+\frac{a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{15 d}-\frac{3 a^3 b \cot ^4(c+d x)}{5 d}-\frac{a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac{1}{5} \int \cot (c+d x) \left (20 a b \left (a^2-b^2\right )-5 \left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right ) \, dx\\ &=-\left (a^4-6 a^2 b^2+b^4\right ) x-\frac{\left (a^4-6 a^2 b^2+b^4\right ) \cot (c+d x)}{d}+\frac{2 a b \left (a^2-b^2\right ) \cot ^2(c+d x)}{d}+\frac{a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{15 d}-\frac{3 a^3 b \cot ^4(c+d x)}{5 d}-\frac{a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\left (4 a b \left (a^2-b^2\right )\right ) \int \cot (c+d x) \, dx\\ &=-\left (a^4-6 a^2 b^2+b^4\right ) x-\frac{\left (a^4-6 a^2 b^2+b^4\right ) \cot (c+d x)}{d}+\frac{2 a b \left (a^2-b^2\right ) \cot ^2(c+d x)}{d}+\frac{a^2 \left (5 a^2-27 b^2\right ) \cot ^3(c+d x)}{15 d}-\frac{3 a^3 b \cot ^4(c+d x)}{5 d}+\frac{4 a b \left (a^2-b^2\right ) \log (\sin (c+d x))}{d}-\frac{a^2 \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}\\ \end{align*}
Mathematica [C] time = 0.411373, size = 154, normalized size = 0.91 \[ -\frac{-\frac{1}{3} a^2 \left (a^2-6 b^2\right ) \cot ^3(c+d x)+\left (-6 a^2 b^2+a^4+b^4\right ) \cot (c+d x)+a^3 b \cot ^4(c+d x)+\frac{1}{5} a^4 \cot ^5(c+d x)-2 a b (a-b) (a+b) \cot ^2(c+d x)+\frac{1}{2} i (a-i b)^4 \log (-\cot (c+d x)+i)-\frac{1}{2} i (a+i b)^4 \log (\cot (c+d x)+i)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 232, normalized size = 1.4 \begin{align*} -{b}^{4}x-{\frac{{b}^{4}\cot \left ( dx+c \right ) }{d}}-{\frac{{b}^{4}c}{d}}-2\,{\frac{{b}^{3}a \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}-4\,{\frac{{b}^{3}a\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-2\,{\frac{{a}^{2}{b}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{d}}+6\,{a}^{2}{b}^{2}x+6\,{\frac{{a}^{2}{b}^{2}\cot \left ( dx+c \right ) }{d}}+6\,{\frac{{a}^{2}{b}^{2}c}{d}}-{\frac{b{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{d}}+2\,{\frac{b{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}+4\,{\frac{b{a}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{{a}^{4}\cot \left ( dx+c \right ) }{d}}-{a}^{4}x-{\frac{{a}^{4}c}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.5249, size = 230, normalized size = 1.35 \begin{align*} -\frac{15 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )}{\left (d x + c\right )} + 30 \,{\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 \,{\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac{15 \, a^{3} b \tan \left (d x + c\right ) + 15 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{4} + 3 \, a^{4} - 30 \,{\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{3} - 5 \,{\left (a^{4} - 6 \, a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{5}}}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03318, size = 433, normalized size = 2.55 \begin{align*} \frac{30 \,{\left (a^{3} b - a b^{3}\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 )^{5} + 15 \,{\left (3 \, a^{3} b - 2 \, a b^{3} -{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{5} - 15 \, a^{3} b \tan \left (d x + c\right ) - 15 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{4} - 3 \, a^{4} + 30 \,{\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{3} + 5 \,{\left (a^{4} - 6 \, a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{15 \, d \tan \left (d x + c\right )^{5}} \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 = 2.6735, size = 562, normalized size = 3.31 \begin{align*} \frac{3 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 30 \, a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 35 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 120 \, a^{2} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 360 \, a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 240 \, a b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 330 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1800 \, a^{2} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 240 \, b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 480 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )}{\left (d x + c\right )} - 1920 \,{\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) + 1920 \,{\left (a^{3} b - a b^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - \frac{4384 \, a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 4384 \, a b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 330 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 1800 \, a^{2} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 240 \, b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 360 \, a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 240 \, a b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 35 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 120 \, a^{2} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 30 \, 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 )^{5}}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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