Optimal. Leaf size=63 \[ -\frac{a^2 \coth ^5(c+d x)}{5 d}-\frac{a (a+2 b) \coth ^3(c+d x)}{3 d}-\frac{(a+b)^2 \coth (c+d x)}{d}+x (a+b)^2 \]
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Rubi [A] time = 0.0771258, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3670, 461, 207} \[ -\frac{a^2 \coth ^5(c+d x)}{5 d}-\frac{a (a+2 b) \coth ^3(c+d x)}{3 d}-\frac{(a+b)^2 \coth (c+d x)}{d}+x (a+b)^2 \]
Antiderivative was successfully verified.
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Rule 3670
Rule 461
Rule 207
Rubi steps
\begin{align*} \int \coth ^6(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^2}{x^6 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^2}{x^6}+\frac{a (a+2 b)}{x^4}+\frac{(a+b)^2}{x^2}-\frac{(a+b)^2}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{(a+b)^2 \coth (c+d x)}{d}-\frac{a (a+2 b) \coth ^3(c+d x)}{3 d}-\frac{a^2 \coth ^5(c+d x)}{5 d}-\frac{(a+b)^2 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=(a+b)^2 x-\frac{(a+b)^2 \coth (c+d x)}{d}-\frac{a (a+2 b) \coth ^3(c+d x)}{3 d}-\frac{a^2 \coth ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [C] time = 0.0794081, size = 98, normalized size = 1.56 \[ -\frac{a^2 \coth ^5(c+d x) \, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};\tanh ^2(c+d x)\right )}{5 d}-\frac{2 a b \coth ^3(c+d x) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\tanh ^2(c+d x)\right )}{3 d}-\frac{b^2 \coth (c+d x) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\tanh ^2(c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 87, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( dx+c-{\rm coth} \left (dx+c\right )-{\frac{ \left ({\rm coth} \left (dx+c\right ) \right ) ^{3}}{3}}-{\frac{ \left ({\rm coth} \left (dx+c\right ) \right ) ^{5}}{5}} \right ) +2\,ab \left ( dx+c-{\rm coth} \left (dx+c\right )-1/3\, \left ({\rm coth} \left (dx+c\right ) \right ) ^{3} \right ) +{b}^{2} \left ( dx+c-{\rm coth} \left (dx+c\right ) \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.06904, size = 312, normalized size = 4.95 \begin{align*} \frac{1}{15} \, a^{2}{\left (15 \, x + \frac{15 \, c}{d} - \frac{2 \,{\left (70 \, e^{\left (-2 \, d x - 2 \, c\right )} - 140 \, e^{\left (-4 \, d x - 4 \, c\right )} + 90 \, e^{\left (-6 \, d x - 6 \, c\right )} - 45 \, e^{\left (-8 \, d x - 8 \, c\right )} - 23\right )}}{d{\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}}\right )} + \frac{2}{3} \, a b{\left (3 \, x + \frac{3 \, c}{d} - \frac{4 \,{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} - 2\right )}}{d{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + b^{2}{\left (x + \frac{c}{d} + \frac{2}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.04268, size = 1206, normalized size = 19.14 \begin{align*} -\frac{{\left (23 \, a^{2} + 40 \, a b + 15 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} + 5 \,{\left (23 \, a^{2} + 40 \, a b + 15 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} -{\left (15 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 23 \, a^{2} + 40 \, a b + 15 \, b^{2}\right )} \sinh \left (d x + c\right )^{5} - 5 \,{\left (5 \, a^{2} + 16 \, a b + 9 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 5 \,{\left (15 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} d x - 2 \,{\left (15 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 23 \, a^{2} + 40 \, a b + 15 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 23 \, a^{2} + 40 \, a b + 15 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} + 5 \,{\left (2 \,{\left (23 \, a^{2} + 40 \, a b + 15 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - 3 \,{\left (5 \, a^{2} + 16 \, a b + 9 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \,{\left (5 \, a^{2} + 4 \, a b + 3 \, b^{2}\right )} \cosh \left (d x + c\right ) - 5 \,{\left ({\left (15 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 23 \, a^{2} + 40 \, a b + 15 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 30 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} d x - 3 \,{\left (15 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 23 \, a^{2} + 40 \, a b + 15 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 46 \, a^{2} + 80 \, a b + 30 \, b^{2}\right )} \sinh \left (d x + c\right )}{15 \,{\left (d \sinh \left (d x + c\right )^{5} + 5 \,{\left (2 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{3} + 5 \,{\left (d \cosh \left (d x + c\right )^{4} - 3 \, d \cosh \left (d x + c\right )^{2} + 2 \, d\right )} \sinh \left (d x + c\right )\right )}} \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 = 1.28937, size = 294, normalized size = 4.67 \begin{align*} \frac{{\left (a^{2} + 2 \, a b + b^{2}\right )}{\left (d x + c\right )}}{d} - \frac{2 \,{\left (45 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} + 60 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 15 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 90 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} - 180 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 60 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 140 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 220 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 90 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 70 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 140 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 60 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 23 \, a^{2} + 40 \, a b + 15 \, b^{2}\right )}}{15 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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