Optimal. Leaf size=110 \[ -\frac{b^2 \left (6 a^2+4 a b+b^2\right ) \coth ^3(c+d x)}{3 d}-\frac{b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \coth (c+d x)}{d}-\frac{b^3 (4 a+b) \coth ^5(c+d x)}{5 d}+x (a+b)^4-\frac{b^4 \coth ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.0677726, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3661, 390, 206} \[ -\frac{b^2 \left (6 a^2+4 a b+b^2\right ) \coth ^3(c+d x)}{3 d}-\frac{b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \coth (c+d x)}{d}-\frac{b^3 (4 a+b) \coth ^5(c+d x)}{5 d}+x (a+b)^4-\frac{b^4 \coth ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 3661
Rule 390
Rule 206
Rubi steps
\begin{align*} \int \left (a+b \coth ^2(c+d x)\right )^4 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^4}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-b (2 a+b) \left (2 a^2+2 a b+b^2\right )-b^2 \left (6 a^2+4 a b+b^2\right ) x^2-b^3 (4 a+b) x^4-b^4 x^6+\frac{(a+b)^4}{1-x^2}\right ) \, dx,x,\coth (c+d x)\right )}{d}\\ &=-\frac{b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \coth (c+d x)}{d}-\frac{b^2 \left (6 a^2+4 a b+b^2\right ) \coth ^3(c+d x)}{3 d}-\frac{b^3 (4 a+b) \coth ^5(c+d x)}{5 d}-\frac{b^4 \coth ^7(c+d x)}{7 d}+\frac{(a+b)^4 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=(a+b)^4 x-\frac{b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \coth (c+d x)}{d}-\frac{b^2 \left (6 a^2+4 a b+b^2\right ) \coth ^3(c+d x)}{3 d}-\frac{b^3 (4 a+b) \coth ^5(c+d x)}{5 d}-\frac{b^4 \coth ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 1.98046, size = 127, normalized size = 1.15 \[ -\frac{\coth (c+d x) \left (b \left (35 b \left (6 a^2+4 a b+b^2\right ) \coth ^2(c+d x)+105 \left (6 a^2 b+4 a^3+4 a b^2+b^3\right )+21 b^2 (4 a+b) \coth ^4(c+d x)+15 b^3 \coth ^6(c+d x)\right )-105 (a+b)^4 \tanh ^{-1}\left (\sqrt{\tanh ^2(c+d x)}\right ) \sqrt{\tanh ^2(c+d x)}\right )}{105 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 344, normalized size = 3.1 \begin{align*} -{\frac{4\, \left ({\rm coth} \left (dx+c\right ) \right ) ^{5}a{b}^{3}}{5\,d}}-2\,{\frac{ \left ({\rm coth} \left (dx+c\right ) \right ) ^{3}{a}^{2}{b}^{2}}{d}}-{\frac{4\, \left ({\rm coth} \left (dx+c\right ) \right ) ^{3}a{b}^{3}}{3\,d}}-{\frac{\ln \left ({\rm coth} \left (dx+c\right )-1 \right ){a}^{4}}{2\,d}}-2\,{\frac{\ln \left ({\rm coth} \left (dx+c\right )-1 \right ){a}^{3}b}{d}}-3\,{\frac{\ln \left ({\rm coth} \left (dx+c\right )-1 \right ){a}^{2}{b}^{2}}{d}}-2\,{\frac{\ln \left ({\rm coth} \left (dx+c\right )-1 \right ) a{b}^{3}}{d}}-{\frac{\ln \left ({\rm coth} \left (dx+c\right )-1 \right ){b}^{4}}{2\,d}}-{\frac{ \left ({\rm coth} \left (dx+c\right ) \right ) ^{5}{b}^{4}}{5\,d}}-{\frac{{b}^{4} \left ({\rm coth} \left (dx+c\right ) \right ) ^{3}}{3\,d}}-{\frac{{\rm coth} \left (dx+c\right ){b}^{4}}{d}}-{\frac{{b}^{4} \left ({\rm coth} \left (dx+c\right ) \right ) ^{7}}{7\,d}}-4\,{\frac{{a}^{3}b{\rm coth} \left (dx+c\right )}{d}}-6\,{\frac{{a}^{2}{\rm coth} \left (dx+c\right ){b}^{2}}{d}}-4\,{\frac{a{b}^{3}{\rm coth} \left (dx+c\right )}{d}}+{\frac{\ln \left ({\rm coth} \left (dx+c\right )+1 \right ){a}^{4}}{2\,d}}+2\,{\frac{\ln \left ({\rm coth} \left (dx+c\right )+1 \right ){a}^{3}b}{d}}+3\,{\frac{\ln \left ({\rm coth} \left (dx+c\right )+1 \right ){a}^{2}{b}^{2}}{d}}+2\,{\frac{\ln \left ({\rm coth} \left (dx+c\right )+1 \right ) a{b}^{3}}{d}}+{\frac{\ln \left ({\rm coth} \left (dx+c\right )+1 \right ){b}^{4}}{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.26051, size = 554, normalized size = 5.04 \begin{align*} \frac{1}{105} \, b^{4}{\left (105 \, x + \frac{105 \, c}{d} - \frac{8 \,{\left (203 \, e^{\left (-2 \, d x - 2 \, c\right )} - 609 \, e^{\left (-4 \, d x - 4 \, c\right )} + 770 \, e^{\left (-6 \, d x - 6 \, c\right )} - 770 \, e^{\left (-8 \, d x - 8 \, c\right )} + 315 \, e^{\left (-10 \, d x - 10 \, c\right )} - 105 \, e^{\left (-12 \, d x - 12 \, c\right )} - 44\right )}}{d{\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} - 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} - 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} - 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} - 1\right )}}\right )} + \frac{4}{15} \, a b^{3}{\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 )} + 2 \, a^{2} b^{2}{\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 )} + 4 \, a^{3} b{\left (x + \frac{c}{d} + \frac{2}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} + a^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.08123, size = 2903, normalized size = 26.39 \begin{align*} \text{result too large to display} \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 = 1.15767, size = 603, normalized size = 5.48 \begin{align*} \frac{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )}{\left (d x + c\right )}}{d} - \frac{8 \,{\left (105 \, a^{3} b e^{\left (12 \, d x + 12 \, c\right )} + 315 \, a^{2} b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 315 \, a b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 105 \, b^{4} e^{\left (12 \, d x + 12 \, c\right )} - 630 \, a^{3} b e^{\left (10 \, d x + 10 \, c\right )} - 1575 \, a^{2} b^{2} e^{\left (10 \, d x + 10 \, c\right )} - 1260 \, a b^{3} e^{\left (10 \, d x + 10 \, c\right )} - 315 \, b^{4} e^{\left (10 \, d x + 10 \, c\right )} + 1575 \, a^{3} b e^{\left (8 \, d x + 8 \, c\right )} + 3360 \, a^{2} b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 2555 \, a b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 770 \, b^{4} e^{\left (8 \, d x + 8 \, c\right )} - 2100 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} - 3990 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 3080 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 770 \, b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 1575 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + 2835 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 2121 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 609 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} - 630 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} - 1155 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 812 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 203 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 105 \, a^{3} b + 210 \, a^{2} b^{2} + 161 \, a b^{3} + 44 \, b^{4}\right )}}{105 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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