Optimal. Leaf size=40 \[ -\frac {a^2 \coth ^3(c+d x)}{3 d}+\frac {a (a-2 b) \coth (c+d x)}{d}+b^2 x \]
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Rubi [A] time = 0.07, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3187, 461, 207} \[ -\frac {a^2 \coth ^3(c+d x)}{3 d}+\frac {a (a-2 b) \coth (c+d x)}{d}+b^2 x \]
Antiderivative was successfully verified.
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Rule 207
Rule 461
Rule 3187
Rubi steps
\begin {align*} \int \text {csch}^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a-(a-b) x^2\right )^2}{x^4 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^2}{x^4}-\frac {a (a-2 b)}{x^2}-\frac {b^2}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {a (a-2 b) \coth (c+d x)}{d}-\frac {a^2 \coth ^3(c+d x)}{3 d}-\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=b^2 x+\frac {a (a-2 b) \coth (c+d x)}{d}-\frac {a^2 \coth ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [B] time = 0.72, size = 85, normalized size = 2.12 \[ \frac {4 \sinh ^4(c+d x) \left (a \text {csch}^2(c+d x)+b\right )^2 \left (3 b^2 (c+d x)-a \coth (c+d x) \left (a \text {csch}^2(c+d x)-2 a+6 b\right )\right )}{3 d (2 a+b \cosh (2 (c+d x))-b)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.39, size = 174, normalized size = 4.35 \[ \frac {2 \, {\left (a^{2} - 3 \, a b\right )} \cosh \left (d x + c\right )^{3} + 6 \, {\left (a^{2} - 3 \, a b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (3 \, b^{2} d x - 2 \, a^{2} + 6 \, a b\right )} \sinh \left (d x + c\right )^{3} - 6 \, {\left (a^{2} - a b\right )} \cosh \left (d x + c\right ) - 3 \, {\left (3 \, b^{2} d x - {\left (3 \, b^{2} d x - 2 \, a^{2} + 6 \, a b\right )} \cosh \left (d x + c\right )^{2} - 2 \, a^{2} + 6 \, a b\right )} \sinh \left (d x + c\right )}{3 \, {\left (d \sinh \left (d x + c\right )^{3} + 3 \, {\left (d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 81, normalized size = 2.02 \[ \frac {3 \, {\left (d x + c\right )} b^{2} - \frac {4 \, {\left (3 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 6 \, a b e^{\left (2 \, d x + 2 \, c\right )} - a^{2} + 3 \, a b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 47, normalized size = 1.18 \[ \frac {a^{2} \left (\frac {2}{3}-\frac {\mathrm {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )-2 a b \coth \left (d x +c \right )+b^{2} \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 121, normalized size = 3.02 \[ b^{2} x + \frac {4}{3} \, a^{2} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + \frac {4 \, a b}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.62, size = 166, normalized size = 4.15 \[ b^2\,x-\frac {\frac {4\,a\,b}{3\,d}-\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a\,b-a^2\right )}{3\,d}+\frac {4\,a\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{3\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1}+\frac {\frac {4\,\left (a\,b-a^2\right )}{3\,d}-\frac {4\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{3\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {4\,a\,b}{3\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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