Optimal. Leaf size=82 \[ \frac {3 b^2 x}{8}+\frac {2 a b \cosh (c+d x)}{d}-\frac {a^2 \coth (c+d x)}{d}-\frac {3 b^2 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {b^2 \cosh (c+d x) \sinh ^3(c+d x)}{4 d} \]
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Rubi [A]
time = 0.07, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3299, 3852, 8,
2718, 2715} \begin {gather*} -\frac {a^2 \coth (c+d x)}{d}+\frac {2 a b \cosh (c+d x)}{d}+\frac {b^2 \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac {3 b^2 \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac {3 b^2 x}{8} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 2718
Rule 3299
Rule 3852
Rubi steps
\begin {align*} \int \text {csch}^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx &=-\int \left (-a^2 \text {csch}^2(c+d x)-2 a b \sinh (c+d x)-b^2 \sinh ^4(c+d x)\right ) \, dx\\ &=a^2 \int \text {csch}^2(c+d x) \, dx+(2 a b) \int \sinh (c+d x) \, dx+b^2 \int \sinh ^4(c+d x) \, dx\\ &=\frac {2 a b \cosh (c+d x)}{d}+\frac {b^2 \cosh (c+d x) \sinh ^3(c+d x)}{4 d}-\frac {1}{4} \left (3 b^2\right ) \int \sinh ^2(c+d x) \, dx-\frac {\left (i a^2\right ) \text {Subst}(\int 1 \, dx,x,-i \coth (c+d x))}{d}\\ &=\frac {2 a b \cosh (c+d x)}{d}-\frac {a^2 \coth (c+d x)}{d}-\frac {3 b^2 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {b^2 \cosh (c+d x) \sinh ^3(c+d x)}{4 d}+\frac {1}{8} \left (3 b^2\right ) \int 1 \, dx\\ &=\frac {3 b^2 x}{8}+\frac {2 a b \cosh (c+d x)}{d}-\frac {a^2 \coth (c+d x)}{d}-\frac {3 b^2 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {b^2 \cosh (c+d x) \sinh ^3(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 92, normalized size = 1.12 \begin {gather*} \frac {3 b^2 (c+d x)}{8 d}+\frac {2 a b \cosh (c) \cosh (d x)}{d}-\frac {a^2 \coth (c+d x)}{d}+\frac {2 a b \sinh (c) \sinh (d x)}{d}-\frac {b^2 \sinh (2 (c+d x))}{4 d}+\frac {b^2 \sinh (4 (c+d x))}{32 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.99, size = 124, normalized size = 1.51
method | result | size |
risch | \(\frac {3 b^{2} x}{8}+\frac {{\mathrm e}^{4 d x +4 c} b^{2}}{64 d}-\frac {{\mathrm e}^{2 d x +2 c} b^{2}}{8 d}+\frac {a b \,{\mathrm e}^{d x +c}}{d}+\frac {a b \,{\mathrm e}^{-d x -c}}{d}+\frac {{\mathrm e}^{-2 d x -2 c} b^{2}}{8 d}-\frac {{\mathrm e}^{-4 d x -4 c} b^{2}}{64 d}-\frac {2 a^{2}}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )}\) | \(124\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 113, normalized size = 1.38 \begin {gather*} \frac {1}{64} \, b^{2} {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + a b {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {2 \, a^{2}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 142, normalized size = 1.73 \begin {gather*} \frac {b^{2} \cosh \left (d x + c\right )^{5} + 5 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} - 9 \, b^{2} \cosh \left (d x + c\right )^{3} + {\left (10 \, b^{2} \cosh \left (d x + c\right )^{3} - 27 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 8 \, {\left (8 \, a^{2} - b^{2}\right )} \cosh \left (d x + c\right ) + 8 \, {\left (3 \, b^{2} d x + 16 \, a b \cosh \left (d x + c\right ) + 8 \, a^{2}\right )} \sinh \left (d x + c\right )}{64 \, d \sinh \left (d x + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 149, normalized size = 1.82 \begin {gather*} \frac {24 \, {\left (d x + c\right )} b^{2} + b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 64 \, a b e^{\left (d x + c\right )} + \frac {{\left (64 \, a b e^{\left (5 \, d x + 5 \, c\right )} - 64 \, a b e^{\left (3 \, d x + 3 \, c\right )} - 9 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + b^{2} - 8 \, {\left (16 \, a^{2} - b^{2}\right )} e^{\left (4 \, d x + 4 \, c\right )}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{{\left (e^{\left (d x + c\right )} + 1\right )} {\left (e^{\left (d x + c\right )} - 1\right )}}}{64 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.74, size = 123, normalized size = 1.50 \begin {gather*} \frac {3\,b^2\,x}{8}-\frac {2\,a^2}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}+\frac {b^2\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,d}-\frac {b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,d}-\frac {b^2\,{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,d}+\frac {b^2\,{\mathrm {e}}^{4\,c+4\,d\,x}}{64\,d}+\frac {a\,b\,{\mathrm {e}}^{c+d\,x}}{d}+\frac {a\,b\,{\mathrm {e}}^{-c-d\,x}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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