Optimal. Leaf size=76 \[ -\frac {b^2 x}{2}-\frac {2 a b \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {a^2 \coth (c+d x)}{d}-\frac {a^2 \coth ^3(c+d x)}{3 d}+\frac {b^2 \cosh (c+d x) \sinh (c+d x)}{2 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3299, 3855,
3852, 2715, 8} \begin {gather*} -\frac {a^2 \coth ^3(c+d x)}{3 d}+\frac {a^2 \coth (c+d x)}{d}-\frac {2 a b \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {b^2 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac {b^2 x}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 3299
Rule 3852
Rule 3855
Rubi steps
\begin {align*} \int \text {csch}^4(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx &=\int \left (2 a b \text {csch}(c+d x)+a^2 \text {csch}^4(c+d x)+b^2 \sinh ^2(c+d x)\right ) \, dx\\ &=a^2 \int \text {csch}^4(c+d x) \, dx+(2 a b) \int \text {csch}(c+d x) \, dx+b^2 \int \sinh ^2(c+d x) \, dx\\ &=-\frac {2 a b \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {b^2 \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac {1}{2} b^2 \int 1 \, dx+\frac {\left (i a^2\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \coth (c+d x)\right )}{d}\\ &=-\frac {b^2 x}{2}-\frac {2 a b \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {a^2 \coth (c+d x)}{d}-\frac {a^2 \coth ^3(c+d x)}{3 d}+\frac {b^2 \cosh (c+d x) \sinh (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 81, normalized size = 1.07 \begin {gather*} \frac {-4 a^2 \coth (c+d x) \left (-2+\text {csch}^2(c+d x)\right )+3 b \left (-2 \left (b c+b d x+4 a \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )-4 a \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )+b \sinh (2 (c+d x))\right )}{12 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.25, size = 108, normalized size = 1.42
method | result | size |
risch | \(-\frac {b^{2} x}{2}+\frac {{\mathrm e}^{2 d x +2 c} b^{2}}{8 d}-\frac {{\mathrm e}^{-2 d x -2 c} b^{2}}{8 d}-\frac {4 a^{2} \left (3 \,{\mathrm e}^{2 d x +2 c}-1\right )}{3 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}+\frac {2 a b \ln \left ({\mathrm e}^{d x +c}-1\right )}{d}-\frac {2 a b \ln \left ({\mathrm e}^{d x +c}+1\right )}{d}\) | \(108\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 170 vs.
\(2 (70) = 140\).
time = 0.29, size = 170, normalized size = 2.24 \begin {gather*} -\frac {1}{8} \, b^{2} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - 2 \, a b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d}\right )} + \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1748 vs.
\(2 (70) = 140\).
time = 0.48, size = 1748, normalized size = 23.00 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 151 vs.
\(2 (70) = 140\).
time = 0.44, size = 151, normalized size = 1.99 \begin {gather*} -\frac {12 \, {\left (d x + c\right )} b^{2} - 3 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 48 \, a b \log \left (e^{\left (d x + c\right )} + 1\right ) - 48 \, a b \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) + \frac {{\left (3 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 3 \, b^{2} + 3 \, {\left (32 \, a^{2} - 3 \, b^{2}\right )} e^{\left (4 \, d x + 4 \, c\right )} - {\left (32 \, a^{2} - 9 \, b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (e^{\left (d x + c\right )} + 1\right )}^{3} {\left (e^{\left (d x + c\right )} - 1\right )}^{3}}}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 163, normalized size = 2.14 \begin {gather*} \frac {b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,d}-\frac {4\,a^2}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {4\,\mathrm {atan}\left (\frac {a\,b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^2\,b^2}}\right )\,\sqrt {a^2\,b^2}}{\sqrt {-d^2}}-\frac {b^2\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,d}-\frac {b^2\,x}{2}-\frac {8\,a^2}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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