Optimal. Leaf size=77 \[ 2 a b x+\frac {a^2 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {b^2 \cosh (c+d x)}{d}+\frac {b^2 \cosh ^3(c+d x)}{3 d}-\frac {a^2 \coth (c+d x) \text {csch}(c+d x)}{2 d} \]
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Rubi [A]
time = 0.07, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3299, 3853,
3855, 2713} \begin {gather*} \frac {a^2 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {a^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}+2 a b x+\frac {b^2 \cosh ^3(c+d x)}{3 d}-\frac {b^2 \cosh (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rule 3299
Rule 3853
Rule 3855
Rubi steps
\begin {align*} \int \text {csch}^3(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx &=-\left (i \int \left (2 i a b+i a^2 \text {csch}^3(c+d x)+i b^2 \sinh ^3(c+d x)\right ) \, dx\right )\\ &=2 a b x+a^2 \int \text {csch}^3(c+d x) \, dx+b^2 \int \sinh ^3(c+d x) \, dx\\ &=2 a b x-\frac {a^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {1}{2} a^2 \int \text {csch}(c+d x) \, dx-\frac {b^2 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=2 a b x+\frac {a^2 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {b^2 \cosh (c+d x)}{d}+\frac {b^2 \cosh ^3(c+d x)}{3 d}-\frac {a^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 105, normalized size = 1.36 \begin {gather*} 2 a b x-\frac {3 b^2 \cosh (c+d x)}{4 d}+\frac {b^2 \cosh (3 (c+d x))}{12 d}-\frac {a^2 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a^2 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {a^2 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(143\) vs.
\(2(71)=142\).
time = 2.16, size = 144, normalized size = 1.87
method | result | size |
risch | \(2 a b x +\frac {{\mathrm e}^{3 d x +3 c} b^{2}}{24 d}-\frac {3 \,{\mathrm e}^{d x +c} b^{2}}{8 d}-\frac {3 \,{\mathrm e}^{-d x -c} b^{2}}{8 d}+\frac {{\mathrm e}^{-3 d x -3 c} b^{2}}{24 d}-\frac {a^{2} {\mathrm e}^{d x +c} \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {a^{2} \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d}-\frac {a^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d}\) | \(144\) |
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. 152 vs.
\(2 (71) = 142\).
time = 0.27, size = 152, normalized size = 1.97 \begin {gather*} 2 \, a b x + \frac {1}{24} \, b^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {1}{2} \, a^{2} {\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} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, 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. 1616 vs.
\(2 (71) = 142\).
time = 0.48, size = 1616, normalized size = 20.99 \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(-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 [B] Leaf count of result is larger than twice the leaf count of optimal. 162 vs.
\(2 (71) = 142\).
time = 0.44, size = 162, normalized size = 2.10 \begin {gather*} \frac {48 \, {\left (d x + c\right )} a b + b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 9 \, b^{2} e^{\left (d x + c\right )} + 12 \, a^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) - 12 \, a^{2} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac {{\left (11 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - b^{2} + 3 \, {\left (8 \, a^{2} + 3 \, b^{2}\right )} e^{\left (6 \, d x + 6 \, c\right )} + {\left (24 \, a^{2} - 19 \, b^{2}\right )} e^{\left (4 \, d x + 4 \, c\right )}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{{\left (e^{\left (d x + c\right )} + 1\right )}^{2} {\left (e^{\left (d x + c\right )} - 1\right )}^{2}}}{24 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.71, size = 175, normalized size = 2.27 \begin {gather*} \frac {\mathrm {atan}\left (\frac {a^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^4}}\right )\,\sqrt {a^4}}{\sqrt {-d^2}}-\frac {3\,b^2\,{\mathrm {e}}^{c+d\,x}}{8\,d}+2\,a\,b\,x-\frac {3\,b^2\,{\mathrm {e}}^{-c-d\,x}}{8\,d}+\frac {b^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}+\frac {b^2\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}-\frac {a^2\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,a^2\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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