Optimal. Leaf size=41 \[ -\frac {a \coth ^3(c+d x)}{3 d}+\frac {a \coth (c+d x)}{d}-\frac {b \tanh ^{-1}(\cosh (c+d x))}{d} \]
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Rubi [A] time = 0.06, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3220, 3770, 3767} \[ -\frac {a \coth ^3(c+d x)}{3 d}+\frac {a \coth (c+d x)}{d}-\frac {b \tanh ^{-1}(\cosh (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3220
Rule 3767
Rule 3770
Rubi steps
\begin {align*} \int \text {csch}^4(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx &=\int \left (b \text {csch}(c+d x)+a \text {csch}^4(c+d x)\right ) \, dx\\ &=a \int \text {csch}^4(c+d x) \, dx+b \int \text {csch}(c+d x) \, dx\\ &=-\frac {b \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {(i a) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \coth (c+d x)\right )}{d}\\ &=-\frac {b \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {a \coth (c+d x)}{d}-\frac {a \coth ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 76, normalized size = 1.85 \[ \frac {2 a \coth (c+d x)}{3 d}-\frac {a \coth (c+d x) \text {csch}^2(c+d x)}{3 d}+\frac {b \log \left (\sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}-\frac {b \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.93, size = 652, normalized size = 15.90 \[ -\frac {12 \, a \cosh \left (d x + c\right )^{2} + 24 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + 12 \, a \sinh \left (d x + c\right )^{2} + 3 \, {\left (b \cosh \left (d x + c\right )^{6} + 6 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + b \sinh \left (d x + c\right )^{6} - 3 \, b \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, b \cosh \left (d x + c\right )^{2} - b\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, b \cosh \left (d x + c\right )^{3} - 3 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, b \cosh \left (d x + c\right )^{4} - 6 \, b \cosh \left (d x + c\right )^{2} + b\right )} \sinh \left (d x + c\right )^{2} + 6 \, {\left (b \cosh \left (d x + c\right )^{5} - 2 \, b \cosh \left (d x + c\right )^{3} + b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - b\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) - 3 \, {\left (b \cosh \left (d x + c\right )^{6} + 6 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + b \sinh \left (d x + c\right )^{6} - 3 \, b \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, b \cosh \left (d x + c\right )^{2} - b\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, b \cosh \left (d x + c\right )^{3} - 3 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, b \cosh \left (d x + c\right )^{4} - 6 \, b \cosh \left (d x + c\right )^{2} + b\right )} \sinh \left (d x + c\right )^{2} + 6 \, {\left (b \cosh \left (d x + c\right )^{5} - 2 \, b \cosh \left (d x + c\right )^{3} + b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - b\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) - 4 \, a}{3 \, {\left (d \cosh \left (d x + c\right )^{6} + 6 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + d \sinh \left (d x + c\right )^{6} - 3 \, d \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, d \cosh \left (d x + c\right )^{3} - 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, d \cosh \left (d x + c\right )^{4} - 6 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )^{2} + 6 \, {\left (d \cosh \left (d x + c\right )^{5} - 2 \, d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 62, normalized size = 1.51 \[ -\frac {3 \, b \log \left (e^{\left (d x + c\right )} + 1\right ) - 3 \, b \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) + \frac {4 \, {\left (3 \, a e^{\left (2 \, d x + 2 \, c\right )} - a\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 36, normalized size = 0.88 \[ \frac {a \left (\frac {2}{3}-\frac {\mathrm {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )-2 b \arctanh \left ({\mathrm e}^{d x +c}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 131, normalized size = 3.20 \[ -b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d}\right )} + \frac {4}{3} \, a {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.69, size = 110, normalized size = 2.68 \[ -\frac {4\,a}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {8\,a}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {2\,\mathrm {atan}\left (\frac {b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {b^2}}\right )\,\sqrt {b^2}}{\sqrt {-d^2}} \]
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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