Optimal. Leaf size=39 \[ \frac {a \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {a \coth (c+d x) \text {csch}(c+d x)}{2 d}+b x \]
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Rubi [A] time = 0.06, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3220, 3768, 3770} \[ \frac {a \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {a \coth (c+d x) \text {csch}(c+d x)}{2 d}+b x \]
Antiderivative was successfully verified.
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Rule 3220
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \text {csch}^3(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx &=-\left (i \int \left (i b+i a \text {csch}^3(c+d x)\right ) \, dx\right )\\ &=b x+a \int \text {csch}^3(c+d x) \, dx\\ &=b x-\frac {a \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {1}{2} a \int \text {csch}(c+d x) \, dx\\ &=b x+\frac {a \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {a \coth (c+d x) \text {csch}(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 63, normalized size = 1.62 \[ -\frac {a \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+b x \]
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 521, normalized size = 13.36 \[ \frac {2 \, b d x \cosh \left (d x + c\right )^{4} + 2 \, b d x \sinh \left (d x + c\right )^{4} - 4 \, b d x \cosh \left (d x + c\right )^{2} - 2 \, a \cosh \left (d x + c\right )^{3} + 2 \, {\left (4 \, b d x \cosh \left (d x + c\right ) - a\right )} \sinh \left (d x + c\right )^{3} + 2 \, b d x + 2 \, {\left (6 \, b d x \cosh \left (d x + c\right )^{2} - 2 \, b d x - 3 \, a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 2 \, a \cosh \left (d x + c\right ) + {\left (a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} - 2 \, a \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} - a\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a \cosh \left (d x + c\right )^{3} - a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) - {\left (a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} - 2 \, a \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} - a\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a \cosh \left (d x + c\right )^{3} - a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 2 \, {\left (4 \, b d x \cosh \left (d x + c\right )^{3} - 4 \, b d x \cosh \left (d x + c\right ) - 3 \, a \cosh \left (d x + c\right )^{2} - a\right )} \sinh \left (d x + c\right )}{2 \, {\left (d \cosh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{4} - 2 \, d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (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 [B] time = 0.15, size = 73, normalized size = 1.87 \[ \frac {2 \, {\left (d x + c\right )} b + a \log \left (e^{\left (d x + c\right )} + 1\right ) - a \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac {2 \, {\left (a e^{\left (3 \, d x + 3 \, c\right )} + a e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 37, normalized size = 0.95 \[ \frac {a \left (-\frac {\mathrm {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\arctanh \left ({\mathrm e}^{d x +c}\right )\right )+\left (d x +c \right ) b}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 91, normalized size = 2.33 \[ b x + \frac {1}{2} \, a {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 102, normalized size = 2.62 \[ b\,x+\frac {\mathrm {atan}\left (\frac {a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^2}}\right )\,\sqrt {a^2}}{\sqrt {-d^2}}-\frac {a\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,a\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
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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