Optimal. Leaf size=25 \[ \frac {b \cosh (c+d x)}{d}-\frac {a \tanh ^{-1}(\cosh (c+d x))}{d} \]
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Rubi [A] time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3014, 3770} \[ \frac {b \cosh (c+d x)}{d}-\frac {a \tanh ^{-1}(\cosh (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3014
Rule 3770
Rubi steps
\begin {align*} \int \text {csch}(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx &=\frac {b \cosh (c+d x)}{d}+a \int \text {csch}(c+d x) \, dx\\ &=-\frac {a \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {b \cosh (c+d x)}{d}\\ \end {align*}
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Mathematica [B] time = 0.03, size = 62, normalized size = 2.48 \[ \frac {a \log \left (\sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}-\frac {a \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {b \sinh (c) \sinh (d x)}{d}+\frac {b \cosh (c) \cosh (d x)}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 126, normalized size = 5.04 \[ \frac {b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} - 2 \, {\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + 2 \, {\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + b}{2 \, {\left (d \cosh \left (d x + c\right ) + d \sinh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 50, normalized size = 2.00 \[ \frac {b e^{\left (d x + c\right )} + b e^{\left (-d x - c\right )} - 2 \, a \log \left (e^{\left (d x + c\right )} + 1\right ) + 2 \, a \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 24, normalized size = 0.96 \[ \frac {-2 a \arctanh \left ({\mathrm e}^{d x +c}\right )+b \cosh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 43, normalized size = 1.72 \[ \frac {1}{2} \, b {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {a \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 66, normalized size = 2.64 \[ \frac {b\,{\mathrm {e}}^{-c-d\,x}}{2\,d}+\frac {b\,{\mathrm {e}}^{c+d\,x}}{2\,d}-\frac {2\,\mathrm {atan}\left (\frac {a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^2}}\right )\,\sqrt {a^2}}{\sqrt {-d^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sinh ^{2}{\left (c + d x \right )}\right ) \operatorname {csch}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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