Optimal. Leaf size=26 \[ -\frac {a \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {b \text {sech}(c+d x)}{d} \]
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Rubi [A] time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3664, 388, 207} \[ -\frac {a \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {b \text {sech}(c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 207
Rule 388
Rule 3664
Rubi steps
\begin {align*} \int \text {csch}(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b-b x^2}{-1+x^2} \, dx,x,\text {sech}(c+d x)\right )}{d}\\ &=-\frac {b \text {sech}(c+d x)}{d}+\frac {a \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\text {sech}(c+d x)\right )}{d}\\ &=-\frac {a \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {b \text {sech}(c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 52, normalized size = 2.00 \[ \frac {a \log \left (\sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}-\frac {a \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}-\frac {b \text {sech}(c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 167, normalized size = 6.42 \[ -\frac {2 \, b \cosh \left (d x + c\right ) + {\left (a \cosh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a \sinh \left (d x + c\right )^{2} + a\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) - {\left (a \cosh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a \sinh \left (d x + c\right )^{2} + a\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 2 \, b \sinh \left (d x + c\right )}{d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2} + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 52, normalized size = 2.00 \[ -\frac {a \log \left (e^{\left (d x + c\right )} + 1\right ) - a \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) + \frac {2 \, b e^{\left (d x + c\right )}}{e^{\left (2 \, d x + 2 \, c\right )} + 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 27, normalized size = 1.04 \[ \frac {-2 a \arctanh \left ({\mathrm e}^{d x +c}\right )-\frac {b}{\cosh \left (d x +c \right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 40, normalized size = 1.54 \[ \frac {a \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} - \frac {2 \, b}{d {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 64, normalized size = 2.46 \[ -\frac {2\,\mathrm {atan}\left (\frac {a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^2}}\right )\,\sqrt {a^2}}{\sqrt {-d^2}}-\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right ) \operatorname {csch}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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