Optimal. Leaf size=34 \[ a^2 x-\frac {2 a b \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {b^2 \coth (c+d x)}{d} \]
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Rubi [A] time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3773, 3770, 3767, 8} \[ a^2 x-\frac {2 a b \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {b^2 \coth (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3773
Rubi steps
\begin {align*} \int (a+b \text {csch}(c+d x))^2 \, dx &=a^2 x+(2 a b) \int \text {csch}(c+d x) \, dx+b^2 \int \text {csch}^2(c+d x) \, dx\\ &=a^2 x-\frac {2 a b \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {\left (i b^2\right ) \operatorname {Subst}(\int 1 \, dx,x,-i \coth (c+d x))}{d}\\ &=a^2 x-\frac {2 a b \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {b^2 \coth (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 61, normalized size = 1.79 \[ -\frac {-2 a \left (a c+a d x+2 b \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )\right )+b^2 \tanh \left (\frac {1}{2} (c+d x)\right )+b^2 \coth \left (\frac {1}{2} (c+d x)\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 222, normalized size = 6.53 \[ \frac {a^{2} d x \cosh \left (d x + c\right )^{2} + 2 \, a^{2} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} d x \sinh \left (d x + c\right )^{2} - a^{2} d x - 2 \, b^{2} - 2 \, {\left (a b \cosh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a b \sinh \left (d x + c\right )^{2} - a b\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + 2 \, {\left (a b \cosh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a b \sinh \left (d x + c\right )^{2} - a b\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right )}{d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2} - d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 59, normalized size = 1.74 \[ \frac {{\left (d x + c\right )} a^{2} - 2 \, a b \log \left (e^{\left (d x + c\right )} + 1\right ) + 2 \, a b \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac {2 \, b^{2}}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 37, normalized size = 1.09 \[ \frac {a^{2} \left (d x +c \right )-4 a b \arctanh \left ({\mathrm e}^{d x +c}\right )-b^{2} \coth \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 44, normalized size = 1.29 \[ a^{2} x + \frac {2 \, a b \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {2 \, b^{2}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.48, size = 74, normalized size = 2.18 \[ a^2\,x-\frac {2\,b^2}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {4\,\mathrm {atan}\left (\frac {a\,b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^2\,b^2}}\right )\,\sqrt {a^2\,b^2}}{\sqrt {-d^2}} \]
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 \[ \int \left (a + b \operatorname {csch}{\left (c + d x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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