Optimal. Leaf size=52 \[ -\frac {a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {b (2 a-b) \cosh (c+d x)}{d}+\frac {b^2 \cosh ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.07, antiderivative size = 52, 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 = {3186, 390, 206} \[ -\frac {a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {b (2 a-b) \cosh (c+d x)}{d}+\frac {b^2 \cosh ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 390
Rule 3186
Rubi steps
\begin {align*} \int \text {csch}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (a-b+b x^2\right )^2}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (-(2 a-b) b-b^2 x^2+\frac {a^2}{1-x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {(2 a-b) b \cosh (c+d x)}{d}+\frac {b^2 \cosh ^3(c+d x)}{3 d}-\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {(2 a-b) b \cosh (c+d x)}{d}+\frac {b^2 \cosh ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 104, normalized size = 2.00 \[ \frac {a^2 \log \left (\sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}-\frac {a^2 \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {2 a b \sinh (c) \sinh (d x)}{d}+\frac {2 a b \cosh (c) \cosh (d x)}{d}-\frac {3 b^2 \cosh (c+d x)}{4 d}+\frac {b^2 \cosh (3 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 492, normalized size = 9.46 \[ \frac {b^{2} \cosh \left (d x + c\right )^{6} + 6 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + b^{2} \sinh \left (d x + c\right )^{6} + 3 \, {\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, b^{2} \cosh \left (d x + c\right )^{2} + 8 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, b^{2} \cosh \left (d x + c\right )^{3} + 3 \, {\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, b^{2} \cosh \left (d x + c\right )^{4} + 6 \, {\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 8 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} + b^{2} - 24 \, {\left (a^{2} \cosh \left (d x + c\right )^{3} + 3 \, a^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a^{2} \sinh \left (d x + c\right )^{3}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + 24 \, {\left (a^{2} \cosh \left (d x + c\right )^{3} + 3 \, a^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a^{2} \sinh \left (d x + c\right )^{3}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 6 \, {\left (b^{2} \cosh \left (d x + c\right )^{5} + 2 \, {\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{24 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + d \sinh \left (d x + c\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 110, normalized size = 2.12 \[ \frac {b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 24 \, a b e^{\left (d x + c\right )} - 9 \, b^{2} e^{\left (d x + c\right )} - 24 \, a^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) + 24 \, a^{2} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) + {\left (24 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 9 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 50, normalized size = 0.96 \[ \frac {-2 a^{2} \arctanh \left ({\mathrm e}^{d x +c}\right )+2 a b \cosh \left (d x +c \right )+b^{2} \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\left (d x +c \right )\right )}{3}\right ) \cosh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.60, size = 102, normalized size = 1.96 \[ \frac {1}{24} \, b^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + a b {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {a^{2} \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 116, normalized size = 2.23 \[ \frac {b^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}-\frac {2\,\mathrm {atan}\left (\frac {a^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^4}}\right )\,\sqrt {a^4}}{\sqrt {-d^2}}+\frac {b^2\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}+\frac {b\,{\mathrm {e}}^{-c-d\,x}\,\left (8\,a-3\,b\right )}{8\,d}+\frac {b\,{\mathrm {e}}^{c+d\,x}\,\left (8\,a-3\,b\right )}{8\,d} \]
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 \[ \int \left (a + b \sinh ^{2}{\left (c + d x \right )}\right )^{2} \operatorname {csch}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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