Optimal. Leaf size=55 \[ \frac {a^2 \log (a+b \sinh (c+d x))}{b^3 d}-\frac {a \sinh (c+d x)}{b^2 d}+\frac {\sinh ^2(c+d x)}{2 b d} \]
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Rubi [A] time = 0.08, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2833, 12, 43} \[ \frac {a^2 \log (a+b \sinh (c+d x))}{b^3 d}-\frac {a \sinh (c+d x)}{b^2 d}+\frac {\sinh ^2(c+d x)}{2 b d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rubi steps
\begin {align*} \int \frac {\cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{b^2 (a+x)} \, dx,x,b \sinh (c+d x)\right )}{b d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{a+x} \, dx,x,b \sinh (c+d x)\right )}{b^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-a+x+\frac {a^2}{a+x}\right ) \, dx,x,b \sinh (c+d x)\right )}{b^3 d}\\ &=\frac {a^2 \log (a+b \sinh (c+d x))}{b^3 d}-\frac {a \sinh (c+d x)}{b^2 d}+\frac {\sinh ^2(c+d x)}{2 b d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 49, normalized size = 0.89 \[ \frac {2 a^2 \log (a+b \sinh (c+d x))-2 a b \sinh (c+d x)+b^2 \sinh ^2(c+d x)}{2 b^3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 309, normalized size = 5.62 \[ -\frac {8 \, a^{2} d x \cosh \left (d x + c\right )^{2} - b^{2} \cosh \left (d x + c\right )^{4} - b^{2} \sinh \left (d x + c\right )^{4} + 4 \, a b \cosh \left (d x + c\right )^{3} - 4 \, {\left (b^{2} \cosh \left (d x + c\right ) - a b\right )} \sinh \left (d x + c\right )^{3} - 4 \, a b \cosh \left (d x + c\right ) + 2 \, {\left (4 \, a^{2} d x - 3 \, b^{2} \cosh \left (d x + c\right )^{2} + 6 \, a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - b^{2} - 8 \, {\left (a^{2} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} \sinh \left (d x + c\right )^{2}\right )} \log \left (\frac {2 \, {\left (b \sinh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 4 \, {\left (4 \, a^{2} d x \cosh \left (d x + c\right ) - b^{2} \cosh \left (d x + c\right )^{3} + 3 \, a b \cosh \left (d x + c\right )^{2} - a b\right )} \sinh \left (d x + c\right )}{8 \, {\left (b^{3} d \cosh \left (d x + c\right )^{2} + 2 \, b^{3} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{3} d \sinh \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 88, normalized size = 1.60 \[ \frac {\frac {8 \, a^{2} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{b^{3}} + \frac {b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 4 \, a {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{b^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 54, normalized size = 0.98 \[ \frac {a^{2} \ln \left (a +b \sinh \left (d x +c \right )\right )}{b^{3} d}-\frac {a \sinh \left (d x +c \right )}{b^{2} d}+\frac {\sinh ^{2}\left (d x +c \right )}{2 b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 119, normalized size = 2.16 \[ \frac {{\left (d x + c\right )} a^{2}}{b^{3} d} - \frac {{\left (4 \, a e^{\left (-d x - c\right )} - b\right )} e^{\left (2 \, d x + 2 \, c\right )}}{8 \, b^{2} d} + \frac {a^{2} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{b^{3} d} + \frac {4 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 46, normalized size = 0.84 \[ \frac {a^2\,\ln \left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )+\frac {b^2\,{\mathrm {sinh}\left (c+d\,x\right )}^2}{2}-a\,b\,\mathrm {sinh}\left (c+d\,x\right )}{b^3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.34, size = 87, normalized size = 1.58 \[ \begin {cases} \frac {x \sinh ^{2}{\relax (c )} \cosh {\relax (c )}}{a} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\sinh ^{3}{\left (c + d x \right )}}{3 a d} & \text {for}\: b = 0 \\\frac {x \sinh ^{2}{\relax (c )} \cosh {\relax (c )}}{a + b \sinh {\relax (c )}} & \text {for}\: d = 0 \\\frac {a^{2} \log {\left (\frac {a}{b} + \sinh {\left (c + d x \right )} \right )}}{b^{3} d} - \frac {a \sinh {\left (c + d x \right )}}{b^{2} d} + \frac {\cosh ^{2}{\left (c + d x \right )}}{2 b d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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