Optimal. Leaf size=49 \[ \frac {a^2 \sinh (c+d x)}{d}+\frac {2 a b \sinh ^3(c+d x)}{3 d}+\frac {b^2 \sinh ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3190, 194} \[ \frac {a^2 \sinh (c+d x)}{d}+\frac {2 a b \sinh ^3(c+d x)}{3 d}+\frac {b^2 \sinh ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 194
Rule 3190
Rubi steps
\begin {align*} \int \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b x^2\right )^2 \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2+2 a b x^2+b^2 x^4\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {a^2 \sinh (c+d x)}{d}+\frac {2 a b \sinh ^3(c+d x)}{3 d}+\frac {b^2 \sinh ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 44, normalized size = 0.90 \[ \frac {a^2 \sinh (c+d x)+\frac {2}{3} a b \sinh ^3(c+d x)+\frac {1}{5} b^2 \sinh ^5(c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.96, size = 106, normalized size = 2.16 \[ \frac {3 \, b^{2} \sinh \left (d x + c\right )^{5} + 5 \, {\left (6 \, b^{2} \cosh \left (d x + c\right )^{2} + 8 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} + 15 \, {\left (b^{2} \cosh \left (d x + c\right )^{4} + {\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 16 \, a^{2} - 8 \, a b + 2 \, b^{2}\right )} \sinh \left (d x + c\right )}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 134, normalized size = 2.73 \[ \frac {b^{2} e^{\left (5 \, d x + 5 \, c\right )}}{160 \, d} - \frac {b^{2} e^{\left (-5 \, d x - 5 \, c\right )}}{160 \, d} + \frac {{\left (8 \, a b - 3 \, b^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{96 \, d} + \frac {{\left (8 \, a^{2} - 4 \, a b + b^{2}\right )} e^{\left (d x + c\right )}}{16 \, d} - \frac {{\left (8 \, a^{2} - 4 \, a b + b^{2}\right )} e^{\left (-d x - c\right )}}{16 \, d} - \frac {{\left (8 \, a b - 3 \, b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 41, normalized size = 0.84 \[ \frac {\frac {b^{2} \left (\sinh ^{5}\left (d x +c \right )\right )}{5}+\frac {2 a \left (\sinh ^{3}\left (d x +c \right )\right ) b}{3}+a^{2} \sinh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 45, normalized size = 0.92 \[ \frac {b^{2} \sinh \left (d x + c\right )^{5}}{5 \, d} + \frac {2 \, a b \sinh \left (d x + c\right )^{3}}{3 \, d} + \frac {a^{2} \sinh \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 42, normalized size = 0.86 \[ \frac {\mathrm {sinh}\left (c+d\,x\right )\,\left (15\,a^2+10\,a\,b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+3\,b^2\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}{15\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.72, size = 58, normalized size = 1.18 \[ \begin {cases} \frac {a^{2} \sinh {\left (c + d x \right )}}{d} + \frac {2 a b \sinh ^{3}{\left (c + d x \right )}}{3 d} + \frac {b^{2} \sinh ^{5}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\relax (c )}\right )^{2} \cosh {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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