Optimal. Leaf size=67 \[ \frac{a^2 b \sinh ^3(c+d x)}{d}+\frac{a^3 \sinh (c+d x)}{d}+\frac{3 a b^2 \sinh ^5(c+d x)}{5 d}+\frac{b^3 \sinh ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.0454635, antiderivative size = 67, normalized size of antiderivative = 1., 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} \[ \frac{a^2 b \sinh ^3(c+d x)}{d}+\frac{a^3 \sinh (c+d x)}{d}+\frac{3 a b^2 \sinh ^5(c+d x)}{5 d}+\frac{b^3 \sinh ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 3190
Rule 194
Rubi steps
\begin{align*} \int \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b x^2\right )^3 \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^3+3 a^2 b x^2+3 a b^2 x^4+b^3 x^6\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{a^3 \sinh (c+d x)}{d}+\frac{a^2 b \sinh ^3(c+d x)}{d}+\frac{3 a b^2 \sinh ^5(c+d x)}{5 d}+\frac{b^3 \sinh ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.11578, size = 59, normalized size = 0.88 \[ \frac{a^2 b \sinh ^3(c+d x)+a^3 \sinh (c+d x)+\frac{3}{5} a b^2 \sinh ^5(c+d x)+\frac{1}{7} b^3 \sinh ^7(c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 56, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({\frac{{b}^{3} \left ( \sinh \left ( dx+c \right ) \right ) ^{7}}{7}}+{\frac{3\,a{b}^{2} \left ( \sinh \left ( dx+c \right ) \right ) ^{5}}{5}}+{a}^{2}b \left ( \sinh \left ( dx+c \right ) \right ) ^{3}+{a}^{3}\sinh \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05374, size = 85, normalized size = 1.27 \begin{align*} \frac{b^{3} \sinh \left (d x + c\right )^{7}}{7 \, d} + \frac{3 \, a b^{2} \sinh \left (d x + c\right )^{5}}{5 \, d} + \frac{a^{2} b \sinh \left (d x + c\right )^{3}}{d} + \frac{a^{3} \sinh \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.54785, size = 512, normalized size = 7.64 \begin{align*} \frac{5 \, b^{3} \sinh \left (d x + c\right )^{7} + 7 \,{\left (15 \, b^{3} \cosh \left (d x + c\right )^{2} + 12 \, a b^{2} - 5 \, b^{3}\right )} \sinh \left (d x + c\right )^{5} + 35 \,{\left (5 \, b^{3} \cosh \left (d x + c\right )^{4} + 16 \, a^{2} b - 12 \, a b^{2} + 3 \, b^{3} + 2 \,{\left (12 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 35 \,{\left (b^{3} \cosh \left (d x + c\right )^{6} +{\left (12 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 64 \, a^{3} - 48 \, a^{2} b + 24 \, a b^{2} - 5 \, b^{3} + 3 \,{\left (16 \, a^{2} b - 12 \, a b^{2} + 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{2240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.6694, size = 75, normalized size = 1.12 \begin{align*} \begin{cases} \frac{a^{3} \sinh{\left (c + d x \right )}}{d} + \frac{a^{2} b \sinh ^{3}{\left (c + d x \right )}}{d} + \frac{3 a b^{2} \sinh ^{5}{\left (c + d x \right )}}{5 d} + \frac{b^{3} \sinh ^{7}{\left (c + d x \right )}}{7 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right )^{3} \cosh{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2839, size = 387, normalized size = 5.78 \begin{align*} \frac{5 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} + 84 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 35 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 560 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} - 420 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 105 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 2240 \, a^{3} e^{\left (d x + c\right )} - 1680 \, a^{2} b e^{\left (d x + c\right )} + 840 \, a b^{2} e^{\left (d x + c\right )} - 175 \, b^{3} e^{\left (d x + c\right )} -{\left (2240 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 1680 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 840 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 175 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 560 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 420 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 105 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 84 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 35 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 5 \, b^{3}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{4480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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