Optimal. Leaf size=90 \[ \frac{2 b \left (4 a^2+b^2\right ) \sinh (c+d x)}{3 d}+\frac{1}{2} a x \left (2 a^2+3 b^2\right )+\frac{5 a b^2 \sinh (c+d x) \cosh (c+d x)}{6 d}+\frac{b \sinh (c+d x) (a+b \cosh (c+d x))^2}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0682165, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2656, 2734} \[ \frac{2 b \left (4 a^2+b^2\right ) \sinh (c+d x)}{3 d}+\frac{1}{2} a x \left (2 a^2+3 b^2\right )+\frac{5 a b^2 \sinh (c+d x) \cosh (c+d x)}{6 d}+\frac{b \sinh (c+d x) (a+b \cosh (c+d x))^2}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2656
Rule 2734
Rubi steps
\begin{align*} \int (a+b \cosh (c+d x))^3 \, dx &=\frac{b (a+b \cosh (c+d x))^2 \sinh (c+d x)}{3 d}+\frac{1}{3} \int (a+b \cosh (c+d x)) \left (3 a^2+2 b^2+5 a b \cosh (c+d x)\right ) \, dx\\ &=\frac{1}{2} a \left (2 a^2+3 b^2\right ) x+\frac{2 b \left (4 a^2+b^2\right ) \sinh (c+d x)}{3 d}+\frac{5 a b^2 \cosh (c+d x) \sinh (c+d x)}{6 d}+\frac{b (a+b \cosh (c+d x))^2 \sinh (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.124111, size = 80, normalized size = 0.89 \[ \frac{9 b \left (4 a^2+b^2\right ) \sinh (c+d x)+12 a^3 c+12 a^3 d x+9 a b^2 \sinh (2 (c+d x))+18 a b^2 c+18 a b^2 d x+b^3 \sinh (3 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.013, size = 77, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({b}^{3} \left ({\frac{2}{3}}+{\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \sinh \left ( dx+c \right ) +3\,a{b}^{2} \left ( 1/2\,\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +3\,{a}^{2}b\sinh \left ( dx+c \right ) +{a}^{3} \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.05084, size = 157, normalized size = 1.74 \begin{align*} \frac{3}{8} \, a b^{2}{\left (4 \, x + \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + a^{3} x + \frac{1}{24} \, b^{3}{\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 )} + \frac{3 \, a^{2} b \sinh \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.21433, size = 188, normalized size = 2.09 \begin{align*} \frac{b^{3} \sinh \left (d x + c\right )^{3} + 6 \,{\left (2 \, a^{3} + 3 \, a b^{2}\right )} d x + 3 \,{\left (b^{3} \cosh \left (d x + c\right )^{2} + 6 \, a b^{2} \cosh \left (d x + c\right ) + 12 \, a^{2} b + 3 \, b^{3}\right )} \sinh \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.665417, size = 128, normalized size = 1.42 \begin{align*} \begin{cases} a^{3} x + \frac{3 a^{2} b \sinh{\left (c + d x \right )}}{d} - \frac{3 a b^{2} x \sinh ^{2}{\left (c + d x \right )}}{2} + \frac{3 a b^{2} x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac{3 a b^{2} \sinh{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{2 d} - \frac{2 b^{3} \sinh ^{3}{\left (c + d x \right )}}{3 d} + \frac{b^{3} \sinh{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \cosh{\left (c \right )}\right )^{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1551, size = 167, normalized size = 1.86 \begin{align*} \frac{b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 9 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 36 \, a^{2} b e^{\left (d x + c\right )} + 9 \, b^{3} e^{\left (d x + c\right )} + 12 \,{\left (2 \, a^{3} + 3 \, a b^{2}\right )}{\left (d x + c\right )} -{\left (9 \, a b^{2} e^{\left (d x + c\right )} + b^{3} + 9 \,{\left (4 \, a^{2} b + b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]