Optimal. Leaf size=60 \[ \frac{3 \cosh (c+d x) (b \sinh (c+d x))^{4/3} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};-\sinh ^2(c+d x)\right )}{4 b d \sqrt{\cosh ^2(c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0163367, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2643} \[ \frac{3 \cosh (c+d x) (b \sinh (c+d x))^{4/3} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};-\sinh ^2(c+d x)\right )}{4 b d \sqrt{\cosh ^2(c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2643
Rubi steps
\begin{align*} \int \sqrt [3]{b \sinh (c+d x)} \, dx &=\frac{3 \cosh (c+d x) \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};-\sinh ^2(c+d x)\right ) (b \sinh (c+d x))^{4/3}}{4 b d \sqrt{\cosh ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0342486, size = 57, normalized size = 0.95 \[ \frac{3 \sqrt{\cosh ^2(c+d x)} \tanh (c+d x) \sqrt [3]{b \sinh (c+d x)} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};-\sinh ^2(c+d x)\right )}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{b\sinh \left ( dx+c \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sinh \left (d x + c\right )\right )^{\frac{1}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (b \sinh \left (d x + c\right )\right )^{\frac{1}{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{b \sinh{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sinh \left (d x + c\right )\right )^{\frac{1}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]