Optimal. Leaf size=87 \[ \frac{20 i \text{EllipticF}\left (\frac{1}{2} i (a+b x),2\right )}{147 b^2}-\frac{4 \sinh (a+b x) \cosh ^{\frac{5}{2}}(a+b x)}{49 b^2}-\frac{20 \sinh (a+b x) \sqrt{\cosh (a+b x)}}{147 b^2}+\frac{2 x \cosh ^{\frac{7}{2}}(a+b x)}{7 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0578226, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5373, 2635, 2641} \[ \frac{20 i F\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{147 b^2}-\frac{4 \sinh (a+b x) \cosh ^{\frac{5}{2}}(a+b x)}{49 b^2}-\frac{20 \sinh (a+b x) \sqrt{\cosh (a+b x)}}{147 b^2}+\frac{2 x \cosh ^{\frac{7}{2}}(a+b x)}{7 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5373
Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int x \cosh ^{\frac{5}{2}}(a+b x) \sinh (a+b x) \, dx &=\frac{2 x \cosh ^{\frac{7}{2}}(a+b x)}{7 b}-\frac{2 \int \cosh ^{\frac{7}{2}}(a+b x) \, dx}{7 b}\\ &=\frac{2 x \cosh ^{\frac{7}{2}}(a+b x)}{7 b}-\frac{4 \cosh ^{\frac{5}{2}}(a+b x) \sinh (a+b x)}{49 b^2}-\frac{10 \int \cosh ^{\frac{3}{2}}(a+b x) \, dx}{49 b}\\ &=\frac{2 x \cosh ^{\frac{7}{2}}(a+b x)}{7 b}-\frac{20 \sqrt{\cosh (a+b x)} \sinh (a+b x)}{147 b^2}-\frac{4 \cosh ^{\frac{5}{2}}(a+b x) \sinh (a+b x)}{49 b^2}-\frac{10 \int \frac{1}{\sqrt{\cosh (a+b x)}} \, dx}{147 b}\\ &=\frac{2 x \cosh ^{\frac{7}{2}}(a+b x)}{7 b}+\frac{20 i F\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{147 b^2}-\frac{20 \sqrt{\cosh (a+b x)} \sinh (a+b x)}{147 b^2}-\frac{4 \cosh ^{\frac{5}{2}}(a+b x) \sinh (a+b x)}{49 b^2}\\ \end{align*}
Mathematica [A] time = 0.343983, size = 77, normalized size = 0.89 \[ \frac{\sqrt{\cosh (a+b x)} (-46 \sinh (a+b x)-6 \sinh (3 (a+b x))+63 b x \cosh (a+b x)+21 b x \cosh (3 (a+b x)))+40 i \text{EllipticF}\left (\frac{1}{2} i (a+b x),2\right )}{294 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.028, size = 0, normalized size = 0. \begin{align*} \int x \left ( \cosh \left ( bx+a \right ) \right ) ^{{\frac{5}{2}}}\sinh \left ( bx+a \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 x \cosh \left (b x + a\right )^{\frac{5}{2}} \sinh \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 x \cosh \left (b x + a\right )^{\frac{5}{2}} \sinh \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]