Optimal. Leaf size=301 \[ \frac{2 i \sqrt{2} a \left (4 a^2+b^2\right ) \sqrt{\frac{2 a+b \sinh (2 c+2 d x)}{2 a-i b}} F\left (\frac{1}{2} \left (2 i c+2 i d x-\frac{\pi }{2}\right )|\frac{2 b}{2 i a+b}\right )}{15 d \sqrt{2 a+b \sinh (2 c+2 d x)}}-\frac{i \left (92 a^2-9 b^2\right ) \sqrt{2 a+b \sinh (2 c+2 d x)} E\left (\frac{1}{2} \left (2 i c+2 i d x-\frac{\pi }{2}\right )|\frac{2 b}{2 i a+b}\right )}{60 \sqrt{2} d \sqrt{\frac{2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}+\frac{b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt{2} d}+\frac{2 \sqrt{2} a b \cosh (2 c+2 d x) \sqrt{2 a+b \sinh (2 c+2 d x)}}{15 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.39189, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2666, 2656, 2753, 2752, 2663, 2661, 2655, 2653} \[ \frac{2 i \sqrt{2} a \left (4 a^2+b^2\right ) \sqrt{\frac{2 a+b \sinh (2 c+2 d x)}{2 a-i b}} F\left (\frac{1}{2} \left (2 i c+2 i d x-\frac{\pi }{2}\right )|\frac{2 b}{2 i a+b}\right )}{15 d \sqrt{2 a+b \sinh (2 c+2 d x)}}-\frac{i \left (92 a^2-9 b^2\right ) \sqrt{2 a+b \sinh (2 c+2 d x)} E\left (\frac{1}{2} \left (2 i c+2 i d x-\frac{\pi }{2}\right )|\frac{2 b}{2 i a+b}\right )}{60 \sqrt{2} d \sqrt{\frac{2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}+\frac{b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt{2} d}+\frac{2 \sqrt{2} a b \cosh (2 c+2 d x) \sqrt{2 a+b \sinh (2 c+2 d x)}}{15 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2666
Rule 2656
Rule 2753
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int (a+b \cosh (c+d x) \sinh (c+d x))^{5/2} \, dx &=\int \left (a+\frac{1}{2} b \sinh (2 c+2 d x)\right )^{5/2} \, dx\\ &=\frac{b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt{2} d}+\frac{2}{5} \int \sqrt{a+\frac{1}{2} b \sinh (2 c+2 d x)} \left (\frac{1}{8} \left (20 a^2-3 b^2\right )+2 a b \sinh (2 c+2 d x)\right ) \, dx\\ &=\frac{2 \sqrt{2} a b \cosh (2 c+2 d x) \sqrt{2 a+b \sinh (2 c+2 d x)}}{15 d}+\frac{b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt{2} d}+\frac{4}{15} \int \frac{\frac{1}{16} a \left (60 a^2-17 b^2\right )+\frac{1}{32} b \left (92 a^2-9 b^2\right ) \sinh (2 c+2 d x)}{\sqrt{a+\frac{1}{2} b \sinh (2 c+2 d x)}} \, dx\\ &=\frac{2 \sqrt{2} a b \cosh (2 c+2 d x) \sqrt{2 a+b \sinh (2 c+2 d x)}}{15 d}+\frac{b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt{2} d}+\frac{1}{60} \left (92 a^2-9 b^2\right ) \int \sqrt{a+\frac{1}{2} b \sinh (2 c+2 d x)} \, dx-\frac{1}{15} \left (2 a \left (4 a^2+b^2\right )\right ) \int \frac{1}{\sqrt{a+\frac{1}{2} b \sinh (2 c+2 d x)}} \, dx\\ &=\frac{2 \sqrt{2} a b \cosh (2 c+2 d x) \sqrt{2 a+b \sinh (2 c+2 d x)}}{15 d}+\frac{b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt{2} d}+\frac{\left (\left (92 a^2-9 b^2\right ) \sqrt{a+\frac{1}{2} b \sinh (2 c+2 d x)}\right ) \int \sqrt{\frac{a}{a-\frac{i b}{2}}+\frac{b \sinh (2 c+2 d x)}{2 \left (a-\frac{i b}{2}\right )}} \, dx}{60 \sqrt{\frac{a+\frac{1}{2} b \sinh (2 c+2 d x)}{a-\frac{i b}{2}}}}-\frac{\left (2 a \left (4 a^2+b^2\right ) \sqrt{\frac{a+\frac{1}{2} b \sinh (2 c+2 d x)}{a-\frac{i b}{2}}}\right ) \int \frac{1}{\sqrt{\frac{a}{a-\frac{i b}{2}}+\frac{b \sinh (2 c+2 d x)}{2 \left (a-\frac{i b}{2}\right )}}} \, dx}{15 \sqrt{a+\frac{1}{2} b \sinh (2 c+2 d x)}}\\ &=\frac{2 \sqrt{2} a b \cosh (2 c+2 d x) \sqrt{2 a+b \sinh (2 c+2 d x)}}{15 d}+\frac{b \cosh (2 c+2 d x) (2 a+b \sinh (2 c+2 d x))^{3/2}}{20 \sqrt{2} d}-\frac{i \left (92 a^2-9 b^2\right ) E\left (\frac{1}{2} \left (2 i c-\frac{\pi }{2}+2 i d x\right )|\frac{2 b}{2 i a+b}\right ) \sqrt{2 a+b \sinh (2 c+2 d x)}}{60 \sqrt{2} d \sqrt{\frac{2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}+\frac{2 i \sqrt{2} a \left (4 a^2+b^2\right ) F\left (\frac{1}{2} \left (2 i c-\frac{\pi }{2}+2 i d x\right )|\frac{2 b}{2 i a+b}\right ) \sqrt{\frac{2 a+b \sinh (2 c+2 d x)}{2 a-i b}}}{15 d \sqrt{2 a+b \sinh (2 c+2 d x)}}\\ \end{align*}
Mathematica [A] time = 1.37497, size = 239, normalized size = 0.79 \[ \frac{-32 i a \left (4 a^2+b^2\right ) \sqrt{\frac{2 a+b \sinh (2 (c+d x))}{2 a-i b}} F\left (\frac{1}{4} (-4 i c-4 i d x+\pi )|-\frac{2 i b}{2 a-i b}\right )+2 \left (92 a^2 b+184 i a^3-18 i a b^2-9 b^3\right ) \sqrt{\frac{2 a+b \sinh (2 (c+d x))}{2 a-i b}} E\left (\frac{1}{4} (-4 i c-4 i d x+\pi )|-\frac{2 i b}{2 a-i b}\right )+b \left (88 a^2 \cosh (2 (c+d x))+b \sinh (4 (c+d x)) (28 a+3 b \sinh (2 (c+d x)))\right )}{120 d \sqrt{4 a+2 b \sinh (2 (c+d x))}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.542, size = 1260, normalized size = 4.2 \begin{align*} \text{result too large to display} \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 \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a\right )}^{\frac{5}{2}}\,{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^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2}\right )} \sqrt{b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a}, x\right ) \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{\left (b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]