Optimal. Leaf size=69 \[ -\frac{10 i \text{EllipticF}\left (\frac{1}{2} i (a+b x),2\right )}{21 b}+\frac{2 \sinh (a+b x) \cosh ^{\frac{5}{2}}(a+b x)}{7 b}+\frac{10 \sinh (a+b x) \sqrt{\cosh (a+b x)}}{21 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0328581, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2635, 2641} \[ -\frac{10 i F\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{21 b}+\frac{2 \sinh (a+b x) \cosh ^{\frac{5}{2}}(a+b x)}{7 b}+\frac{10 \sinh (a+b x) \sqrt{\cosh (a+b x)}}{21 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int \cosh ^{\frac{7}{2}}(a+b x) \, dx &=\frac{2 \cosh ^{\frac{5}{2}}(a+b x) \sinh (a+b x)}{7 b}+\frac{5}{7} \int \cosh ^{\frac{3}{2}}(a+b x) \, dx\\ &=\frac{10 \sqrt{\cosh (a+b x)} \sinh (a+b x)}{21 b}+\frac{2 \cosh ^{\frac{5}{2}}(a+b x) \sinh (a+b x)}{7 b}+\frac{5}{21} \int \frac{1}{\sqrt{\cosh (a+b x)}} \, dx\\ &=-\frac{10 i F\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{21 b}+\frac{10 \sqrt{\cosh (a+b x)} \sinh (a+b x)}{21 b}+\frac{2 \cosh ^{\frac{5}{2}}(a+b x) \sinh (a+b x)}{7 b}\\ \end{align*}
Mathematica [A] time = 0.106611, size = 55, normalized size = 0.8 \[ \frac{(23 \sinh (a+b x)+3 \sinh (3 (a+b x))) \sqrt{\cosh (a+b x)}-20 i \text{EllipticF}\left (\frac{1}{2} i (a+b x),2\right )}{42 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.114, size = 201, normalized size = 2.9 \begin{align*}{\frac{2}{21\,b}\sqrt{ \left ( 2\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) \left ( \sinh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}} \left ( 48\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{9}-120\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{7}+128\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{5}-72\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{3}+5\,\sqrt{- \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cosh \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) +16\,\cosh \left ( 1/2\,bx+a/2 \right ) \right ){\frac{1}{\sqrt{2\, \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+ \left ( \sinh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}}}} \left ( \sinh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}}}} \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 \cosh \left (b x + a\right )^{\frac{7}{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 (\cosh \left (b x + a\right )^{\frac{7}{2}}, 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 \cosh \left (b x + a\right )^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]