Optimal. Leaf size=301 \[ -\frac{\sqrt{\frac{\pi }{2}} \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )}{15 b^{7/2} d x}+\frac{\sqrt{\frac{\pi }{2}} \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )}{15 b^{7/2} d x}-\frac{x}{15 b^2 \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^{3/2}}-\frac{\sqrt{d x^2} \sqrt{d x^2+2}}{15 b^3 d x \sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}-\frac{d x^4+2 x^2}{5 b x \sqrt{d x^2} \sqrt{d x^2+2} \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0774556, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {5889, 5885} \[ -\frac{\sqrt{\frac{\pi }{2}} \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )}{15 b^{7/2} d x}+\frac{\sqrt{\frac{\pi }{2}} \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )}{15 b^{7/2} d x}-\frac{x}{15 b^2 \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^{3/2}}-\frac{\sqrt{d x^2} \sqrt{d x^2+2}}{15 b^3 d x \sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}-\frac{d x^4+2 x^2}{5 b x \sqrt{d x^2} \sqrt{d x^2+2} \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5889
Rule 5885
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^{7/2}} \, dx &=-\frac{2 x^2+d x^4}{5 b x \sqrt{d x^2} \sqrt{2+d x^2} \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^{5/2}}-\frac{x}{15 b^2 \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^{3/2}}+\frac{\int \frac{1}{\left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^{3/2}} \, dx}{15 b^2}\\ &=-\frac{2 x^2+d x^4}{5 b x \sqrt{d x^2} \sqrt{2+d x^2} \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^{5/2}}-\frac{x}{15 b^2 \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^{3/2}}-\frac{\sqrt{d x^2} \sqrt{2+d x^2}}{15 b^3 d x \sqrt{a+b \cosh ^{-1}\left (1+d x^2\right )}}+\frac{\sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (1+d x^2\right )}}{\sqrt{2} \sqrt{b}}\right ) \left (\cosh \left (\frac{a}{2 b}\right )-\sinh \left (\frac{a}{2 b}\right )\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )}{15 b^{7/2} d x}-\frac{\sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (1+d x^2\right )}}{\sqrt{2} \sqrt{b}}\right ) \left (\cosh \left (\frac{a}{2 b}\right )+\sinh \left (\frac{a}{2 b}\right )\right ) \sinh \left (\frac{1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )}{15 b^{7/2} d x}\\ \end{align*}
Mathematica [A] time = 1.22105, size = 291, normalized size = 0.97 \[ -\frac{x \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \left (4 \sqrt{b} \left (\cosh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \left (\left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^2+3 b^2\right )+b \sinh \left (\frac{1}{2} \cosh ^{-1}\left (d x^2+1\right )\right ) \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )\right )+\sqrt{2 \pi } \left (\sinh \left (\frac{a}{2 b}\right )+\cosh \left (\frac{a}{2 b}\right )\right ) \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^{5/2} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )+\sqrt{2 \pi } \left (\sinh \left (\frac{a}{2 b}\right )-\cosh \left (\frac{a}{2 b}\right )\right ) \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^{5/2} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}\left (d x^2+1\right )}}{\sqrt{2} \sqrt{b}}\right )\right )}{30 b^{7/2} \sqrt{d x^2} \sqrt{\frac{d x^2}{d x^2+2}} \sqrt{d x^2+2} \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.063, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\rm arccosh} \left (d{x}^{2}+1\right ) \right ) ^{-{\frac{7}{2}}}\, 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 \frac{1}{{\left (b \operatorname{arcosh}\left (d x^{2} + 1\right ) + a\right )}^{\frac{7}{2}}}\,{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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]