Optimal. Leaf size=439 \[ \frac{\sqrt{1-c x} \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^3 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^3 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \cosh \left (\frac{6 a}{b}\right ) \text{Chi}\left (\frac{6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^3 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left (\frac{8 a}{b}\right ) \text{Chi}\left (\frac{8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 b c^3 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^3 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^3 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \sinh \left (\frac{6 a}{b}\right ) \text{Shi}\left (\frac{6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^3 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left (\frac{8 a}{b}\right ) \text{Shi}\left (\frac{8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 b c^3 \sqrt{c x-1}}-\frac{5 \sqrt{1-c x} \log \left (a+b \cosh ^{-1}(c x)\right )}{128 b c^3 \sqrt{c x-1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.979051, antiderivative size = 556, normalized size of antiderivative = 1.27, number of steps used = 16, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5798, 5781, 5448, 3303, 3298, 3301} \[ \frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{32 b c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c x)\right )}{32 b c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{6 a}{b}\right ) \text{Chi}\left (\frac{6 a}{b}+6 \cosh ^{-1}(c x)\right )}{32 b c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{8 a}{b}\right ) \text{Chi}\left (\frac{8 a}{b}+8 \cosh ^{-1}(c x)\right )}{128 b c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{32 b c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c x)\right )}{32 b c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{6 a}{b}\right ) \text{Shi}\left (\frac{6 a}{b}+6 \cosh ^{-1}(c x)\right )}{32 b c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{8 a}{b}\right ) \text{Shi}\left (\frac{8 a}{b}+8 \cosh ^{-1}(c x)\right )}{128 b c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 \sqrt{1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{128 b c^3 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5798
Rule 5781
Rule 5448
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{x^2 \left (1-c^2 x^2\right )^{5/2}}{a+b \cosh ^{-1}(c x)} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{x^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{a+b \cosh ^{-1}(c x)} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh ^6(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int \left (-\frac{5}{128 (a+b x)}+\frac{\cosh (2 x)}{32 (a+b x)}+\frac{\cosh (4 x)}{32 (a+b x)}-\frac{\cosh (6 x)}{32 (a+b x)}+\frac{\cosh (8 x)}{128 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{5 \sqrt{1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{128 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int \frac{\cosh (8 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{128 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{1-c^2 x^2} \operatorname{Subst}\left (\int \frac{\cosh (6 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{5 \sqrt{1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{128 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (\sqrt{1-c^2 x^2} \cosh \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (\sqrt{1-c^2 x^2} \cosh \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (\sqrt{1-c^2 x^2} \cosh \left (\frac{6 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{6 a}{b}+6 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (\sqrt{1-c^2 x^2} \cosh \left (\frac{8 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{8 a}{b}+8 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{128 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (\sqrt{1-c^2 x^2} \sinh \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (\sqrt{1-c^2 x^2} \sinh \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (\sqrt{1-c^2 x^2} \sinh \left (\frac{6 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{6 a}{b}+6 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (\sqrt{1-c^2 x^2} \sinh \left (\frac{8 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{8 a}{b}+8 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{128 c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{32 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c x)\right )}{32 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{6 a}{b}\right ) \text{Chi}\left (\frac{6 a}{b}+6 \cosh ^{-1}(c x)\right )}{32 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{8 a}{b}\right ) \text{Chi}\left (\frac{8 a}{b}+8 \cosh ^{-1}(c x)\right )}{128 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5 \sqrt{1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{128 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{32 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c x)\right )}{32 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{6 a}{b}\right ) \text{Shi}\left (\frac{6 a}{b}+6 \cosh ^{-1}(c x)\right )}{32 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{8 a}{b}\right ) \text{Shi}\left (\frac{8 a}{b}+8 \cosh ^{-1}(c x)\right )}{128 b c^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 1.16922, size = 233, normalized size = 0.53 \[ \frac{\sqrt{1-c^2 x^2} \left (4 \cosh \left (\frac{2 a}{b}\right ) \text{Chi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+4 \cosh \left (\frac{4 a}{b}\right ) \text{Chi}\left (4 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-4 \cosh \left (\frac{6 a}{b}\right ) \text{Chi}\left (6 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+\cosh \left (\frac{8 a}{b}\right ) \text{Chi}\left (8 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-4 \sinh \left (\frac{2 a}{b}\right ) \text{Shi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-4 \sinh \left (\frac{4 a}{b}\right ) \text{Shi}\left (4 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+4 \sinh \left (\frac{6 a}{b}\right ) \text{Shi}\left (6 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-\sinh \left (\frac{8 a}{b}\right ) \text{Shi}\left (8 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-5 \log \left (a+b \cosh ^{-1}(c x)\right )\right )}{128 c^3 \sqrt{\frac{c x-1}{c x+1}} (b c x+b)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.262, size = 773, normalized size = 1.8 \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 \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{5}{2}} x^{2}}{b \operatorname{arcosh}\left (c x\right ) + a}\,{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 (\frac{{\left (c^{4} x^{6} - 2 \, c^{2} x^{4} + x^{2}\right )} \sqrt{-c^{2} x^{2} + 1}}{b \operatorname{arcosh}\left (c x\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 \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{5}{2}} x^{2}}{b \operatorname{arcosh}\left (c x\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]