Optimal. Leaf size=336 \[ -\frac{5 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b x \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{10 b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{b^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.968666, antiderivative size = 351, normalized size of antiderivative = 1.04, number of steps used = 17, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {5798, 5752, 5718, 5694, 4182, 2279, 2391, 5750, 74} \[ -\frac{5 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b x \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2}}-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{10 b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5798
Rule 5752
Rule 5718
Rule 5694
Rule 4182
Rule 2279
Rule 2391
Rule 5750
Rule 74
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{\left (2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )^2}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (2 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\left (-1+c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{\left (b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (4 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{\left (b \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (4 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{10 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (4 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (4 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{10 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (4 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (4 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{10 b \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 b^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 b^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 4.46191, size = 341, normalized size = 1.01 \[ \frac{-b^2 \left (20 \left (\frac{c x-1}{c x+1}\right )^{3/2} (c x+1)^3 \text{PolyLog}\left (2,-e^{-\cosh ^{-1}(c x)}\right )-20 \left (\frac{c x-1}{c x+1}\right )^{3/2} (c x+1)^3 \text{PolyLog}\left (2,e^{-\cosh ^{-1}(c x)}\right )+2 \cosh ^{-1}(c x)^2-2 \left (3 \cosh ^{-1}(c x)^2+1\right ) \cosh \left (2 \cosh ^{-1}(c x)\right )-15 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \left (\log \left (1-e^{-\cosh ^{-1}(c x)}\right )-\log \left (e^{-\cosh ^{-1}(c x)}+1\right )\right )-2 \cosh ^{-1}(c x) \sinh \left (2 \cosh ^{-1}(c x)\right )+5 \cosh ^{-1}(c x) \log \left (1-e^{-\cosh ^{-1}(c x)}\right ) \sinh \left (3 \cosh ^{-1}(c x)\right )-5 \cosh ^{-1}(c x) \log \left (e^{-\cosh ^{-1}(c x)}+1\right ) \sinh \left (3 \cosh ^{-1}(c x)\right )+2\right )+4 a^2 \left (3 c^2 x^2-2\right )-a b \left (\cosh ^{-1}(c x) \left (4-12 \cosh \left (2 \cosh ^{-1}(c x)\right )\right )-2 \sinh \left (2 \cosh ^{-1}(c x)\right )+5 \left (\sinh \left (3 \cosh ^{-1}(c x)\right )-3 \sqrt{\frac{c x-1}{c x+1}} (c x+1)\right ) \log \left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )\right )}{12 c^4 d \left (d-c^2 d x^2\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.408, size = 835, normalized size = 2.5 \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*} \frac{1}{6} \, a b c{\left (\frac{2 \, \sqrt{-d} x}{c^{6} d^{3} x^{2} - c^{4} d^{3}} + \frac{5 \, \sqrt{-d} \log \left (c x + 1\right )}{c^{5} d^{3}} - \frac{5 \, \sqrt{-d} \log \left (c x - 1\right )}{c^{5} d^{3}}\right )} + \frac{2}{3} \, a b{\left (\frac{3 \, x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} c^{2} d} - \frac{2}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} c^{4} d}\right )} \operatorname{arcosh}\left (c x\right ) + \frac{1}{3} \, a^{2}{\left (\frac{3 \, x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} c^{2} d} - \frac{2}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} c^{4} d}\right )} + b^{2} \int \frac{x^{3} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )^{2}}{{\left (-c^{2} d x^{2} + d\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 (-\frac{{\left (b^{2} x^{3} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b x^{3} \operatorname{arcosh}\left (c x\right ) + a^{2} x^{3}\right )} \sqrt{-c^{2} d x^{2} + d}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{5}{2}}}\, dx \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 (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2} x^{3}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]