Optimal. Leaf size=177 \[ -\frac{3 b \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{2 c^5 d^2}+\frac{3 b \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{2 c^5 d^2}+\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{3 x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d^2}-\frac{3 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^5 d^2}-\frac{b x^2}{2 c^3 d^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \sqrt{c x-1} \sqrt{c x+1}}{2 c^5 d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.225377, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {5750, 98, 21, 74, 5766, 5694, 4182, 2279, 2391} \[ -\frac{3 b \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{2 c^5 d^2}+\frac{3 b \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{2 c^5 d^2}+\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{3 x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d^2}-\frac{3 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^5 d^2}-\frac{b x^2}{2 c^3 d^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \sqrt{c x-1} \sqrt{c x+1}}{2 c^5 d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5750
Rule 98
Rule 21
Rule 74
Rule 5766
Rule 5694
Rule 4182
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{b \int \frac{x^3}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 c d^2}-\frac{3 \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{2 c^2 d}\\ &=-\frac{b x^2}{2 c^3 d^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d^2}+\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{b \int \frac{x (-2-2 c x)}{\sqrt{-1+c x} (1+c x)^{3/2}} \, dx}{2 c^3 d^2}-\frac{(3 b) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 c^3 d^2}-\frac{3 \int \frac{a+b \cosh ^{-1}(c x)}{d-c^2 d x^2} \, dx}{2 c^4 d}\\ &=-\frac{b x^2}{2 c^3 d^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{3 b \sqrt{-1+c x} \sqrt{1+c x}}{2 c^5 d^2}+\frac{3 x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d^2}+\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac{3 \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^5 d^2}+\frac{b \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{c^3 d^2}\\ &=-\frac{b x^2}{2 c^3 d^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b \sqrt{-1+c x} \sqrt{1+c x}}{2 c^5 d^2}+\frac{3 x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d^2}+\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^5 d^2}-\frac{(3 b) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^5 d^2}+\frac{(3 b) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^5 d^2}\\ &=-\frac{b x^2}{2 c^3 d^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b \sqrt{-1+c x} \sqrt{1+c x}}{2 c^5 d^2}+\frac{3 x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d^2}+\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^5 d^2}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 c^5 d^2}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 c^5 d^2}\\ &=-\frac{b x^2}{2 c^3 d^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b \sqrt{-1+c x} \sqrt{1+c x}}{2 c^5 d^2}+\frac{3 x \left (a+b \cosh ^{-1}(c x)\right )}{2 c^4 d^2}+\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac{3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^5 d^2}-\frac{3 b \text{Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{2 c^5 d^2}+\frac{3 b \text{Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{2 c^5 d^2}\\ \end{align*}
Mathematica [A] time = 1.04641, size = 244, normalized size = 1.38 \[ \frac{-6 b \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )+6 b \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )-\frac{2 a c x}{c^2 x^2-1}+4 a c x+3 a \log (1-c x)-3 a \log (c x+1)-4 b c x \sqrt{\frac{c x-1}{c x+1}}+\frac{b c x \sqrt{\frac{c x-1}{c x+1}}}{1-c x}+\frac{b \sqrt{\frac{c x-1}{c x+1}}}{1-c x}-3 b \sqrt{\frac{c x-1}{c x+1}}+4 b c x \cosh ^{-1}(c x)+\frac{b \cosh ^{-1}(c x)}{1-c x}-\frac{b \cosh ^{-1}(c x)}{c x+1}+6 b \cosh ^{-1}(c x) \log \left (1-e^{\cosh ^{-1}(c x)}\right )-6 b \cosh ^{-1}(c x) \log \left (e^{\cosh ^{-1}(c x)}+1\right )}{4 c^5 d^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.249, size = 300, normalized size = 1.7 \begin{align*}{\frac{ax}{{d}^{2}{c}^{4}}}-{\frac{a}{4\,{c}^{5}{d}^{2} \left ( cx-1 \right ) }}+{\frac{3\,a\ln \left ( cx-1 \right ) }{4\,{c}^{5}{d}^{2}}}-{\frac{a}{4\,{c}^{5}{d}^{2} \left ( cx+1 \right ) }}-{\frac{3\,a\ln \left ( cx+1 \right ) }{4\,{c}^{5}{d}^{2}}}+{\frac{b{\rm arccosh} \left (cx\right )x}{{d}^{2}{c}^{4}}}-{\frac{b}{{c}^{5}{d}^{2}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{\rm arccosh} \left (cx\right )x}{2\,{d}^{2}{c}^{4} \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{b}{2\,{c}^{5}{d}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{cx-1}\sqrt{cx+1}}-{\frac{3\,b{\rm arccosh} \left (cx\right )}{2\,{c}^{5}{d}^{2}}\ln \left ( 1+cx+\sqrt{cx-1}\sqrt{cx+1} \right ) }-{\frac{3\,b}{2\,{c}^{5}{d}^{2}}{\it polylog} \left ( 2,-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{3\,b{\rm arccosh} \left (cx\right )}{2\,{c}^{5}{d}^{2}}\ln \left ( 1-cx-\sqrt{cx-1}\sqrt{cx+1} \right ) }+{\frac{3\,b}{2\,{c}^{5}{d}^{2}}{\it polylog} \left ( 2,cx+\sqrt{cx-1}\sqrt{cx+1} \right ) } \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*} \text{result too large to display} \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{b x^{4} \operatorname{arcosh}\left (c x\right ) + a x^{4}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, 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*} \frac{\int \frac{a x^{4}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac{b x^{4} \operatorname{acosh}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \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 )} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]