Optimal. Leaf size=197 \[ \frac{3 \sqrt{c x-1} \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{4 b c^4 \sqrt{1-c x}}+\frac{\sqrt{c x-1} \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{4 b c^4 \sqrt{1-c x}}-\frac{3 \sqrt{c x-1} \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{4 b c^4 \sqrt{1-c x}}-\frac{\sqrt{c x-1} \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{4 b c^4 \sqrt{1-c x}} \]
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Rubi [A] time = 0.689944, antiderivative size = 245, normalized size of antiderivative = 1.24, number of steps used = 10, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5798, 5781, 3312, 3303, 3298, 3301} \[ \frac{3 \sqrt{c x-1} \sqrt{c x+1} \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{4 b c^4 \sqrt{1-c^2 x^2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b c^4 \sqrt{1-c^2 x^2}}-\frac{3 \sqrt{c x-1} \sqrt{c x+1} \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{4 b c^4 \sqrt{1-c^2 x^2}}-\frac{\sqrt{c x-1} \sqrt{c x+1} \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b c^4 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5781
Rule 3312
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^3}{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt{1-c^2 x^2}}\\ &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\cosh ^3(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c^4 \sqrt{1-c^2 x^2}}\\ &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \left (\frac{3 \cosh (x)}{4 (a+b x)}+\frac{\cosh (3 x)}{4 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^4 \sqrt{1-c^2 x^2}}\\ &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^4 \sqrt{1-c^2 x^2}}+\frac{\left (3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^4 \sqrt{1-c^2 x^2}}\\ &=\frac{\left (3 \sqrt{-1+c x} \sqrt{1+c x} \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^4 \sqrt{1-c^2 x^2}}+\frac{\left (\sqrt{-1+c x} \sqrt{1+c x} \cosh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^4 \sqrt{1-c^2 x^2}}-\frac{\left (3 \sqrt{-1+c x} \sqrt{1+c x} \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^4 \sqrt{1-c^2 x^2}}-\frac{\left (\sqrt{-1+c x} \sqrt{1+c x} \sinh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 c^4 \sqrt{1-c^2 x^2}}\\ &=\frac{3 \sqrt{-1+c x} \sqrt{1+c x} \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{4 b c^4 \sqrt{1-c^2 x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b c^4 \sqrt{1-c^2 x^2}}-\frac{3 \sqrt{-1+c x} \sqrt{1+c x} \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{4 b c^4 \sqrt{1-c^2 x^2}}-\frac{\sqrt{-1+c x} \sqrt{1+c x} \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b c^4 \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.279478, size = 130, normalized size = 0.66 \[ \frac{\sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (3 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )+\cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-3 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )-\sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )\right )}{4 b c^4 \sqrt{-(c x-1) (c x+1)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.192, size = 349, normalized size = 1.8 \begin{align*}{\frac{1}{8\,{c}^{4} \left ({c}^{2}{x}^{2}-1 \right ) b}\sqrt{-{c}^{2}{x}^{2}+1} \left ( \sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ){\it Ei} \left ( 1,3\,{\rm arccosh} \left (cx\right )+3\,{\frac{a}{b}} \right ){{\rm e}^{-{\frac{b{\rm arccosh} \left (cx\right )-3\,a}{b}}}}}+{\frac{1}{8\,{c}^{4} \left ({c}^{2}{x}^{2}-1 \right ) b}\sqrt{-{c}^{2}{x}^{2}+1} \left ( \sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ){\it Ei} \left ( 1,-3\,{\rm arccosh} \left (cx\right )-3\,{\frac{a}{b}} \right ){{\rm e}^{-{\frac{b{\rm arccosh} \left (cx\right )+3\,a}{b}}}}}+{\frac{3}{8\,{c}^{4} \left ({c}^{2}{x}^{2}-1 \right ) b}\sqrt{-{c}^{2}{x}^{2}+1} \left ( \sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ){\it Ei} \left ( 1,-{\rm arccosh} \left (cx\right )-{\frac{a}{b}} \right ){{\rm e}^{-{\frac{a+b{\rm arccosh} \left (cx\right )}{b}}}}}+{\frac{3}{8\,{c}^{4} \left ({c}^{2}{x}^{2}-1 \right ) b}\sqrt{-{c}^{2}{x}^{2}+1} \left ( \sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ){\it Ei} \left ( 1,{\rm arccosh} \left (cx\right )+{\frac{a}{b}} \right ){{\rm e}^{-{\frac{b{\rm arccosh} \left (cx\right )-a}{b}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{-c^{2} x^{2} + 1}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} x^{2} + 1} x^{3}}{a c^{2} x^{2} +{\left (b c^{2} x^{2} - b\right )} \operatorname{arcosh}\left (c x\right ) - a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\sqrt{-c^{2} x^{2} + 1}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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