Optimal. Leaf size=237 \[ -\frac{3 \sqrt{c x-1} \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{4 b^2 c^4 \sqrt{1-c x}}-\frac{3 \sqrt{c x-1} \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^4 \sqrt{1-c x}}+\frac{3 \sqrt{c x-1} \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{4 b^2 c^4 \sqrt{1-c x}}+\frac{3 \sqrt{c x-1} \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^4 \sqrt{1-c x}}-\frac{x^3 \sqrt{c x-1}}{b c \sqrt{1-c x} \left (a+b \cosh ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.767886, antiderivative size = 298, normalized size of antiderivative = 1.26, number of steps used = 11, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5798, 5775, 5670, 5448, 3303, 3298, 3301} \[ -\frac{3 \sqrt{c x-1} \sqrt{c x+1} \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{4 b^2 c^4 \sqrt{1-c^2 x^2}}-\frac{3 \sqrt{c x-1} \sqrt{c x+1} \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b^2 c^4 \sqrt{1-c^2 x^2}}+\frac{3 \sqrt{c x-1} \sqrt{c x+1} \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{4 b^2 c^4 \sqrt{1-c^2 x^2}}+\frac{3 \sqrt{c x-1} \sqrt{c x+1} \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b^2 c^4 \sqrt{1-c^2 x^2}}-\frac{x^3 \sqrt{c x-1} \sqrt{c x+1}}{b c \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 5798
Rule 5775
Rule 5670
Rule 5448
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 )^2} \, 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 )^2} \, dx}{\sqrt{1-c^2 x^2}}\\ &=-\frac{x^3 \sqrt{-1+c x} \sqrt{1+c x}}{b c \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}+\frac{\left (3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^2}{a+b \cosh ^{-1}(c x)} \, dx}{b c \sqrt{1-c^2 x^2}}\\ &=-\frac{x^3 \sqrt{-1+c x} \sqrt{1+c x}}{b c \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}+\frac{\left (3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^4 \sqrt{1-c^2 x^2}}\\ &=-\frac{x^3 \sqrt{-1+c x} \sqrt{1+c x}}{b c \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}+\frac{\left (3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \left (\frac{\sinh (x)}{4 (a+b x)}+\frac{\sinh (3 x)}{4 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^4 \sqrt{1-c^2 x^2}}\\ &=-\frac{x^3 \sqrt{-1+c x} \sqrt{1+c x}}{b c \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}+\frac{\left (3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b 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{\sinh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^4 \sqrt{1-c^2 x^2}}\\ &=-\frac{x^3 \sqrt{-1+c x} \sqrt{1+c x}}{b c \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}+\frac{\left (3 \sqrt{-1+c x} \sqrt{1+c x} \cosh \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 b c^4 \sqrt{1-c^2 x^2}}+\frac{\left (3 \sqrt{-1+c x} \sqrt{1+c x} \cosh \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 b 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{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^4 \sqrt{1-c^2 x^2}}-\frac{\left (3 \sqrt{-1+c x} \sqrt{1+c x} \sinh \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 b c^4 \sqrt{1-c^2 x^2}}\\ &=-\frac{x^3 \sqrt{-1+c x} \sqrt{1+c x}}{b c \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )}-\frac{3 \sqrt{-1+c x} \sqrt{1+c x} \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac{a}{b}\right )}{4 b^2 c^4 \sqrt{1-c^2 x^2}}-\frac{3 \sqrt{-1+c x} \sqrt{1+c x} \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right ) \sinh \left (\frac{3 a}{b}\right )}{4 b^2 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{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{4 b^2 c^4 \sqrt{1-c^2 x^2}}+\frac{3 \sqrt{-1+c x} \sqrt{1+c x} \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{4 b^2 c^4 \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.47508, size = 144, normalized size = 0.61 \[ \frac{\sqrt{1-c^2 x^2} \left (\frac{4 b c^3 x^3}{a+b \cosh ^{-1}(c x)}+3 \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )+3 \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-3 \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )-3 \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )\right )}{4 b^2 c^4 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.303, size = 634, normalized size = 2.7 \begin{align*} -{\frac{1}{ \left ( 8\,{c}^{2}{x}^{2}-8 \right ){c}^{4}b \left ( a+b{\rm arccosh} \left (cx\right ) \right ) }\sqrt{-{c}^{2}{x}^{2}+1} \left ( -4\,\sqrt{cx+1}\sqrt{cx-1}{x}^{3}{c}^{3}+4\,{c}^{4}{x}^{4}+3\,\sqrt{cx+1}\sqrt{cx-1}xc-5\,{c}^{2}{x}^{2}+1 \right ) }-{\frac{3}{8\,{c}^{4} \left ({c}^{2}{x}^{2}-1 \right ){b}^{2}} \left ( \sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ) \sqrt{-{c}^{2}{x}^{2}+1}{\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}{ \left ( 8\,{c}^{2}{x}^{2}-8 \right ){c}^{4}{b}^{2} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) }\sqrt{-{c}^{2}{x}^{2}+1}\sqrt{cx-1}\sqrt{cx+1} \left ( 4\,\sqrt{cx+1}\sqrt{cx-1}{x}^{2}b{c}^{2}+4\,{x}^{3}b{c}^{3}+3\,{\rm arccosh} \left (cx\right ){{\rm e}^{-3\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-3\,{\rm arccosh} \left (cx\right )-3\,{\frac{a}{b}} \right ) b+3\,{{\rm e}^{-3\,{\frac{a}{b}}}}{\it Ei} \left ( 1,-3\,{\rm arccosh} \left (cx\right )-3\,{\frac{a}{b}} \right ) a-\sqrt{cx+1}\sqrt{cx-1}b-3\,xbc \right ) }+{\frac{3}{ \left ( 8\,{c}^{2}{x}^{2}-8 \right ){c}^{4}{b}^{2} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) }\sqrt{-{c}^{2}{x}^{2}+1}\sqrt{cx-1}\sqrt{cx+1} \left ({\rm arccosh} \left (cx\right ){{\rm e}^{-{\frac{a}{b}}}}{\it Ei} \left ( 1,-{\rm arccosh} \left (cx\right )-{\frac{a}{b}} \right ) b+\sqrt{cx+1}\sqrt{cx-1}b+{{\rm e}^{-{\frac{a}{b}}}}{\it Ei} \left ( 1,-{\rm arccosh} \left (cx\right )-{\frac{a}{b}} \right ) a+xbc \right ) }-{\frac{3}{ \left ( 8\,{c}^{2}{x}^{2}-8 \right ){c}^{4}b \left ( a+b{\rm arccosh} \left (cx\right ) \right ) }\sqrt{-{c}^{2}{x}^{2}+1} \left ( -\sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ) }-{\frac{3}{8\,{c}^{4} \left ({c}^{2}{x}^{2}-1 \right ){b}^{2}} \left ( \sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ) \sqrt{-{c}^{2}{x}^{2}+1}{\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*} -\frac{c^{3} x^{6} - c x^{4} +{\left (c^{2} x^{5} - x^{3}\right )} \sqrt{c x + 1} \sqrt{c x - 1}}{{\left ({\left (c x + 1\right )} \sqrt{c x - 1} b^{2} c^{2} x +{\left (b^{2} c^{3} x^{2} - b^{2} c\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) +{\left ({\left (c x + 1\right )} \sqrt{c x - 1} a b c^{2} x +{\left (a b c^{3} x^{2} - a b c\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1}} + \int \frac{3 \, c^{5} x^{7} - 7 \, c^{3} x^{5} + 4 \, c x^{3} +{\left (3 \, c^{3} x^{5} - 2 \, c x^{3}\right )}{\left (c x + 1\right )}{\left (c x - 1\right )} + 3 \,{\left (2 \, c^{4} x^{6} - 3 \, c^{2} x^{4} + x^{2}\right )} \sqrt{c x + 1} \sqrt{c x - 1}}{{\left ({\left (c x + 1\right )}^{\frac{3}{2}}{\left (c x - 1\right )} b^{2} c^{3} x^{2} + 2 \,{\left (b^{2} c^{4} x^{3} - b^{2} c^{2} x\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (b^{2} c^{5} x^{4} - 2 \, b^{2} c^{3} x^{2} + b^{2} c\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right ) +{\left ({\left (c x + 1\right )}^{\frac{3}{2}}{\left (c x - 1\right )} a b c^{3} x^{2} + 2 \,{\left (a b c^{4} x^{3} - a b c^{2} x\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (a b c^{5} x^{4} - 2 \, a b c^{3} x^{2} + a b c\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1}}\,{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^{2} c^{2} x^{2} +{\left (b^{2} c^{2} x^{2} - b^{2}\right )} \operatorname{arcosh}\left (c x\right )^{2} - a^{2} + 2 \,{\left (a b c^{2} x^{2} - a b\right )} \operatorname{arcosh}\left (c x\right )}, 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 )^{2}}\, 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 )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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