Optimal. Leaf size=136 \[ -\frac {\sqrt {c x-1} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{b^2 c^3 \sqrt {1-c x}}+\frac {\sqrt {c x-1} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{b^2 c^3 \sqrt {1-c x}}-\frac {x^2 \sqrt {c x-1}}{b c \sqrt {1-c x} \left (a+b \cosh ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.63, antiderivative size = 175, normalized size of antiderivative = 1.29, number of steps used = 8, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5798, 5775, 5670, 5448, 12, 3303, 3298, 3301} \[ -\frac {\sqrt {c x-1} \sqrt {c x+1} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{b^2 c^3 \sqrt {1-c^2 x^2}}+\frac {\sqrt {c x-1} \sqrt {c x+1} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{b^2 c^3 \sqrt {1-c^2 x^2}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3298
Rule 3301
Rule 3303
Rule 5448
Rule 5670
Rule 5775
Rule 5798
Rubi steps
\begin {align*} \int \frac {x^2}{\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^2}{\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^2 \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 (2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{a+b \cosh ^{-1}(c x)} \, dx}{b c \sqrt {1-c^2 x^2}}\\ &=-\frac {x^2 \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 (2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3 \sqrt {1-c^2 x^2}}\\ &=-\frac {x^2 \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 (2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 (a+b x)} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3 \sqrt {1-c^2 x^2}}\\ &=-\frac {x^2 \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 (\sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3 \sqrt {1-c^2 x^2}}\\ &=-\frac {x^2 \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 (\sqrt {-1+c x} \sqrt {1+c x} \cosh \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 )}{b c^3 \sqrt {1-c^2 x^2}}-\frac {\left (\sqrt {-1+c x} \sqrt {1+c x} \sinh \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 )}{b c^3 \sqrt {1-c^2 x^2}}\\ &=-\frac {x^2 \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 {\sqrt {-1+c x} \sqrt {1+c x} \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{b^2 c^3 \sqrt {1-c^2 x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{b^2 c^3 \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 117, normalized size = 0.86 \[ \frac {\sqrt {1-c^2 x^2} \left (\sinh \left (\frac {2 a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-\cosh \left (\frac {2 a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+b c^2 x^2\right )}{b^2 c^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{a^{2} c^{2} x^{2} + {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \operatorname {arcosh}\left (c x\right )^{2} - a^{2} + 2 \, {\left (a b c^{2} x^{2} - a b\right )} \operatorname {arcosh}\left (c x\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.54, size = 377, normalized size = 2.77 \[ -\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-2 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right )}{4 \left (c^{2} x^{2}-1\right ) c^{3} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right ) b}-\frac {\left (\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}\, \Ei \left (1, 2 \,\mathrm {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {b \,\mathrm {arccosh}\left (c x \right )-2 a}{b}}}{2 b^{2} \left (c^{2} x^{2}-1\right ) c^{3}}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (2 \sqrt {c x +1}\, \sqrt {c x -1}\, x b c +2 x^{2} b \,c^{2}+2 \,\mathrm {arccosh}\left (c x \right ) \Ei \left (1, -2 \,\mathrm {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {2 a}{b}} b +2 \Ei \left (1, -2 \,\mathrm {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {2 a}{b}} a -b \right )}{4 \left (c^{2} x^{2}-1\right ) c^{3} b^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {c x -1}\, \sqrt {c x +1}}{2 \left (c^{2} x^{2}-1\right ) c^{3} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right ) b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {c^{3} x^{5} - c x^{3} + {\left (c^{2} x^{4} - x^{2}\right )} \sqrt {c x + 1} \sqrt {c x - 1}}{{\left ({\left (c x + 1\right )} \sqrt {c x - 1} b^{2} c^{2} x + {\left (b^{2} c^{3} x^{2} - b^{2} c\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left ({\left (c x + 1\right )} \sqrt {c x - 1} a b c^{2} x + {\left (a b c^{3} x^{2} - a b c\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}} + \int \frac {2 \, c^{5} x^{6} - 5 \, c^{3} x^{4} + {\left (2 \, c^{3} x^{4} - c x^{2}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} + 3 \, c x^{2} + 2 \, {\left (2 \, c^{4} x^{5} - 3 \, c^{2} x^{3} + x\right )} \sqrt {c x + 1} \sqrt {c x - 1}}{{\left ({\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} b^{2} c^{3} x^{2} + 2 \, {\left (b^{2} c^{4} x^{3} - b^{2} c^{2} x\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (b^{2} c^{5} x^{4} - 2 \, b^{2} c^{3} x^{2} + b^{2} c\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left ({\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} a b c^{3} x^{2} + 2 \, {\left (a b c^{4} x^{3} - a b c^{2} x\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (a b c^{5} x^{4} - 2 \, a b c^{3} x^{2} + a b c\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {1-c^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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