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 = 0.77, antiderivative size = 298, normalized size of antiderivative = 1.26, number of steps used = 11, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 3298
Rule 3301
Rule 3303
Rule 5448
Rule 5670
Rule 5775
Rule 5798
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*}
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Mathematica [A] time = 0.48, 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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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} x^{2} + 1} x^{3}}{a^{2} c^{2} x^{2} + {\left (b^{2} c^{2} x^{2} - b^{2}\right )} \operatorname {arcosh}\left (c x\right )^{2} - a^{2} + 2 \, {\left (a b c^{2} x^{2} - a b\right )} \operatorname {arcosh}\left (c x\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.65, size = 634, normalized size = 2.68 \[ -\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-4 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}+4 c^{4} x^{4}+3 \sqrt {c x +1}\, \sqrt {c x -1}\, x c -5 c^{2} x^{2}+1\right )}{8 \left (c^{2} x^{2}-1\right ) c^{4} b \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}-\frac {3 \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, 3 \,\mathrm {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {b \,\mathrm {arccosh}\left (c x \right )-3 a}{b}}}{8 b^{2} \left (c^{2} x^{2}-1\right ) c^{4}}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (4 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} b \,c^{2}+4 x^{3} b \,c^{3}+3 \,\mathrm {arccosh}\left (c x \right ) \Ei \left (1, -3 \,\mathrm {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {3 a}{b}} b +3 \Ei \left (1, -3 \,\mathrm {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {3 a}{b}} a -\sqrt {c x +1}\, \sqrt {c x -1}\, b -3 x b c \right )}{8 \left (c^{2} x^{2}-1\right ) c^{4} b^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}+\frac {3 \sqrt {-c^{2} x^{2}+1}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (\mathrm {arccosh}\left (c x \right ) {\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\mathrm {arccosh}\left (c x \right )-\frac {a}{b}\right ) b +\sqrt {c x +1}\, \sqrt {c x -1}\, b +{\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\mathrm {arccosh}\left (c x \right )-\frac {a}{b}\right ) a +x b c \right )}{8 \left (c^{2} x^{2}-1\right ) c^{4} b^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}-\frac {3 \sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right )}{8 \left (c^{2} x^{2}-1\right ) c^{4} b \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}-\frac {3 \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, \mathrm {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{-\frac {b \,\mathrm {arccosh}\left (c x \right )-a}{b}}}{8 b^{2} \left (c^{2} x^{2}-1\right ) c^{4}} \]
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 \[ -\frac {c^{3} x^{6} - c x^{4} + {\left (c^{2} x^{5} - x^{3}\right )} \sqrt {c x + 1} \sqrt {c x - 1}}{{\left ({\left (c x + 1\right )} \sqrt {c x - 1} b^{2} c^{2} x + {\left (b^{2} c^{3} x^{2} - b^{2} c\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left ({\left (c x + 1\right )} \sqrt {c x - 1} a b c^{2} x + {\left (a b c^{3} x^{2} - a b c\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}} + \int \frac {3 \, c^{5} x^{7} - 7 \, c^{3} x^{5} + 4 \, c x^{3} + {\left (3 \, c^{3} x^{5} - 2 \, c x^{3}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} + 3 \, {\left (2 \, c^{4} x^{6} - 3 \, c^{2} x^{4} + x^{2}\right )} \sqrt {c x + 1} \sqrt {c x - 1}}{{\left ({\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} b^{2} c^{3} x^{2} + 2 \, {\left (b^{2} c^{4} x^{3} - b^{2} c^{2} x\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (b^{2} c^{5} x^{4} - 2 \, b^{2} c^{3} x^{2} + b^{2} c\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left ({\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} a b c^{3} x^{2} + 2 \, {\left (a b c^{4} x^{3} - a b c^{2} x\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (a b c^{5} x^{4} - 2 \, a b c^{3} x^{2} + a b c\right )} \sqrt {c x + 1}\right )} \sqrt {-c x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\sqrt {1-c^2\,x^2}} \,d x \]
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 \[ \int \frac {x^{3}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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