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 = 0.69, 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 3298
Rule 3301
Rule 3303
Rule 3312
Rule 5781
Rule 5798
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*}
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Mathematica [A] time = 0.29, 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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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} x^{2} + 1} x^{3}}{a c^{2} x^{2} + {\left (b c^{2} x^{2} - b\right )} \operatorname {arcosh}\left (c x\right ) - a}, x\right ) \]
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 = 0.45, size = 349, normalized size = 1.77 \[ \frac {\sqrt {-c^{2} x^{2}+1}\, \left (\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \Ei \left (1, 3 \,\mathrm {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {b \,\mathrm {arccosh}\left (c x \right )-3 a}{b}}}{8 b \left (c^{2} x^{2}-1\right ) c^{4}}+\frac {\sqrt {-c^{2} x^{2}+1}\, \left (\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \Ei \left (1, -3 \,\mathrm {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {b \,\mathrm {arccosh}\left (c x \right )+3 a}{b}}}{8 b \left (c^{2} x^{2}-1\right ) c^{4}}+\frac {3 \sqrt {-c^{2} x^{2}+1}\, \left (\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \Ei \left (1, -\mathrm {arccosh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {a +b \,\mathrm {arccosh}\left (c x \right )}{b}}}{8 b \left (c^{2} x^{2}-1\right ) c^{4}}+\frac {3 \sqrt {-c^{2} x^{2}+1}\, \left (\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \Ei \left (1, \mathrm {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{-\frac {b \,\mathrm {arccosh}\left (c x \right )-a}{b}}}{8 b \left (c^{2} x^{2}-1\right ) c^{4}} \]
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 \[ \int \frac {x^{3}}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}\,{d x} \]
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 \[ \int \frac {x^3}{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {1-c^2\,x^2}} \,d x \]
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 \[ \int \frac {x^{3}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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