Optimal. Leaf size=231 \[ -\frac {3 a x^2 \sqrt {c-a^2 c x^2}}{8 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^4}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{4 \sqrt {a x-1} \sqrt {a x+1}}+\frac {3 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {3}{4} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x) \]
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Rubi [A] time = 0.54, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5713, 5683, 5676, 5662, 5759, 30} \[ -\frac {3 a x^2 \sqrt {c-a^2 c x^2}}{8 \sqrt {a x-1} \sqrt {a x+1}}-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^4}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{4 \sqrt {a x-1} \sqrt {a x+1}}+\frac {3 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{8 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {3}{4} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 30
Rule 5662
Rule 5676
Rule 5683
Rule 5713
Rule 5759
Rubi steps
\begin {align*} \int \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3 \, dx &=\frac {\sqrt {c-a^2 c x^2} \int \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3 \, dx}{\sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac {\sqrt {c-a^2 c x^2} \int \frac {\cosh ^{-1}(a x)^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{2 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (3 a \sqrt {c-a^2 c x^2}\right ) \int x \cosh ^{-1}(a x)^2 \, dx}{2 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{4 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^4}{8 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 a^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {x^2 \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{2 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {3}{4} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{4 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^4}{8 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 \sqrt {c-a^2 c x^2}\right ) \int \frac {\cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{4 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (3 a \sqrt {c-a^2 c x^2}\right ) \int x \, dx}{4 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=-\frac {3 a x^2 \sqrt {c-a^2 c x^2}}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{4} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)+\frac {3 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{8 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {3 a x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{4 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac {\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^4}{8 a \sqrt {-1+a x} \sqrt {1+a x}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 98, normalized size = 0.42 \[ -\frac {\sqrt {-c (a x-1) (a x+1)} \left (2 \cosh ^{-1}(a x)^4+\left (6 \cosh ^{-1}(a x)^2+3\right ) \cosh \left (2 \cosh ^{-1}(a x)\right )-2 \left (2 \cosh ^{-1}(a x)^2+3\right ) \cosh ^{-1}(a x) \sinh \left (2 \cosh ^{-1}(a x)\right )\right )}{16 a \sqrt {\frac {a x-1}{a x+1}} (a x+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-a^{2} c x^{2} + c} \operatorname {arcosh}\left (a x\right )^{3}, x\right ) \]
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 [A] time = 0.33, size = 256, normalized size = 1.11 \[ -\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (a x \right )^{4}}{8 \sqrt {a x -1}\, \sqrt {a x +1}\, a}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 x^{3} a^{3}-2 a x +2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \mathrm {arccosh}\left (a x \right )^{3}-6 \mathrm {arccosh}\left (a x \right )^{2}+6 \,\mathrm {arccosh}\left (a x \right )-3\right )}{32 \left (a x -1\right ) \left (a x +1\right ) a}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 x^{3} a^{3}-2 a x -2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \mathrm {arccosh}\left (a x \right )^{3}+6 \mathrm {arccosh}\left (a x \right )^{2}+6 \,\mathrm {arccosh}\left (a x \right )+3\right )}{32 \left (a x -1\right ) \left (a x +1\right ) a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int {\mathrm {acosh}\left (a\,x\right )}^3\,\sqrt {c-a^2\,c\,x^2} \,d x \]
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 \[ \int \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {acosh}^{3}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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