Optimal. Leaf size=402 \[ -\frac {51 a c x^2 \sqrt {c-a^2 c x^2}}{128 \sqrt {a x-1} \sqrt {a x+1}}-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{16 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \cosh ^{-1}(a x)^3+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac {45}{64} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)+\frac {3}{32} c x (1-a x) (a x+1) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^4}{32 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {3 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{16 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {27 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{128 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {3 a^3 c x^4 \sqrt {c-a^2 c x^2}}{128 \sqrt {a x-1} \sqrt {a x+1}} \]
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Rubi [A] time = 0.95, antiderivative size = 414, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {5713, 5685, 5683, 5676, 5662, 5759, 30, 5716, 14} \[ \frac {3 a^3 c x^4 \sqrt {c-a^2 c x^2}}{128 \sqrt {a x-1} \sqrt {a x+1}}-\frac {51 a c x^2 \sqrt {c-a^2 c x^2}}{128 \sqrt {a x-1} \sqrt {a x+1}}-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{16 \sqrt {a x-1} \sqrt {a x+1}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac {1}{4} c x (1-a x) (a x+1) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac {45}{64} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)+\frac {3}{32} c x (1-a x) (a x+1) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^4}{32 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {3 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{16 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {27 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{128 a \sqrt {a x-1} \sqrt {a x+1}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 5662
Rule 5676
Rule 5683
Rule 5685
Rule 5713
Rule 5716
Rule 5759
Rubi steps
\begin {align*} \int \left (c-a^2 c x^2\right )^{3/2} \cosh ^{-1}(a x)^3 \, dx &=-\frac {\left (c \sqrt {c-a^2 c x^2}\right ) \int (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^3 \, dx}{\sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \int \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3 \, dx}{4 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 a c \sqrt {c-a^2 c x^2}\right ) \int x \left (-1+a^2 x^2\right ) \cosh ^{-1}(a x)^2 \, dx}{4 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {3 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{16 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \int (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x) \, dx}{8 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\cosh ^{-1}(a x)^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{8 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (9 a c \sqrt {c-a^2 c x^2}\right ) \int x \cosh ^{-1}(a x)^2 \, dx}{8 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {3}{32} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{16 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{16 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^4}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (9 c \sqrt {c-a^2 c x^2}\right ) \int \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \, dx}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 a c \sqrt {c-a^2 c x^2}\right ) \int x \left (-1+a^2 x^2\right ) \, dx}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (9 a^2 c \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}{8 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {45}{64} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)+\frac {3}{32} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{16 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{16 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^4}{32 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (9 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{64 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (9 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{16 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 a c \sqrt {c-a^2 c x^2}\right ) \int \left (-x+a^2 x^3\right ) \, dx}{32 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (9 a c \sqrt {c-a^2 c x^2}\right ) \int x \, dx}{64 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (9 a c \sqrt {c-a^2 c x^2}\right ) \int x \, dx}{16 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=-\frac {51 a c x^2 \sqrt {c-a^2 c x^2}}{128 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3 a^3 c x^4 \sqrt {c-a^2 c x^2}}{128 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {45}{64} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)+\frac {3}{32} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)+\frac {27 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{128 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{16 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{16 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac {1}{4} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^4}{32 a \sqrt {-1+a x} \sqrt {1+a x}}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 148, normalized size = 0.37 \[ -\frac {c \sqrt {c-a^2 c x^2} \left (96 \cosh ^{-1}(a x)^4-24 \left (\cosh \left (4 \cosh ^{-1}(a x)\right )-16 \cosh \left (2 \cosh ^{-1}(a x)\right )\right ) \cosh ^{-1}(a x)^2-3 \left (\cosh \left (4 \cosh ^{-1}(a x)\right )-64 \cosh \left (2 \cosh ^{-1}(a x)\right )\right )+32 \cosh ^{-1}(a x)^3 \left (\sinh \left (4 \cosh ^{-1}(a x)\right )-8 \sinh \left (2 \cosh ^{-1}(a x)\right )\right )+12 \cosh ^{-1}(a x) \left (\sinh \left (4 \cosh ^{-1}(a x)\right )-32 \sinh \left (2 \cosh ^{-1}(a x)\right )\right )\right )}{1024 a \sqrt {\frac {a x-1}{a x+1}} (a x+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} c x^{2} - c\right )} \sqrt {-a^{2} c x^{2} + c} \operatorname {arcosh}\left (a x\right )^{3}, x\right ) \]
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 = 0.35, size = 536, normalized size = 1.33 \[ -\frac {3 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (a x \right )^{4} c}{32 \sqrt {a x -1}\, \sqrt {a x +1}\, a}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (8 x^{5} a^{5}-12 x^{3} a^{3}+8 \sqrt {a x +1}\, \sqrt {a x -1}\, x^{4} a^{4}+4 a x -8 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (32 \mathrm {arccosh}\left (a x \right )^{3}-24 \mathrm {arccosh}\left (a x \right )^{2}+12 \,\mathrm {arccosh}\left (a x \right )-3\right ) c}{2048 \left (a x -1\right ) \left (a x +1\right ) a}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 x^{3} a^{3}-2 a x +2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \mathrm {arccosh}\left (a x \right )^{3}-6 \mathrm {arccosh}\left (a x \right )^{2}+6 \,\mathrm {arccosh}\left (a x \right )-3\right ) c}{32 \left (a x -1\right ) \left (a x +1\right ) a}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 x^{3} a^{3}-2 a x -2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \mathrm {arccosh}\left (a x \right )^{3}+6 \mathrm {arccosh}\left (a x \right )^{2}+6 \,\mathrm {arccosh}\left (a x \right )+3\right ) c}{32 \left (a x -1\right ) \left (a x +1\right ) a}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (8 x^{5} a^{5}-12 x^{3} a^{3}-8 \sqrt {a x +1}\, \sqrt {a x -1}\, x^{4} a^{4}+4 a x +8 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (32 \mathrm {arccosh}\left (a x \right )^{3}+24 \mathrm {arccosh}\left (a x \right )^{2}+12 \,\mathrm {arccosh}\left (a x \right )+3\right ) c}{2048 \left (a x -1\right ) \left (a x +1\right ) a} \]
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 \[ \int {\mathrm {acosh}\left (a\,x\right )}^3\,{\left (c-a^2\,c\,x^2\right )}^{3/2} \,d x \]
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 \[ \int \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {acosh}^{3}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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