Optimal. Leaf size=402 \[ -\frac {51 a c x^2 \sqrt {c-a^2 c x^2}}{128 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3 a^3 c x^4 \sqrt {c-a^2 c x^2}}{128 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {45}{64} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)+\frac {3}{32} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)+\frac {27 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{128 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{16 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{16 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \cosh ^{-1}(a x)^3-\frac {3 c \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^4}{32 a \sqrt {-1+a x} \sqrt {1+a x}} \]
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Rubi [A]
time = 0.52, antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {5897, 5895,
5893, 5883, 5939, 30, 5912, 5914, 5898, 5896, 74, 14} \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 74
Rule 5883
Rule 5893
Rule 5895
Rule 5896
Rule 5897
Rule 5898
Rule 5912
Rule 5914
Rule 5939
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 = 0.33, size = 148, normalized size = 0.37 \begin {gather*} -\frac {c \sqrt {c-a^2 c x^2} \left (96 \cosh ^{-1}(a x)^4-3 \left (-64 \cosh \left (2 \cosh ^{-1}(a x)\right )+\cosh \left (4 \cosh ^{-1}(a x)\right )\right )-24 \cosh ^{-1}(a x)^2 \left (-16 \cosh \left (2 \cosh ^{-1}(a x)\right )+\cosh \left (4 \cosh ^{-1}(a x)\right )\right )+12 \cosh ^{-1}(a x) \left (-32 \sinh \left (2 \cosh ^{-1}(a x)\right )+\sinh \left (4 \cosh ^{-1}(a x)\right )\right )+32 \cosh ^{-1}(a x)^3 \left (-8 \sinh \left (2 \cosh ^{-1}(a x)\right )+\sinh \left (4 \cosh ^{-1}(a x)\right )\right )\right )}{1024 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 2.55, size = 536, normalized size = 1.33
method | result | size |
default | \(-\frac {3 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (a x \right )^{4} c}{32 \sqrt {a x -1}\, \sqrt {a x +1}\, a}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (8 a^{5} x^{5}-12 a^{3} x^{3}+8 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{4} x^{4}+4 a x -8 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (32 \mathrm {arccosh}\left (a x \right )^{3}-24 \mathrm {arccosh}\left (a x \right )^{2}+12 \,\mathrm {arccosh}\left (a x \right )-3\right ) c}{2048 \left (a x -1\right ) \left (a x +1\right ) a}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 a^{3} x^{3}-2 a x +2 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \mathrm {arccosh}\left (a x \right )^{3}-6 \mathrm {arccosh}\left (a x \right )^{2}+6 \,\mathrm {arccosh}\left (a x \right )-3\right ) c}{32 \left (a x -1\right ) \left (a x +1\right ) a}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 a^{3} x^{3}-2 a x -2 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \mathrm {arccosh}\left (a x \right )^{3}+6 \mathrm {arccosh}\left (a x \right )^{2}+6 \,\mathrm {arccosh}\left (a x \right )+3\right ) c}{32 \left (a x -1\right ) \left (a x +1\right ) a}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-8 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{4} x^{4}+8 a^{5} x^{5}+8 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}-12 a^{3} x^{3}-\sqrt {a x -1}\, \sqrt {a x +1}+4 a x \right ) \left (32 \mathrm {arccosh}\left (a x \right )^{3}+24 \mathrm {arccosh}\left (a x \right )^{2}+12 \,\mathrm {arccosh}\left (a x \right )+3\right ) c}{2048 \left (a x -1\right ) \left (a x +1\right ) a}\) | \(536\) |
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int {\mathrm {acosh}\left (a\,x\right )}^3\,{\left (c-a^2\,c\,x^2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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