Optimal. Leaf size=246 \[ \frac {2 \sqrt {a x-1} \sqrt {a x+1}}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1}}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}-\frac {4 \sqrt {a x-1} \sqrt {a x+1} \log \left (1-a^2 x^2\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {4 x \cosh ^{-1}(a x)}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {x \cosh ^{-1}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}} \]
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Rubi [A] time = 0.34, antiderivative size = 276, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5713, 5691, 5688, 260, 261} \[ \frac {2 \sqrt {a x-1} \sqrt {a x+1}}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1}}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}-\frac {4 \sqrt {a x-1} \sqrt {a x+1} \log \left (1-a^2 x^2\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {4 x \cosh ^{-1}(a x)}{15 c^3 (1-a x) (a x+1) \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)}{5 c^3 (1-a x)^2 (a x+1)^2 \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 260
Rule 261
Rule 5688
Rule 5691
Rule 5713
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx &=-\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)}{(-1+a x)^{7/2} (1+a x)^{7/2}} \, dx}{c^3 \sqrt {c-a^2 c x^2}}\\ &=\frac {x \cosh ^{-1}(a x)}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt {c-a^2 c x^2}}+\frac {\left (4 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)}{(-1+a x)^{5/2} (1+a x)^{5/2}} \, dx}{5 c^3 \sqrt {c-a^2 c x^2}}-\frac {\left (a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{\left (-1+a^2 x^2\right )^3} \, dx}{5 c^3 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {-1+a x} \sqrt {1+a x}}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt {c-a^2 c x^2}}+\frac {4 x \cosh ^{-1}(a x)}{15 c^3 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}-\frac {\left (8 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {\left (4 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{\left (-1+a^2 x^2\right )^2} \, dx}{15 c^3 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {-1+a x} \sqrt {1+a x}}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt {c-a^2 c x^2}}+\frac {4 x \cosh ^{-1}(a x)}{15 c^3 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}+\frac {\left (8 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{1-a^2 x^2} \, dx}{15 c^3 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {-1+a x} \sqrt {1+a x}}{20 a c^3 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{15 a c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}+\frac {8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {x \cosh ^{-1}(a x)}{5 c^3 (1-a x)^2 (1+a x)^2 \sqrt {c-a^2 c x^2}}+\frac {4 x \cosh ^{-1}(a x)}{15 c^3 (1-a x) (1+a x) \sqrt {c-a^2 c x^2}}-\frac {4 \sqrt {-1+a x} \sqrt {1+a x} \log \left (1-a^2 x^2\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 116, normalized size = 0.47 \[ \frac {\sqrt {a x-1} \sqrt {a x+1} \left (-8 a^2 x^2-16 \left (a^2 x^2-1\right )^2 \log \left (1-a^2 x^2\right )+11\right )+4 a x \left (8 a^4 x^4-20 a^2 x^2+15\right ) \cosh ^{-1}(a x)}{60 a c^3 \left (a^2 x^2-1\right )^2 \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} c x^{2} + c} \operatorname {arcosh}\left (a x\right )}{a^{8} c^{4} x^{8} - 4 \, a^{6} c^{4} x^{6} + 6 \, a^{4} c^{4} x^{4} - 4 \, a^{2} c^{4} x^{2} + c^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 4.41, size = 142, normalized size = 0.58 \[ \frac {1}{60} \, \sqrt {-c} {\left (\frac {16 \, \log \left ({\left | a^{2} x^{2} - 1 \right |}\right )}{a c^{4}} - \frac {24 \, a^{4} x^{4} - 56 \, a^{2} x^{2} + 35}{{\left (a^{2} x^{2} - 1\right )}^{2} a c^{4}}\right )} - \frac {\sqrt {-a^{2} c x^{2} + c} {\left (4 \, {\left (\frac {2 \, a^{4} x^{2}}{c} - \frac {5 \, a^{2}}{c}\right )} x^{2} + \frac {15}{c}\right )} x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{15 \, {\left (a^{2} c x^{2} - c\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 419, normalized size = 1.70 \[ -\frac {16 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )}{15 c^{4} a \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (8 x^{5} a^{5}-20 x^{3} a^{3}-8 \sqrt {a x +1}\, \sqrt {a x -1}\, x^{4} a^{4}+15 a x +16 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-8 \sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (-64 \sqrt {a x +1}\, \sqrt {a x -1}\, x^{7} a^{7}-64 x^{8} a^{8}+248 \sqrt {a x +1}\, \sqrt {a x -1}\, x^{5} a^{5}+280 x^{6} a^{6}+160 a^{4} x^{4} \mathrm {arccosh}\left (a x \right )-340 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}-456 x^{4} a^{4}-380 a^{2} x^{2} \mathrm {arccosh}\left (a x \right )+165 \sqrt {a x +1}\, \sqrt {a x -1}\, a x +328 a^{2} x^{2}+256 \,\mathrm {arccosh}\left (a x \right )-88\right )}{60 \left (40 a^{10} x^{10}-215 x^{8} a^{8}+469 x^{6} a^{6}-517 x^{4} a^{4}+287 a^{2} x^{2}-64\right ) a \,c^{4}}+\frac {8 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \ln \left (\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}-1\right )}{15 c^{4} a \left (a^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 191, normalized size = 0.78 \[ -\frac {1}{60} \, a {\left (\frac {16 \, \sqrt {-\frac {1}{a^{4} c}} \log \left (x^{2} - \frac {1}{a^{2}}\right )}{c^{3}} + \frac {3}{{\left (a^{6} c^{3} x^{4} \sqrt {-\frac {1}{c}} - 2 \, a^{4} c^{3} x^{2} \sqrt {-\frac {1}{c}} + a^{2} c^{3} \sqrt {-\frac {1}{c}}\right )} c} - \frac {8}{{\left (a^{4} c^{2} x^{2} \sqrt {-\frac {1}{c}} - a^{2} c^{2} \sqrt {-\frac {1}{c}}\right )} c^{2}}\right )} + \frac {1}{15} \, {\left (\frac {8 \, x}{\sqrt {-a^{2} c x^{2} + c} c^{3}} + \frac {4 \, x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{2}} + \frac {3 \, x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c}\right )} \operatorname {arcosh}\left (a x\right ) \]
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 \[ \int \frac {\mathrm {acosh}\left (a\,x\right )}{{\left (c-a^2\,c\,x^2\right )}^{7/2}} \,d x \]
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 \[ \int \frac {\operatorname {acosh}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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