Optimal. Leaf size=94 \[ -\frac {c^3 \left (a^2 x^2+1\right )^{7/2}}{a \sinh ^{-1}(a x)}+\frac {35 c^3 \text {Shi}\left (\sinh ^{-1}(a x)\right )}{64 a}+\frac {63 c^3 \text {Shi}\left (3 \sinh ^{-1}(a x)\right )}{64 a}+\frac {35 c^3 \text {Shi}\left (5 \sinh ^{-1}(a x)\right )}{64 a}+\frac {7 c^3 \text {Shi}\left (7 \sinh ^{-1}(a x)\right )}{64 a} \]
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Rubi [A] time = 0.18, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {5696, 5779, 5448, 3298} \[ -\frac {c^3 \left (a^2 x^2+1\right )^{7/2}}{a \sinh ^{-1}(a x)}+\frac {35 c^3 \text {Shi}\left (\sinh ^{-1}(a x)\right )}{64 a}+\frac {63 c^3 \text {Shi}\left (3 \sinh ^{-1}(a x)\right )}{64 a}+\frac {35 c^3 \text {Shi}\left (5 \sinh ^{-1}(a x)\right )}{64 a}+\frac {7 c^3 \text {Shi}\left (7 \sinh ^{-1}(a x)\right )}{64 a} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 5448
Rule 5696
Rule 5779
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^3}{\sinh ^{-1}(a x)^2} \, dx &=-\frac {c^3 \left (1+a^2 x^2\right )^{7/2}}{a \sinh ^{-1}(a x)}+\left (7 a c^3\right ) \int \frac {x \left (1+a^2 x^2\right )^{5/2}}{\sinh ^{-1}(a x)} \, dx\\ &=-\frac {c^3 \left (1+a^2 x^2\right )^{7/2}}{a \sinh ^{-1}(a x)}+\frac {\left (7 c^3\right ) \operatorname {Subst}\left (\int \frac {\cosh ^6(x) \sinh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a}\\ &=-\frac {c^3 \left (1+a^2 x^2\right )^{7/2}}{a \sinh ^{-1}(a x)}+\frac {\left (7 c^3\right ) \operatorname {Subst}\left (\int \left (\frac {5 \sinh (x)}{64 x}+\frac {9 \sinh (3 x)}{64 x}+\frac {5 \sinh (5 x)}{64 x}+\frac {\sinh (7 x)}{64 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a}\\ &=-\frac {c^3 \left (1+a^2 x^2\right )^{7/2}}{a \sinh ^{-1}(a x)}+\frac {\left (7 c^3\right ) \operatorname {Subst}\left (\int \frac {\sinh (7 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a}+\frac {\left (35 c^3\right ) \operatorname {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a}+\frac {\left (35 c^3\right ) \operatorname {Subst}\left (\int \frac {\sinh (5 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a}+\frac {\left (63 c^3\right ) \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a}\\ &=-\frac {c^3 \left (1+a^2 x^2\right )^{7/2}}{a \sinh ^{-1}(a x)}+\frac {35 c^3 \text {Shi}\left (\sinh ^{-1}(a x)\right )}{64 a}+\frac {63 c^3 \text {Shi}\left (3 \sinh ^{-1}(a x)\right )}{64 a}+\frac {35 c^3 \text {Shi}\left (5 \sinh ^{-1}(a x)\right )}{64 a}+\frac {7 c^3 \text {Shi}\left (7 \sinh ^{-1}(a x)\right )}{64 a}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 82, normalized size = 0.87 \[ \frac {c^3 \left (-64 \left (a^2 x^2+1\right )^{7/2}+35 \sinh ^{-1}(a x) \text {Shi}\left (\sinh ^{-1}(a x)\right )+63 \sinh ^{-1}(a x) \text {Shi}\left (3 \sinh ^{-1}(a x)\right )+35 \sinh ^{-1}(a x) \text {Shi}\left (5 \sinh ^{-1}(a x)\right )+7 \sinh ^{-1}(a x) \text {Shi}\left (7 \sinh ^{-1}(a x)\right )\right )}{64 a \sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}}{\operatorname {arsinh}\left (a x\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a^{2} c x^{2} + c\right )}^{3}}{\operatorname {arsinh}\left (a x\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 106, normalized size = 1.13 \[ \frac {c^{3} \left (35 \Shi \left (5 \arcsinh \left (a x \right )\right ) \arcsinh \left (a x \right )+7 \Shi \left (7 \arcsinh \left (a x \right )\right ) \arcsinh \left (a x \right )+35 \Shi \left (\arcsinh \left (a x \right )\right ) \arcsinh \left (a x \right )+63 \Shi \left (3 \arcsinh \left (a x \right )\right ) \arcsinh \left (a x \right )-35 \sqrt {a^{2} x^{2}+1}-\cosh \left (7 \arcsinh \left (a x \right )\right )-21 \cosh \left (3 \arcsinh \left (a x \right )\right )-7 \cosh \left (5 \arcsinh \left (a x \right )\right )\right )}{64 a \arcsinh \left (a x \right )} \]
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 \[ -\frac {a^{9} c^{3} x^{9} + 4 \, a^{7} c^{3} x^{7} + 6 \, a^{5} c^{3} x^{5} + 4 \, a^{3} c^{3} x^{3} + a c^{3} x + {\left (a^{8} c^{3} x^{8} + 4 \, a^{6} c^{3} x^{6} + 6 \, a^{4} c^{3} x^{4} + 4 \, a^{2} c^{3} x^{2} + c^{3}\right )} \sqrt {a^{2} x^{2} + 1}}{{\left (a^{3} x^{2} + \sqrt {a^{2} x^{2} + 1} a^{2} x + a\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )} + \int \frac {7 \, a^{10} c^{3} x^{10} + 29 \, a^{8} c^{3} x^{8} + 46 \, a^{6} c^{3} x^{6} + 34 \, a^{4} c^{3} x^{4} + 11 \, a^{2} c^{3} x^{2} + c^{3} + {\left (7 \, a^{8} c^{3} x^{8} + 20 \, a^{6} c^{3} x^{6} + 18 \, a^{4} c^{3} x^{4} + 4 \, a^{2} c^{3} x^{2} - c^{3}\right )} {\left (a^{2} x^{2} + 1\right )} + 7 \, {\left (2 \, a^{9} c^{3} x^{9} + 7 \, a^{7} c^{3} x^{7} + 9 \, a^{5} c^{3} x^{5} + 5 \, a^{3} c^{3} x^{3} + a c^{3} x\right )} \sqrt {a^{2} x^{2} + 1}}{{\left (a^{4} x^{4} + {\left (a^{2} x^{2} + 1\right )} a^{2} x^{2} + 2 \, a^{2} x^{2} + 2 \, {\left (a^{3} x^{3} + a x\right )} \sqrt {a^{2} x^{2} + 1} + 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}\,{d x} \]
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 \[ \int \frac {{\left (c\,a^2\,x^2+c\right )}^3}{{\mathrm {asinh}\left (a\,x\right )}^2} \,d x \]
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 \[ c^{3} \left (\int \frac {3 a^{2} x^{2}}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx + \int \frac {3 a^{4} x^{4}}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx + \int \frac {a^{6} x^{6}}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {asinh}^{2}{\left (a x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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