Optimal. Leaf size=67 \[ \frac {35 c^3 \text {Shi}\left (\cosh ^{-1}(a x)\right )}{64 a}-\frac {21 c^3 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{64 a}+\frac {7 c^3 \text {Shi}\left (5 \cosh ^{-1}(a x)\right )}{64 a}-\frac {c^3 \text {Shi}\left (7 \cosh ^{-1}(a x)\right )}{64 a} \]
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Rubi [A] time = 0.14, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5700, 3312, 3298} \[ \frac {35 c^3 \text {Shi}\left (\cosh ^{-1}(a x)\right )}{64 a}-\frac {21 c^3 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{64 a}+\frac {7 c^3 \text {Shi}\left (5 \cosh ^{-1}(a x)\right )}{64 a}-\frac {c^3 \text {Shi}\left (7 \cosh ^{-1}(a x)\right )}{64 a} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3312
Rule 5700
Rubi steps
\begin {align*} \int \frac {\left (c-a^2 c x^2\right )^3}{\cosh ^{-1}(a x)} \, dx &=-\frac {c^3 \operatorname {Subst}\left (\int \frac {\sinh ^7(x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=-\frac {\left (i c^3\right ) \operatorname {Subst}\left (\int \left (\frac {35 i \sinh (x)}{64 x}-\frac {21 i \sinh (3 x)}{64 x}+\frac {7 i \sinh (5 x)}{64 x}-\frac {i \sinh (7 x)}{64 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=-\frac {c^3 \operatorname {Subst}\left (\int \frac {\sinh (7 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a}+\frac {\left (7 c^3\right ) \operatorname {Subst}\left (\int \frac {\sinh (5 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a}-\frac {\left (21 c^3\right ) \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a}+\frac {\left (35 c^3\right ) \operatorname {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a}\\ &=\frac {35 c^3 \text {Shi}\left (\cosh ^{-1}(a x)\right )}{64 a}-\frac {21 c^3 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{64 a}+\frac {7 c^3 \text {Shi}\left (5 \cosh ^{-1}(a x)\right )}{64 a}-\frac {c^3 \text {Shi}\left (7 \cosh ^{-1}(a x)\right )}{64 a}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 45, normalized size = 0.67 \[ \frac {c^3 \left (35 \text {Shi}\left (\cosh ^{-1}(a x)\right )-21 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )+7 \text {Shi}\left (5 \cosh ^{-1}(a x)\right )-\text {Shi}\left (7 \cosh ^{-1}(a x)\right )\right )}{64 a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}}{\operatorname {arcosh}\left (a x\right )}, x\right ) \]
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 \[ \int -\frac {{\left (a^{2} c x^{2} - c\right )}^{3}}{\operatorname {arcosh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 44, normalized size = 0.66 \[ \frac {c^{3} \left (35 \Shi \left (\mathrm {arccosh}\left (a x \right )\right )-21 \Shi \left (3 \,\mathrm {arccosh}\left (a x \right )\right )+7 \Shi \left (5 \,\mathrm {arccosh}\left (a x \right )\right )-\Shi \left (7 \,\mathrm {arccosh}\left (a x \right )\right )\right )}{64 a} \]
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 \[ -\int \frac {{\left (a^{2} c x^{2} - c\right )}^{3}}{\operatorname {arcosh}\left (a x\right )}\,{d x} \]
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 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^3}{\mathrm {acosh}\left (a\,x\right )} \,d x \]
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 \[ - c^{3} \left (\int \frac {3 a^{2} x^{2}}{\operatorname {acosh}{\left (a x \right )}}\, dx + \int \left (- \frac {3 a^{4} x^{4}}{\operatorname {acosh}{\left (a x \right )}}\right )\, dx + \int \frac {a^{6} x^{6}}{\operatorname {acosh}{\left (a x \right )}}\, dx + \int \left (- \frac {1}{\operatorname {acosh}{\left (a x \right )}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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