Optimal. Leaf size=50 \[ \frac {5 c^2 \text {Shi}\left (\cosh ^{-1}(a x)\right )}{8 a}-\frac {5 c^2 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{16 a}+\frac {c^2 \text {Shi}\left (5 \cosh ^{-1}(a x)\right )}{16 a} \]
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Rubi [A]
time = 0.08, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5906, 3393,
3379} \begin {gather*} \frac {5 c^2 \text {Shi}\left (\cosh ^{-1}(a x)\right )}{8 a}-\frac {5 c^2 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{16 a}+\frac {c^2 \text {Shi}\left (5 \cosh ^{-1}(a x)\right )}{16 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3393
Rule 5906
Rubi steps
\begin {align*} \int \frac {\left (c-a^2 c x^2\right )^2}{\cosh ^{-1}(a x)} \, dx &=\frac {c^2 \text {Subst}\left (\int \frac {\sinh ^5(x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=-\frac {\left (i c^2\right ) \text {Subst}\left (\int \left (\frac {5 i \sinh (x)}{8 x}-\frac {5 i \sinh (3 x)}{16 x}+\frac {i \sinh (5 x)}{16 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=\frac {c^2 \text {Subst}\left (\int \frac {\sinh (5 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a}-\frac {\left (5 c^2\right ) \text {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a}+\frac {\left (5 c^2\right ) \text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a}\\ &=\frac {5 c^2 \text {Shi}\left (\cosh ^{-1}(a x)\right )}{8 a}-\frac {5 c^2 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{16 a}+\frac {c^2 \text {Shi}\left (5 \cosh ^{-1}(a x)\right )}{16 a}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 34, normalized size = 0.68 \begin {gather*} \frac {c^2 \left (10 \text {Shi}\left (\cosh ^{-1}(a x)\right )-5 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )+\text {Shi}\left (5 \cosh ^{-1}(a x)\right )\right )}{16 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.77, size = 35, normalized size = 0.70
method | result | size |
derivativedivides | \(-\frac {c^{2} \left (-10 \hyperbolicSineIntegral \left (\mathrm {arccosh}\left (a x \right )\right )+5 \hyperbolicSineIntegral \left (3 \,\mathrm {arccosh}\left (a x \right )\right )-\hyperbolicSineIntegral \left (5 \,\mathrm {arccosh}\left (a x \right )\right )\right )}{16 a}\) | \(35\) |
default | \(-\frac {c^{2} \left (-10 \hyperbolicSineIntegral \left (\mathrm {arccosh}\left (a x \right )\right )+5 \hyperbolicSineIntegral \left (3 \,\mathrm {arccosh}\left (a x \right )\right )-\hyperbolicSineIntegral \left (5 \,\mathrm {arccosh}\left (a x \right )\right )\right )}{16 a}\) | \(35\) |
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 \begin {gather*} \text {Failed to integrate} \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*} c^{2} \left (\int \left (- \frac {2 a^{2} x^{2}}{\operatorname {acosh}{\left (a x \right )}}\right )\, dx + \int \frac {a^{4} x^{4}}{\operatorname {acosh}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {acosh}{\left (a x \right )}}\, dx\right ) \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 = -0.02 \begin {gather*} \int \frac {{\left (c-a^2\,c\,x^2\right )}^2}{\mathrm {acosh}\left (a\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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