Optimal. Leaf size=29 \[ \frac {3 c \text {Shi}\left (\cosh ^{-1}(a x)\right )}{4 a}-\frac {c \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{4 a} \]
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Rubi [A] time = 0.08, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5700, 3312, 3298} \[ \frac {3 c \text {Shi}\left (\cosh ^{-1}(a x)\right )}{4 a}-\frac {c \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{4 a} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3312
Rule 5700
Rubi steps
\begin {align*} \int \frac {c-a^2 c x^2}{\cosh ^{-1}(a x)} \, dx &=-\frac {c \operatorname {Subst}\left (\int \frac {\sinh ^3(x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=-\frac {(i c) \operatorname {Subst}\left (\int \left (\frac {3 i \sinh (x)}{4 x}-\frac {i \sinh (3 x)}{4 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=-\frac {c \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a}+\frac {(3 c) \operatorname {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a}\\ &=\frac {3 c \text {Shi}\left (\cosh ^{-1}(a x)\right )}{4 a}-\frac {c \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{4 a}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 25, normalized size = 0.86 \[ \frac {c \left (3 \text {Shi}\left (\cosh ^{-1}(a x)\right )-\text {Shi}\left (3 \cosh ^{-1}(a x)\right )\right )}{4 a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {a^{2} c x^{2} - c}{\operatorname {arcosh}\left (a x\right )}, x\right ) \]
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 \[ \int -\frac {a^{2} c x^{2} - c}{\operatorname {arcosh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 24, normalized size = 0.83 \[ \frac {c \left (3 \Shi \left (\mathrm {arccosh}\left (a x \right )\right )-\Shi \left (3 \,\mathrm {arccosh}\left (a x \right )\right )\right )}{4 a} \]
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 \[ -\int \frac {a^{2} c x^{2} - c}{\operatorname {arcosh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {c-a^2\,c\,x^2}{\mathrm {acosh}\left (a\,x\right )} \,d x \]
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 \[ - c \left (\int \frac {a^{2} x^{2}}{\operatorname {acosh}{\left (a x \right )}}\, dx + \int \left (- \frac {1}{\operatorname {acosh}{\left (a x \right )}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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