Optimal. Leaf size=50 \[ \frac {5 c^2 \text {Shi}\left (\cosh ^{-1}(a x)\right )}{8 a}-\frac {5 c^2 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{16 a}+\frac {c^2 \text {Shi}\left (5 \cosh ^{-1}(a x)\right )}{16 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, 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 = {5700, 3312, 3298} \[ \frac {5 c^2 \text {Shi}\left (\cosh ^{-1}(a x)\right )}{8 a}-\frac {5 c^2 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )}{16 a}+\frac {c^2 \text {Shi}\left (5 \cosh ^{-1}(a x)\right )}{16 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3298
Rule 3312
Rule 5700
Rubi steps
\begin {align*} \int \frac {\left (c-a^2 c x^2\right )^2}{\cosh ^{-1}(a x)} \, dx &=\frac {c^2 \operatorname {Subst}\left (\int \frac {\sinh ^5(x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a}\\ &=-\frac {\left (i c^2\right ) \operatorname {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 \operatorname {Subst}\left (\int \frac {\sinh (5 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a}-\frac {\left (5 c^2\right ) \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{16 a}+\frac {\left (5 c^2\right ) \operatorname {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*}
________________________________________________________________________________________
Mathematica [A] time = 0.17, size = 34, normalized size = 0.68 \[ \frac {c^2 \left (10 \text {Shi}\left (\cosh ^{-1}(a x)\right )-5 \text {Shi}\left (3 \cosh ^{-1}(a x)\right )+\text {Shi}\left (5 \cosh ^{-1}(a x)\right )\right )}{16 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}{\operatorname {arcosh}\left (a x\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a^{2} c x^{2} - c\right )}^{2}}{\operatorname {arcosh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 33, normalized size = 0.66 \[ \frac {c^{2} \left (10 \Shi \left (\mathrm {arccosh}\left (a x \right )\right )-5 \Shi \left (3 \,\mathrm {arccosh}\left (a x \right )\right )+\Shi \left (5 \,\mathrm {arccosh}\left (a x \right )\right )\right )}{16 a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a^{2} c x^{2} - c\right )}^{2}}{\operatorname {arcosh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^2}{\mathrm {acosh}\left (a\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ c^{2} \left (\int \left (- \frac {2 a^{2} x^{2}}{\operatorname {acosh}{\left (a x \right )}}\right )\, dx + \int \frac {a^{4} x^{4}}{\operatorname {acosh}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {acosh}{\left (a x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________