Optimal. Leaf size=98 \[ \frac {\sqrt {-1+a x} \text {Chi}\left (2 \cosh ^{-1}(a x)\right )}{2 a^5 \sqrt {1-a x}}+\frac {\sqrt {-1+a x} \text {Chi}\left (4 \cosh ^{-1}(a x)\right )}{8 a^5 \sqrt {1-a x}}+\frac {3 \sqrt {-1+a x} \log \left (\cosh ^{-1}(a x)\right )}{8 a^5 \sqrt {1-a x}} \]
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Rubi [A]
time = 0.12, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5952, 3393,
3382} \begin {gather*} \frac {\sqrt {a x-1} \text {Chi}\left (2 \cosh ^{-1}(a x)\right )}{2 a^5 \sqrt {1-a x}}+\frac {\sqrt {a x-1} \text {Chi}\left (4 \cosh ^{-1}(a x)\right )}{8 a^5 \sqrt {1-a x}}+\frac {3 \sqrt {a x-1} \log \left (\cosh ^{-1}(a x)\right )}{8 a^5 \sqrt {1-a x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3382
Rule 3393
Rule 5952
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt {1-a^2 x^2} \cosh ^{-1}(a x)} \, dx &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x^4}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \frac {\cosh ^4(x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5 \sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \left (\frac {3}{8 x}+\frac {\cosh (2 x)}{2 x}+\frac {\cosh (4 x)}{8 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^5 \sqrt {1-a^2 x^2}}\\ &=\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \log \left (\cosh ^{-1}(a x)\right )}{8 a^5 \sqrt {1-a^2 x^2}}+\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \frac {\cosh (4 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^5 \sqrt {1-a^2 x^2}}+\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \text {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^5 \sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {Chi}\left (2 \cosh ^{-1}(a x)\right )}{2 a^5 \sqrt {1-a^2 x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {Chi}\left (4 \cosh ^{-1}(a x)\right )}{8 a^5 \sqrt {1-a^2 x^2}}+\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \log \left (\cosh ^{-1}(a x)\right )}{8 a^5 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 69, normalized size = 0.70 \begin {gather*} \frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \left (4 \text {Chi}\left (2 \cosh ^{-1}(a x)\right )+\text {Chi}\left (4 \cosh ^{-1}(a x)\right )+3 \log \left (\cosh ^{-1}(a x)\right )\right )}{8 a^5 \sqrt {-((-1+a x) (1+a x))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(248\) vs.
\(2(80)=160\).
time = 4.81, size = 249, normalized size = 2.54
method | result | size |
default | \(\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \expIntegral \left (1, 4 \,\mathrm {arccosh}\left (a x \right )\right )}{16 a^{5} \left (a^{2} x^{2}-1\right )}+\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \expIntegral \left (1, -4 \,\mathrm {arccosh}\left (a x \right )\right )}{16 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \ln \left (\mathrm {arccosh}\left (a x \right )\right )}{8 a^{5} \left (a^{2} x^{2}-1\right )}+\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \expIntegral \left (1, 2 \,\mathrm {arccosh}\left (a x \right )\right )}{4 a^{5} \left (a^{2} x^{2}-1\right )}+\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \expIntegral \left (1, -2 \,\mathrm {arccosh}\left (a x \right )\right )}{4 a^{5} \left (a^{2} x^{2}-1\right )}\) | \(249\) |
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*} \int \frac {x^{4}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {acosh}{\left (a x \right )}}\, dx \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 = -0.01 \begin {gather*} \int \frac {x^4}{\mathrm {acosh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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