Optimal. Leaf size=183 \[ \frac {a^2 \cosh ^{-1}(a x)}{x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x^2}-\frac {\cosh ^{-1}(a x)^3}{3 x^3}+a^3 \cosh ^{-1}(a x)^2 \text {ArcTan}\left (e^{\cosh ^{-1}(a x)}\right )-a^3 \text {ArcTan}\left (\sqrt {-1+a x} \sqrt {1+a x}\right )-i a^3 \cosh ^{-1}(a x) \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )+i a^3 \cosh ^{-1}(a x) \text {PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )+i a^3 \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )-i a^3 \text {PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.38, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5883, 5933,
5947, 4265, 2611, 2320, 6724, 94, 211} \begin {gather*} -a^3 \text {ArcTan}\left (\sqrt {a x-1} \sqrt {a x+1}\right )+a^3 \cosh ^{-1}(a x)^2 \text {ArcTan}\left (e^{\cosh ^{-1}(a x)}\right )-i a^3 \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+i a^3 \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+i a^3 \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )-i a^3 \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )+\frac {a^2 \cosh ^{-1}(a x)}{x}-\frac {\cosh ^{-1}(a x)^3}{3 x^3}+\frac {a \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 94
Rule 211
Rule 2320
Rule 2611
Rule 4265
Rule 5883
Rule 5933
Rule 5947
Rule 6724
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)^3}{x^4} \, dx &=-\frac {\cosh ^{-1}(a x)^3}{3 x^3}+a \int \frac {\cosh ^{-1}(a x)^2}{x^3 \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x^2}-\frac {\cosh ^{-1}(a x)^3}{3 x^3}-a^2 \int \frac {\cosh ^{-1}(a x)}{x^2} \, dx+\frac {1}{2} a^3 \int \frac {\cosh ^{-1}(a x)^2}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {a^2 \cosh ^{-1}(a x)}{x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x^2}-\frac {\cosh ^{-1}(a x)^3}{3 x^3}+\frac {1}{2} a^3 \text {Subst}\left (\int x^2 \text {sech}(x) \, dx,x,\cosh ^{-1}(a x)\right )-a^3 \int \frac {1}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {a^2 \cosh ^{-1}(a x)}{x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x^2}-\frac {\cosh ^{-1}(a x)^3}{3 x^3}+a^3 \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-\left (i a^3\right ) \text {Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )+\left (i a^3\right ) \text {Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )-a^4 \text {Subst}\left (\int \frac {1}{a+a x^2} \, dx,x,\sqrt {-1+a x} \sqrt {1+a x}\right )\\ &=\frac {a^2 \cosh ^{-1}(a x)}{x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x^2}-\frac {\cosh ^{-1}(a x)^3}{3 x^3}+a^3 \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-a^3 \tan ^{-1}\left (\sqrt {-1+a x} \sqrt {1+a x}\right )-i a^3 \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+i a^3 \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+\left (i a^3\right ) \text {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )-\left (i a^3\right ) \text {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=\frac {a^2 \cosh ^{-1}(a x)}{x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x^2}-\frac {\cosh ^{-1}(a x)^3}{3 x^3}+a^3 \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-a^3 \tan ^{-1}\left (\sqrt {-1+a x} \sqrt {1+a x}\right )-i a^3 \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+i a^3 \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+\left (i a^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )-\left (i a^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )\\ &=\frac {a^2 \cosh ^{-1}(a x)}{x}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x^2}-\frac {\cosh ^{-1}(a x)^3}{3 x^3}+a^3 \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-a^3 \tan ^{-1}\left (\sqrt {-1+a x} \sqrt {1+a x}\right )-i a^3 \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+i a^3 \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )+i a^3 \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )-i a^3 \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.40, size = 201, normalized size = 1.10 \begin {gather*} \frac {1}{6} \left (\frac {6 a^2 \cosh ^{-1}(a x)}{x}+\frac {3 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \cosh ^{-1}(a x)^2}{x^2}-\frac {2 \cosh ^{-1}(a x)^3}{x^3}-3 i a^3 \left (-4 i \text {ArcTan}\left (\tanh \left (\frac {1}{2} \cosh ^{-1}(a x)\right )\right )+\cosh ^{-1}(a x)^2 \log \left (1-i e^{-\cosh ^{-1}(a x)}\right )-\cosh ^{-1}(a x)^2 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )+2 \cosh ^{-1}(a x) \text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(a x)}\right )-2 \cosh ^{-1}(a x) \text {PolyLog}\left (2,i e^{-\cosh ^{-1}(a x)}\right )+2 \text {PolyLog}\left (3,-i e^{-\cosh ^{-1}(a x)}\right )-2 \text {PolyLog}\left (3,i e^{-\cosh ^{-1}(a x)}\right )\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 3.05, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {arccosh}\left (a x \right )^{3}}{x^{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________