Optimal. Leaf size=228 \[ -\frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{12 a^5}-\frac{3 \sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{8 a^5}-\frac{5 \sqrt{5 \pi } \text{Erf}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{24 a^5}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{12 a^5}+\frac{3 \sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{8 a^5}+\frac{5 \sqrt{5 \pi } \text{Erfi}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{24 a^5}+\frac{16 x^3}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{20 x^5}{3 \sqrt{\cosh ^{-1}(a x)}}-\frac{2 x^4 \sqrt{a x-1} \sqrt{a x+1}}{3 a \cosh ^{-1}(a x)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.864105, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 34, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {5668, 5775, 5670, 5448, 3308, 2180, 2204, 2205} \[ -\frac{\sqrt{\pi } \text{Erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{12 a^5}-\frac{3 \sqrt{3 \pi } \text{Erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{8 a^5}-\frac{5 \sqrt{5 \pi } \text{Erf}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{24 a^5}+\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{12 a^5}+\frac{3 \sqrt{3 \pi } \text{Erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{8 a^5}+\frac{5 \sqrt{5 \pi } \text{Erfi}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{24 a^5}+\frac{16 x^3}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{20 x^5}{3 \sqrt{\cosh ^{-1}(a x)}}-\frac{2 x^4 \sqrt{a x-1} \sqrt{a x+1}}{3 a \cosh ^{-1}(a x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5668
Rule 5775
Rule 5670
Rule 5448
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{x^4}{\cosh ^{-1}(a x)^{5/2}} \, dx &=-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}-\frac{8 \int \frac{x^3}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}} \, dx}{3 a}+\frac{1}{3} (10 a) \int \frac{x^5}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac{16 x^3}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{20 x^5}{3 \sqrt{\cosh ^{-1}(a x)}}+\frac{100}{3} \int \frac{x^4}{\sqrt{\cosh ^{-1}(a x)}} \, dx-\frac{16 \int \frac{x^2}{\sqrt{\cosh ^{-1}(a x)}} \, dx}{a^2}\\ &=-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac{16 x^3}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{20 x^5}{3 \sqrt{\cosh ^{-1}(a x)}}-\frac{16 \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}+\frac{100 \operatorname{Subst}\left (\int \frac{\cosh ^4(x) \sinh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a^5}\\ &=-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac{16 x^3}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{20 x^5}{3 \sqrt{\cosh ^{-1}(a x)}}-\frac{16 \operatorname{Subst}\left (\int \left (\frac{\sinh (x)}{4 \sqrt{x}}+\frac{\sinh (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}+\frac{100 \operatorname{Subst}\left (\int \left (\frac{\sinh (x)}{8 \sqrt{x}}+\frac{3 \sinh (3 x)}{16 \sqrt{x}}+\frac{\sinh (5 x)}{16 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{3 a^5}\\ &=-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac{16 x^3}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{20 x^5}{3 \sqrt{\cosh ^{-1}(a x)}}+\frac{25 \operatorname{Subst}\left (\int \frac{\sinh (5 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{12 a^5}-\frac{4 \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}-\frac{4 \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{6 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a^5}\\ &=-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac{16 x^3}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{20 x^5}{3 \sqrt{\cosh ^{-1}(a x)}}-\frac{25 \operatorname{Subst}\left (\int \frac{e^{-5 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{24 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{e^{5 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{24 a^5}+\frac{2 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}+\frac{2 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}-\frac{2 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}-\frac{2 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}-\frac{25 \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{12 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{12 a^5}-\frac{25 \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^5}\\ &=-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac{16 x^3}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{20 x^5}{3 \sqrt{\cosh ^{-1}(a x)}}-\frac{25 \operatorname{Subst}\left (\int e^{-5 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{12 a^5}+\frac{25 \operatorname{Subst}\left (\int e^{5 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{12 a^5}+\frac{4 \operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{a^5}+\frac{4 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{a^5}-\frac{4 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{a^5}-\frac{4 \operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{a^5}-\frac{25 \operatorname{Subst}\left (\int e^{-x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{6 a^5}+\frac{25 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{6 a^5}-\frac{25 \operatorname{Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{4 a^5}+\frac{25 \operatorname{Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{4 a^5}\\ &=-\frac{2 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^{3/2}}+\frac{16 x^3}{3 a^2 \sqrt{\cosh ^{-1}(a x)}}-\frac{20 x^5}{3 \sqrt{\cosh ^{-1}(a x)}}-\frac{\sqrt{\pi } \text{erf}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{12 a^5}-\frac{3 \sqrt{3 \pi } \text{erf}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{8 a^5}-\frac{5 \sqrt{5 \pi } \text{erf}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{24 a^5}+\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{\cosh ^{-1}(a x)}\right )}{12 a^5}+\frac{3 \sqrt{3 \pi } \text{erfi}\left (\sqrt{3} \sqrt{\cosh ^{-1}(a x)}\right )}{8 a^5}+\frac{5 \sqrt{5 \pi } \text{erfi}\left (\sqrt{5} \sqrt{\cosh ^{-1}(a x)}\right )}{24 a^5}\\ \end{align*}
Mathematica [A] time = 1.5425, size = 278, normalized size = 1.22 \[ -\frac{2 \left (-\cosh ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\cosh ^{-1}(a x)\right )-2 \cosh ^{-1}(a x)^{3/2} \text{Gamma}\left (\frac{1}{2},\cosh ^{-1}(a x)\right )+5 \cosh ^{-1}(a x) \left (-\sqrt{5} \sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-5 \cosh ^{-1}(a x)\right )-\sqrt{5} \sqrt{\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},5 \cosh ^{-1}(a x)\right )+e^{-5 \cosh ^{-1}(a x)}+e^{5 \cosh ^{-1}(a x)}\right )+3 \left (3 \sqrt{3} \left (-\cosh ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-3 \cosh ^{-1}(a x)\right )-3 \sqrt{3} \cosh ^{-1}(a x)^{3/2} \text{Gamma}\left (\frac{1}{2},3 \cosh ^{-1}(a x)\right )+3 e^{-3 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)+3 e^{3 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)+\sinh \left (3 \cosh ^{-1}(a x)\right )\right )+2 \sqrt{\frac{a x-1}{a x+1}} (a x+1)+2 e^{-\cosh ^{-1}(a x)} \cosh ^{-1}(a x)+2 e^{\cosh ^{-1}(a x)} \cosh ^{-1}(a x)+\sinh \left (5 \cosh ^{-1}(a x)\right )}{24 a^5 \cosh ^{-1}(a x)^{3/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.174, size = 0, normalized size = 0. \begin{align*} \int{{x}^{4} \left ({\rm arccosh} \left (ax\right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{arcosh}\left (a x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{acosh}^{\frac{5}{2}}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]