Optimal. Leaf size=165 \[ -\frac{287 a^3 \sqrt [4]{1-a x}}{24 \sqrt [4]{a x+1}}-\frac{61 a^2 \sqrt [4]{1-a x}}{24 x \sqrt [4]{a x+1}}-\frac{55}{8} a^3 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+\frac{55}{8} a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+\frac{13 a \sqrt [4]{1-a x}}{12 x^2 \sqrt [4]{a x+1}}-\frac{\sqrt [4]{1-a x}}{3 x^3 \sqrt [4]{a x+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0754489, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {6126, 98, 151, 155, 12, 93, 298, 203, 206} \[ -\frac{287 a^3 \sqrt [4]{1-a x}}{24 \sqrt [4]{a x+1}}-\frac{61 a^2 \sqrt [4]{1-a x}}{24 x \sqrt [4]{a x+1}}-\frac{55}{8} a^3 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+\frac{55}{8} a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+\frac{13 a \sqrt [4]{1-a x}}{12 x^2 \sqrt [4]{a x+1}}-\frac{\sqrt [4]{1-a x}}{3 x^3 \sqrt [4]{a x+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6126
Rule 98
Rule 151
Rule 155
Rule 12
Rule 93
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{-\frac{5}{2} \tanh ^{-1}(a x)}}{x^4} \, dx &=\int \frac{(1-a x)^{5/4}}{x^4 (1+a x)^{5/4}} \, dx\\ &=-\frac{\sqrt [4]{1-a x}}{3 x^3 \sqrt [4]{1+a x}}-\frac{1}{3} \int \frac{\frac{13 a}{2}-6 a^2 x}{x^3 (1-a x)^{3/4} (1+a x)^{5/4}} \, dx\\ &=-\frac{\sqrt [4]{1-a x}}{3 x^3 \sqrt [4]{1+a x}}+\frac{13 a \sqrt [4]{1-a x}}{12 x^2 \sqrt [4]{1+a x}}+\frac{1}{6} \int \frac{\frac{61 a^2}{4}-13 a^3 x}{x^2 (1-a x)^{3/4} (1+a x)^{5/4}} \, dx\\ &=-\frac{\sqrt [4]{1-a x}}{3 x^3 \sqrt [4]{1+a x}}+\frac{13 a \sqrt [4]{1-a x}}{12 x^2 \sqrt [4]{1+a x}}-\frac{61 a^2 \sqrt [4]{1-a x}}{24 x \sqrt [4]{1+a x}}-\frac{1}{6} \int \frac{\frac{165 a^3}{8}-\frac{61 a^4 x}{4}}{x (1-a x)^{3/4} (1+a x)^{5/4}} \, dx\\ &=-\frac{287 a^3 \sqrt [4]{1-a x}}{24 \sqrt [4]{1+a x}}-\frac{\sqrt [4]{1-a x}}{3 x^3 \sqrt [4]{1+a x}}+\frac{13 a \sqrt [4]{1-a x}}{12 x^2 \sqrt [4]{1+a x}}-\frac{61 a^2 \sqrt [4]{1-a x}}{24 x \sqrt [4]{1+a x}}-\frac{\int \frac{165 a^4}{16 x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx}{3 a}\\ &=-\frac{287 a^3 \sqrt [4]{1-a x}}{24 \sqrt [4]{1+a x}}-\frac{\sqrt [4]{1-a x}}{3 x^3 \sqrt [4]{1+a x}}+\frac{13 a \sqrt [4]{1-a x}}{12 x^2 \sqrt [4]{1+a x}}-\frac{61 a^2 \sqrt [4]{1-a x}}{24 x \sqrt [4]{1+a x}}-\frac{1}{16} \left (55 a^3\right ) \int \frac{1}{x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac{287 a^3 \sqrt [4]{1-a x}}{24 \sqrt [4]{1+a x}}-\frac{\sqrt [4]{1-a x}}{3 x^3 \sqrt [4]{1+a x}}+\frac{13 a \sqrt [4]{1-a x}}{12 x^2 \sqrt [4]{1+a x}}-\frac{61 a^2 \sqrt [4]{1-a x}}{24 x \sqrt [4]{1+a x}}-\frac{1}{4} \left (55 a^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{-1+x^4} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=-\frac{287 a^3 \sqrt [4]{1-a x}}{24 \sqrt [4]{1+a x}}-\frac{\sqrt [4]{1-a x}}{3 x^3 \sqrt [4]{1+a x}}+\frac{13 a \sqrt [4]{1-a x}}{12 x^2 \sqrt [4]{1+a x}}-\frac{61 a^2 \sqrt [4]{1-a x}}{24 x \sqrt [4]{1+a x}}+\frac{1}{8} \left (55 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac{1}{8} \left (55 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=-\frac{287 a^3 \sqrt [4]{1-a x}}{24 \sqrt [4]{1+a x}}-\frac{\sqrt [4]{1-a x}}{3 x^3 \sqrt [4]{1+a x}}+\frac{13 a \sqrt [4]{1-a x}}{12 x^2 \sqrt [4]{1+a x}}-\frac{61 a^2 \sqrt [4]{1-a x}}{24 x \sqrt [4]{1+a x}}-\frac{55}{8} a^3 \tan ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+\frac{55}{8} a^3 \tanh ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0231751, size = 78, normalized size = 0.47 \[ \frac{\sqrt [4]{1-a x} \left (330 a^3 x^3 \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},\frac{1-a x}{a x+1}\right )-287 a^3 x^3-61 a^2 x^2+26 a x-8\right )}{24 x^3 \sqrt [4]{a x+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.127, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \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{1}{x^{4} \left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.79842, size = 448, normalized size = 2.72 \begin{align*} -\frac{2 \,{\left (287 \, a^{3} x^{3} + 61 \, a^{2} x^{2} - 26 \, a x + 8\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 330 \,{\left (a^{4} x^{4} + a^{3} x^{3}\right )} \arctan \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 165 \,{\left (a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) + 165 \,{\left (a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right )}{48 \,{\left (a x^{4} + x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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*} \int \frac{1}{x^{4} \left (\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]