Optimal. Leaf size=194 \[ -\frac{113 a^2 \sqrt [4]{1-a x}}{96 x^2 \sqrt [4]{a x+1}}+\frac{2467 a^4 \sqrt [4]{1-a x}}{192 \sqrt [4]{a x+1}}+\frac{521 a^3 \sqrt [4]{1-a x}}{192 x \sqrt [4]{a x+1}}+\frac{475}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{475}{64} a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+\frac{17 a \sqrt [4]{1-a x}}{24 x^3 \sqrt [4]{a x+1}}-\frac{\sqrt [4]{1-a x}}{4 x^4 \sqrt [4]{a x+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0983551, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ -\frac{113 a^2 \sqrt [4]{1-a x}}{96 x^2 \sqrt [4]{a x+1}}+\frac{2467 a^4 \sqrt [4]{1-a x}}{192 \sqrt [4]{a x+1}}+\frac{521 a^3 \sqrt [4]{1-a x}}{192 x \sqrt [4]{a x+1}}+\frac{475}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{475}{64} a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+\frac{17 a \sqrt [4]{1-a x}}{24 x^3 \sqrt [4]{a x+1}}-\frac{\sqrt [4]{1-a x}}{4 x^4 \sqrt [4]{a x+1}} \]
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^5} \, dx &=\int \frac{(1-a x)^{5/4}}{x^5 (1+a x)^{5/4}} \, dx\\ &=-\frac{\sqrt [4]{1-a x}}{4 x^4 \sqrt [4]{1+a x}}-\frac{1}{4} \int \frac{\frac{17 a}{2}-8 a^2 x}{x^4 (1-a x)^{3/4} (1+a x)^{5/4}} \, dx\\ &=-\frac{\sqrt [4]{1-a x}}{4 x^4 \sqrt [4]{1+a x}}+\frac{17 a \sqrt [4]{1-a x}}{24 x^3 \sqrt [4]{1+a x}}+\frac{1}{12} \int \frac{\frac{113 a^2}{4}-\frac{51 a^3 x}{2}}{x^3 (1-a x)^{3/4} (1+a x)^{5/4}} \, dx\\ &=-\frac{\sqrt [4]{1-a x}}{4 x^4 \sqrt [4]{1+a x}}+\frac{17 a \sqrt [4]{1-a x}}{24 x^3 \sqrt [4]{1+a x}}-\frac{113 a^2 \sqrt [4]{1-a x}}{96 x^2 \sqrt [4]{1+a x}}-\frac{1}{24} \int \frac{\frac{521 a^3}{8}-\frac{113 a^4 x}{2}}{x^2 (1-a x)^{3/4} (1+a x)^{5/4}} \, dx\\ &=-\frac{\sqrt [4]{1-a x}}{4 x^4 \sqrt [4]{1+a x}}+\frac{17 a \sqrt [4]{1-a x}}{24 x^3 \sqrt [4]{1+a x}}-\frac{113 a^2 \sqrt [4]{1-a x}}{96 x^2 \sqrt [4]{1+a x}}+\frac{521 a^3 \sqrt [4]{1-a x}}{192 x \sqrt [4]{1+a x}}+\frac{1}{24} \int \frac{\frac{1425 a^4}{16}-\frac{521 a^5 x}{8}}{x (1-a x)^{3/4} (1+a x)^{5/4}} \, dx\\ &=\frac{2467 a^4 \sqrt [4]{1-a x}}{192 \sqrt [4]{1+a x}}-\frac{\sqrt [4]{1-a x}}{4 x^4 \sqrt [4]{1+a x}}+\frac{17 a \sqrt [4]{1-a x}}{24 x^3 \sqrt [4]{1+a x}}-\frac{113 a^2 \sqrt [4]{1-a x}}{96 x^2 \sqrt [4]{1+a x}}+\frac{521 a^3 \sqrt [4]{1-a x}}{192 x \sqrt [4]{1+a x}}+\frac{\int \frac{1425 a^5}{32 x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx}{12 a}\\ &=\frac{2467 a^4 \sqrt [4]{1-a x}}{192 \sqrt [4]{1+a x}}-\frac{\sqrt [4]{1-a x}}{4 x^4 \sqrt [4]{1+a x}}+\frac{17 a \sqrt [4]{1-a x}}{24 x^3 \sqrt [4]{1+a x}}-\frac{113 a^2 \sqrt [4]{1-a x}}{96 x^2 \sqrt [4]{1+a x}}+\frac{521 a^3 \sqrt [4]{1-a x}}{192 x \sqrt [4]{1+a x}}+\frac{1}{128} \left (475 a^4\right ) \int \frac{1}{x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=\frac{2467 a^4 \sqrt [4]{1-a x}}{192 \sqrt [4]{1+a x}}-\frac{\sqrt [4]{1-a x}}{4 x^4 \sqrt [4]{1+a x}}+\frac{17 a \sqrt [4]{1-a x}}{24 x^3 \sqrt [4]{1+a x}}-\frac{113 a^2 \sqrt [4]{1-a x}}{96 x^2 \sqrt [4]{1+a x}}+\frac{521 a^3 \sqrt [4]{1-a x}}{192 x \sqrt [4]{1+a x}}+\frac{1}{32} \left (475 a^4\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{2467 a^4 \sqrt [4]{1-a x}}{192 \sqrt [4]{1+a x}}-\frac{\sqrt [4]{1-a x}}{4 x^4 \sqrt [4]{1+a x}}+\frac{17 a \sqrt [4]{1-a x}}{24 x^3 \sqrt [4]{1+a x}}-\frac{113 a^2 \sqrt [4]{1-a x}}{96 x^2 \sqrt [4]{1+a x}}+\frac{521 a^3 \sqrt [4]{1-a x}}{192 x \sqrt [4]{1+a x}}-\frac{1}{64} \left (475 a^4\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}{64} \left (475 a^4\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{2467 a^4 \sqrt [4]{1-a x}}{192 \sqrt [4]{1+a x}}-\frac{\sqrt [4]{1-a x}}{4 x^4 \sqrt [4]{1+a x}}+\frac{17 a \sqrt [4]{1-a x}}{24 x^3 \sqrt [4]{1+a x}}-\frac{113 a^2 \sqrt [4]{1-a x}}{96 x^2 \sqrt [4]{1+a x}}+\frac{521 a^3 \sqrt [4]{1-a x}}{192 x \sqrt [4]{1+a x}}+\frac{475}{64} a^4 \tan ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac{475}{64} a^4 \tanh ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0319339, size = 86, normalized size = 0.44 \[ \frac{\sqrt [4]{1-a x} \left (-2850 a^4 x^4 \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},\frac{1-a x}{a x+1}\right )+2467 a^4 x^4+521 a^3 x^3-226 a^2 x^2+136 a x-48\right )}{192 x^4 \sqrt [4]{a x+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.118, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5}} \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^{5} \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 = 1.79816, size = 477, normalized size = 2.46 \begin{align*} \frac{2 \,{\left (2467 \, a^{4} x^{4} + 521 \, a^{3} x^{3} - 226 \, a^{2} x^{2} + 136 \, a x - 48\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 2850 \,{\left (a^{5} x^{5} + a^{4} x^{4}\right )} \arctan \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 1425 \,{\left (a^{5} x^{5} + a^{4} x^{4}\right )} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) + 1425 \,{\left (a^{5} x^{5} + a^{4} x^{4}\right )} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right )}{384 \,{\left (a x^{5} + x^{4}\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^{5} \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]