Optimal. Leaf size=233 \[ \frac{\sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a} \log ^2\left (c x^n\right )+\sqrt{b}\right )}{4 \sqrt{2} a^{5/4} n}-\frac{\sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a} \log ^2\left (c x^n\right )+\sqrt{b}\right )}{4 \sqrt{2} a^{5/4} n}+\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}\right )}{2 \sqrt{2} a^{5/4} n}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}+1\right )}{2 \sqrt{2} a^{5/4} n}+\frac{\log (x)}{a} \]
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Rubi [A] time = 0.182802, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {321, 211, 1165, 628, 1162, 617, 204} \[ \frac{\sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a} \log ^2\left (c x^n\right )+\sqrt{b}\right )}{4 \sqrt{2} a^{5/4} n}-\frac{\sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a} \log ^2\left (c x^n\right )+\sqrt{b}\right )}{4 \sqrt{2} a^{5/4} n}+\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}\right )}{2 \sqrt{2} a^{5/4} n}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}+1\right )}{2 \sqrt{2} a^{5/4} n}+\frac{\log (x)}{a} \]
Antiderivative was successfully verified.
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Rule 321
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{a x+\frac{b x}{\log ^4\left (c x^n\right )}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4}{b+a x^4} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\log (x)}{a}-\frac{b \operatorname{Subst}\left (\int \frac{1}{b+a x^4} \, dx,x,\log \left (c x^n\right )\right )}{a n}\\ &=\frac{\log (x)}{a}-\frac{\sqrt{b} \operatorname{Subst}\left (\int \frac{\sqrt{b}-\sqrt{a} x^2}{b+a x^4} \, dx,x,\log \left (c x^n\right )\right )}{2 a n}-\frac{\sqrt{b} \operatorname{Subst}\left (\int \frac{\sqrt{b}+\sqrt{a} x^2}{b+a x^4} \, dx,x,\log \left (c x^n\right )\right )}{2 a n}\\ &=\frac{\log (x)}{a}+\frac{\sqrt [4]{b} \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{b}}{\sqrt [4]{a}}+2 x}{-\frac{\sqrt{b}}{\sqrt{a}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}-x^2} \, dx,x,\log \left (c x^n\right )\right )}{4 \sqrt{2} a^{5/4} n}+\frac{\sqrt [4]{b} \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{b}}{\sqrt [4]{a}}-2 x}{-\frac{\sqrt{b}}{\sqrt{a}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}-x^2} \, dx,x,\log \left (c x^n\right )\right )}{4 \sqrt{2} a^{5/4} n}-\frac{\sqrt{b} \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{b}}{\sqrt{a}}-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+x^2} \, dx,x,\log \left (c x^n\right )\right )}{4 a^{3/2} n}-\frac{\sqrt{b} \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{b}}{\sqrt{a}}+\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a}}+x^2} \, dx,x,\log \left (c x^n\right )\right )}{4 a^{3/2} n}\\ &=\frac{\log (x)}{a}+\frac{\sqrt [4]{b} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a} \log ^2\left (c x^n\right )\right )}{4 \sqrt{2} a^{5/4} n}-\frac{\sqrt [4]{b} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a} \log ^2\left (c x^n\right )\right )}{4 \sqrt{2} a^{5/4} n}-\frac{\sqrt [4]{b} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}\right )}{2 \sqrt{2} a^{5/4} n}+\frac{\sqrt [4]{b} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}\right )}{2 \sqrt{2} a^{5/4} n}\\ &=\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}\right )}{2 \sqrt{2} a^{5/4} n}-\frac{\sqrt [4]{b} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}\right )}{2 \sqrt{2} a^{5/4} n}+\frac{\log (x)}{a}+\frac{\sqrt [4]{b} \log \left (\sqrt{b}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a} \log ^2\left (c x^n\right )\right )}{4 \sqrt{2} a^{5/4} n}-\frac{\sqrt [4]{b} \log \left (\sqrt{b}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a} \log ^2\left (c x^n\right )\right )}{4 \sqrt{2} a^{5/4} n}\\ \end{align*}
Mathematica [A] time = 0.0664152, size = 211, normalized size = 0.91 \[ \frac{\sqrt{2} \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a} \log ^2\left (c x^n\right )+\sqrt{b}\right )-\sqrt{2} \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a} \log ^2\left (c x^n\right )+\sqrt{b}\right )+2 \sqrt{2} \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}\right )-2 \sqrt{2} \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} \log \left (c x^n\right )}{\sqrt [4]{b}}+1\right )+8 \sqrt [4]{a} \log \left (c x^n\right )}{8 a^{5/4} n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 181, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( c{x}^{n} \right ) }{na}}-{\frac{\sqrt{2}}{8\,na}\sqrt [4]{{\frac{b}{a}}}\ln \left ({ \left ( \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{2}+\sqrt [4]{{\frac{b}{a}}}\ln \left ( c{x}^{n} \right ) \sqrt{2}+\sqrt{{\frac{b}{a}}} \right ) \left ( \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{2}-\sqrt [4]{{\frac{b}{a}}}\ln \left ( c{x}^{n} \right ) \sqrt{2}+\sqrt{{\frac{b}{a}}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}}{4\,na}\sqrt [4]{{\frac{b}{a}}}\arctan \left ({\ln \left ( c{x}^{n} \right ) \sqrt{2}{\frac{1}{\sqrt [4]{{\frac{b}{a}}}}}}+1 \right ) }+{\frac{\sqrt{2}}{4\,na}\sqrt [4]{{\frac{b}{a}}}\arctan \left ( -{\ln \left ( c{x}^{n} \right ) \sqrt{2}{\frac{1}{\sqrt [4]{{\frac{b}{a}}}}}}+1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -b \int \frac{1}{4 \, a^{2} x \log \left (c\right )^{3} \log \left (x^{n}\right ) + 6 \, a^{2} x \log \left (c\right )^{2} \log \left (x^{n}\right )^{2} + 4 \, a^{2} x \log \left (c\right ) \log \left (x^{n}\right )^{3} + a^{2} x \log \left (x^{n}\right )^{4} +{\left (a^{2} \log \left (c\right )^{4} + a b\right )} x}\,{d x} + \frac{\log \left (x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97476, size = 505, normalized size = 2.17 \begin{align*} -\frac{4 \, a \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{a^{2} n^{2} \sqrt{-\frac{b}{a^{5} n^{4}}} + n^{2} \log \left (x\right )^{2} + 2 \, n \log \left (c\right ) \log \left (x\right ) + \log \left (c\right )^{2}} a^{4} n^{3} \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{3}{4}} -{\left (a^{4} n^{4} \log \left (x\right ) + a^{4} n^{3} \log \left (c\right )\right )} \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{3}{4}}}{b}\right ) + a \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{1}{4}} \log \left (a n \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{1}{4}} + n \log \left (x\right ) + \log \left (c\right )\right ) - a \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{1}{4}} \log \left (-a n \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{1}{4}} + n \log \left (x\right ) + \log \left (c\right )\right ) - 4 \, \log \left (x\right )}{4 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3349, size = 365, normalized size = 1.57 \begin{align*} \frac{1}{4} \, i \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{1}{4}} \log \left (a^{2} i n^{5} \log \left (x\right ) + a^{2} i n^{4} \log \left (c\right ) - \left (-a^{3} b\right )^{\frac{1}{4}} a n^{4}\right ) - \frac{1}{4} \, i \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{1}{4}} \log \left (-a^{2} i n^{5} \log \left (x\right ) - a^{2} i n^{4} \log \left (c\right ) - \left (-a^{3} b\right )^{\frac{1}{4}} a n^{4}\right ) + \frac{1}{8} \, \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{1}{4}} \log \left (\frac{1}{4} \,{\left (\pi a^{2} n^{5}{\left (\mathrm{sgn}\left (x\right ) - 1\right )} + \pi a^{2} n^{4}{\left (\mathrm{sgn}\left (c\right ) - 1\right )}\right )}^{2} +{\left (a^{2} n^{5} \log \left ({\left | x \right |}\right ) + a^{2} n^{4} \log \left ({\left | c \right |}\right ) + \left (-a^{3} b\right )^{\frac{1}{4}} a n^{4}\right )}^{2}\right ) - \frac{1}{8} \, \left (-\frac{b}{a^{5} n^{4}}\right )^{\frac{1}{4}} \log \left (\frac{1}{4} \,{\left (\pi a^{2} n^{5}{\left (\mathrm{sgn}\left (x\right ) - 1\right )} + \pi a^{2} n^{4}{\left (\mathrm{sgn}\left (c\right ) - 1\right )}\right )}^{2} +{\left (a^{2} n^{5} \log \left ({\left | x \right |}\right ) + a^{2} n^{4} \log \left ({\left | c \right |}\right ) - \left (-a^{3} b\right )^{\frac{1}{4}} a n^{4}\right )}^{2}\right ) + \frac{\log \left (x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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