Optimal. Leaf size=227 \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a}+\sqrt{b} \log ^2\left (c x^n\right )\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} n}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a}+\sqrt{b} \log ^2\left (c x^n\right )\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} n}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} n}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} n} \]
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Rubi [A] time = 0.162432, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {211, 1165, 628, 1162, 617, 204} \[ -\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a}+\sqrt{b} \log ^2\left (c x^n\right )\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} n}+\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a}+\sqrt{b} \log ^2\left (c x^n\right )\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} n}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} n}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} n} \]
Antiderivative was successfully verified.
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Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{a x+b x \log ^4\left (c x^n\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{a+b x^4} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\log \left (c x^n\right )\right )}{2 \sqrt{a} n}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\log \left (c x^n\right )\right )}{2 \sqrt{a} n}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\log \left (c x^n\right )\right )}{4 \sqrt{a} \sqrt{b} n}+\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\log \left (c x^n\right )\right )}{4 \sqrt{a} \sqrt{b} n}-\frac{\operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\log \left (c x^n\right )\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} n}-\frac{\operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\log \left (c x^n\right )\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} n}\\ &=-\frac{\log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{b} \log ^2\left (c x^n\right )\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} n}+\frac{\log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{b} \log ^2\left (c x^n\right )\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} n}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} n}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} n}\\ &=-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} n}+\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{b} n}-\frac{\log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{b} \log ^2\left (c x^n\right )\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} n}+\frac{\log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{b} \log ^2\left (c x^n\right )\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} n}\\ \end{align*}
Mathematica [A] time = 0.06405, size = 167, normalized size = 0.74 \[ \frac{-\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a}+\sqrt{b} \log ^2\left (c x^n\right )\right )+\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \log \left (c x^n\right )+\sqrt{a}+\sqrt{b} \log ^2\left (c x^n\right )\right )-2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \log \left (c x^n\right )}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{b} n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 168, normalized size = 0.7 \begin{align*}{\frac{\sqrt{2}}{8\,na}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{2}+\sqrt [4]{{\frac{a}{b}}}\ln \left ( c{x}^{n} \right ) \sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{2}-\sqrt [4]{{\frac{a}{b}}}\ln \left ( c{x}^{n} \right ) \sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}}{4\,na}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\ln \left ( c{x}^{n} \right ) \sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{\sqrt{2}}{4\,na}\sqrt [4]{{\frac{a}{b}}}\arctan \left ( -{\ln \left ( c{x}^{n} \right ) \sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+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*} \int \frac{1}{b x \log \left (c x^{n}\right )^{4} + a x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01281, size = 502, normalized size = 2.21 \begin{align*} \left (-\frac{1}{a^{3} b n^{4}}\right )^{\frac{1}{4}} \arctan \left (\sqrt{a^{2} n^{2} \sqrt{-\frac{1}{a^{3} b 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^{2} b n^{3} \left (-\frac{1}{a^{3} b n^{4}}\right )^{\frac{3}{4}} -{\left (a^{2} b n^{4} \log \left (x\right ) + a^{2} b n^{3} \log \left (c\right )\right )} \left (-\frac{1}{a^{3} b n^{4}}\right )^{\frac{3}{4}}\right ) + \frac{1}{4} \, \left (-\frac{1}{a^{3} b n^{4}}\right )^{\frac{1}{4}} \log \left (a n \left (-\frac{1}{a^{3} b n^{4}}\right )^{\frac{1}{4}} + n \log \left (x\right ) + \log \left (c\right )\right ) - \frac{1}{4} \, \left (-\frac{1}{a^{3} b n^{4}}\right )^{\frac{1}{4}} \log \left (-a n \left (-\frac{1}{a^{3} b n^{4}}\right )^{\frac{1}{4}} + n \log \left (x\right ) + \log \left (c\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 66.9285, size = 257, normalized size = 1.13 \begin{align*} \begin{cases} \frac{\tilde{\infty } \log{\left (x \right )}}{\log{\left (c \right )}^{4}} & \text{for}\: a = 0 \wedge b = 0 \wedge n = 0 \\- \frac{1}{b \left (3 n^{4} \log{\left (x \right )}^{3} + 9 n^{3} \log{\left (c \right )} \log{\left (x \right )}^{2} + 9 n^{2} \log{\left (c \right )}^{2} \log{\left (x \right )} + 3 n \log{\left (c \right )}^{3}\right )} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{a + b \log{\left (c \right )}^{4}} & \text{for}\: n = 0 \\- \frac{\sqrt [4]{-1} \log{\left (- \sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + n \log{\left (x \right )} + \log{\left (c \right )} \right )}}{4 a^{\frac{3}{4}} b^{3} n \left (\frac{1}{b}\right )^{\frac{11}{4}}} + \frac{\sqrt [4]{-1} \log{\left (\sqrt [4]{-1} \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + n \log{\left (x \right )} + \log{\left (c \right )} \right )}}{4 a^{\frac{3}{4}} b^{3} n \left (\frac{1}{b}\right )^{\frac{11}{4}}} - \frac{\sqrt [4]{-1} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} n \log{\left (x \right )}}{\sqrt [4]{a} \sqrt [4]{\frac{1}{b}}} + \frac{\left (-1\right )^{\frac{3}{4}} \log{\left (c \right )}}{\sqrt [4]{a} \sqrt [4]{\frac{1}{b}}} \right )}}{2 a^{\frac{3}{4}} b^{3} n \left (\frac{1}{b}\right )^{\frac{11}{4}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.43187, size = 271, normalized size = 1.19 \begin{align*} \frac{1}{4} \, i \left (-\frac{1}{a^{3} b n^{4}}\right )^{\frac{1}{4}} \log \left (b i n \log \left (x\right ) + b i \log \left (c\right ) - \left (-a b^{3}\right )^{\frac{1}{4}}\right ) - \frac{1}{4} \, i \left (-\frac{1}{a^{3} b n^{4}}\right )^{\frac{1}{4}} \log \left (-b i n \log \left (x\right ) - b i \log \left (c\right ) - \left (-a b^{3}\right )^{\frac{1}{4}}\right ) + \frac{1}{8} \, \left (-\frac{1}{a^{3} b n^{4}}\right )^{\frac{1}{4}} \log \left (\frac{1}{4} \,{\left (\pi b n{\left (\mathrm{sgn}\left (x\right ) - 1\right )} + \pi b{\left (\mathrm{sgn}\left (c\right ) - 1\right )}\right )}^{2} +{\left (b n \log \left ({\left | x \right |}\right ) + b \log \left ({\left | c \right |}\right ) + \left (-a b^{3}\right )^{\frac{1}{4}}\right )}^{2}\right ) - \frac{1}{8} \, \left (-\frac{1}{a^{3} b n^{4}}\right )^{\frac{1}{4}} \log \left (\frac{1}{4} \,{\left (\pi b n{\left (\mathrm{sgn}\left (x\right ) - 1\right )} + \pi b{\left (\mathrm{sgn}\left (c\right ) - 1\right )}\right )}^{2} +{\left (b n \log \left ({\left | x \right |}\right ) + b \log \left ({\left | c \right |}\right ) - \left (-a b^{3}\right )^{\frac{1}{4}}\right )}^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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