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 = 0.18, antiderivative size = 233, normalized size of antiderivative = 1.00, 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 204
Rule 211
Rule 321
Rule 617
Rule 628
Rule 1162
Rule 1165
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*}
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Mathematica [A] time = 0.07, 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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fricas [A] time = 0.47, size = 192, normalized size = 0.82 \[ -\frac {4 \, a \left (-\frac {b}{a^{5} n^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {a^{2} n^{2} \sqrt {-\frac {b}{a^{5} n^{4}}} + n^{2} \log \relax (x)^{2} + 2 \, n \log \relax (c) \log \relax (x) + \log \relax (c)^{2}} a^{4} n^{3} \left (-\frac {b}{a^{5} n^{4}}\right )^{\frac {3}{4}} - {\left (a^{4} n^{4} \log \relax (x) + a^{4} n^{3} \log \relax (c)\right )} \left (-\frac {b}{a^{5} n^{4}}\right )^{\frac {3}{4}}}{b}\right ) + a \left (-\frac {b}{a^{5} n^{4}}\right )^{\frac {1}{4}} \log \left (a n \left (-\frac {b}{a^{5} n^{4}}\right )^{\frac {1}{4}} + n \log \relax (x) + \log \relax (c)\right ) - a \left (-\frac {b}{a^{5} n^{4}}\right )^{\frac {1}{4}} \log \left (-a n \left (-\frac {b}{a^{5} n^{4}}\right )^{\frac {1}{4}} + n \log \relax (x) + \log \relax (c)\right ) - 4 \, \log \relax (x)}{4 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 178, normalized size = 0.76 \[ \frac {\log \relax (x)}{a} - \frac {4 \, \left (-\frac {b n^{12}}{a}\right )^{\frac {1}{4}} \arctan \left (\frac {\pi a {\left (\mathrm {sgn}\relax (c) - 1\right )} - 2 \, \left (-a^{3} b\right )^{\frac {1}{4}}}{2 \, {\left (a n \log \relax (x) + a \log \left ({\left | c \right |}\right )\right )}}\right ) + \left (-\frac {b n^{12}}{a}\right )^{\frac {1}{4}} \log \left (\frac {1}{4} \, {\left (\pi a n {\left (\mathrm {sgn}\relax (x) - 1\right )} + \pi a {\left (\mathrm {sgn}\relax (c) - 1\right )}\right )}^{2} + {\left (a n \log \left ({\left | x \right |}\right ) + a \log \left ({\left | c \right |}\right ) + \left (-a^{3} b\right )^{\frac {1}{4}}\right )}^{2}\right ) - \left (-\frac {b n^{12}}{a}\right )^{\frac {1}{4}} \log \left (\frac {1}{4} \, {\left (\pi a n {\left (\mathrm {sgn}\relax (x) - 1\right )} + \pi a {\left (\mathrm {sgn}\relax (c) - 1\right )}\right )}^{2} + {\left (a n \log \left ({\left | x \right |}\right ) + a \log \left ({\left | c \right |}\right ) - \left (-a^{3} b\right )^{\frac {1}{4}}\right )}^{2}\right )}{8 \, a n^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 181, normalized size = 0.78 \[ \frac {\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \ln \left (c \,x^{n}\right )}{\left (\frac {b}{a}\right )^{\frac {1}{4}}}+1\right )}{4 a n}-\frac {\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \ln \left (c \,x^{n}\right )}{\left (\frac {b}{a}\right )^{\frac {1}{4}}}+1\right )}{4 a n}+\frac {\ln \left (c \,x^{n}\right )}{a n}-\frac {\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {\ln \left (c \,x^{n}\right )^{2}+\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (c \,x^{n}\right )+\sqrt {\frac {b}{a}}}{\ln \left (c \,x^{n}\right )^{2}-\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (c \,x^{n}\right )+\sqrt {\frac {b}{a}}}\right )}{8 a n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -b \int \frac {1}{4 \, a^{2} x \log \relax (c)^{3} \log \left (x^{n}\right ) + 6 \, a^{2} x \log \relax (c)^{2} \log \left (x^{n}\right )^{2} + 4 \, a^{2} x \log \relax (c) \log \left (x^{n}\right )^{3} + a^{2} x \log \left (x^{n}\right )^{4} + {\left (a^{2} \log \relax (c)^{4} + a b\right )} x}\,{d x} + \frac {\log \relax (x)}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.21, size = 176, normalized size = 0.76 \[ \frac {\ln \relax (x)}{a}+\frac {{\left (-b\right )}^{1/4}\,\left (\ln \left (-\frac {{\left (-b\right )}^{5/2}}{a^{11/2}\,x^3}-\frac {{\left (-b\right )}^{9/4}\,\ln \left (c\,x^n\right )\,1{}\mathrm {i}}{a^{21/4}\,x^3}\right )\,1{}\mathrm {i}-\ln \left (-\frac {{\left (-b\right )}^{5/2}}{a^{11/2}\,x^3}+\frac {{\left (-b\right )}^{9/4}\,\ln \left (c\,x^n\right )\,1{}\mathrm {i}}{a^{21/4}\,x^3}\right )\,1{}\mathrm {i}\right )}{4\,a^{5/4}\,n}-\frac {{\left (-b\right )}^{1/4}\,\ln \left (\frac {{\left (-b\right )}^{5/2}+a^{1/4}\,{\left (-b\right )}^{9/4}\,\ln \left (c\,x^n\right )}{x^3}\right )}{4\,a^{5/4}\,n}+\frac {{\left (-b\right )}^{1/4}\,\ln \left (\frac {{\left (-b\right )}^{5/2}-a^{1/4}\,{\left (-b\right )}^{9/4}\,\ln \left (c\,x^n\right )}{x^3}\right )}{4\,a^{5/4}\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 36.99, size = 270, normalized size = 1.16 \[ \begin {cases} \tilde {\infty } \log {\relax (c )}^{4} \log {\relax (x )} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\frac {\log {\relax (c )}^{4} \log {\relax (x )}}{a \log {\relax (c )}^{4} + b} & \text {for}\: n = 0 \\\frac {\begin {cases} \frac {\log {\left (c x^{n} \right )}^{5}}{5 n} & \text {for}\: \left |{c x^{n}}\right | < 1 \\- \frac {\log {\left (\frac {x^{- n}}{c} \right )}^{5}}{5 n} & \text {for}\: \frac {1}{\left |{c x^{n}}\right |} < 1 \\- \frac {24 {G_{6, 6}^{6, 0}\left (\begin {matrix} & 1, 1, 1, 1, 1, 1 \\0, 0, 0, 0, 0, 0 & \end {matrix} \middle | {c x^{n}} \right )}}{n} + \frac {24 {G_{6, 6}^{0, 6}\left (\begin {matrix} 1, 1, 1, 1, 1, 1 & \\ & 0, 0, 0, 0, 0, 0 \end {matrix} \middle | {c x^{n}} \right )}}{n} & \text {otherwise} \end {cases}}{b} & \text {for}\: a = 0 \\\frac {\log {\relax (x )}}{a} & \text {for}\: b = 0 \\\frac {\sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac {1}{a}} \log {\left (- \sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac {1}{a}} + n \log {\relax (x )} + \log {\relax (c )} \right )}}{4 a n} - \frac {\sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac {1}{a}} \log {\left (\sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac {1}{a}} + n \log {\relax (x )} + \log {\relax (c )} \right )}}{4 a n} + \frac {\sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac {1}{a}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} n \log {\relax (x )}}{\sqrt [4]{b} \sqrt [4]{\frac {1}{a}}} + \frac {\left (-1\right )^{\frac {3}{4}} \log {\relax (c )}}{\sqrt [4]{b} \sqrt [4]{\frac {1}{a}}} \right )}}{2 a n} + \frac {\log {\relax (x )}}{a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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