Optimal. Leaf size=18 \[ -\frac {1}{a x+b \log ^2\left (c x^n\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2561, 2544} \[ -\frac {1}{a x+b \log ^2\left (c x^n\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2544
Rule 2561
Rubi steps
\begin {align*} \int \frac {a x^2+2 b n x \log \left (c x^n\right )}{\left (a x^2+b x \log ^2\left (c x^n\right )\right )^2} \, dx &=\int \frac {x \left (a x+2 b n \log \left (c x^n\right )\right )}{\left (a x^2+b x \log ^2\left (c x^n\right )\right )^2} \, dx\\ &=\int \frac {a x+2 b n \log \left (c x^n\right )}{x \left (a x+b \log ^2\left (c x^n\right )\right )^2} \, dx\\ &=-\frac {1}{a x+b \log ^2\left (c x^n\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 18, normalized size = 1.00 \[ -\frac {1}{a x+b \log ^2\left (c x^n\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 31, normalized size = 1.72 \[ -\frac {1}{b n^{2} \log \relax (x)^{2} + 2 \, b n \log \relax (c) \log \relax (x) + b \log \relax (c)^{2} + a x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 31, normalized size = 1.72 \[ -\frac {1}{b n^{2} \log \relax (x)^{2} + 2 \, b n \log \relax (c) \log \relax (x) + b \log \relax (c)^{2} + a x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.31, size = 451, normalized size = 25.06 \[ -\frac {4}{-\pi ^{2} b \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2 \pi ^{2} b \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-\pi ^{2} b \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 \pi ^{2} b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}-\pi ^{2} b \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}-\pi ^{2} b \mathrm {csgn}\left (i c \,x^{n}\right )^{6}-4 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \relax (c )-4 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \left (x^{n}\right )+4 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )+4 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (x^{n}\right )+4 i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )+4 i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (x^{n}\right )-4 i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (c )-4 i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (x^{n}\right )+4 b \ln \relax (c )^{2}+8 b \ln \relax (c ) \ln \left (x^{n}\right )+4 b \ln \left (x^{n}\right )^{2}+4 a x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.65, size = 31, normalized size = 1.72 \[ -\frac {1}{b \log \relax (c)^{2} + 2 \, b \log \relax (c) \log \left (x^{n}\right ) + b \log \left (x^{n}\right )^{2} + a x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.26, size = 18, normalized size = 1.00 \[ -\frac {1}{b\,{\ln \left (c\,x^n\right )}^2+a\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________