Optimal. Leaf size=148 \[ \frac {1}{2} b m n x^2-\frac {1}{2} m x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {b n \left (e+f x^2\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 f}-\frac {b e n \log \left (-\frac {f x^2}{e}\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 f}+\frac {\left (e+f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{2 f}-\frac {b e m n \text {Li}_2\left (1+\frac {f x^2}{e}\right )}{4 f} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.15, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2504, 2436,
2332, 2423, 2525, 2458, 45, 2393, 2354, 2438} \begin {gather*} -\frac {b e m n \text {PolyLog}\left (2,\frac {f x^2}{e}+1\right )}{4 f}+\frac {\left (e+f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{2 f}-\frac {1}{2} m x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {b n \left (e+f x^2\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 f}-\frac {b e n \log \left (-\frac {f x^2}{e}\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 f}+\frac {1}{2} b m n x^2 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 2332
Rule 2354
Rule 2393
Rule 2423
Rule 2436
Rule 2438
Rule 2458
Rule 2504
Rule 2525
Rubi steps
\begin {align*} \int x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right ) \, dx &=-\frac {1}{2} m x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {\left (e+f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{2 f}-(b n) \int \left (-\frac {m x}{2}+\frac {\left (e+f x^2\right ) \log \left (d \left (e+f x^2\right )^m\right )}{2 f x}\right ) \, dx\\ &=\frac {1}{4} b m n x^2-\frac {1}{2} m x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {\left (e+f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{2 f}-\frac {(b n) \int \frac {\left (e+f x^2\right ) \log \left (d \left (e+f x^2\right )^m\right )}{x} \, dx}{2 f}\\ &=\frac {1}{4} b m n x^2-\frac {1}{2} m x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {\left (e+f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{2 f}-\frac {(b n) \text {Subst}\left (\int \frac {(e+f x) \log \left (d (e+f x)^m\right )}{x} \, dx,x,x^2\right )}{4 f}\\ &=\frac {1}{4} b m n x^2-\frac {1}{2} m x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {\left (e+f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{2 f}-\frac {(b n) \text {Subst}\left (\int \frac {x \log \left (d x^m\right )}{-\frac {e}{f}+\frac {x}{f}} \, dx,x,e+f x^2\right )}{4 f^2}\\ &=\frac {1}{4} b m n x^2-\frac {1}{2} m x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {\left (e+f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{2 f}-\frac {(b n) \text {Subst}\left (\int \left (f \log \left (d x^m\right )-\frac {e f \log \left (d x^m\right )}{e-x}\right ) \, dx,x,e+f x^2\right )}{4 f^2}\\ &=\frac {1}{4} b m n x^2-\frac {1}{2} m x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {\left (e+f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{2 f}-\frac {(b n) \text {Subst}\left (\int \log \left (d x^m\right ) \, dx,x,e+f x^2\right )}{4 f}+\frac {(b e n) \text {Subst}\left (\int \frac {\log \left (d x^m\right )}{e-x} \, dx,x,e+f x^2\right )}{4 f}\\ &=\frac {1}{2} b m n x^2-\frac {1}{2} m x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {b n \left (e+f x^2\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 f}-\frac {b e n \log \left (-\frac {f x^2}{e}\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 f}+\frac {\left (e+f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{2 f}+\frac {(b e m n) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{e}\right )}{x} \, dx,x,e+f x^2\right )}{4 f}\\ &=\frac {1}{2} b m n x^2-\frac {1}{2} m x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {b n \left (e+f x^2\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 f}-\frac {b e n \log \left (-\frac {f x^2}{e}\right ) \log \left (d \left (e+f x^2\right )^m\right )}{4 f}+\frac {\left (e+f x^2\right ) \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )}{2 f}-\frac {b e m n \text {Li}_2\left (1+\frac {f x^2}{e}\right )}{4 f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.06, size = 266, normalized size = 1.80 \begin {gather*} \frac {-2 a f m x^2+2 b f m n x^2-2 b f m x^2 \log \left (c x^n\right )+2 b e m n \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 b e m n \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )-b e m n \log \left (e+f x^2\right )-2 b e m n \log (x) \log \left (e+f x^2\right )+2 b e m \log \left (c x^n\right ) \log \left (e+f x^2\right )+2 a e \log \left (d \left (e+f x^2\right )^m\right )+2 a f x^2 \log \left (d \left (e+f x^2\right )^m\right )-b f n x^2 \log \left (d \left (e+f x^2\right )^m\right )+2 b f x^2 \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )+2 b e m n \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+2 b e m n \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{4 f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.29, size = 2068, normalized size = 13.97
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2068\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x\,\ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________