Optimal. Leaf size=180 \[ -\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{4 d f}+\frac {1}{4} x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {b n \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}+\frac {b n \log \left (d f x^2+1\right )}{16 d^2 f^2}-\frac {3 b n x^2}{16 d f}-\frac {1}{16} b n x^4 \log \left (d f x^2+1\right )+\frac {1}{16} b n x^4 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2454, 2395, 43, 2376, 2391} \[ -\frac {b n \text {PolyLog}\left (2,-d f x^2\right )}{8 d^2 f^2}-\frac {\log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {b n \log \left (d f x^2+1\right )}{16 d^2 f^2}-\frac {3 b n x^2}{16 d f}-\frac {1}{16} b n x^4 \log \left (d f x^2+1\right )+\frac {1}{16} b n x^4 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 2376
Rule 2391
Rule 2395
Rule 2454
Rubi steps
\begin {align*} \int x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx &=\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-(b n) \int \left (\frac {x}{4 d f}-\frac {x^3}{8}-\frac {\log \left (1+d f x^2\right )}{4 d^2 f^2 x}+\frac {1}{4} x^3 \log \left (1+d f x^2\right )\right ) \, dx\\ &=-\frac {b n x^2}{8 d f}+\frac {1}{32} b n x^4+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac {1}{4} (b n) \int x^3 \log \left (1+d f x^2\right ) \, dx+\frac {(b n) \int \frac {\log \left (1+d f x^2\right )}{x} \, dx}{4 d^2 f^2}\\ &=-\frac {b n x^2}{8 d f}+\frac {1}{32} b n x^4+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac {b n \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}-\frac {1}{8} (b n) \operatorname {Subst}\left (\int x \log (1+d f x) \, dx,x,x^2\right )\\ &=-\frac {b n x^2}{8 d f}+\frac {1}{32} b n x^4+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{16} b n x^4 \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac {b n \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}+\frac {1}{16} (b d f n) \operatorname {Subst}\left (\int \frac {x^2}{1+d f x} \, dx,x,x^2\right )\\ &=-\frac {b n x^2}{8 d f}+\frac {1}{32} b n x^4+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{16} b n x^4 \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac {b n \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}+\frac {1}{16} (b d f n) \operatorname {Subst}\left (\int \left (-\frac {1}{d^2 f^2}+\frac {x}{d f}+\frac {1}{d^2 f^2 (1+d f x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {3 b n x^2}{16 d f}+\frac {1}{16} b n x^4+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{4 d f}-\frac {1}{8} x^4 \left (a+b \log \left (c x^n\right )\right )+\frac {b n \log \left (1+d f x^2\right )}{16 d^2 f^2}-\frac {1}{16} b n x^4 \log \left (1+d f x^2\right )-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )}{4 d^2 f^2}+\frac {1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac {b n \text {Li}_2\left (-d f x^2\right )}{8 d^2 f^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.11, size = 348, normalized size = 1.93 \[ -\frac {a \log \left (d f x^2+1\right )}{4 d^2 f^2}+\frac {a x^2}{4 d f}+\frac {1}{4} a x^4 \log \left (d f x^2+1\right )-\frac {a x^4}{8}+\frac {b \left (n-4 \left (\log \left (c x^n\right )-n \log (x)\right )\right ) \log \left (d f x^2+1\right )}{16 d^2 f^2}+\frac {b x^2 \left (4 \left (\log \left (c x^n\right )-n \log (x)\right )-n\right )}{16 d f}+\frac {1}{16} b x^4 \left (4 \left (\log \left (c x^n\right )-n \log (x)\right )+4 n \log (x)-n\right ) \log \left (d f x^2+1\right )+\frac {1}{32} b x^4 \left (n-4 \left (\log \left (c x^n\right )-n \log (x)\right )\right )-\frac {1}{2} b d f n \left (\frac {\text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+\log (x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )}{2 d^3 f^3}+\frac {\text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )+\log (x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )}{2 d^3 f^3}-\frac {\frac {1}{2} x^2 \log (x)-\frac {x^2}{4}}{d^2 f^2}+\frac {\frac {1}{4} x^4 \log (x)-\frac {x^4}{16}}{d f}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b x^{3} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a x^{3} \log \left (d f x^{2} + 1\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.31, size = 827, normalized size = 4.59 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{16} \, {\left (4 \, b x^{4} \log \left (x^{n}\right ) - {\left (b {\left (n - 4 \, \log \relax (c)\right )} - 4 \, a\right )} x^{4}\right )} \log \left (d f x^{2} + 1\right ) - \int \frac {4 \, b d f x^{5} \log \left (x^{n}\right ) + {\left (4 \, a d f - {\left (d f n - 4 \, d f \log \relax (c)\right )} b\right )} x^{5}}{8 \, {\left (d f x^{2} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
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]
________________________________________________________________________________________