Optimal. Leaf size=80 \[ \frac {b p \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}-\frac {\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}+\frac {b p^2 \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2504, 2444,
2441, 2352} \begin {gather*} \frac {b p^2 \text {PolyLog}\left (2,\frac {b x^2}{a}+1\right )}{a}-\frac {\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}+\frac {b p \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2352
Rule 2441
Rule 2444
Rule 2504
Rubi steps
\begin {align*} \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\log ^2\left (c (a+b x)^p\right )}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}+\frac {(b p) \text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx,x,x^2\right )}{a}\\ &=\frac {b p \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}-\frac {\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}-\frac {\left (b^2 p^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx,x,x^2\right )}{a}\\ &=\frac {b p \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}-\frac {\left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a x^2}+\frac {b p^2 \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 93, normalized size = 1.16 \begin {gather*} \frac {b p \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}-\frac {b \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{2 x^2}+\frac {b p^2 \text {Li}_2\left (\frac {a+b x^2}{a}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.32, size = 841, normalized size = 10.51
method | result | size |
risch | \(-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right )^{2}}{2 x^{2}}+\frac {2 p b \ln \left (\left (b \,x^{2}+a \right )^{p}\right ) \ln \left (x \right )}{a}-\frac {p b \ln \left (\left (b \,x^{2}+a \right )^{p}\right ) \ln \left (b \,x^{2}+a \right )}{a}-\frac {2 p^{2} b \ln \left (x \right ) \ln \left (\frac {-b x +\sqrt {-b a}}{\sqrt {-b a}}\right )}{a}-\frac {2 p^{2} b \ln \left (x \right ) \ln \left (\frac {b x +\sqrt {-b a}}{\sqrt {-b a}}\right )}{a}-\frac {2 p^{2} b \dilog \left (\frac {-b x +\sqrt {-b a}}{\sqrt {-b a}}\right )}{a}-\frac {2 p^{2} b \dilog \left (\frac {b x +\sqrt {-b a}}{\sqrt {-b a}}\right )}{a}+\frac {p^{2} b \ln \left (b \,x^{2}+a \right )^{2}}{2 a}-\frac {i \ln \left (\left (b \,x^{2}+a \right )^{p}\right ) \pi \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}}{2 x^{2}}-\frac {i p b \ln \left (b \,x^{2}+a \right ) \pi \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}}{2 a}-\frac {i p b \ln \left (x \right ) \pi \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3}}{a}+\frac {i p b \ln \left (b \,x^{2}+a \right ) \pi \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3}}{2 a}-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) \ln \left (c \right )}{x^{2}}+\frac {i \ln \left (\left (b \,x^{2}+a \right )^{p}\right ) \pi \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )}{2 x^{2}}+\frac {i p b \ln \left (x \right ) \pi \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{a}+\frac {i p b \ln \left (b \,x^{2}+a \right ) \pi \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )}{2 a}-\frac {i \ln \left (\left (b \,x^{2}+a \right )^{p}\right ) \pi \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{2 x^{2}}+\frac {2 p b \ln \left (x \right ) \ln \left (c \right )}{a}-\frac {i p b \ln \left (x \right ) \pi \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )}{a}-\frac {i p b \ln \left (b \,x^{2}+a \right ) \pi \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{2 a}+\frac {i \ln \left (\left (b \,x^{2}+a \right )^{p}\right ) \pi \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3}}{2 x^{2}}+\frac {i p b \ln \left (x \right ) \pi \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}}{a}-\frac {p b \ln \left (b \,x^{2}+a \right ) \ln \left (c \right )}{a}-\frac {\left (i \pi \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}-i \pi \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )-i \pi \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3}+i \pi \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )+2 \ln \left (c \right )\right )^{2}}{8 x^{2}}\) | \(841\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 118, normalized size = 1.48 \begin {gather*} \frac {1}{2} \, b^{2} p^{2} {\left (\frac {\log \left (b x^{2} + a\right )^{2}}{a b} - \frac {2 \, {\left (2 \, \log \left (\frac {b x^{2}}{a} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {b x^{2}}{a}\right )\right )}}{a b}\right )} - b p {\left (\frac {\log \left (b x^{2} + a\right )}{a} - \frac {\log \left (x^{2}\right )}{a}\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{2 \, x^{2}} \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{x^{3}}\, dx \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 \frac {{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________