Optimal. Leaf size=51 \[ \frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \text {Ei}\left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{p}\right )}{2 b p} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2504, 2436,
2337, 2209} \begin {gather*} \frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \text {Ei}\left (\frac {\log \left (c \left (b x^2+a\right )^p\right )}{p}\right )}{2 b p} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2209
Rule 2337
Rule 2436
Rule 2504
Rubi steps
\begin {align*} \int \frac {x}{\log \left (c \left (a+b x^2\right )^p\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{\log \left (c (a+b x)^p\right )} \, dx,x,x^2\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{\log \left (c x^p\right )} \, dx,x,a+b x^2\right )}{2 b}\\ &=\frac {\left (\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{p}}}{x} \, dx,x,\log \left (c \left (a+b x^2\right )^p\right )\right )}{2 b p}\\ &=\frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \text {Ei}\left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{p}\right )}{2 b p}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 51, normalized size = 1.00 \begin {gather*} \frac {\left (a+b x^2\right ) \left (c \left (a+b x^2\right )^p\right )^{-1/p} \text {Ei}\left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{p}\right )}{2 b p} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 2.79, size = 272, normalized size = 5.33
method | result | size |
risch | \(-\frac {\left (b \,x^{2}+a \right ) \left (\left (b \,x^{2}+a \right )^{p}\right )^{-\frac {1}{p}} c^{-\frac {1}{p}} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \left (\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right )-\mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )\right ) \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )\right )}{2 p}} \expIntegral \left (1, -\ln \left (b \,x^{2}+a \right )-\frac {i \pi \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}-i \pi \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )-i \pi \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3}+i \pi \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )+2 \ln \left (c \right )+2 \ln \left (\left (b \,x^{2}+a \right )^{p}\right )-2 p \ln \left (b \,x^{2}+a \right )}{2 p}\right )}{2 b p}\) | \(272\) |
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 [A]
time = 0.41, size = 29, normalized size = 0.57 \begin {gather*} \frac {\operatorname {log\_integral}\left ({\left (b x^{2} + a\right )} c^{\left (\frac {1}{p}\right )}\right )}{2 \, b c^{\left (\frac {1}{p}\right )} p} \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 {x}{\log {\left (c \left (a + b x^{2}\right )^{p} \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 4.09, size = 31, normalized size = 0.61 \begin {gather*} \frac {{\rm Ei}\left (\frac {\log \left (c\right )}{p} + \log \left (b x^{2} + a\right )\right )}{2 \, b c^{\left (\frac {1}{p}\right )} p} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x}{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________