Optimal. Leaf size=145 \[ \frac {x^3}{3}-\frac {x^2}{d \log (f)}+\frac {x^2}{d \left (1+f^{c+d x}\right ) \log (f)}+\frac {2 x \log \left (1+f^{c+d x}\right )}{d^2 \log ^2(f)}-\frac {x^2 \log \left (1+f^{c+d x}\right )}{d \log (f)}+\frac {2 \text {Li}_2\left (-f^{c+d x}\right )}{d^3 \log ^3(f)}-\frac {2 x \text {Li}_2\left (-f^{c+d x}\right )}{d^2 \log ^2(f)}+\frac {2 \text {Li}_3\left (-f^{c+d x}\right )}{d^3 \log ^3(f)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.29, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 10, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {6820, 2216,
2215, 2221, 2611, 2320, 6724, 2222, 2317, 2438} \begin {gather*} \frac {2 \text {PolyLog}\left (2,-f^{c+d x}\right )}{d^3 \log ^3(f)}+\frac {2 \text {PolyLog}\left (3,-f^{c+d x}\right )}{d^3 \log ^3(f)}-\frac {2 x \text {PolyLog}\left (2,-f^{c+d x}\right )}{d^2 \log ^2(f)}+\frac {2 x \log \left (f^{c+d x}+1\right )}{d^2 \log ^2(f)}-\frac {x^2 \log \left (f^{c+d x}+1\right )}{d \log (f)}+\frac {x^2}{d \log (f) \left (f^{c+d x}+1\right )}-\frac {x^2}{d \log (f)}+\frac {x^3}{3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2215
Rule 2216
Rule 2221
Rule 2222
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 6724
Rule 6820
Rubi steps
\begin {align*} \int \frac {x^2}{1+2 f^{c+d x}+f^{2 c+2 d x}} \, dx &=\int \frac {x^2}{\left (1+f^{c+d x}\right )^2} \, dx\\ &=-\int \frac {f^{c+d x} x^2}{\left (1+f^{c+d x}\right )^2} \, dx+\int \frac {x^2}{1+f^{c+d x}} \, dx\\ &=\frac {x^3}{3}+\frac {x^2}{d \left (1+f^{c+d x}\right ) \log (f)}-\frac {2 \int \frac {x}{1+f^{c+d x}} \, dx}{d \log (f)}-\int \frac {f^{c+d x} x^2}{1+f^{c+d x}} \, dx\\ &=\frac {x^3}{3}-\frac {x^2}{d \log (f)}+\frac {x^2}{d \left (1+f^{c+d x}\right ) \log (f)}-\frac {x^2 \log \left (1+f^{c+d x}\right )}{d \log (f)}+\frac {2 \int \frac {f^{c+d x} x}{1+f^{c+d x}} \, dx}{d \log (f)}+\frac {2 \int x \log \left (1+f^{c+d x}\right ) \, dx}{d \log (f)}\\ &=\frac {x^3}{3}-\frac {x^2}{d \log (f)}+\frac {x^2}{d \left (1+f^{c+d x}\right ) \log (f)}+\frac {2 x \log \left (1+f^{c+d x}\right )}{d^2 \log ^2(f)}-\frac {x^2 \log \left (1+f^{c+d x}\right )}{d \log (f)}-\frac {2 x \text {Li}_2\left (-f^{c+d x}\right )}{d^2 \log ^2(f)}-\frac {2 \int \log \left (1+f^{c+d x}\right ) \, dx}{d^2 \log ^2(f)}+\frac {2 \int \text {Li}_2\left (-f^{c+d x}\right ) \, dx}{d^2 \log ^2(f)}\\ &=\frac {x^3}{3}-\frac {x^2}{d \log (f)}+\frac {x^2}{d \left (1+f^{c+d x}\right ) \log (f)}+\frac {2 x \log \left (1+f^{c+d x}\right )}{d^2 \log ^2(f)}-\frac {x^2 \log \left (1+f^{c+d x}\right )}{d \log (f)}-\frac {2 x \text {Li}_2\left (-f^{c+d x}\right )}{d^2 \log ^2(f)}-\frac {2 \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,f^{c+d x}\right )}{d^3 \log ^3(f)}+\frac {2 \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,f^{c+d x}\right )}{d^3 \log ^3(f)}\\ &=\frac {x^3}{3}-\frac {x^2}{d \log (f)}+\frac {x^2}{d \left (1+f^{c+d x}\right ) \log (f)}+\frac {2 x \log \left (1+f^{c+d x}\right )}{d^2 \log ^2(f)}-\frac {x^2 \log \left (1+f^{c+d x}\right )}{d \log (f)}+\frac {2 \text {Li}_2\left (-f^{c+d x}\right )}{d^3 \log ^3(f)}-\frac {2 x \text {Li}_2\left (-f^{c+d x}\right )}{d^2 \log ^2(f)}+\frac {2 \text {Li}_3\left (-f^{c+d x}\right )}{d^3 \log ^3(f)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.20, size = 123, normalized size = 0.85 \begin {gather*} \frac {d^3 x^3 \log ^3(f)+6 d x \log (f) \log \left (1+f^{c+d x}\right )-\frac {3 d^2 x^2 \log ^2(f) \left (f^{c+d x}+\left (1+f^{c+d x}\right ) \log \left (1+f^{c+d x}\right )\right )}{1+f^{c+d x}}+(6-6 d x \log (f)) \text {Li}_2\left (-f^{c+d x}\right )+6 \text {Li}_3\left (-f^{c+d x}\right )}{3 d^3 \log ^3(f)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.04, size = 232, normalized size = 1.60
method | result | size |
risch | \(\frac {x^{2}}{d \left (1+f^{d x +c}\right ) \ln \left (f \right )}+\frac {x^{3}}{3}-\frac {c^{2} x}{d^{2}}-\frac {2 c^{3}}{3 d^{3}}-\frac {\ln \left (1+f^{d x} f^{c}\right ) x^{2}}{d \ln \left (f \right )}-\frac {2 \polylog \left (2, -f^{d x} f^{c}\right ) x}{d^{2} \ln \left (f \right )^{2}}+\frac {2 \polylog \left (3, -f^{d x} f^{c}\right )}{d^{3} \ln \left (f \right )^{3}}+\frac {c^{2} \ln \left (f^{d x} f^{c}\right )}{d^{3} \ln \left (f \right )}-\frac {x^{2}}{d \ln \left (f \right )}-\frac {2 c x}{d^{2} \ln \left (f \right )}-\frac {c^{2}}{d^{3} \ln \left (f \right )}+\frac {2 \ln \left (1+f^{d x} f^{c}\right ) x}{d^{2} \ln \left (f \right )^{2}}+\frac {2 \polylog \left (2, -f^{d x} f^{c}\right )}{d^{3} \ln \left (f \right )^{3}}+\frac {2 c \ln \left (f^{d x} f^{c}\right )}{d^{3} \ln \left (f \right )^{2}}\) | \(232\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 159, normalized size = 1.10 \begin {gather*} \frac {x^{2}}{d f^{d x} f^{c} \log \left (f\right ) + d \log \left (f\right )} + \frac {d^{3} x^{3} \log \left (f\right )^{3} - 3 \, d^{2} x^{2} \log \left (f\right )^{2}}{3 \, d^{3} \log \left (f\right )^{3}} - \frac {d^{2} x^{2} \log \left (f^{d x} f^{c} + 1\right ) \log \left (f\right )^{2} + 2 \, d x {\rm Li}_2\left (-f^{d x} f^{c}\right ) \log \left (f\right ) - 2 \, {\rm Li}_{3}(-f^{d x} f^{c})}{d^{3} \log \left (f\right )^{3}} + \frac {2 \, {\left (d x \log \left (f^{d x} f^{c} + 1\right ) \log \left (f\right ) + {\rm Li}_2\left (-f^{d x} f^{c}\right )\right )}}{d^{3} \log \left (f\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.41, size = 210, normalized size = 1.45 \begin {gather*} \frac {3 \, c^{2} \log \left (f\right )^{2} + {\left (d^{3} x^{3} + c^{3}\right )} \log \left (f\right )^{3} + {\left ({\left (d^{3} x^{3} + c^{3}\right )} \log \left (f\right )^{3} - 3 \, {\left (d^{2} x^{2} - c^{2}\right )} \log \left (f\right )^{2}\right )} f^{d x + c} - 6 \, {\left (d x \log \left (f\right ) + {\left (d x \log \left (f\right ) - 1\right )} f^{d x + c} - 1\right )} {\rm Li}_2\left (-f^{d x + c}\right ) - 3 \, {\left (d^{2} x^{2} \log \left (f\right )^{2} - 2 \, d x \log \left (f\right ) + {\left (d^{2} x^{2} \log \left (f\right )^{2} - 2 \, d x \log \left (f\right )\right )} f^{d x + c}\right )} \log \left (f^{d x + c} + 1\right ) + 6 \, {\left (f^{d x + c} + 1\right )} {\rm polylog}\left (3, -f^{d x + c}\right )}{3 \, {\left (d^{3} f^{d x + c} \log \left (f\right )^{3} + d^{3} \log \left (f\right )^{3}\right )}} \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*} \frac {x^{2}}{d f^{c + d x} \log {\left (f \right )} + d \log {\left (f \right )}} + \frac {\int \left (- \frac {2 x}{e^{c \log {\left (f \right )}} e^{d x \log {\left (f \right )}} + 1}\right )\, dx + \int \frac {d x^{2} \log {\left (f \right )}}{e^{c \log {\left (f \right )}} e^{d x \log {\left (f \right )}} + 1}\, dx}{d \log {\left (f \right )}} \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 {x^2}{f^{2\,c+2\,d\,x}+2\,f^{c+d\,x}+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________