Optimal. Leaf size=203 \[ \frac {\text {Li}_2\left (-\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{d^2 \log ^2(f) \sqrt {a^2-4 b c}}-\frac {\text {Li}_2\left (-\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{d^2 \log ^2(f) \sqrt {a^2-4 b c}}+\frac {x \log \left (\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}+1\right )}{d \log (f) \sqrt {a^2-4 b c}}-\frac {x \log \left (\frac {2 c f^{c+d x}}{\sqrt {a^2-4 b c}+a}+1\right )}{d \log (f) \sqrt {a^2-4 b c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.41, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2267, 2264, 2190, 2279, 2391} \[ \frac {\text {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{d^2 \log ^2(f) \sqrt {a^2-4 b c}}-\frac {\text {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{\sqrt {a^2-4 b c}+a}\right )}{d^2 \log ^2(f) \sqrt {a^2-4 b c}}+\frac {x \log \left (\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}+1\right )}{d \log (f) \sqrt {a^2-4 b c}}-\frac {x \log \left (\frac {2 c f^{c+d x}}{\sqrt {a^2-4 b c}+a}+1\right )}{d \log (f) \sqrt {a^2-4 b c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2190
Rule 2264
Rule 2267
Rule 2279
Rule 2391
Rubi steps
\begin {align*} \int \frac {x}{a+b f^{-c-d x}+c f^{c+d x}} \, dx &=\int \frac {f^{c+d x} x}{b+a f^{c+d x}+c f^{2 (c+d x)}} \, dx\\ &=\frac {(2 c) \int \frac {f^{c+d x} x}{a-\sqrt {a^2-4 b c}+2 c f^{c+d x}} \, dx}{\sqrt {a^2-4 b c}}-\frac {(2 c) \int \frac {f^{c+d x} x}{a+\sqrt {a^2-4 b c}+2 c f^{c+d x}} \, dx}{\sqrt {a^2-4 b c}}\\ &=\frac {x \log \left (1+\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}-\frac {x \log \left (1+\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}-\frac {\int \log \left (1+\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right ) \, dx}{\sqrt {a^2-4 b c} d \log (f)}+\frac {\int \log \left (1+\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right ) \, dx}{\sqrt {a^2-4 b c} d \log (f)}\\ &=\frac {x \log \left (1+\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}-\frac {x \log \left (1+\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}-\frac {\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{a-\sqrt {a^2-4 b c}}\right )}{x} \, dx,x,f^{c+d x}\right )}{\sqrt {a^2-4 b c} d^2 \log ^2(f)}+\frac {\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 c x}{a+\sqrt {a^2-4 b c}}\right )}{x} \, dx,x,f^{c+d x}\right )}{\sqrt {a^2-4 b c} d^2 \log ^2(f)}\\ &=\frac {x \log \left (1+\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}-\frac {x \log \left (1+\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}+\frac {\text {Li}_2\left (-\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d^2 \log ^2(f)}-\frac {\text {Li}_2\left (-\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d^2 \log ^2(f)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.46, size = 0, normalized size = 0.00 \[ \int \frac {x}{a+b f^{-c-d x}+c f^{c+d x}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 353, normalized size = 1.74 \[ \frac {b c \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} \log \left (2 \, c f^{d x + c} + b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} + a\right ) \log \relax (f) - b c \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} \log \left (2 \, c f^{d x + c} - b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} + a\right ) \log \relax (f) + {\left (b d x + b c\right )} \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} \log \relax (f) \log \left (\frac {{\left (b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} + a\right )} f^{d x + c} + 2 \, b}{2 \, b}\right ) - {\left (b d x + b c\right )} \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} \log \relax (f) \log \left (-\frac {{\left (b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} - a\right )} f^{d x + c} - 2 \, b}{2 \, b}\right ) + b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} {\rm Li}_2\left (-\frac {{\left (b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} + a\right )} f^{d x + c} + 2 \, b}{2 \, b} + 1\right ) - b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} {\rm Li}_2\left (\frac {{\left (b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} - a\right )} f^{d x + c} - 2 \, b}{2 \, b} + 1\right )}{{\left (a^{2} - 4 \, b c\right )} d^{2} \log \relax (f)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{c f^{d x + c} + b f^{-d x - c} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.09, size = 433, normalized size = 2.13 \[ -\frac {x \ln \left (\frac {-2 b \,f^{-c} f^{-d x}-a +\sqrt {a^{2}-4 b c}}{-a +\sqrt {a^{2}-4 b c}}\right )}{\sqrt {a^{2}-4 b c}\, d \ln \relax (f )}+\frac {x \ln \left (\frac {2 b \,f^{-c} f^{-d x}+a +\sqrt {a^{2}-4 b c}}{a +\sqrt {a^{2}-4 b c}}\right )}{\sqrt {a^{2}-4 b c}\, d \ln \relax (f )}+\frac {2 c \arctan \left (\frac {2 b \,f^{-c} f^{-d x}+a}{\sqrt {-a^{2}+4 b c}}\right )}{\sqrt {-a^{2}+4 b c}\, d^{2} \ln \relax (f )}-\frac {c \ln \left (\frac {-2 b \,f^{-c} f^{-d x}-a +\sqrt {a^{2}-4 b c}}{-a +\sqrt {a^{2}-4 b c}}\right )}{\sqrt {a^{2}-4 b c}\, d^{2} \ln \relax (f )}+\frac {c \ln \left (\frac {2 b \,f^{-c} f^{-d x}+a +\sqrt {a^{2}-4 b c}}{a +\sqrt {a^{2}-4 b c}}\right )}{\sqrt {a^{2}-4 b c}\, d^{2} \ln \relax (f )}+\frac {\dilog \left (\frac {-2 b \,f^{-c} f^{-d x}-a +\sqrt {a^{2}-4 b c}}{-a +\sqrt {a^{2}-4 b c}}\right )}{\sqrt {a^{2}-4 b c}\, d^{2} \ln \relax (f )^{2}}-\frac {\dilog \left (\frac {2 b \,f^{-c} f^{-d x}+a +\sqrt {a^{2}-4 b c}}{a +\sqrt {a^{2}-4 b c}}\right )}{\sqrt {a^{2}-4 b c}\, d^{2} \ln \relax (f )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x}{a+c\,f^{c+d\,x}+\frac {b}{f^{c+d\,x}}} \,d x \]
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]
________________________________________________________________________________________