Optimal. Leaf size=47 \[ -\frac {2 \tanh ^{-1}\left (\frac {a+2 c f^{c+d x}}{\sqrt {a^2-4 b c}}\right )}{d \log (f) \sqrt {a^2-4 b c}} \]
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Rubi [A] time = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2282, 1386, 618, 206} \[ -\frac {2 \tanh ^{-1}\left (\frac {a+2 c f^{c+d x}}{\sqrt {a^2-4 b c}}\right )}{d \log (f) \sqrt {a^2-4 b c}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 1386
Rule 2282
Rubi steps
\begin {align*} \int \frac {1}{a+b f^{-c-d x}+c f^{c+d x}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \left (a+\frac {b}{x}+c x\right )} \, dx,x,f^{c+d x}\right )}{d \log (f)}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{b+a x+c x^2} \, dx,x,f^{c+d x}\right )}{d \log (f)}\\ &=-\frac {2 \operatorname {Subst}\left (\int \frac {1}{a^2-4 b c-x^2} \, dx,x,a+2 c f^{c+d x}\right )}{d \log (f)}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {a+2 c f^{c+d x}}{\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 47, normalized size = 1.00 \[ -\frac {2 \tanh ^{-1}\left (\frac {a+2 c f^{c+d x}}{\sqrt {a^2-4 b c}}\right )}{d \log (f) \sqrt {a^2-4 b c}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 189, normalized size = 4.02 \[ \left [\frac {\log \left (\frac {2 \, c^{2} f^{2 \, d x + 2 \, c} + a^{2} - 2 \, b c + 2 \, {\left (a c - \sqrt {a^{2} - 4 \, b c} c\right )} f^{d x + c} - \sqrt {a^{2} - 4 \, b c} a}{c f^{2 \, d x + 2 \, c} + a f^{d x + c} + b}\right )}{\sqrt {a^{2} - 4 \, b c} d \log \relax (f)}, -\frac {2 \, \sqrt {-a^{2} + 4 \, b c} \arctan \left (-\frac {2 \, \sqrt {-a^{2} + 4 \, b c} c f^{d x + c} + \sqrt {-a^{2} + 4 \, b c} a}{a^{2} - 4 \, b c}\right )}{{\left (a^{2} - 4 \, b c\right )} d \log \relax (f)}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 48, normalized size = 1.02 \[ \frac {2 \, \arctan \left (\frac {2 \, c f^{d x} f^{c} + a}{\sqrt {-a^{2} + 4 \, b c}}\right )}{\sqrt {-a^{2} + 4 \, b c} d \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 135, normalized size = 2.87 \[ -\frac {\ln \left (f^{-d x -c}+\frac {-a^{2}+4 b c +\sqrt {a^{2}-4 b c}\, a}{2 \sqrt {a^{2}-4 b c}\, b}\right )}{\sqrt {a^{2}-4 b c}\, d \ln \relax (f )}+\frac {\ln \left (f^{-d x -c}+\frac {a^{2}-4 b c +\sqrt {a^{2}-4 b c}\, a}{2 \sqrt {a^{2}-4 b c}\, b}\right )}{\sqrt {a^{2}-4 b c}\, d \ln \relax (f )} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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mupad [B] time = 3.64, size = 47, normalized size = 1.00 \[ \frac {2\,\mathrm {atan}\left (\frac {a+2\,c\,f^{c+d\,x}}{\sqrt {4\,b\,c-a^2}}\right )}{d\,\ln \relax (f)\,\sqrt {4\,b\,c-a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.38, size = 66, normalized size = 1.40 \[ \operatorname {RootSum} {\left (z^{2} \left (a^{2} d^{2} \log {\relax (f )}^{2} - 4 b c d^{2} \log {\relax (f )}^{2}\right ) - 1, \left (i \mapsto i \log {\left (f^{c + d x} + \frac {- i a^{2} d \log {\relax (f )} + 4 i b c d \log {\relax (f )} + a}{2 c} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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