Optimal. Leaf size=46 \[ \frac{\left (\frac{c d}{a}-e\right ) \log (a+c x)}{2 c^2}-\frac{\left (\frac{c d}{a}+e\right ) \log (a-c x)}{2 c^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0576315, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{\left (\frac{c d}{a}-e\right ) \log (a+c x)}{2 c^2}-\frac{\left (\frac{c d}{a}+e\right ) \log (a-c x)}{2 c^2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)/(a^2 - c^2*x^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 7.50794, size = 41, normalized size = 0.89 \[ - \frac{\left (a e - c d\right ) \log{\left (a + c x \right )}}{2 a c^{2}} - \frac{\left (a e + c d\right ) \log{\left (a - c x \right )}}{2 a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)/(-c**2*x**2+a**2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0120563, size = 37, normalized size = 0.8 \[ \frac{d \tanh ^{-1}\left (\frac{c x}{a}\right )}{a c}-\frac{e \log \left (a^2-c^2 x^2\right )}{2 c^2} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)/(a^2 - c^2*x^2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 60, normalized size = 1.3 \[ -{\frac{\ln \left ( cx+a \right ) e}{2\,{c}^{2}}}+{\frac{\ln \left ( cx+a \right ) d}{2\,ac}}-{\frac{\ln \left ( cx-a \right ) e}{2\,{c}^{2}}}-{\frac{\ln \left ( cx-a \right ) d}{2\,ac}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)/(-c^2*x^2+a^2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.684199, size = 62, normalized size = 1.35 \[ \frac{{\left (c d - a e\right )} \log \left (c x + a\right )}{2 \, a c^{2}} - \frac{{\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(e*x + d)/(c^2*x^2 - a^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.287713, size = 55, normalized size = 1.2 \[ \frac{{\left (c d - a e\right )} \log \left (c x + a\right ) -{\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(e*x + d)/(c^2*x^2 - a^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.859247, size = 71, normalized size = 1.54 \[ - \frac{\left (a e - c d\right ) \log{\left (x + \frac{a^{2} e - a \left (a e - c d\right )}{c^{2} d} \right )}}{2 a c^{2}} - \frac{\left (a e + c d\right ) \log{\left (x + \frac{a^{2} e - a \left (a e + c d\right )}{c^{2} d} \right )}}{2 a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)/(-c**2*x**2+a**2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.274294, size = 68, normalized size = 1.48 \[ \frac{{\left (c d - a e\right )}{\rm ln}\left ({\left | c x + a \right |}\right )}{2 \, a c^{2}} - \frac{{\left (c d + a e\right )}{\rm ln}\left ({\left | c x - a \right |}\right )}{2 \, a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(e*x + d)/(c^2*x^2 - a^2),x, algorithm="giac")
[Out]