Optimal. Leaf size=59 \[ -\frac{(a e+c d) \log (a-c x)}{2 a^3}+\frac{(c d-a e) \log (a+c x)}{2 a^3}-\frac{d}{a^2 x}+\frac{e \log (x)}{a^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0539944, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {801} \[ -\frac{(a e+c d) \log (a-c x)}{2 a^3}+\frac{(c d-a e) \log (a+c x)}{2 a^3}-\frac{d}{a^2 x}+\frac{e \log (x)}{a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 801
Rubi steps
\begin{align*} \int \frac{d+e x}{x^2 \left (a^2-c^2 x^2\right )} \, dx &=\int \left (\frac{d}{a^2 x^2}+\frac{e}{a^2 x}+\frac{c (c d+a e)}{2 a^3 (a-c x)}-\frac{c (-c d+a e)}{2 a^3 (a+c x)}\right ) \, dx\\ &=-\frac{d}{a^2 x}+\frac{e \log (x)}{a^2}-\frac{(c d+a e) \log (a-c x)}{2 a^3}+\frac{(c d-a e) \log (a+c x)}{2 a^3}\\ \end{align*}
Mathematica [A] time = 0.0128025, size = 51, normalized size = 0.86 \[ -\frac{e \log \left (a^2-c^2 x^2\right )}{2 a^2}+\frac{c d \tanh ^{-1}\left (\frac{c x}{a}\right )}{a^3}-\frac{d}{a^2 x}+\frac{e \log (x)}{a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.012, size = 72, normalized size = 1.2 \begin{align*} -{\frac{d}{{a}^{2}x}}+{\frac{e\ln \left ( x \right ) }{{a}^{2}}}-{\frac{\ln \left ( cx+a \right ) e}{2\,{a}^{2}}}+{\frac{\ln \left ( cx+a \right ) cd}{2\,{a}^{3}}}-{\frac{\ln \left ( cx-a \right ) e}{2\,{a}^{2}}}-{\frac{\ln \left ( cx-a \right ) cd}{2\,{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.08224, size = 76, normalized size = 1.29 \begin{align*} \frac{e \log \left (x\right )}{a^{2}} + \frac{{\left (c d - a e\right )} \log \left (c x + a\right )}{2 \, a^{3}} - \frac{{\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, a^{3}} - \frac{d}{a^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.6317, size = 130, normalized size = 2.2 \begin{align*} \frac{2 \, a e x \log \left (x\right ) +{\left (c d - a e\right )} x \log \left (c x + a\right ) -{\left (c d + a e\right )} x \log \left (c x - a\right ) - 2 \, a d}{2 \, a^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 1.78309, size = 221, normalized size = 3.75 \begin{align*} - \frac{d}{a^{2} x} + \frac{e \log{\left (x \right )}}{a^{2}} - \frac{\left (a e - c d\right ) \log{\left (x + \frac{6 a^{4} e^{3} - 3 a^{3} e^{2} \left (a e - c d\right ) + 2 a^{2} c^{2} d^{2} e - 3 a^{2} e \left (a e - c d\right )^{2} + a c^{2} d^{2} \left (a e - c d\right )}{9 a^{2} c^{2} d e^{2} - c^{4} d^{3}} \right )}}{2 a^{3}} - \frac{\left (a e + c d\right ) \log{\left (x + \frac{6 a^{4} e^{3} - 3 a^{3} e^{2} \left (a e + c d\right ) + 2 a^{2} c^{2} d^{2} e - 3 a^{2} e \left (a e + c d\right )^{2} + a c^{2} d^{2} \left (a e + c d\right )}{9 a^{2} c^{2} d e^{2} - c^{4} d^{3}} \right )}}{2 a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14551, size = 100, normalized size = 1.69 \begin{align*} \frac{e \log \left ({\left | x \right |}\right )}{a^{2}} - \frac{d}{a^{2} x} + \frac{{\left (c^{2} d - a c e\right )} \log \left ({\left | c x + a \right |}\right )}{2 \, a^{3} c} - \frac{{\left (c^{2} d + a c e\right )} \log \left ({\left | c x - a \right |}\right )}{2 \, a^{3} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]