Optimal. Leaf size=45 \[ \frac {d+e x}{2 c^2 \left (a^2-c^2 x^2\right )}-\frac {e \tanh ^{-1}\left (\frac {c x}{a}\right )}{2 a c^3} \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {778, 208} \begin {gather*} \frac {d+e x}{2 c^2 \left (a^2-c^2 x^2\right )}-\frac {e \tanh ^{-1}\left (\frac {c x}{a}\right )}{2 a c^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 208
Rule 778
Rubi steps
\begin {align*} \int \frac {x (d+e x)}{\left (a^2-c^2 x^2\right )^2} \, dx &=\frac {d+e x}{2 c^2 \left (a^2-c^2 x^2\right )}-\frac {e \int \frac {1}{a^2-c^2 x^2} \, dx}{2 c^2}\\ &=\frac {d+e x}{2 c^2 \left (a^2-c^2 x^2\right )}-\frac {e \tanh ^{-1}\left (\frac {c x}{a}\right )}{2 a c^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 42, normalized size = 0.93 \begin {gather*} \frac {\frac {c (d+e x)}{a^2-c^2 x^2}-\frac {e \tanh ^{-1}\left (\frac {c x}{a}\right )}{a}}{2 c^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x (d+e x)}{\left (a^2-c^2 x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 80, normalized size = 1.78 \begin {gather*} -\frac {2 \, a c e x + 2 \, a c d + {\left (c^{2} e x^{2} - a^{2} e\right )} \log \left (c x + a\right ) - {\left (c^{2} e x^{2} - a^{2} e\right )} \log \left (c x - a\right )}{4 \, {\left (a c^{5} x^{2} - a^{3} c^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 63, normalized size = 1.40 \begin {gather*} -\frac {x e + d}{2 \, {\left (c^{2} x^{2} - a^{2}\right )} c^{2}} - \frac {e \log \left ({\left | c x + a \right |}\right )}{4 \, a c^{3}} + \frac {e \log \left ({\left | c x - a \right |}\right )}{4 \, a c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.05, size = 96, normalized size = 2.13 \begin {gather*} \frac {d}{4 \left (c x +a \right ) a \,c^{2}}-\frac {d}{4 \left (c x -a \right ) a \,c^{2}}+\frac {e \ln \left (c x -a \right )}{4 a \,c^{3}}-\frac {e \ln \left (c x +a \right )}{4 a \,c^{3}}-\frac {e}{4 \left (c x +a \right ) c^{3}}-\frac {e}{4 \left (c x -a \right ) c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.51, size = 58, normalized size = 1.29 \begin {gather*} -\frac {e x + d}{2 \, {\left (c^{4} x^{2} - a^{2} c^{2}\right )}} - \frac {e \log \left (c x + a\right )}{4 \, a c^{3}} + \frac {e \log \left (c x - a\right )}{4 \, a c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.06, size = 46, normalized size = 1.02 \begin {gather*} \frac {\frac {d}{2\,c^2}+\frac {e\,x}{2\,c^2}}{a^2-c^2\,x^2}-\frac {e\,\mathrm {atanh}\left (\frac {c\,x}{a}\right )}{2\,a\,c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.36, size = 46, normalized size = 1.02 \begin {gather*} \frac {- d - e x}{- 2 a^{2} c^{2} + 2 c^{4} x^{2}} + \frac {e \left (\frac {\log {\left (- \frac {a}{c} + x \right )}}{4} - \frac {\log {\left (\frac {a}{c} + x \right )}}{4}\right )}{a c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________