3.301 \(\int \frac {d+e x}{a^2-c^2 x^2} \, dx\)

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]

-1/2*(c*d/a+e)*ln(-c*x+a)/c^2+1/2*(c*d/a-e)*ln(c*x+a)/c^2

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Rubi [A]  time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {633, 31} \[ \frac {\left (\frac {c d}{a}-e\right ) \log (a+c x)}{2 c^2}-\frac {\left (\frac {c d}{a}+e\right ) \log (a-c x)}{2 c^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(a^2 - c^2*x^2),x]

[Out]

-(((c*d)/a + e)*Log[a - c*x])/(2*c^2) + (((c*d)/a - e)*Log[a + c*x])/(2*c^2)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 633

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q),
Int[1/(-q + c*x), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x), x], x]] /; FreeQ[{a, c, d, e}, x] && NiceS
qrtQ[-(a*c)]

Rubi steps

\begin {align*} \int \frac {d+e x}{a^2-c^2 x^2} \, dx &=\frac {1}{2} \left (-\frac {c d}{a}+e\right ) \int \frac {1}{-a c-c^2 x} \, dx+\frac {1}{2} \left (\frac {c d}{a}+e\right ) \int \frac {1}{a c-c^2 x} \, dx\\ &=-\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}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 37, normalized size = 0.80 \[ \frac {d \tanh ^{-1}\left (\frac {c x}{a}\right )}{a c}-\frac {e \log \left (a^2-c^2 x^2\right )}{2 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)/(a^2 - c^2*x^2),x]

[Out]

(d*ArcTanh[(c*x)/a])/(a*c) - (e*Log[a^2 - c^2*x^2])/(2*c^2)

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fricas [A]  time = 0.77, size = 41, normalized size = 0.89 \[ \frac {{\left (c d - a e\right )} \log \left (c x + a\right ) - {\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, a c^{2}} \]

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]

1/2*((c*d - a*e)*log(c*x + a) - (c*d + a*e)*log(c*x - a))/(a*c^2)

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giac [A]  time = 0.15, size = 50, normalized size = 1.09 \[ \frac {{\left (c d - a e\right )} \log \left ({\left | c x + a \right |}\right )}{2 \, a c^{2}} - \frac {{\left (c d + a e\right )} \log \left ({\left | c x - a \right |}\right )}{2 \, a c^{2}} \]

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]

1/2*(c*d - a*e)*log(abs(c*x + a))/(a*c^2) - 1/2*(c*d + a*e)*log(abs(c*x - a))/(a*c^2)

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maple [A]  time = 0.05, size = 60, normalized size = 1.30 \[ -\frac {d \ln \left (c x -a \right )}{2 a c}+\frac {d \ln \left (c x +a \right )}{2 a c}-\frac {e \ln \left (c x -a \right )}{2 c^{2}}-\frac {e \ln \left (c x +a \right )}{2 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/(-c^2*x^2+a^2),x)

[Out]

-1/2/c^2*ln(c*x+a)*e+1/2/a/c*ln(c*x+a)*d-1/2/c^2*ln(c*x-a)*e-1/2/a/c*ln(c*x-a)*d

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maxima [A]  time = 0.54, size = 46, normalized size = 1.00 \[ \frac {{\left (c d - a e\right )} \log \left (c x + a\right )}{2 \, a c^{2}} - \frac {{\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, a c^{2}} \]

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]

1/2*(c*d - a*e)*log(c*x + a)/(a*c^2) - 1/2*(c*d + a*e)*log(c*x - a)/(a*c^2)

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mupad [B]  time = 0.09, size = 45, normalized size = 0.98 \[ -\frac {\ln \left (a+c\,x\right )\,\left (a\,e-c\,d\right )}{2\,a\,c^2}-\frac {\ln \left (a-c\,x\right )\,\left (a\,e+c\,d\right )}{2\,a\,c^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(a^2 - c^2*x^2),x)

[Out]

- (log(a + c*x)*(a*e - c*d))/(2*a*c^2) - (log(a - c*x)*(a*e + c*d))/(2*a*c^2)

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sympy [A]  time = 0.33, 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]

-(a*e - c*d)*log(x + (a**2*e - a*(a*e - c*d))/(c**2*d))/(2*a*c**2) - (a*e + c*d)*log(x + (a**2*e - a*(a*e + c*
d))/(c**2*d))/(2*a*c**2)

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