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

Optimal. Leaf size=55 \[ \frac{d \tanh ^{-1}\left (\frac{c x}{a}\right )}{2 a^3 c}+\frac{a^2 e+c^2 d x}{2 a^2 c^2 \left (a^2-c^2 x^2\right )} \]

[Out]

(a^2*e + c^2*d*x)/(2*a^2*c^2*(a^2 - c^2*x^2)) + (d*ArcTanh[(c*x)/a])/(2*a^3*c)

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Rubi [A]  time = 0.0432374, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{d \tanh ^{-1}\left (\frac{c x}{a}\right )}{2 a^3 c}+\frac{a^2 e+c^2 d x}{2 a^2 c^2 \left (a^2-c^2 x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*e + c^2*d*x)/(2*a^2*c^2*(a^2 - c^2*x^2)) + (d*ArcTanh[(c*x)/a])/(2*a^3*c)

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Rubi in Sympy [A]  time = 11.7774, size = 44, normalized size = 0.8 \[ \frac{a^{2} e + c^{2} d x}{2 a^{2} c^{2} \left (a^{2} - c^{2} x^{2}\right )} + \frac{d \operatorname{atanh}{\left (\frac{c x}{a} \right )}}{2 a^{3} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)/(-c**2*x**2+a**2)**2,x)

[Out]

(a**2*e + c**2*d*x)/(2*a**2*c**2*(a**2 - c**2*x**2)) + d*atanh(c*x/a)/(2*a**3*c)

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Mathematica [A]  time = 0.0236813, size = 58, normalized size = 1.05 \[ \frac{d \tanh ^{-1}\left (\frac{c x}{a}\right )}{2 a^3 c}+\frac{a^2 (-e)-c^2 d x}{2 a^2 c^2 \left (c^2 x^2-a^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-(a^2*e) - c^2*d*x)/(2*a^2*c^2*(-a^2 + c^2*x^2)) + (d*ArcTanh[(c*x)/a])/(2*a^3*
c)

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Maple [A]  time = 0.016, size = 102, normalized size = 1.9 \[{\frac{d\ln \left ( cx+a \right ) }{4\,{a}^{3}c}}+{\frac{e}{4\,a{c}^{2} \left ( cx+a \right ) }}-{\frac{d}{4\,{a}^{2}c \left ( cx+a \right ) }}-{\frac{d\ln \left ( cx-a \right ) }{4\,{a}^{3}c}}-{\frac{e}{4\,a{c}^{2} \left ( cx-a \right ) }}-{\frac{d}{4\,{a}^{2}c \left ( cx-a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(-c^2*x^2+a^2)^2,x)

[Out]

1/4*d/a^3/c*ln(c*x+a)+1/4/a/c^2*e/(c*x+a)-1/4/a^2/c/(c*x+a)*d-1/4*d/a^3/c*ln(c*x
-a)-1/4/a/c^2*e/(c*x-a)-1/4/a^2/c/(c*x-a)*d

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Maxima [A]  time = 0.687744, size = 92, normalized size = 1.67 \[ -\frac{c^{2} d x + a^{2} e}{2 \,{\left (a^{2} c^{4} x^{2} - a^{4} c^{2}\right )}} + \frac{d \log \left (c x + a\right )}{4 \, a^{3} c} - \frac{d \log \left (c x - a\right )}{4 \, a^{3} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(c^2*x^2 - a^2)^2,x, algorithm="maxima")

[Out]

-1/2*(c^2*d*x + a^2*e)/(a^2*c^4*x^2 - a^4*c^2) + 1/4*d*log(c*x + a)/(a^3*c) - 1/
4*d*log(c*x - a)/(a^3*c)

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Fricas [A]  time = 0.290459, size = 117, normalized size = 2.13 \[ -\frac{2 \, a c^{2} d x + 2 \, a^{3} e -{\left (c^{3} d x^{2} - a^{2} c d\right )} \log \left (c x + a\right ) +{\left (c^{3} d x^{2} - a^{2} c d\right )} \log \left (c x - a\right )}{4 \,{\left (a^{3} c^{4} x^{2} - a^{5} c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(c^2*x^2 - a^2)^2,x, algorithm="fricas")

[Out]

-1/4*(2*a*c^2*d*x + 2*a^3*e - (c^3*d*x^2 - a^2*c*d)*log(c*x + a) + (c^3*d*x^2 -
a^2*c*d)*log(c*x - a))/(a^3*c^4*x^2 - a^5*c^2)

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Sympy [A]  time = 1.7989, size = 54, normalized size = 0.98 \[ - \frac{a^{2} e + c^{2} d x}{- 2 a^{4} c^{2} + 2 a^{2} c^{4} x^{2}} + \frac{d \left (- \frac{\log{\left (- \frac{a}{c} + x \right )}}{4} + \frac{\log{\left (\frac{a}{c} + x \right )}}{4}\right )}{a^{3} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)/(-c**2*x**2+a**2)**2,x)

[Out]

-(a**2*e + c**2*d*x)/(-2*a**4*c**2 + 2*a**2*c**4*x**2) + d*(-log(-a/c + x)/4 + l
og(a/c + x)/4)/(a**3*c)

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GIAC/XCAS [A]  time = 0.272085, size = 96, normalized size = 1.75 \[ \frac{d{\rm ln}\left ({\left | c x + a \right |}\right )}{4 \, a^{3} c} - \frac{d{\rm ln}\left ({\left | c x - a \right |}\right )}{4 \, a^{3} c} - \frac{c^{2} d x + a^{2} e}{2 \,{\left (c^{2} x^{2} - a^{2}\right )} a^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(c^2*x^2 - a^2)^2,x, algorithm="giac")

[Out]

1/4*d*ln(abs(c*x + a))/(a^3*c) - 1/4*d*ln(abs(c*x - a))/(a^3*c) - 1/2*(c^2*d*x +
a^2*e)/((c^2*x^2 - a^2)*a^2*c^2)