3.234 \(\int \frac{1}{x \left (d+e x^2\right ) \left (a+c x^4\right )} \, dx\)

Optimal. Leaf size=114 \[ -\frac{c d \log \left (a+c x^4\right )}{4 a \left (a e^2+c d^2\right )}-\frac{e^2 \log \left (d+e x^2\right )}{2 d \left (a e^2+c d^2\right )}-\frac{\sqrt{c} e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \left (a e^2+c d^2\right )}+\frac{\log (x)}{a d} \]

[Out]

-(Sqrt[c]*e*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*Sqrt[a]*(c*d^2 + a*e^2)) + Log[x]/
(a*d) - (e^2*Log[d + e*x^2])/(2*d*(c*d^2 + a*e^2)) - (c*d*Log[a + c*x^4])/(4*a*(
c*d^2 + a*e^2))

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Rubi [A]  time = 0.263241, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{c d \log \left (a+c x^4\right )}{4 a \left (a e^2+c d^2\right )}-\frac{e^2 \log \left (d+e x^2\right )}{2 d \left (a e^2+c d^2\right )}-\frac{\sqrt{c} e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \left (a e^2+c d^2\right )}+\frac{\log (x)}{a d} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(d + e*x^2)*(a + c*x^4)),x]

[Out]

-(Sqrt[c]*e*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*Sqrt[a]*(c*d^2 + a*e^2)) + Log[x]/
(a*d) - (e^2*Log[d + e*x^2])/(2*d*(c*d^2 + a*e^2)) - (c*d*Log[a + c*x^4])/(4*a*(
c*d^2 + a*e^2))

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Rubi in Sympy [A]  time = 43.9538, size = 99, normalized size = 0.87 \[ - \frac{e^{2} \log{\left (d + e x^{2} \right )}}{2 d \left (a e^{2} + c d^{2}\right )} - \frac{c d \log{\left (a + c x^{4} \right )}}{4 a \left (a e^{2} + c d^{2}\right )} + \frac{\log{\left (x^{2} \right )}}{2 a d} - \frac{\sqrt{c} e \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{a} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(e*x**2+d)/(c*x**4+a),x)

[Out]

-e**2*log(d + e*x**2)/(2*d*(a*e**2 + c*d**2)) - c*d*log(a + c*x**4)/(4*a*(a*e**2
+ c*d**2)) + log(x**2)/(2*a*d) - sqrt(c)*e*atan(sqrt(c)*x**2/sqrt(a))/(2*sqrt(a
)*(a*e**2 + c*d**2))

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Mathematica [A]  time = 0.118285, size = 134, normalized size = 1.18 \[ \frac{-c d^2 \log \left (a+c x^4\right )+2 \sqrt{a} \sqrt{c} d e \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \sqrt{a} \sqrt{c} d e \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )-2 a e^2 \log \left (d+e x^2\right )+4 a e^2 \log (x)+4 c d^2 \log (x)}{4 a^2 d e^2+4 a c d^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(d + e*x^2)*(a + c*x^4)),x]

[Out]

(2*Sqrt[a]*Sqrt[c]*d*e*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 2*Sqrt[a]*Sqrt[
c]*d*e*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 4*c*d^2*Log[x] + 4*a*e^2*Log[x]
- 2*a*e^2*Log[d + e*x^2] - c*d^2*Log[a + c*x^4])/(4*a*c*d^3 + 4*a^2*d*e^2)

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Maple [A]  time = 0.014, size = 101, normalized size = 0.9 \[{\frac{\ln \left ( x \right ) }{ad}}-{\frac{cd\ln \left ( c{x}^{4}+a \right ) }{4\, \left ( a{e}^{2}+c{d}^{2} \right ) a}}-{\frac{ce}{2\,a{e}^{2}+2\,c{d}^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{{e}^{2}\ln \left ( e{x}^{2}+d \right ) }{2\,d \left ( a{e}^{2}+c{d}^{2} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(e*x^2+d)/(c*x^4+a),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)*(e*x^2 + d)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 7.64565, size = 1, normalized size = 0.01 \[ \left [\frac{a d e \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{4} - 2 \, a x^{2} \sqrt{-\frac{c}{a}} - a}{c x^{4} + a}\right ) - c d^{2} \log \left (c x^{4} + a\right ) - 2 \, a e^{2} \log \left (e x^{2} + d\right ) + 4 \,{\left (c d^{2} + a e^{2}\right )} \log \left (x\right )}{4 \,{\left (a c d^{3} + a^{2} d e^{2}\right )}}, \frac{2 \, a d e \sqrt{\frac{c}{a}} \arctan \left (\frac{a \sqrt{\frac{c}{a}}}{c x^{2}}\right ) - c d^{2} \log \left (c x^{4} + a\right ) - 2 \, a e^{2} \log \left (e x^{2} + d\right ) + 4 \,{\left (c d^{2} + a e^{2}\right )} \log \left (x\right )}{4 \,{\left (a c d^{3} + a^{2} d e^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)*(e*x^2 + d)*x),x, algorithm="fricas")

[Out]

[1/4*(a*d*e*sqrt(-c/a)*log((c*x^4 - 2*a*x^2*sqrt(-c/a) - a)/(c*x^4 + a)) - c*d^2
*log(c*x^4 + a) - 2*a*e^2*log(e*x^2 + d) + 4*(c*d^2 + a*e^2)*log(x))/(a*c*d^3 +
a^2*d*e^2), 1/4*(2*a*d*e*sqrt(c/a)*arctan(a*sqrt(c/a)/(c*x^2)) - c*d^2*log(c*x^4
+ a) - 2*a*e^2*log(e*x^2 + d) + 4*(c*d^2 + a*e^2)*log(x))/(a*c*d^3 + a^2*d*e^2)
]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(e*x**2+d)/(c*x**4+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.274363, size = 138, normalized size = 1.21 \[ -\frac{c d{\rm ln}\left (c x^{4} + a\right )}{4 \,{\left (a c d^{2} + a^{2} e^{2}\right )}} - \frac{c \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right ) e}{2 \,{\left (c d^{2} + a e^{2}\right )} \sqrt{a c}} - \frac{e^{3}{\rm ln}\left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c d^{3} e + a d e^{3}\right )}} + \frac{{\rm ln}\left (x^{2}\right )}{2 \, a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)*(e*x^2 + d)*x),x, algorithm="giac")

[Out]

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