3.6.26 \(\int \frac {e-4 f x^3}{e^2+4 d f x^2+4 e f x^3+4 f^2 x^6} \, dx\) [526]

Optimal. Leaf size=38 \[ \frac {\tan ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x}{e+2 f x^3}\right )}{2 \sqrt {d} \sqrt {f}} \]

[Out]

1/2*arctan(2*x*d^(1/2)*f^(1/2)/(2*f*x^3+e))/d^(1/2)/f^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2118, 211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {2 \sqrt {d} \sqrt {f} x}{e+2 f x^3}\right )}{2 \sqrt {d} \sqrt {f}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e - 4*f*x^3)/(e^2 + 4*d*f*x^2 + 4*e*f*x^3 + 4*f^2*x^6),x]

[Out]

ArcTan[(2*Sqrt[d]*Sqrt[f]*x)/(e + 2*f*x^3)]/(2*Sqrt[d]*Sqrt[f])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2118

Int[((A_) + (B_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^(n_) + (d_.)*(x_)^(n2_)), x_Symbol] :> Dist[A^2
*(n - 1), Subst[Int[1/(a + A^2*b*(n - 1)^2*x^2), x], x, x/(A*(n - 1) - B*x^n)], x] /; FreeQ[{a, b, c, d, A, B,
n}, x] && EqQ[n2, 2*n] && NeQ[n, 2] && EqQ[a*B^2 - A^2*d*(n - 1)^2, 0] && EqQ[B*c + 2*A*d*(n - 1), 0]

Rubi steps

\begin {align*} \int \frac {e-4 f x^3}{e^2+4 d f x^2+4 e f x^3+4 f^2 x^6} \, dx &=\left (2 e^2\right ) \text {Subst}\left (\int \frac {1}{e^2+16 d e^2 f x^2} \, dx,x,\frac {x}{2 e+4 f x^3}\right )\\ &=\frac {\tan ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x}{e+2 f x^3}\right )}{2 \sqrt {d} \sqrt {f}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.04, size = 87, normalized size = 2.29 \begin {gather*} -\frac {\text {RootSum}\left [e^2+4 d f \text {$\#$1}^2+4 e f \text {$\#$1}^3+4 f^2 \text {$\#$1}^6\&,\frac {-e \log (x-\text {$\#$1})+4 f \log (x-\text {$\#$1}) \text {$\#$1}^3}{2 d \text {$\#$1}+3 e \text {$\#$1}^2+6 f \text {$\#$1}^5}\&\right ]}{4 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(e - 4*f*x^3)/(e^2 + 4*d*f*x^2 + 4*e*f*x^3 + 4*f^2*x^6),x]

[Out]

-1/4*RootSum[e^2 + 4*d*f*#1^2 + 4*e*f*#1^3 + 4*f^2*#1^6 & , (-(e*Log[x - #1]) + 4*f*Log[x - #1]*#1^3)/(2*d*#1
+ 3*e*#1^2 + 6*f*#1^5) & ]/f

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.04, size = 70, normalized size = 1.84

method	result	size
default	\(\frac {\munderset {\textit {\_R} =\RootOf \left (4 f^{2} \textit {\_Z}^{6}+4 e f \,\textit {\_Z}^{3}+4 d f \,\textit {\_Z}^{2}+e^{2}\right )}{\sum }\frac {\left (-4 \textit {\_R}^{3} f +e \right ) \ln \left (x -\textit {\_R} \right )}{6 f \,\textit {\_R}^{5}+3 e \,\textit {\_R}^{2}+2 d \textit {\_R}}}{4 f}\)	\(70\)
risch	\(-\frac {\ln \left (\left (32 f \left (-d f \right )^{\frac {3}{2}} d +54 e^{2} f^{2} d \right ) x^{3}+\left (54 e^{2} \left (-d f \right )^{\frac {3}{2}}-32 d^{3} f^{2}\right ) x +16 e \left (-d f \right )^{\frac {3}{2}} d +27 f \,e^{3} d \right )}{4 \sqrt {-d f}}+\frac {\ln \left (\left (32 f \left (-d f \right )^{\frac {3}{2}} d -54 e^{2} f^{2} d \right ) x^{3}+\left (54 e^{2} \left (-d f \right )^{\frac {3}{2}}+32 d^{3} f^{2}\right ) x +16 e \left (-d f \right )^{\frac {3}{2}} d -27 f \,e^{3} d \right )}{4 \sqrt {-d f}}\)	\(148\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*f*x^3+e)/(4*f^2*x^6+4*e*f*x^3+4*d*f*x^2+e^2),x,method=_RETURNVERBOSE)

[Out]

1/4/f*sum((-4*_R^3*f+e)/(6*_R^5*f+3*_R^2*e+2*_R*d)*ln(x-_R),_R=RootOf(4*_Z^6*f^2+4*_Z^3*e*f+4*_Z^2*d*f+e^2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*f*x^3+e)/(4*f^2*x^6+4*e*f*x^3+4*d*f*x^2+e^2),x, algorithm="maxima")

[Out]

-integrate((4*f*x^3 - e)/(4*f^2*x^6 + 4*f*x^3*e + 4*d*f*x^2 + e^2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (29) = 58\).
time = 0.43, size = 154, normalized size = 4.05 \begin {gather*} \left [-\frac {\sqrt {-d f} \log \left (\frac {4 \, f^{2} x^{6} + 4 \, f x^{3} e - 4 \, d f x^{2} + 4 \, {\left (2 \, f x^{4} + x e\right )} \sqrt {-d f} + e^{2}}{4 \, f^{2} x^{6} + 4 \, f x^{3} e + 4 \, d f x^{2} + e^{2}}\right )}{4 \, d f}, -\frac {\sqrt {d f} \arctan \left (\frac {\sqrt {d f} x^{2}}{d}\right ) - \sqrt {d f} \arctan \left (\frac {{\left (2 \, f x^{5} + x^{2} e + 2 \, d x\right )} \sqrt {d f} e^{\left (-1\right )}}{d}\right )}{2 \, d f}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*f*x^3+e)/(4*f^2*x^6+4*e*f*x^3+4*d*f*x^2+e^2),x, algorithm="fricas")

[Out]

[-1/4*sqrt(-d*f)*log((4*f^2*x^6 + 4*f*x^3*e - 4*d*f*x^2 + 4*(2*f*x^4 + x*e)*sqrt(-d*f) + e^2)/(4*f^2*x^6 + 4*f
*x^3*e + 4*d*f*x^2 + e^2))/(d*f), -1/2*(sqrt(d*f)*arctan(sqrt(d*f)*x^2/d) - sqrt(d*f)*arctan((2*f*x^5 + x^2*e
+ 2*d*x)*sqrt(d*f)*e^(-1)/d))/(d*f)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (34) = 68\).
time = 0.37, size = 70, normalized size = 1.84 \begin {gather*} \frac {\sqrt {- \frac {1}{d f}} \log {\left (- d x \sqrt {- \frac {1}{d f}} + \frac {e}{2 f} + x^{3} \right )}}{4} - \frac {\sqrt {- \frac {1}{d f}} \log {\left (d x \sqrt {- \frac {1}{d f}} + \frac {e}{2 f} + x^{3} \right )}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*f*x**3+e)/(4*f**2*x**6+4*e*f*x**3+4*d*f*x**2+e**2),x)

[Out]

sqrt(-1/(d*f))*log(-d*x*sqrt(-1/(d*f)) + e/(2*f) + x**3)/4 - sqrt(-1/(d*f))*log(d*x*sqrt(-1/(d*f)) + e/(2*f) +
x**3)/4

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (29) = 58\).
time = 4.31, size = 69, normalized size = 1.82 \begin {gather*} -\frac {\sqrt {-d f} \log \left ({\left | 2 \, f x^{3} + 2 \, \sqrt {-d f} x + e \right |}\right )}{4 \, d f} + \frac {\sqrt {-d f} \log \left ({\left | 2 \, f x^{3} - 2 \, \sqrt {-d f} x + e \right |}\right )}{4 \, d f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*f*x^3+e)/(4*f^2*x^6+4*e*f*x^3+4*d*f*x^2+e^2),x, algorithm="giac")

[Out]

-1/4*sqrt(-d*f)*log(abs(2*f*x^3 + 2*sqrt(-d*f)*x + e))/(d*f) + 1/4*sqrt(-d*f)*log(abs(2*f*x^3 - 2*sqrt(-d*f)*x
+ e))/(d*f)

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Mupad [B]
time = 3.32, size = 54, normalized size = 1.42 \begin {gather*} \frac {\mathrm {atan}\left (\frac {2\,f^{3/2}\,x^5+2\,d\,\sqrt {f}\,x+e\,\sqrt {f}\,x^2}{\sqrt {d}\,e}\right )-\mathrm {atan}\left (\frac {\sqrt {f}\,x^2}{\sqrt {d}}\right )}{2\,\sqrt {d}\,\sqrt {f}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e - 4*f*x^3)/(e^2 + 4*f^2*x^6 + 4*d*f*x^2 + 4*e*f*x^3),x)

[Out]

(atan((2*f^(3/2)*x^5 + 2*d*f^(1/2)*x + e*f^(1/2)*x^2)/(d^(1/2)*e)) - atan((f^(1/2)*x^2)/d^(1/2)))/(2*d^(1/2)*f
^(1/2))

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