3.1.43 \(\int \frac {1}{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx\) [43]

Optimal. Leaf size=153 \[ \frac {2 \tanh ^{-1}\left (\frac {d+4 e x}{\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}\right )}{\sqrt {d^4-64 a e^3} \sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}-\frac {2 \tanh ^{-1}\left (\frac {d+4 e x}{\sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}}\right )}{\sqrt {d^4-64 a e^3} \sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}} \]

[Out]

2*arctanh((4*e*x+d)/(3*d^2-2*(-64*a*e^3+d^4)^(1/2))^(1/2))/(-64*a*e^3+d^4)^(1/2)/(3*d^2-2*(-64*a*e^3+d^4)^(1/2
))^(1/2)-2*arctanh((4*e*x+d)/(3*d^2+2*(-64*a*e^3+d^4)^(1/2))^(1/2))/(-64*a*e^3+d^4)^(1/2)/(3*d^2+2*(-64*a*e^3+
d^4)^(1/2))^(1/2)

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Rubi [A]
time = 0.20, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {1120, 1107, 214} \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {d+4 e x}{\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}\right )}{\sqrt {d^4-64 a e^3} \sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}-\frac {2 \tanh ^{-1}\left (\frac {d+4 e x}{\sqrt {2 \sqrt {d^4-64 a e^3}+3 d^2}}\right )}{\sqrt {d^4-64 a e^3} \sqrt {2 \sqrt {d^4-64 a e^3}+3 d^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^(-1),x]

[Out]

(2*ArcTanh[(d + 4*e*x)/Sqrt[3*d^2 - 2*Sqrt[d^4 - 64*a*e^3]]])/(Sqrt[d^4 - 64*a*e^3]*Sqrt[3*d^2 - 2*Sqrt[d^4 -
64*a*e^3]]) - (2*ArcTanh[(d + 4*e*x)/Sqrt[3*d^2 + 2*Sqrt[d^4 - 64*a*e^3]]])/(Sqrt[d^4 - 64*a*e^3]*Sqrt[3*d^2 +
2*Sqrt[d^4 - 64*a*e^3]])

Rule 214

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

Rule 1107

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/
2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && PosQ[b^2 - 4*a*c]

Rule 1120

Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4
, x, 3], e = Coeff[P4, x, 4]}, Subst[Int[SimplifyIntegrand[(a + d^4/(256*e^3) - b*(d/(8*e)) + (c - 3*(d^2/(8*e
)))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2, 0] && NeQ[d, 0]] /; FreeQ[p, x] &&
PolyQ[P4, x, 4] && NeQ[p, 2] && NeQ[p, 3]

Rubi steps

\begin {align*} \int \frac {1}{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx &=\text {Subst}\left (\int \frac {1}{\frac {1}{32} \left (\frac {5 d^4}{e}+256 a e^2\right )-3 d^2 e x^2+8 e^3 x^4} \, dx,x,\frac {d}{4 e}+x\right )\\ &=\frac {\left (4 e^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {3 d^2 e}{2}-e \sqrt {d^4-64 a e^3}+8 e^3 x^2} \, dx,x,\frac {d}{4 e}+x\right )}{\sqrt {d^4-64 a e^3}}-\frac {\left (4 e^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {3 d^2 e}{2}+e \sqrt {d^4-64 a e^3}+8 e^3 x^2} \, dx,x,\frac {d}{4 e}+x\right )}{\sqrt {d^4-64 a e^3}}\\ &=\frac {2 \tanh ^{-1}\left (\frac {d+4 e x}{\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}\right )}{\sqrt {d^4-64 a e^3} \sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}}-\frac {2 \tanh ^{-1}\left (\frac {d+4 e x}{\sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}}\right )}{\sqrt {d^4-64 a e^3} \sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.02, size = 71, normalized size = 0.46 \begin {gather*} -\text {RootSum}\left [8 a e^2-d^3 \text {$\#$1}+8 d e^2 \text {$\#$1}^3+8 e^3 \text {$\#$1}^4\&,\frac {\log (x-\text {$\#$1})}{d^3-24 d e^2 \text {$\#$1}^2-32 e^3 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^(-1),x]

[Out]

-RootSum[8*a*e^2 - d^3*#1 + 8*d*e^2*#1^3 + 8*e^3*#1^4 & , Log[x - #1]/(d^3 - 24*d*e^2*#1^2 - 32*e^3*#1^3) & ]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.08, size = 67, normalized size = 0.44

method	result	size
default	\(\munderset {\textit {\_R} =\RootOf \left (8 e^{3} \textit {\_Z}^{4}+8 d \,e^{2} \textit {\_Z}^{3}-d^{3} \textit {\_Z} +8 a \,e^{2}\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3} e^{3}+24 \textit {\_R}^{2} d \,e^{2}-d^{3}}\)	\(67\)
risch	\(\munderset {\textit {\_R} =\RootOf \left (8 e^{3} \textit {\_Z}^{4}+8 d \,e^{2} \textit {\_Z}^{3}-d^{3} \textit {\_Z} +8 a \,e^{2}\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3} e^{3}+24 \textit {\_R}^{2} d \,e^{2}-d^{3}}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2),x,method=_RETURNVERBOSE)

[Out]

sum(1/(32*_R^3*e^3+24*_R^2*d*e^2-d^3)*ln(x-_R),_R=RootOf(8*_Z^4*e^3+8*_Z^3*d*e^2-_Z*d^3+8*a*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(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2),x, algorithm="maxima")

[Out]

integrate(1/(8*x^4*e^3 + 8*d*x^3*e^2 - d^3*x + 8*a*e^2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1115 vs. \(2 (133) = 266\).
time = 0.41, size = 1115, normalized size = 7.29 \begin {gather*} -\sqrt {\frac {3 \, d^{2} + \frac {2 \, {\left (5 \, d^{8} - 64 \, a d^{4} e^{3} - 16384 \, a^{2} e^{6}\right )}}{\sqrt {25 \, d^{12} + 960 \, a d^{8} e^{3} - 98304 \, a^{2} d^{4} e^{6} - 4194304 \, a^{3} e^{9}}}}{5 \, d^{8} - 64 \, a d^{4} e^{3} - 16384 \, a^{2} e^{6}}} \log \left (8 \, e x + 2 \, {\left (2 \, d^{4} - 128 \, a e^{3} - \frac {3 \, {\left (5 \, d^{10} - 64 \, a d^{6} e^{3} - 16384 \, a^{2} d^{2} e^{6}\right )}}{\sqrt {25 \, d^{12} + 960 \, a d^{8} e^{3} - 98304 \, a^{2} d^{4} e^{6} - 4194304 \, a^{3} e^{9}}}\right )} \sqrt {\frac {3 \, d^{2} + \frac {2 \, {\left (5 \, d^{8} - 64 \, a d^{4} e^{3} - 16384 \, a^{2} e^{6}\right )}}{\sqrt {25 \, d^{12} + 960 \, a d^{8} e^{3} - 98304 \, a^{2} d^{4} e^{6} - 4194304 \, a^{3} e^{9}}}}{5 \, d^{8} - 64 \, a d^{4} e^{3} - 16384 \, a^{2} e^{6}}} + 2 \, d\right ) + \sqrt {\frac {3 \, d^{2} + \frac {2 \, {\left (5 \, d^{8} - 64 \, a d^{4} e^{3} - 16384 \, a^{2} e^{6}\right )}}{\sqrt {25 \, d^{12} + 960 \, a d^{8} e^{3} - 98304 \, a^{2} d^{4} e^{6} - 4194304 \, a^{3} e^{9}}}}{5 \, d^{8} - 64 \, a d^{4} e^{3} - 16384 \, a^{2} e^{6}}} \log \left (8 \, e x - 2 \, {\left (2 \, d^{4} - 128 \, a e^{3} - \frac {3 \, {\left (5 \, d^{10} - 64 \, a d^{6} e^{3} - 16384 \, a^{2} d^{2} e^{6}\right )}}{\sqrt {25 \, d^{12} + 960 \, a d^{8} e^{3} - 98304 \, a^{2} d^{4} e^{6} - 4194304 \, a^{3} e^{9}}}\right )} \sqrt {\frac {3 \, d^{2} + \frac {2 \, {\left (5 \, d^{8} - 64 \, a d^{4} e^{3} - 16384 \, a^{2} e^{6}\right )}}{\sqrt {25 \, d^{12} + 960 \, a d^{8} e^{3} - 98304 \, a^{2} d^{4} e^{6} - 4194304 \, a^{3} e^{9}}}}{5 \, d^{8} - 64 \, a d^{4} e^{3} - 16384 \, a^{2} e^{6}}} + 2 \, d\right ) - \sqrt {\frac {3 \, d^{2} - \frac {2 \, {\left (5 \, d^{8} - 64 \, a d^{4} e^{3} - 16384 \, a^{2} e^{6}\right )}}{\sqrt {25 \, d^{12} + 960 \, a d^{8} e^{3} - 98304 \, a^{2} d^{4} e^{6} - 4194304 \, a^{3} e^{9}}}}{5 \, d^{8} - 64 \, a d^{4} e^{3} - 16384 \, a^{2} e^{6}}} \log \left (8 \, e x + 2 \, {\left (2 \, d^{4} - 128 \, a e^{3} + \frac {3 \, {\left (5 \, d^{10} - 64 \, a d^{6} e^{3} - 16384 \, a^{2} d^{2} e^{6}\right )}}{\sqrt {25 \, d^{12} + 960 \, a d^{8} e^{3} - 98304 \, a^{2} d^{4} e^{6} - 4194304 \, a^{3} e^{9}}}\right )} \sqrt {\frac {3 \, d^{2} - \frac {2 \, {\left (5 \, d^{8} - 64 \, a d^{4} e^{3} - 16384 \, a^{2} e^{6}\right )}}{\sqrt {25 \, d^{12} + 960 \, a d^{8} e^{3} - 98304 \, a^{2} d^{4} e^{6} - 4194304 \, a^{3} e^{9}}}}{5 \, d^{8} - 64 \, a d^{4} e^{3} - 16384 \, a^{2} e^{6}}} + 2 \, d\right ) + \sqrt {\frac {3 \, d^{2} - \frac {2 \, {\left (5 \, d^{8} - 64 \, a d^{4} e^{3} - 16384 \, a^{2} e^{6}\right )}}{\sqrt {25 \, d^{12} + 960 \, a d^{8} e^{3} - 98304 \, a^{2} d^{4} e^{6} - 4194304 \, a^{3} e^{9}}}}{5 \, d^{8} - 64 \, a d^{4} e^{3} - 16384 \, a^{2} e^{6}}} \log \left (8 \, e x - 2 \, {\left (2 \, d^{4} - 128 \, a e^{3} + \frac {3 \, {\left (5 \, d^{10} - 64 \, a d^{6} e^{3} - 16384 \, a^{2} d^{2} e^{6}\right )}}{\sqrt {25 \, d^{12} + 960 \, a d^{8} e^{3} - 98304 \, a^{2} d^{4} e^{6} - 4194304 \, a^{3} e^{9}}}\right )} \sqrt {\frac {3 \, d^{2} - \frac {2 \, {\left (5 \, d^{8} - 64 \, a d^{4} e^{3} - 16384 \, a^{2} e^{6}\right )}}{\sqrt {25 \, d^{12} + 960 \, a d^{8} e^{3} - 98304 \, a^{2} d^{4} e^{6} - 4194304 \, a^{3} e^{9}}}}{5 \, d^{8} - 64 \, a d^{4} e^{3} - 16384 \, a^{2} e^{6}}} + 2 \, d\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2),x, algorithm="fricas")

[Out]

-sqrt((3*d^2 + 2*(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)/sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*a^2*d^4*e^6 - 419
4304*a^3*e^9))/(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6))*log(8*e*x + 2*(2*d^4 - 128*a*e^3 - 3*(5*d^10 - 64*a*d^6
*e^3 - 16384*a^2*d^2*e^6)/sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3*e^9))*sqrt((3*d^2 + 2
*(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)/sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3*e^9))/(
5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)) + 2*d) + sqrt((3*d^2 + 2*(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)/sqrt(25
*d^12 + 960*a*d^8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3*e^9))/(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6))*log(8*e*
x - 2*(2*d^4 - 128*a*e^3 - 3*(5*d^10 - 64*a*d^6*e^3 - 16384*a^2*d^2*e^6)/sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*
a^2*d^4*e^6 - 4194304*a^3*e^9))*sqrt((3*d^2 + 2*(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)/sqrt(25*d^12 + 960*a*d^
8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3*e^9))/(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)) + 2*d) - sqrt((3*d^2 - 2
*(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)/sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3*e^9))/(
5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6))*log(8*e*x + 2*(2*d^4 - 128*a*e^3 + 3*(5*d^10 - 64*a*d^6*e^3 - 16384*a^2
*d^2*e^6)/sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3*e^9))*sqrt((3*d^2 - 2*(5*d^8 - 64*a*d
^4*e^3 - 16384*a^2*e^6)/sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3*e^9))/(5*d^8 - 64*a*d^4
*e^3 - 16384*a^2*e^6)) + 2*d) + sqrt((3*d^2 - 2*(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)/sqrt(25*d^12 + 960*a*d^
8*e^3 - 98304*a^2*d^4*e^6 - 4194304*a^3*e^9))/(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6))*log(8*e*x - 2*(2*d^4 - 1
28*a*e^3 + 3*(5*d^10 - 64*a*d^6*e^3 - 16384*a^2*d^2*e^6)/sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*a^2*d^4*e^6 - 41
94304*a^3*e^9))*sqrt((3*d^2 - 2*(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)/sqrt(25*d^12 + 960*a*d^8*e^3 - 98304*a^
2*d^4*e^6 - 4194304*a^3*e^9))/(5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)) + 2*d)

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Sympy [A]
time = 0.99, size = 122, normalized size = 0.80 \begin {gather*} \operatorname {RootSum} {\left (t^{4} \cdot \left (1048576 a^{3} e^{9} - 12288 a^{2} d^{4} e^{6} - 384 a d^{8} e^{3} + 5 d^{12}\right ) + t^{2} \cdot \left (384 a d^{2} e^{3} - 6 d^{6}\right ) + 1, \left ( t \mapsto t \log {\left (x + \frac {- 49152 t^{3} a^{2} d^{2} e^{6} - 192 t^{3} a d^{6} e^{3} + 15 t^{3} d^{10} + 256 t a e^{3} - 13 t d^{4} + 2 d}{8 e} \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(8*e**3*x**4+8*d*e**2*x**3-d**3*x+8*a*e**2),x)

[Out]

RootSum(_t**4*(1048576*a**3*e**9 - 12288*a**2*d**4*e**6 - 384*a*d**8*e**3 + 5*d**12) + _t**2*(384*a*d**2*e**3
- 6*d**6) + 1, Lambda(_t, _t*log(x + (-49152*_t**3*a**2*d**2*e**6 - 192*_t**3*a*d**6*e**3 + 15*_t**3*d**10 + 2
56*_t*a*e**3 - 13*_t*d**4 + 2*d)/(8*e))))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 505 vs. \(2 (129) = 258\).
time = 4.60, size = 505, normalized size = 3.30 \begin {gather*} -\frac {2 \, \log \left (\frac {1}{4} \, d e^{\left (-1\right )} + \frac {1}{4} \, \sqrt {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )} + x\right )}{{\left (d e^{\left (-1\right )} + \sqrt {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )}\right )}^{3} e^{3} - 3 \, {\left (d e^{\left (-1\right )} + \sqrt {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )}\right )}^{2} d e^{2} + 2 \, d^{3}} - \frac {2 \, \log \left (\frac {1}{4} \, d e^{\left (-1\right )} - \frac {1}{4} \, \sqrt {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )} + x\right )}{{\left (d e^{\left (-1\right )} - \sqrt {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )}\right )}^{3} e^{3} - 3 \, {\left (d e^{\left (-1\right )} - \sqrt {3 \, d^{2} e^{2} + 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )}\right )}^{2} d e^{2} + 2 \, d^{3}} - \frac {2 \, \log \left (\frac {1}{4} \, d e^{\left (-1\right )} + \frac {1}{4} \, \sqrt {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )} + x\right )}{{\left (d e^{\left (-1\right )} + \sqrt {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )}\right )}^{3} e^{3} - 3 \, {\left (d e^{\left (-1\right )} + \sqrt {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )}\right )}^{2} d e^{2} + 2 \, d^{3}} - \frac {2 \, \log \left (\frac {1}{4} \, d e^{\left (-1\right )} - \frac {1}{4} \, \sqrt {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )} + x\right )}{{\left (d e^{\left (-1\right )} - \sqrt {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )}\right )}^{3} e^{3} - 3 \, {\left (d e^{\left (-1\right )} - \sqrt {3 \, d^{2} e^{2} - 2 \, \sqrt {d^{4} - 64 \, a e^{3}} e^{2}} e^{\left (-2\right )}\right )}^{2} d e^{2} + 2 \, d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2),x, algorithm="giac")

[Out]

-2*log(1/4*d*e^(-1) + 1/4*sqrt(3*d^2*e^2 + 2*sqrt(d^4 - 64*a*e^3)*e^2)*e^(-2) + x)/((d*e^(-1) + sqrt(3*d^2*e^2
+ 2*sqrt(d^4 - 64*a*e^3)*e^2)*e^(-2))^3*e^3 - 3*(d*e^(-1) + sqrt(3*d^2*e^2 + 2*sqrt(d^4 - 64*a*e^3)*e^2)*e^(-
2))^2*d*e^2 + 2*d^3) - 2*log(1/4*d*e^(-1) - 1/4*sqrt(3*d^2*e^2 + 2*sqrt(d^4 - 64*a*e^3)*e^2)*e^(-2) + x)/((d*e
^(-1) - sqrt(3*d^2*e^2 + 2*sqrt(d^4 - 64*a*e^3)*e^2)*e^(-2))^3*e^3 - 3*(d*e^(-1) - sqrt(3*d^2*e^2 + 2*sqrt(d^4
- 64*a*e^3)*e^2)*e^(-2))^2*d*e^2 + 2*d^3) - 2*log(1/4*d*e^(-1) + 1/4*sqrt(3*d^2*e^2 - 2*sqrt(d^4 - 64*a*e^3)*
e^2)*e^(-2) + x)/((d*e^(-1) + sqrt(3*d^2*e^2 - 2*sqrt(d^4 - 64*a*e^3)*e^2)*e^(-2))^3*e^3 - 3*(d*e^(-1) + sqrt(
3*d^2*e^2 - 2*sqrt(d^4 - 64*a*e^3)*e^2)*e^(-2))^2*d*e^2 + 2*d^3) - 2*log(1/4*d*e^(-1) - 1/4*sqrt(3*d^2*e^2 - 2
*sqrt(d^4 - 64*a*e^3)*e^2)*e^(-2) + x)/((d*e^(-1) - sqrt(3*d^2*e^2 - 2*sqrt(d^4 - 64*a*e^3)*e^2)*e^(-2))^3*e^3
- 3*(d*e^(-1) - sqrt(3*d^2*e^2 - 2*sqrt(d^4 - 64*a*e^3)*e^2)*e^(-2))^2*d*e^2 + 2*d^3)

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Mupad [B]
time = 3.73, size = 1264, normalized size = 8.26 \begin {gather*} -\mathrm {atan}\left (\frac {d^3\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}\,3{}\mathrm {i}+d^9\,2{}\mathrm {i}-a\,d^5\,e^3\,256{}\mathrm {i}+a^2\,d\,e^6\,8192{}\mathrm {i}+a^2\,e^7\,x\,32768{}\mathrm {i}+d^8\,e\,x\,8{}\mathrm {i}+d^2\,e\,x\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}\,12{}\mathrm {i}-a\,d^4\,e^4\,x\,1024{}\mathrm {i}}{5\,d^{12}\,\sqrt {\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}+3\,d^6-192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}+1048576\,a^3\,e^9\,\sqrt {\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}+3\,d^6-192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}-384\,a\,d^8\,e^3\,\sqrt {\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}+3\,d^6-192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}-12288\,a^2\,d^4\,e^6\,\sqrt {\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}+3\,d^6-192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}}\right )\,\sqrt {\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}+3\,d^6-192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {d^3\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}\,3{}\mathrm {i}-d^9\,2{}\mathrm {i}+a\,d^5\,e^3\,256{}\mathrm {i}-a^2\,d\,e^6\,8192{}\mathrm {i}-a^2\,e^7\,x\,32768{}\mathrm {i}-d^8\,e\,x\,8{}\mathrm {i}+d^2\,e\,x\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}\,12{}\mathrm {i}+a\,d^4\,e^4\,x\,1024{}\mathrm {i}}{5\,d^{12}\,\sqrt {-\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}-3\,d^6+192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}+1048576\,a^3\,e^9\,\sqrt {-\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}-3\,d^6+192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}-384\,a\,d^8\,e^3\,\sqrt {-\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}-3\,d^6+192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}-12288\,a^2\,d^4\,e^6\,\sqrt {-\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}-3\,d^6+192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}}\right )\,\sqrt {-\frac {2\,\sqrt {-262144\,a^3\,e^9+12288\,a^2\,d^4\,e^6-192\,a\,d^8\,e^3+d^{12}}-3\,d^6+192\,a\,d^2\,e^3}{1048576\,a^3\,e^9-12288\,a^2\,d^4\,e^6-384\,a\,d^8\,e^3+5\,d^{12}}}\,2{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(8*a*e^2 - d^3*x + 8*e^3*x^4 + 8*d*e^2*x^3),x)

[Out]

atan((d^3*(d^12 - 262144*a^3*e^9 - 192*a*d^8*e^3 + 12288*a^2*d^4*e^6)^(1/2)*3i - d^9*2i + a*d^5*e^3*256i - a^2
*d*e^6*8192i - a^2*e^7*x*32768i - d^8*e*x*8i + d^2*e*x*(d^12 - 262144*a^3*e^9 - 192*a*d^8*e^3 + 12288*a^2*d^4*
e^6)^(1/2)*12i + a*d^4*e^4*x*1024i)/(5*d^12*(-(2*(d^12 - 262144*a^3*e^9 - 192*a*d^8*e^3 + 12288*a^2*d^4*e^6)^(
1/2) - 3*d^6 + 192*a*d^2*e^3)/(5*d^12 + 1048576*a^3*e^9 - 384*a*d^8*e^3 - 12288*a^2*d^4*e^6))^(1/2) + 1048576*
a^3*e^9*(-(2*(d^12 - 262144*a^3*e^9 - 192*a*d^8*e^3 + 12288*a^2*d^4*e^6)^(1/2) - 3*d^6 + 192*a*d^2*e^3)/(5*d^1
2 + 1048576*a^3*e^9 - 384*a*d^8*e^3 - 12288*a^2*d^4*e^6))^(1/2) - 384*a*d^8*e^3*(-(2*(d^12 - 262144*a^3*e^9 -
192*a*d^8*e^3 + 12288*a^2*d^4*e^6)^(1/2) - 3*d^6 + 192*a*d^2*e^3)/(5*d^12 + 1048576*a^3*e^9 - 384*a*d^8*e^3 -
12288*a^2*d^4*e^6))^(1/2) - 12288*a^2*d^4*e^6*(-(2*(d^12 - 262144*a^3*e^9 - 192*a*d^8*e^3 + 12288*a^2*d^4*e^6)
^(1/2) - 3*d^6 + 192*a*d^2*e^3)/(5*d^12 + 1048576*a^3*e^9 - 384*a*d^8*e^3 - 12288*a^2*d^4*e^6))^(1/2)))*(-(2*(
d^12 - 262144*a^3*e^9 - 192*a*d^8*e^3 + 12288*a^2*d^4*e^6)^(1/2) - 3*d^6 + 192*a*d^2*e^3)/(5*d^12 + 1048576*a^
3*e^9 - 384*a*d^8*e^3 - 12288*a^2*d^4*e^6))^(1/2)*2i - atan((d^3*(d^12 - 262144*a^3*e^9 - 192*a*d^8*e^3 + 1228
8*a^2*d^4*e^6)^(1/2)*3i + d^9*2i - a*d^5*e^3*256i + a^2*d*e^6*8192i + a^2*e^7*x*32768i + d^8*e*x*8i + d^2*e*x*
(d^12 - 262144*a^3*e^9 - 192*a*d^8*e^3 + 12288*a^2*d^4*e^6)^(1/2)*12i - a*d^4*e^4*x*1024i)/(5*d^12*((2*(d^12 -
262144*a^3*e^9 - 192*a*d^8*e^3 + 12288*a^2*d^4*e^6)^(1/2) + 3*d^6 - 192*a*d^2*e^3)/(5*d^12 + 1048576*a^3*e^9
- 384*a*d^8*e^3 - 12288*a^2*d^4*e^6))^(1/2) + 1048576*a^3*e^9*((2*(d^12 - 262144*a^3*e^9 - 192*a*d^8*e^3 + 122
88*a^2*d^4*e^6)^(1/2) + 3*d^6 - 192*a*d^2*e^3)/(5*d^12 + 1048576*a^3*e^9 - 384*a*d^8*e^3 - 12288*a^2*d^4*e^6))
^(1/2) - 384*a*d^8*e^3*((2*(d^12 - 262144*a^3*e^9 - 192*a*d^8*e^3 + 12288*a^2*d^4*e^6)^(1/2) + 3*d^6 - 192*a*d
^2*e^3)/(5*d^12 + 1048576*a^3*e^9 - 384*a*d^8*e^3 - 12288*a^2*d^4*e^6))^(1/2) - 12288*a^2*d^4*e^6*((2*(d^12 -
262144*a^3*e^9 - 192*a*d^8*e^3 + 12288*a^2*d^4*e^6)^(1/2) + 3*d^6 - 192*a*d^2*e^3)/(5*d^12 + 1048576*a^3*e^9 -
384*a*d^8*e^3 - 12288*a^2*d^4*e^6))^(1/2)))*((2*(d^12 - 262144*a^3*e^9 - 192*a*d^8*e^3 + 12288*a^2*d^4*e^6)^(
1/2) + 3*d^6 - 192*a*d^2*e^3)/(5*d^12 + 1048576*a^3*e^9 - 384*a*d^8*e^3 - 12288*a^2*d^4*e^6))^(1/2)*2i

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