3.29.55 \(\int \frac {1+x^6}{\sqrt [4]{x^3+x^5} (1-x^6)} \, dx\)
Optimal. Leaf size=297 \[ \frac {2}{3} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^5+x^3}}\right )+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^5+x^3}}\right )}{3 \sqrt [4]{2}}+\frac {2}{3} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^5+x^3}}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^5+x^3}}\right )}{3 \sqrt [4]{2}}-\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^5+x^3}}{\sqrt {2} x^2-\sqrt {x^5+x^3}}\right )}{3\ 2^{3/4}}+\frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^5+x^3}}{\sqrt {x^5+x^3}-x^2}\right )+\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^5+x^3}}{2^{3/4}}}{x \sqrt [4]{x^5+x^3}}\right )}{3\ 2^{3/4}}+\frac {1}{3} \sqrt {2} \tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^5+x^3}}{\sqrt {2}}}{x \sqrt [4]{x^5+x^3}}\right ) \]
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Rubi [F] time = 0.00, antiderivative size = 0, normalized size of antiderivative = 0.00,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used =
{} \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
Int[(1 + x^6)/((x^3 + x^5)^(1/4)*(1 - x^6)),x]
[Out]
$Aborted
Rubi steps
Aborted
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Mathematica [F] time = 0.88, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^6}{\sqrt [4]{x^3+x^5} \left (1-x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[(1 + x^6)/((x^3 + x^5)^(1/4)*(1 - x^6)),x]
[Out]
Integrate[(1 + x^6)/((x^3 + x^5)^(1/4)*(1 - x^6)), x]
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IntegrateAlgebraic [A] time = 1.20, size = 297, normalized size = 1.00 \begin {gather*} \frac {2}{3} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^3+x^5}}\right )+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{3 \sqrt [4]{2}}-\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^3+x^5}}{\sqrt {2} x^2-\sqrt {x^3+x^5}}\right )}{3\ 2^{3/4}}+\frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^3+x^5}}{-x^2+\sqrt {x^3+x^5}}\right )+\frac {2}{3} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^3+x^5}}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{3 \sqrt [4]{2}}+\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^3+x^5}}{2^{3/4}}}{x \sqrt [4]{x^3+x^5}}\right )}{3\ 2^{3/4}}+\frac {1}{3} \sqrt {2} \tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^3+x^5}}{\sqrt {2}}}{x \sqrt [4]{x^3+x^5}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(1 + x^6)/((x^3 + x^5)^(1/4)*(1 - x^6)),x]
[Out]
(2*ArcTan[x/(x^3 + x^5)^(1/4)])/3 + ArcTan[(2^(1/4)*x)/(x^3 + x^5)^(1/4)]/(3*2^(1/4)) - ArcTan[(2^(3/4)*x*(x^3
+ x^5)^(1/4))/(Sqrt[2]*x^2 - Sqrt[x^3 + x^5])]/(3*2^(3/4)) + (Sqrt[2]*ArcTan[(Sqrt[2]*x*(x^3 + x^5)^(1/4))/(-
x^2 + Sqrt[x^3 + x^5])])/3 + (2*ArcTanh[x/(x^3 + x^5)^(1/4)])/3 + ArcTanh[(2^(1/4)*x)/(x^3 + x^5)^(1/4)]/(3*2^
(1/4)) + ArcTanh[(x^2/2^(1/4) + Sqrt[x^3 + x^5]/2^(3/4))/(x*(x^3 + x^5)^(1/4))]/(3*2^(3/4)) + (Sqrt[2]*ArcTanh
[(x^2/Sqrt[2] + Sqrt[x^3 + x^5]/Sqrt[2])/(x*(x^3 + x^5)^(1/4))])/3
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fricas [B] time = 56.84, size = 1837, normalized size = 6.19
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^6+1)/(x^5+x^3)^(1/4)/(-x^6+1),x, algorithm="fricas")
[Out]
-1/6*2^(3/4)*arctan(-1/2*(4*2^(3/4)*(x^5 + x^3)^(1/4)*x^2 - 2^(3/4)*(2*2^(3/4)*sqrt(x^5 + x^3)*x + 2^(1/4)*(x^
4 + 2*x^3 + x^2)) + 4*2^(1/4)*(x^5 + x^3)^(3/4))/(x^4 - 2*x^3 + x^2)) + 1/24*2^(3/4)*log((4*sqrt(2)*(x^5 + x^3
)^(1/4)*x^2 + 2^(3/4)*(x^4 + 2*x^3 + x^2) + 4*2^(1/4)*sqrt(x^5 + x^3)*x + 4*(x^5 + x^3)^(3/4))/(x^4 - 2*x^3 +
x^2)) - 1/24*2^(3/4)*log((4*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 - 2^(3/4)*(x^4 + 2*x^3 + x^2) - 4*2^(1/4)*sqrt(x^5 +
x^3)*x + 4*(x^5 + x^3)^(3/4))/(x^4 - 2*x^3 + x^2)) - 1/3*sqrt(2)*arctan(-(x^6 + 2*x^5 + 3*x^4 + 2*x^3 + 2*sqr
t(2)*(x^5 + x^3)^(3/4)*(x^2 - 3*x + 1) + x^2 + 2*sqrt(2)*(x^5 + x^3)^(1/4)*(3*x^4 - x^3 + 3*x^2) + 4*sqrt(x^5
+ x^3)*(x^3 + x^2 + x) - (2*sqrt(2)*sqrt(x^5 + x^3)*(x^3 - 3*x^2 + x) + 16*(x^5 + x^3)^(3/4)*x + sqrt(2)*(x^6
- 8*x^5 + x^4 - 8*x^3 + x^2) + 4*(x^5 + x^3)^(1/4)*(x^4 + x^3 + x^2))*sqrt((x^4 + x^3 + 2*sqrt(2)*(x^5 + x^3)^
(1/4)*x^2 + x^2 + 4*sqrt(x^5 + x^3)*x + 2*sqrt(2)*(x^5 + x^3)^(3/4))/(x^4 + x^3 + x^2)))/(x^6 - 14*x^5 + 3*x^4
- 14*x^3 + x^2)) + 1/3*sqrt(2)*arctan(-(x^6 + 2*x^5 + 3*x^4 + 2*x^3 - 2*sqrt(2)*(x^5 + x^3)^(3/4)*(x^2 - 3*x
+ 1) + x^2 - 2*sqrt(2)*(x^5 + x^3)^(1/4)*(3*x^4 - x^3 + 3*x^2) + 4*sqrt(x^5 + x^3)*(x^3 + x^2 + x) + (2*sqrt(2
)*sqrt(x^5 + x^3)*(x^3 - 3*x^2 + x) - 16*(x^5 + x^3)^(3/4)*x + sqrt(2)*(x^6 - 8*x^5 + x^4 - 8*x^3 + x^2) - 4*(
x^5 + x^3)^(1/4)*(x^4 + x^3 + x^2))*sqrt((x^4 + x^3 - 2*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 + x^2 + 4*sqrt(x^5 + x^3
)*x - 2*sqrt(2)*(x^5 + x^3)^(3/4))/(x^4 + x^3 + x^2)))/(x^6 - 14*x^5 + 3*x^4 - 14*x^3 + x^2)) + 1/12*sqrt(2)*l
og(4*(x^4 + x^3 + 2*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 + x^2 + 4*sqrt(x^5 + x^3)*x + 2*sqrt(2)*(x^5 + x^3)^(3/4))/(
x^4 + x^3 + x^2)) - 1/12*sqrt(2)*log(4*(x^4 + x^3 - 2*sqrt(2)*(x^5 + x^3)^(1/4)*x^2 + x^2 + 4*sqrt(x^5 + x^3)*
x - 2*sqrt(2)*(x^5 + x^3)^(3/4))/(x^4 + x^3 + x^2)) + 1/6*2^(1/4)*arctan(1/2*(2*x^6 + 8*x^5 + 12*x^4 + 8*x^3 +
4*2^(3/4)*(x^5 + x^3)^(3/4)*(x^2 - 6*x + 1) + 8*sqrt(2)*sqrt(x^5 + x^3)*(x^3 + 2*x^2 + x) + 2*x^2 + sqrt(2)*(
32*sqrt(2)*(x^5 + x^3)^(3/4)*x + 2^(3/4)*(x^6 - 16*x^5 - 2*x^4 - 16*x^3 + x^2) + 4*2^(1/4)*sqrt(x^5 + x^3)*(x^
3 - 6*x^2 + x) + 8*(x^5 + x^3)^(1/4)*(x^4 + 2*x^3 + x^2))*sqrt((4*2^(3/4)*(x^5 + x^3)^(1/4)*x^2 + sqrt(2)*(x^4
+ 2*x^3 + x^2) + 8*sqrt(x^5 + x^3)*x + 4*2^(1/4)*(x^5 + x^3)^(3/4))/(x^4 + 2*x^3 + x^2)) + 8*2^(1/4)*(x^5 + x
^3)^(1/4)*(3*x^4 - 2*x^3 + 3*x^2))/(x^6 - 28*x^5 + 6*x^4 - 28*x^3 + x^2)) - 1/6*2^(1/4)*arctan(1/2*(2*x^6 + 8*
x^5 + 12*x^4 + 8*x^3 - 4*2^(3/4)*(x^5 + x^3)^(3/4)*(x^2 - 6*x + 1) + 8*sqrt(2)*sqrt(x^5 + x^3)*(x^3 + 2*x^2 +
x) + 2*x^2 + sqrt(2)*(32*sqrt(2)*(x^5 + x^3)^(3/4)*x - 2^(3/4)*(x^6 - 16*x^5 - 2*x^4 - 16*x^3 + x^2) - 4*2^(1/
4)*sqrt(x^5 + x^3)*(x^3 - 6*x^2 + x) + 8*(x^5 + x^3)^(1/4)*(x^4 + 2*x^3 + x^2))*sqrt(-(4*2^(3/4)*(x^5 + x^3)^(
1/4)*x^2 - sqrt(2)*(x^4 + 2*x^3 + x^2) - 8*sqrt(x^5 + x^3)*x + 4*2^(1/4)*(x^5 + x^3)^(3/4))/(x^4 + 2*x^3 + x^2
)) - 8*2^(1/4)*(x^5 + x^3)^(1/4)*(3*x^4 - 2*x^3 + 3*x^2))/(x^6 - 28*x^5 + 6*x^4 - 28*x^3 + x^2)) + 1/24*2^(1/4
)*log(8*(4*2^(3/4)*(x^5 + x^3)^(1/4)*x^2 + sqrt(2)*(x^4 + 2*x^3 + x^2) + 8*sqrt(x^5 + x^3)*x + 4*2^(1/4)*(x^5
+ x^3)^(3/4))/(x^4 + 2*x^3 + x^2)) - 1/24*2^(1/4)*log(-8*(4*2^(3/4)*(x^5 + x^3)^(1/4)*x^2 - sqrt(2)*(x^4 + 2*x
^3 + x^2) - 8*sqrt(x^5 + x^3)*x + 4*2^(1/4)*(x^5 + x^3)^(3/4))/(x^4 + 2*x^3 + x^2)) + 1/3*arctan(2*((x^5 + x^3
)^(1/4)*x^2 + (x^5 + x^3)^(3/4))/(x^4 - x^3 + x^2)) + 1/3*log((x^4 + x^3 + 2*(x^5 + x^3)^(1/4)*x^2 + x^2 + 2*s
qrt(x^5 + x^3)*x + 2*(x^5 + x^3)^(3/4))/(x^4 - x^3 + x^2))
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {x^{6} + 1}{{\left (x^{6} - 1\right )} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^6+1)/(x^5+x^3)^(1/4)/(-x^6+1),x, algorithm="giac")
[Out]
integrate(-(x^6 + 1)/((x^6 - 1)*(x^5 + x^3)^(1/4)), x)
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maple [C] time = 27.53, size = 1425, normalized size = 4.80
| | |
| method |
result |
size |
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| trager |
\(\text {Expression too large to display}\) |
\(1425\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^6+1)/(x^5+x^3)^(1/4)/(-x^6+1),x,method=_RETURNVERBOSE)
[Out]
-1/12*RootOf(_Z^2+RootOf(_Z^4+8)^2)*ln(((x^5+x^3)^(1/2)*RootOf(_Z^4+8)^2*RootOf(_Z^2+RootOf(_Z^4+8)^2)*x-2*(x^
5+x^3)^(1/4)*RootOf(_Z^4+8)^2*x^2+RootOf(_Z^2+RootOf(_Z^4+8)^2)*x^4-2*RootOf(_Z^2+RootOf(_Z^4+8)^2)*x^3-4*(x^5
+x^3)^(3/4)+RootOf(_Z^2+RootOf(_Z^4+8)^2)*x^2)/(1+x)^2/x^2)+1/12*RootOf(_Z^4+8)*ln((RootOf(_Z^4+8)^3*(x^5+x^3)
^(1/2)*x+2*(x^5+x^3)^(1/4)*RootOf(_Z^4+8)^2*x^2-RootOf(_Z^4+8)*x^4+2*RootOf(_Z^4+8)*x^3-RootOf(_Z^4+8)*x^2-4*(
x^5+x^3)^(3/4))/(1+x)^2/x^2)+1/48*ln(-(RootOf(_Z^4+8)^3*x^4-2*RootOf(_Z^4+8)^3*x^3+8*(x^5+x^3)^(1/4)*RootOf(_Z
^4+8)^2*x^2+RootOf(_Z^4+8)^3*x^2-16*(x^5+x^3)^(1/2)*RootOf(_Z^4+8)*x+16*(x^5+x^3)^(3/4))/x^2/(-1+x)^2)*RootOf(
_Z^4+8)^3+1/48*ln(-(RootOf(_Z^4+8)^3*x^4-2*RootOf(_Z^4+8)^3*x^3+8*(x^5+x^3)^(1/4)*RootOf(_Z^4+8)^2*x^2+RootOf(
_Z^4+8)^3*x^2-16*(x^5+x^3)^(1/2)*RootOf(_Z^4+8)*x+16*(x^5+x^3)^(3/4))/x^2/(-1+x)^2)*RootOf(_Z^4+8)^2*RootOf(_Z
^2+RootOf(_Z^4+8)^2)-1/24*RootOf(_Z^4+8)^2*RootOf(_Z^2+RootOf(_Z^4+8)^2)*ln((RootOf(_Z^4+8)^3*x^4-RootOf(_Z^2+
RootOf(_Z^4+8)^2)*RootOf(_Z^4+8)^2*x^4+2*RootOf(_Z^4+8)^3*x^3-2*RootOf(_Z^2+RootOf(_Z^4+8)^2)*RootOf(_Z^4+8)^2
*x^3+8*(x^5+x^3)^(1/4)*RootOf(_Z^4+8)*RootOf(_Z^2+RootOf(_Z^4+8)^2)*x^2+RootOf(_Z^4+8)^3*x^2-RootOf(_Z^4+8)^2*
RootOf(_Z^2+RootOf(_Z^4+8)^2)*x^2-8*(x^5+x^3)^(1/2)*RootOf(_Z^4+8)*x-8*RootOf(_Z^2+RootOf(_Z^4+8)^2)*(x^5+x^3)
^(1/2)*x+16*(x^5+x^3)^(3/4))/x^2/(-1+x)^2)-1/24*RootOf(_Z^4+8)^3*RootOf(_Z^2+RootOf(_Z^4+8)^2)*ln((-RootOf(_Z^
2+RootOf(_Z^4+8)^2)*RootOf(_Z^4+8)^3*x^4+2*RootOf(_Z^4+8)^3*(x^5+x^3)^(1/2)*RootOf(_Z^2+RootOf(_Z^4+8)^2)*x-Ro
otOf(_Z^2+RootOf(_Z^4+8)^2)*RootOf(_Z^4+8)^3*x^3-RootOf(_Z^2+RootOf(_Z^4+8)^2)*x^2*RootOf(_Z^4+8)^3+16*(x^5+x^
3)^(3/4)-16*(x^5+x^3)^(1/4)*x^2)/x^2/(x^2-x+1))+1/3*ln((x^4+2*(x^5+x^3)^(3/4)+2*(x^5+x^3)^(1/2)*x+2*(x^5+x^3)^
(1/4)*x^2+x^3+x^2)/x^2/(x^2-x+1))+1/12*ln(-(-x^4*RootOf(_Z^4+8)^2+4*RootOf(_Z^4+8)^2*(x^5+x^3)^(1/2)*x-x^3*Roo
tOf(_Z^4+8)^2-RootOf(_Z^4+8)^2*x^2-8*(x^5+x^3)^(3/4)+8*(x^5+x^3)^(1/4)*x^2)/x^2/(x^2+x+1))*RootOf(_Z^4+8)^2+1/
12*ln(-(-x^4*RootOf(_Z^4+8)^2+4*RootOf(_Z^4+8)^2*(x^5+x^3)^(1/2)*x-x^3*RootOf(_Z^4+8)^2-RootOf(_Z^4+8)^2*x^2-8
*(x^5+x^3)^(3/4)+8*(x^5+x^3)^(1/4)*x^2)/x^2/(x^2+x+1))*RootOf(_Z^2+RootOf(_Z^4+8)^2)*RootOf(_Z^4+8)-1/6*RootOf
(_Z^2+RootOf(_Z^4+8)^2)*RootOf(_Z^4+8)*ln(((x^5+x^3)^(1/4)*RootOf(_Z^4+8)^3*RootOf(_Z^2+RootOf(_Z^4+8)^2)*x^2+
x^4*RootOf(_Z^4+8)^2-RootOf(_Z^2+RootOf(_Z^4+8)^2)*RootOf(_Z^4+8)*x^4-2*RootOf(_Z^4+8)^2*(x^5+x^3)^(1/2)*x-2*R
ootOf(_Z^2+RootOf(_Z^4+8)^2)*RootOf(_Z^4+8)*(x^5+x^3)^(1/2)*x-x^3*RootOf(_Z^4+8)^2+RootOf(_Z^2+RootOf(_Z^4+8)^
2)*RootOf(_Z^4+8)*x^3+RootOf(_Z^4+8)^2*x^2-x^2*RootOf(_Z^2+RootOf(_Z^4+8)^2)*RootOf(_Z^4+8)+8*(x^5+x^3)^(3/4))
/x^2/(x^2+x+1))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {x^{6} + 1}{{\left (x^{6} - 1\right )} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^6+1)/(x^5+x^3)^(1/4)/(-x^6+1),x, algorithm="maxima")
[Out]
-integrate((x^6 + 1)/((x^6 - 1)*(x^5 + x^3)^(1/4)), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {x^6+1}{{\left (x^5+x^3\right )}^{1/4}\,\left (x^6-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(x^6 + 1)/((x^3 + x^5)^(1/4)*(x^6 - 1)),x)
[Out]
int(-(x^6 + 1)/((x^3 + x^5)^(1/4)*(x^6 - 1)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x^{6}}{x^{6} \sqrt [4]{x^{5} + x^{3}} - \sqrt [4]{x^{5} + x^{3}}}\, dx - \int \frac {1}{x^{6} \sqrt [4]{x^{5} + x^{3}} - \sqrt [4]{x^{5} + x^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**6+1)/(x**5+x**3)**(1/4)/(-x**6+1),x)
[Out]
-Integral(x**6/(x**6*(x**5 + x**3)**(1/4) - (x**5 + x**3)**(1/4)), x) - Integral(1/(x**6*(x**5 + x**3)**(1/4)
- (x**5 + x**3)**(1/4)), x)
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