3.30.64 \(\int \frac {-b^{10}+a^{10} x^{10}}{\sqrt {b^4+a^4 x^4} (b^{10}+a^{10} x^{10})} \, dx\)

Optimal. Leaf size=366 \[ \frac {1}{5} \text {RootSum}\left [\text {$\#$1}^8-12 \text {$\#$1}^6 a^2 b^2+24 \text {$\#$1}^4 a^4 b^4-48 \text {$\#$1}^2 a^6 b^6+16 a^8 b^8\& ,\frac {-\text {$\#$1}^6 \log \left (-\text {$\#$1} x+\sqrt {a^4 x^4+b^4}+a^2 x^2+b^2\right )+\text {$\#$1}^6 \log (x)-4 \text {$\#$1}^4 a^2 b^2 \log (x)+4 \text {$\#$1}^4 a^2 b^2 \log \left (-\text {$\#$1} x+\sqrt {a^4 x^4+b^4}+a^2 x^2+b^2\right )-8 \text {$\#$1}^2 a^4 b^4 \log (x)+8 \text {$\#$1}^2 a^4 b^4 \log \left (-\text {$\#$1} x+\sqrt {a^4 x^4+b^4}+a^2 x^2+b^2\right )-8 a^6 b^6 \log \left (-\text {$\#$1} x+\sqrt {a^4 x^4+b^4}+a^2 x^2+b^2\right )+8 a^6 b^6 \log (x)}{-\text {$\#$1}^7+9 \text {$\#$1}^5 a^2 b^2-12 \text {$\#$1}^3 a^4 b^4+12 \text {$\#$1} a^6 b^6}\& \right ]-\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {a^4 x^4+b^4}+a^2 x^2+b^2}\right )}{5 a b} \]

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Rubi [F]  time = 2.79, 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*} \int \frac {-b^{10}+a^{10} x^{10}}{\sqrt {b^4+a^4 x^4} \left (b^{10}+a^{10} x^{10}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-b^10 + a^10*x^10)/(Sqrt[b^4 + a^4*x^4]*(b^10 + a^10*x^10)),x]

[Out]

-1/5*ArcTan[(Sqrt[2]*a*b*x)/Sqrt[b^4 + a^4*x^4]]/(Sqrt[2]*a*b) + (2*(b^2 + a^2*x^2)*Sqrt[(b^4 + a^4*x^4)/(b^2
+ a^2*x^2)^2]*EllipticF[2*ArcTan[(a*x)/b], 1/2])/(5*a*b*Sqrt[b^4 + a^4*x^4]) - (8*b^8*Defer[Int][1/(Sqrt[b^4 +
a^4*x^4]*(b^8 - a^2*b^6*x^2 + a^4*b^4*x^4 - a^6*b^2*x^6 + a^8*x^8)), x])/5 + (6*a^2*b^6*Defer[Int][x^2/(Sqrt[
b^4 + a^4*x^4]*(b^8 - a^2*b^6*x^2 + a^4*b^4*x^4 - a^6*b^2*x^6 + a^8*x^8)), x])/5 - (4*a^4*b^4*Defer[Int][x^4/(
Sqrt[b^4 + a^4*x^4]*(b^8 - a^2*b^6*x^2 + a^4*b^4*x^4 - a^6*b^2*x^6 + a^8*x^8)), x])/5 + (2*a^6*b^2*Defer[Int][
x^6/(Sqrt[b^4 + a^4*x^4]*(b^8 - a^2*b^6*x^2 + a^4*b^4*x^4 - a^6*b^2*x^6 + a^8*x^8)), x])/5

Rubi steps

\begin {align*} \int \frac {-b^{10}+a^{10} x^{10}}{\sqrt {b^4+a^4 x^4} \left (b^{10}+a^{10} x^{10}\right )} \, dx &=\int \left (\frac {1}{\sqrt {b^4+a^4 x^4}}-\frac {2 b^{10}}{\sqrt {b^4+a^4 x^4} \left (b^{10}+a^{10} x^{10}\right )}\right ) \, dx\\ &=-\left (\left (2 b^{10}\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4} \left (b^{10}+a^{10} x^{10}\right )} \, dx\right )+\int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx\\ &=\frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{2 a b \sqrt {b^4+a^4 x^4}}-\left (2 b^{10}\right ) \int \left (\frac {1}{5 b^8 \left (b^2+a^2 x^2\right ) \sqrt {b^4+a^4 x^4}}+\frac {4 b^6-3 a^2 b^4 x^2+2 a^4 b^2 x^4-a^6 x^6}{5 b^8 \sqrt {b^4+a^4 x^4} \left (b^8-a^2 b^6 x^2+a^4 b^4 x^4-a^6 b^2 x^6+a^8 x^8\right )}\right ) \, dx\\ &=\frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{2 a b \sqrt {b^4+a^4 x^4}}-\frac {1}{5} \left (2 b^2\right ) \int \frac {1}{\left (b^2+a^2 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{5} \left (2 b^2\right ) \int \frac {4 b^6-3 a^2 b^4 x^2+2 a^4 b^2 x^4-a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (b^8-a^2 b^6 x^2+a^4 b^4 x^4-a^6 b^2 x^6+a^8 x^8\right )} \, dx\\ &=\frac {\left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{2 a b \sqrt {b^4+a^4 x^4}}-\frac {1}{5} \int \frac {1}{\sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{5} \int \frac {b^2-a^2 x^2}{\left (b^2+a^2 x^2\right ) \sqrt {b^4+a^4 x^4}} \, dx-\frac {1}{5} \left (2 b^2\right ) \int \left (\frac {4 b^6}{\sqrt {b^4+a^4 x^4} \left (b^8-a^2 b^6 x^2+a^4 b^4 x^4-a^6 b^2 x^6+a^8 x^8\right )}-\frac {3 a^2 b^4 x^2}{\sqrt {b^4+a^4 x^4} \left (b^8-a^2 b^6 x^2+a^4 b^4 x^4-a^6 b^2 x^6+a^8 x^8\right )}+\frac {2 a^4 b^2 x^4}{\sqrt {b^4+a^4 x^4} \left (b^8-a^2 b^6 x^2+a^4 b^4 x^4-a^6 b^2 x^6+a^8 x^8\right )}-\frac {a^6 x^6}{\sqrt {b^4+a^4 x^4} \left (b^8-a^2 b^6 x^2+a^4 b^4 x^4-a^6 b^2 x^6+a^8 x^8\right )}\right ) \, dx\\ &=\frac {2 \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{5 a b \sqrt {b^4+a^4 x^4}}-\frac {1}{5} b^2 \operatorname {Subst}\left (\int \frac {1}{b^2+2 a^2 b^4 x^2} \, dx,x,\frac {x}{\sqrt {b^4+a^4 x^4}}\right )+\frac {1}{5} \left (2 a^6 b^2\right ) \int \frac {x^6}{\sqrt {b^4+a^4 x^4} \left (b^8-a^2 b^6 x^2+a^4 b^4 x^4-a^6 b^2 x^6+a^8 x^8\right )} \, dx-\frac {1}{5} \left (4 a^4 b^4\right ) \int \frac {x^4}{\sqrt {b^4+a^4 x^4} \left (b^8-a^2 b^6 x^2+a^4 b^4 x^4-a^6 b^2 x^6+a^8 x^8\right )} \, dx+\frac {1}{5} \left (6 a^2 b^6\right ) \int \frac {x^2}{\sqrt {b^4+a^4 x^4} \left (b^8-a^2 b^6 x^2+a^4 b^4 x^4-a^6 b^2 x^6+a^8 x^8\right )} \, dx-\frac {1}{5} \left (8 b^8\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4} \left (b^8-a^2 b^6 x^2+a^4 b^4 x^4-a^6 b^2 x^6+a^8 x^8\right )} \, dx\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {2} a b x}{\sqrt {b^4+a^4 x^4}}\right )}{5 \sqrt {2} a b}+\frac {2 \left (b^2+a^2 x^2\right ) \sqrt {\frac {b^4+a^4 x^4}{\left (b^2+a^2 x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {a x}{b}\right )|\frac {1}{2}\right )}{5 a b \sqrt {b^4+a^4 x^4}}+\frac {1}{5} \left (2 a^6 b^2\right ) \int \frac {x^6}{\sqrt {b^4+a^4 x^4} \left (b^8-a^2 b^6 x^2+a^4 b^4 x^4-a^6 b^2 x^6+a^8 x^8\right )} \, dx-\frac {1}{5} \left (4 a^4 b^4\right ) \int \frac {x^4}{\sqrt {b^4+a^4 x^4} \left (b^8-a^2 b^6 x^2+a^4 b^4 x^4-a^6 b^2 x^6+a^8 x^8\right )} \, dx+\frac {1}{5} \left (6 a^2 b^6\right ) \int \frac {x^2}{\sqrt {b^4+a^4 x^4} \left (b^8-a^2 b^6 x^2+a^4 b^4 x^4-a^6 b^2 x^6+a^8 x^8\right )} \, dx-\frac {1}{5} \left (8 b^8\right ) \int \frac {1}{\sqrt {b^4+a^4 x^4} \left (b^8-a^2 b^6 x^2+a^4 b^4 x^4-a^6 b^2 x^6+a^8 x^8\right )} \, dx\\ \end {align*}

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Mathematica [C]  time = 1.34, size = 544, normalized size = 1.49 \begin {gather*} \frac {\sqrt [10]{-1} \sqrt {\frac {a^4 x^4}{b^4}+1} \left (\left (1+\sqrt [5]{-1}\right )^2 \left (1-3 \sqrt [5]{-1}+(-1)^{2/5}\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )+2 \left ((-1)^{2/5} \Pi \left (-i;\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )+\left (1-\sqrt [5]{-1}+2 (-1)^{2/5}-(-1)^{3/5}+(-1)^{4/5}\right ) \Pi \left (-\sqrt [10]{-1};\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )-(-1)^{4/5} \Pi \left ((-1)^{3/10};\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )+(-1)^{3/5} \Pi \left ((-1)^{3/10};\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )+\sqrt [5]{-1} \Pi \left ((-1)^{3/10};\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )-\Pi \left ((-1)^{3/10};\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )+(-1)^{2/5} \Pi \left ((-1)^{7/10};\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )-(-1)^{4/5} \Pi \left (-(-1)^{9/10};\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )+(-1)^{3/5} \Pi \left (-(-1)^{9/10};\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )+\sqrt [5]{-1} \Pi \left (-(-1)^{9/10};\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )-\Pi \left (-(-1)^{9/10};\left .i \sinh ^{-1}\left (\sqrt {\frac {i a^2}{b^2}} x\right )\right |-1\right )\right )\right )}{\left (\sqrt [5]{-1}-1\right )^2 \left (1+\sqrt [5]{-1}\right )^4 \left (1+(-1)^{2/5}\right ) \left (1-\sqrt [5]{-1}+(-1)^{2/5}\right ) \sqrt {\frac {i a^2}{b^2}} \sqrt {a^4 x^4+b^4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(-b^10 + a^10*x^10)/(Sqrt[b^4 + a^4*x^4]*(b^10 + a^10*x^10)),x]

[Out]

((-1)^(1/10)*Sqrt[1 + (a^4*x^4)/b^4]*((1 + (-1)^(1/5))^2*(1 - 3*(-1)^(1/5) + (-1)^(2/5))*EllipticF[I*ArcSinh[S
qrt[(I*a^2)/b^2]*x], -1] + 2*((-1)^(2/5)*EllipticPi[-I, I*ArcSinh[Sqrt[(I*a^2)/b^2]*x], -1] + (1 - (-1)^(1/5)
+ 2*(-1)^(2/5) - (-1)^(3/5) + (-1)^(4/5))*EllipticPi[-(-1)^(1/10), I*ArcSinh[Sqrt[(I*a^2)/b^2]*x], -1] - Ellip
ticPi[(-1)^(3/10), I*ArcSinh[Sqrt[(I*a^2)/b^2]*x], -1] + (-1)^(1/5)*EllipticPi[(-1)^(3/10), I*ArcSinh[Sqrt[(I*
a^2)/b^2]*x], -1] + (-1)^(3/5)*EllipticPi[(-1)^(3/10), I*ArcSinh[Sqrt[(I*a^2)/b^2]*x], -1] - (-1)^(4/5)*Ellipt
icPi[(-1)^(3/10), I*ArcSinh[Sqrt[(I*a^2)/b^2]*x], -1] + (-1)^(2/5)*EllipticPi[(-1)^(7/10), I*ArcSinh[Sqrt[(I*a
^2)/b^2]*x], -1] - EllipticPi[-(-1)^(9/10), I*ArcSinh[Sqrt[(I*a^2)/b^2]*x], -1] + (-1)^(1/5)*EllipticPi[-(-1)^
(9/10), I*ArcSinh[Sqrt[(I*a^2)/b^2]*x], -1] + (-1)^(3/5)*EllipticPi[-(-1)^(9/10), I*ArcSinh[Sqrt[(I*a^2)/b^2]*
x], -1] - (-1)^(4/5)*EllipticPi[-(-1)^(9/10), I*ArcSinh[Sqrt[(I*a^2)/b^2]*x], -1])))/((-1 + (-1)^(1/5))^2*(1 +
(-1)^(1/5))^4*(1 + (-1)^(2/5))*(1 - (-1)^(1/5) + (-1)^(2/5))*Sqrt[(I*a^2)/b^2]*Sqrt[b^4 + a^4*x^4])

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IntegrateAlgebraic [A]  time = 2.38, size = 364, normalized size = 0.99 \begin {gather*} -\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} a b x}{b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}}\right )}{5 a b}+\frac {1}{5} \text {RootSum}\left [16 a^8 b^8-48 a^6 b^6 \text {$\#$1}^2+24 a^4 b^4 \text {$\#$1}^4-12 a^2 b^2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-8 a^6 b^6 \log (x)+8 a^6 b^6 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right )+8 a^4 b^4 \log (x) \text {$\#$1}^2-8 a^4 b^4 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2+4 a^2 b^2 \log (x) \text {$\#$1}^4-4 a^2 b^2 \log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-\log (x) \text {$\#$1}^6+\log \left (b^2+a^2 x^2+\sqrt {b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^6}{-12 a^6 b^6 \text {$\#$1}+12 a^4 b^4 \text {$\#$1}^3-9 a^2 b^2 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-b^10 + a^10*x^10)/(Sqrt[b^4 + a^4*x^4]*(b^10 + a^10*x^10)),x]

[Out]

-1/5*(Sqrt[2]*ArcTan[(Sqrt[2]*a*b*x)/(b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4])])/(a*b) + RootSum[16*a^8*b^8 - 48*a
^6*b^6*#1^2 + 24*a^4*b^4*#1^4 - 12*a^2*b^2*#1^6 + #1^8 & , (-8*a^6*b^6*Log[x] + 8*a^6*b^6*Log[b^2 + a^2*x^2 +
Sqrt[b^4 + a^4*x^4] - x*#1] + 8*a^4*b^4*Log[x]*#1^2 - 8*a^4*b^4*Log[b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4] - x*#1
]*#1^2 + 4*a^2*b^2*Log[x]*#1^4 - 4*a^2*b^2*Log[b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4] - x*#1]*#1^4 - Log[x]*#1^6
+ Log[b^2 + a^2*x^2 + Sqrt[b^4 + a^4*x^4] - x*#1]*#1^6)/(-12*a^6*b^6*#1 + 12*a^4*b^4*#1^3 - 9*a^2*b^2*#1^5 + #
1^7) & ]/5

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fricas [B]  time = 2.42, size = 1303, normalized size = 3.56

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^10*x^10-b^10)/(a^4*x^4+b^4)^(1/2)/(a^10*x^10+b^10),x, algorithm="fricas")

[Out]

-1/20*(sqrt(2)*a*b*sqrt(-(5*sqrt(1/5)*a^2*b^2*sqrt(1/(a^4*b^4)) + 1)/(a^2*b^2))*log(-(sqrt(2)*(3*a^8*x^8 + 5*a
^6*b^2*x^6 + 9*a^4*b^4*x^4 + 5*a^2*b^6*x^2 + 3*b^8 - 5*sqrt(1/5)*(a^10*b^2*x^8 + 3*a^8*b^4*x^6 + 3*a^6*b^6*x^4
+ 3*a^4*b^8*x^2 + a^2*b^10)*sqrt(1/(a^4*b^4)))*sqrt(-(5*sqrt(1/5)*a^2*b^2*sqrt(1/(a^4*b^4)) + 1)/(a^2*b^2)) +
4*(3*a^4*x^5 + a^2*b^2*x^3 + 3*b^4*x - 5*sqrt(1/5)*(a^6*b^2*x^5 + a^4*b^4*x^3 + a^2*b^6*x)*sqrt(1/(a^4*b^4)))
*sqrt(a^4*x^4 + b^4))/(a^8*x^8 - a^6*b^2*x^6 + a^4*b^4*x^4 - a^2*b^6*x^2 + b^8)) - sqrt(2)*a*b*sqrt(-(5*sqrt(1
/5)*a^2*b^2*sqrt(1/(a^4*b^4)) + 1)/(a^2*b^2))*log((sqrt(2)*(3*a^8*x^8 + 5*a^6*b^2*x^6 + 9*a^4*b^4*x^4 + 5*a^2*
b^6*x^2 + 3*b^8 - 5*sqrt(1/5)*(a^10*b^2*x^8 + 3*a^8*b^4*x^6 + 3*a^6*b^6*x^4 + 3*a^4*b^8*x^2 + a^2*b^10)*sqrt(1
/(a^4*b^4)))*sqrt(-(5*sqrt(1/5)*a^2*b^2*sqrt(1/(a^4*b^4)) + 1)/(a^2*b^2)) - 4*(3*a^4*x^5 + a^2*b^2*x^3 + 3*b^4
*x - 5*sqrt(1/5)*(a^6*b^2*x^5 + a^4*b^4*x^3 + a^2*b^6*x)*sqrt(1/(a^4*b^4)))*sqrt(a^4*x^4 + b^4))/(a^8*x^8 - a^
6*b^2*x^6 + a^4*b^4*x^4 - a^2*b^6*x^2 + b^8)) + sqrt(2)*a*b*sqrt((5*sqrt(1/5)*a^2*b^2*sqrt(1/(a^4*b^4)) - 1)/(
a^2*b^2))*log(-(sqrt(2)*(3*a^8*x^8 + 5*a^6*b^2*x^6 + 9*a^4*b^4*x^4 + 5*a^2*b^6*x^2 + 3*b^8 + 5*sqrt(1/5)*(a^10
*b^2*x^8 + 3*a^8*b^4*x^6 + 3*a^6*b^6*x^4 + 3*a^4*b^8*x^2 + a^2*b^10)*sqrt(1/(a^4*b^4)))*sqrt((5*sqrt(1/5)*a^2*
b^2*sqrt(1/(a^4*b^4)) - 1)/(a^2*b^2)) + 4*(3*a^4*x^5 + a^2*b^2*x^3 + 3*b^4*x + 5*sqrt(1/5)*(a^6*b^2*x^5 + a^4*
b^4*x^3 + a^2*b^6*x)*sqrt(1/(a^4*b^4)))*sqrt(a^4*x^4 + b^4))/(a^8*x^8 - a^6*b^2*x^6 + a^4*b^4*x^4 - a^2*b^6*x^
2 + b^8)) - sqrt(2)*a*b*sqrt((5*sqrt(1/5)*a^2*b^2*sqrt(1/(a^4*b^4)) - 1)/(a^2*b^2))*log((sqrt(2)*(3*a^8*x^8 +
5*a^6*b^2*x^6 + 9*a^4*b^4*x^4 + 5*a^2*b^6*x^2 + 3*b^8 + 5*sqrt(1/5)*(a^10*b^2*x^8 + 3*a^8*b^4*x^6 + 3*a^6*b^6*
x^4 + 3*a^4*b^8*x^2 + a^2*b^10)*sqrt(1/(a^4*b^4)))*sqrt((5*sqrt(1/5)*a^2*b^2*sqrt(1/(a^4*b^4)) - 1)/(a^2*b^2))
- 4*(3*a^4*x^5 + a^2*b^2*x^3 + 3*b^4*x + 5*sqrt(1/5)*(a^6*b^2*x^5 + a^4*b^4*x^3 + a^2*b^6*x)*sqrt(1/(a^4*b^4)
))*sqrt(a^4*x^4 + b^4))/(a^8*x^8 - a^6*b^2*x^6 + a^4*b^4*x^4 - a^2*b^6*x^2 + b^8)) + 2*sqrt(2)*arctan(sqrt(2)*
a*b*x/sqrt(a^4*x^4 + b^4)))/(a*b)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{10} x^{10} - b^{10}}{{\left (a^{10} x^{10} + b^{10}\right )} \sqrt {a^{4} x^{4} + b^{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^10*x^10-b^10)/(a^4*x^4+b^4)^(1/2)/(a^10*x^10+b^10),x, algorithm="giac")

[Out]

integrate((a^10*x^10 - b^10)/((a^10*x^10 + b^10)*sqrt(a^4*x^4 + b^4)), x)

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maple [B]  time = 0.31, size = 187, normalized size = 0.51

method	result	size
elliptic	\(\frac {\left (\frac {4 \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{x \sqrt {-a^{2} b^{2}+\sqrt {5}\, \sqrt {a^{4} b^{4}}}}\right )}{5 \sqrt {-a^{2} b^{2}+\sqrt {5}\, \sqrt {a^{4} b^{4}}}}+\frac {4 \arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{x \sqrt {-a^{2} b^{2}-\sqrt {5}\, \sqrt {a^{4} b^{4}}}}\right )}{5 \sqrt {-a^{2} b^{2}-\sqrt {5}\, \sqrt {a^{4} b^{4}}}}+\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}+b^{4}}\, \sqrt {2}}{2 x a b}\right )}{5 a b}\right ) \sqrt {2}}{2}\)	\(187\)
default	\(\frac {\sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, i\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}+b^{4}}}-\frac {b^{2} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{8} \textit {\_Z}^{8}-a^{6} b^{2} \textit {\_Z}^{6}+a^{4} b^{4} \textit {\_Z}^{4}-a^{2} b^{6} \textit {\_Z}^{2}+b^{8}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{6} a^{6}+2 \underline {\hspace {1.25 ex}}\alpha ^{4} a^{4} b^{2}-3 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b^{4}+4 b^{6}\right ) \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{6} a^{6}+\underline {\hspace {1.25 ex}}\alpha ^{4} a^{4} b^{2}-\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b^{4}+a^{2} b^{4} x^{2}+b^{6}\right ) a^{2}}{b^{4} \sqrt {a^{4} \underline {\hspace {1.25 ex}}\alpha ^{4}+b^{4}}\, \sqrt {a^{4} x^{4}+b^{4}}}\right )}{\sqrt {a^{4} \underline {\hspace {1.25 ex}}\alpha ^{4}+b^{4}}}+\frac {2 a^{2} \underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{6} a^{6}-\underline {\hspace {1.25 ex}}\alpha ^{4} a^{4} b^{2}+\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b^{4}-b^{6}\right ) \sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, \frac {i \left (\underline {\hspace {1.25 ex}}\alpha ^{6} a^{6}-\underline {\hspace {1.25 ex}}\alpha ^{4} a^{4} b^{2}+\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b^{4}-b^{6}\right )}{b^{6}}, \frac {\sqrt {-\frac {i a^{2}}{b^{2}}}}{\sqrt {\frac {i a^{2}}{b^{2}}}}\right )}{\sqrt {\frac {i a^{2}}{b^{2}}}\, b^{8} \sqrt {a^{4} x^{4}+b^{4}}}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (4 \underline {\hspace {1.25 ex}}\alpha ^{6} a^{6}-3 \underline {\hspace {1.25 ex}}\alpha ^{4} a^{4} b^{2}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b^{4}-b^{6}\right )}\right )}{10 a^{2}}-\frac {2 \sqrt {1-\frac {i a^{2} x^{2}}{b^{2}}}\, \sqrt {1+\frac {i a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {\frac {i a^{2}}{b^{2}}}, i, \frac {\sqrt {-\frac {i a^{2}}{b^{2}}}}{\sqrt {\frac {i a^{2}}{b^{2}}}}\right )}{5 \sqrt {\frac {i a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}+b^{4}}}\)	\(577\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^10*x^10-b^10)/(a^4*x^4+b^4)^(1/2)/(a^10*x^10+b^10),x,method=_RETURNVERBOSE)

[Out]

1/2*(4/5/(-a^2*b^2+5^(1/2)*(a^4*b^4)^(1/2))^(1/2)*arctan((a^4*x^4+b^4)^(1/2)*2^(1/2)/x/(-a^2*b^2+5^(1/2)*(a^4*
b^4)^(1/2))^(1/2))+4/5/(-a^2*b^2-5^(1/2)*(a^4*b^4)^(1/2))^(1/2)*arctan((a^4*x^4+b^4)^(1/2)*2^(1/2)/x/(-a^2*b^2
-5^(1/2)*(a^4*b^4)^(1/2))^(1/2))+1/5/a/b*arctan(1/2*(a^4*x^4+b^4)^(1/2)*2^(1/2)/x/a/b))*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{10} x^{10} - b^{10}}{{\left (a^{10} x^{10} + b^{10}\right )} \sqrt {a^{4} x^{4} + b^{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^10*x^10-b^10)/(a^4*x^4+b^4)^(1/2)/(a^10*x^10+b^10),x, algorithm="maxima")

[Out]

integrate((a^10*x^10 - b^10)/((a^10*x^10 + b^10)*sqrt(a^4*x^4 + b^4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {b^{10}-a^{10}\,x^{10}}{\sqrt {a^4\,x^4+b^4}\,\left (a^{10}\,x^{10}+b^{10}\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(b^10 - a^10*x^10)/((b^4 + a^4*x^4)^(1/2)*(b^10 + a^10*x^10)),x)

[Out]

int(-(b^10 - a^10*x^10)/((b^4 + a^4*x^4)^(1/2)*(b^10 + a^10*x^10)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**10*x**10-b**10)/(a**4*x**4+b**4)**(1/2)/(a**10*x**10+b**10),x)

[Out]

Timed out

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