3.28.71 \(\int \frac {-b^4+c^2 x^2+a^4 x^4}{\sqrt {-b^4+a^4 x^4} (b^4+a^4 x^4)} \, dx\)

Optimal. Leaf size=263 \[ -\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \left (2 a^2 b^2+i c^2\right ) \tan ^{-1}\left (\frac {(1+i) a b x}{\sqrt {a^4 x^4-b^4}+a^2 x^2+i b^2}\right )}{a^3 b^3}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \left (2 a^2 b^2-i c^2\right ) \tanh ^{-1}\left (\frac {(1-i) \sqrt {a^4 x^4-b^4}+(1-i) a^2 x^2+(1+i) b^2}{2 \sqrt {3-2 \sqrt {2}} a b x}\right )}{a^3 b^3}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (c^2+2 i a^2 b^2\right ) \tanh ^{-1}\left (\frac {(1-i) \sqrt {a^4 x^4-b^4}+(1-i) a^2 x^2+(1+i) b^2}{2 \sqrt {3+2 \sqrt {2}} a b x}\right )}{a^3 b^3} \]

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Rubi [C]  time = 1.08, antiderivative size = 350, normalized size of antiderivative = 1.33, number of steps used = 16, number of rules used = 7, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.137, Rules used = {6725, 224, 221, 1211, 1699, 205, 208} \begin {gather*} \frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {a^4 x^4-b^4}}-\frac {\left (2 \sqrt {-a^4} b^2-c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{4 \sqrt {2} \left (-a^4\right )^{3/4} b^3}-\frac {\left (2 \sqrt {-a^4} b^2+c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{4 \sqrt {2} \left (-a^4\right )^{3/4} b^3}-\frac {\left (2 a^4 b^2-\sqrt {-a^4} c^2\right ) \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a^5 b \sqrt {a^4 x^4-b^4}}-\frac {\left (2 a^4 b^2+\sqrt {-a^4} c^2\right ) \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a^5 b \sqrt {a^4 x^4-b^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-b^4 + c^2*x^2 + a^4*x^4)/(Sqrt[-b^4 + a^4*x^4]*(b^4 + a^4*x^4)),x]

[Out]

-1/4*((2*Sqrt[-a^4]*b^2 - c^2)*ArcTan[(Sqrt[2]*(-a^4)^(1/4)*b*x)/Sqrt[-b^4 + a^4*x^4]])/(Sqrt[2]*(-a^4)^(3/4)*
b^3) - ((2*Sqrt[-a^4]*b^2 + c^2)*ArcTanh[(Sqrt[2]*(-a^4)^(1/4)*b*x)/Sqrt[-b^4 + a^4*x^4]])/(4*Sqrt[2]*(-a^4)^(
3/4)*b^3) + (b*Sqrt[1 - (a^4*x^4)/b^4]*EllipticF[ArcSin[(a*x)/b], -1])/(a*Sqrt[-b^4 + a^4*x^4]) - ((2*a^4*b^2
- Sqrt[-a^4]*c^2)*Sqrt[1 - (a^4*x^4)/b^4]*EllipticF[ArcSin[(a*x)/b], -1])/(4*a^5*b*Sqrt[-b^4 + a^4*x^4]) - ((2
*a^4*b^2 + Sqrt[-a^4]*c^2)*Sqrt[1 - (a^4*x^4)/b^4]*EllipticF[ArcSin[(a*x)/b], -1])/(4*a^5*b*Sqrt[-b^4 + a^4*x^
4])

Rule 205

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

Rule 208

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

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + (b*x^4)/a]/Sqrt[a + b*x^4], Int[1/Sqrt[1 + (b*x^4)
/a], x], x] /; FreeQ[{a, b}, x] && NegQ[b/a] &&  !GtQ[a, 0]

Rule 1211

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[1/(2*d), Int[1/Sqrt[a + c*x^4], x],
x] + Dist[1/(2*d), Int[(d - e*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d
^2 + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0]

Rule 1699

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[A, Subst[Int[1/
(d + 2*a*e*x^2), x], x, x/Sqrt[a + c*x^4]], x] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ
[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {-b^4+c^2 x^2+a^4 x^4}{\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )} \, dx &=\int \left (\frac {1}{\sqrt {-b^4+a^4 x^4}}-\frac {2 b^4-c^2 x^2}{\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )}\right ) \, dx\\ &=\int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx-\int \frac {2 b^4-c^2 x^2}{\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )} \, dx\\ &=\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{\sqrt {-b^4+a^4 x^4}}-\int \left (-\frac {\sqrt {-a^4} \left (2 \sqrt {-a^4} b^4-b^2 c^2\right )}{2 a^4 b^2 \left (b^2-\sqrt {-a^4} x^2\right ) \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {-a^4} \left (-2 \sqrt {-a^4} b^4-b^2 c^2\right )}{2 a^4 b^2 \left (b^2+\sqrt {-a^4} x^2\right ) \sqrt {-b^4+a^4 x^4}}\right ) \, dx\\ &=\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}+\frac {1}{2} \left (-2 b^2-\frac {c^2}{\sqrt {-a^4}}\right ) \int \frac {1}{\left (b^2+\sqrt {-a^4} x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx+\frac {\left (\sqrt {-a^4} \left (2 \sqrt {-a^4} b^4-b^2 c^2\right )\right ) \int \frac {1}{\left (b^2-\sqrt {-a^4} x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{2 a^4 b^2}\\ &=\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {\left (2 b^2+\frac {c^2}{\sqrt {-a^4}}\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx}{4 b^2}-\frac {\left (2 b^2+\frac {c^2}{\sqrt {-a^4}}\right ) \int \frac {b^2-\sqrt {-a^4} x^2}{\left (b^2+\sqrt {-a^4} x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{4 b^2}+\frac {1}{4} \left (-2+\frac {c^2}{\sqrt {-a^4} b^2}\right ) \int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx+\frac {1}{4} \left (-2+\frac {c^2}{\sqrt {-a^4} b^2}\right ) \int \frac {b^2+\sqrt {-a^4} x^2}{\left (b^2-\sqrt {-a^4} x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx\\ &=\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {1}{4} \left (2 b^2+\frac {c^2}{\sqrt {-a^4}}\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-2 \sqrt {-a^4} b^4 x^2} \, dx,x,\frac {x}{\sqrt {-b^4+a^4 x^4}}\right )-\frac {1}{4} \left (b^2 \left (2-\frac {c^2}{\sqrt {-a^4} b^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2+2 \sqrt {-a^4} b^4 x^2} \, dx,x,\frac {x}{\sqrt {-b^4+a^4 x^4}}\right )-\frac {\left (\left (2 b^2+\frac {c^2}{\sqrt {-a^4}}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{4 b^2 \sqrt {-b^4+a^4 x^4}}+\frac {\left (\left (-2+\frac {c^2}{\sqrt {-a^4} b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}\right ) \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{4 \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {\left (2 \sqrt {-a^4} b^2-c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{4 \sqrt {2} \left (-a^4\right )^{3/4} b^3}-\frac {\left (2 \sqrt {-a^4} b^2+c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{4 \sqrt {2} \left (-a^4\right )^{3/4} b^3}+\frac {b \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{a \sqrt {-b^4+a^4 x^4}}-\frac {\left (2 b^2+\frac {c^2}{\sqrt {-a^4}}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a b \sqrt {-b^4+a^4 x^4}}-\frac {b \left (2-\frac {c^2}{\sqrt {-a^4} b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a \sqrt {-b^4+a^4 x^4}}\\ \end {align*}

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Mathematica [C]  time = 0.65, size = 305, normalized size = 1.16 \begin {gather*} \frac {x \left (5 c^2 x^2 \sqrt {1-\frac {a^4 x^4}{b^4}} F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};\frac {a^4 x^4}{b^4},-\frac {a^4 x^4}{b^4}\right )+3 a^4 x^4 \sqrt {1-\frac {a^4 x^4}{b^4}} F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\frac {a^4 x^4}{b^4},-\frac {a^4 x^4}{b^4}\right )-\frac {75 b^{12} F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\frac {a^4 x^4}{b^4},-\frac {a^4 x^4}{b^4}\right )}{\left (a^4 x^4+b^4\right ) \left (5 b^4 F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\frac {a^4 x^4}{b^4},-\frac {a^4 x^4}{b^4}\right )+2 a^4 x^4 \left (F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};\frac {a^4 x^4}{b^4},-\frac {a^4 x^4}{b^4}\right )-2 F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};\frac {a^4 x^4}{b^4},-\frac {a^4 x^4}{b^4}\right )\right )\right )}\right )}{15 b^4 \sqrt {a^4 x^4-b^4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(-b^4 + c^2*x^2 + a^4*x^4)/(Sqrt[-b^4 + a^4*x^4]*(b^4 + a^4*x^4)),x]

[Out]

(x*(5*c^2*x^2*Sqrt[1 - (a^4*x^4)/b^4]*AppellF1[3/4, 1/2, 1, 7/4, (a^4*x^4)/b^4, -((a^4*x^4)/b^4)] + 3*a^4*x^4*
Sqrt[1 - (a^4*x^4)/b^4]*AppellF1[5/4, 1/2, 1, 9/4, (a^4*x^4)/b^4, -((a^4*x^4)/b^4)] - (75*b^12*AppellF1[1/4, 1
/2, 1, 5/4, (a^4*x^4)/b^4, -((a^4*x^4)/b^4)])/((b^4 + a^4*x^4)*(5*b^4*AppellF1[1/4, 1/2, 1, 5/4, (a^4*x^4)/b^4
, -((a^4*x^4)/b^4)] + 2*a^4*x^4*(-2*AppellF1[5/4, 1/2, 2, 9/4, (a^4*x^4)/b^4, -((a^4*x^4)/b^4)] + AppellF1[5/4
, 3/2, 1, 9/4, (a^4*x^4)/b^4, -((a^4*x^4)/b^4)])))))/(15*b^4*Sqrt[-b^4 + a^4*x^4])

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IntegrateAlgebraic [A]  time = 1.54, size = 227, normalized size = 0.86 \begin {gather*} \frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 a^2 b^2-i c^2\right ) \tan ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-b^4+a^4 x^4}}{a b x}\right )}{a^3 b^3}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 a^2 b^2+i c^2\right ) \log \left (i b^2-(1-i) a b x+a^2 x^2+\sqrt {-b^4+a^4 x^4}\right )}{a^3 b^3}-\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 a^2 b^2+i c^2\right ) \log \left (-a^3 b^5+(1+i) a^4 b^4 x+i a^5 b^3 x^2+i a^3 b^3 \sqrt {-b^4+a^4 x^4}\right )}{a^3 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-b^4 + c^2*x^2 + a^4*x^4)/(Sqrt[-b^4 + a^4*x^4]*(b^4 + a^4*x^4)),x]

[Out]

((1/8 + I/8)*(2*a^2*b^2 - I*c^2)*ArcTan[((1/2 + I/2)*Sqrt[-b^4 + a^4*x^4])/(a*b*x)])/(a^3*b^3) + ((1/8 + I/8)*
(2*a^2*b^2 + I*c^2)*Log[I*b^2 - (1 - I)*a*b*x + a^2*x^2 + Sqrt[-b^4 + a^4*x^4]])/(a^3*b^3) - ((1/8 + I/8)*(2*a
^2*b^2 + I*c^2)*Log[-(a^3*b^5) + (1 + I)*a^4*b^4*x + I*a^5*b^3*x^2 + I*a^3*b^3*Sqrt[-b^4 + a^4*x^4]])/(a^3*b^3
)

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fricas [A]  time = 7.60, size = 131, normalized size = 0.50 \begin {gather*} \frac {2 \, {\left (2 \, a^{2} b^{2} + c^{2}\right )} \arctan \left (\frac {\sqrt {a^{4} x^{4} - b^{4}} a x}{a^{2} b x^{2} + b^{3}}\right ) - {\left (2 \, a^{2} b^{2} - c^{2}\right )} \log \left (\frac {a^{4} x^{4} + 2 \, a^{2} b^{2} x^{2} - b^{4} + 2 \, \sqrt {a^{4} x^{4} - b^{4}} a b x}{a^{4} x^{4} + b^{4}}\right )}{8 \, a^{3} b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*(2*a^2*b^2 + c^2)*arctan(sqrt(a^4*x^4 - b^4)*a*x/(a^2*b*x^2 + b^3)) - (2*a^2*b^2 - c^2)*log((a^4*x^4 +
2*a^2*b^2*x^2 - b^4 + 2*sqrt(a^4*x^4 - b^4)*a*b*x)/(a^4*x^4 + b^4)))/(a^3*b^3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{4} x^{4} - b^{4} + c^{2} x^{2}}{{\left (a^{4} x^{4} + b^{4}\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^4*x^4-b^4+c^2*x^2)/(a^4*x^4-b^4)^(1/2)/(a^4*x^4+b^4),x, algorithm="giac")

[Out]

integrate((a^4*x^4 - b^4 + c^2*x^2)/((a^4*x^4 + b^4)*sqrt(a^4*x^4 - b^4)), x)

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maple [C]  time = 0.24, size = 267, normalized size = 1.02

method	result	size
default	\(\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )}{\sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} a^{4}+b^{4}\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} c^{2}-2 b^{4}\right ) \left (-\frac {\sqrt {2}\, \arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}\right ) a^{4}}{\sqrt {-2 b^{4}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{\sqrt {-b^{4}}}+\frac {4 \underline {\hspace {1.25 ex}}\alpha ^{3} a^{4} \sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}}{b^{2}}, \frac {\sqrt {\frac {a^{2}}{b^{2}}}}{\sqrt {-\frac {a^{2}}{b^{2}}}}\right )}{\sqrt {-\frac {a^{2}}{b^{2}}}\, b^{4} \sqrt {a^{4} x^{4}-b^{4}}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{16 a^{4}}\)	\(267\)
elliptic	\(\frac {\left (\frac {c^{2} \left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}\right )}{16 a^{4} b^{4}}+\frac {c^{2} \left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}+1\right )}{8 a^{4} b^{4}}+\frac {c^{2} \left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}-1\right )}{8 a^{4} b^{4}}+\frac {\sqrt {2}\, \ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}\right )}{8 \left (a^{4} b^{4}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}+1\right )}{4 \left (a^{4} b^{4}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}-1\right )}{4 \left (a^{4} b^{4}\right )^{\frac {1}{4}}}\right ) \sqrt {2}}{2}\)	\(476\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(-a^2/b^2)^(1/2)*(a^2*x^2/b^2+1)^(1/2)*(1-a^2*x^2/b^2)^(1/2)/(a^4*x^4-b^4)^(1/2)*EllipticF(x*(-a^2/b^2)^(1/2
),I)+1/16/a^4*sum((_alpha^2*c^2-2*b^4)/_alpha^3*(-2^(1/2)/(-b^4)^(1/2)*arctanh(_alpha^2*(_alpha^2+x^2)*a^4/(-2
*b^4)^(1/2)/(a^4*x^4-b^4)^(1/2))+4/(-a^2/b^2)^(1/2)*_alpha^3*a^4/b^4*(a^2*x^2/b^2+1)^(1/2)*(1-a^2*x^2/b^2)^(1/
2)/(a^4*x^4-b^4)^(1/2)*EllipticPi(x*(-a^2/b^2)^(1/2),_alpha^2*a^2/b^2,(a^2/b^2)^(1/2)/(-a^2/b^2)^(1/2))),_alph
a=RootOf(_Z^4*a^4+b^4))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{4} x^{4} - b^{4} + c^{2} x^{2}}{{\left (a^{4} x^{4} + b^{4}\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^4*x^4-b^4+c^2*x^2)/(a^4*x^4-b^4)^(1/2)/(a^4*x^4+b^4),x, algorithm="maxima")

[Out]

integrate((a^4*x^4 - b^4 + c^2*x^2)/((a^4*x^4 + b^4)*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 {a^4\,x^4-b^4+c^2\,x^2}{\left (a^4\,x^4+b^4\right )\,\sqrt {a^4\,x^4-b^4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{4} x^{4} - b^{4} + c^{2} x^{2}}{\sqrt {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right )} \left (a^{4} x^{4} + b^{4}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a**4*x**4 - b**4 + c**2*x**2)/(sqrt((a*x - b)*(a*x + b)*(a**2*x**2 + b**2))*(a**4*x**4 + b**4)), x)

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