3.29.99 \(\int \frac {x^4}{\sqrt {-b^4+a^4 x^4} (-b^8+a^8 x^8)} \, dx\)

Optimal. Leaf size=319 \[ -\frac {x}{4 a^4 b^4 \sqrt {a^4 x^4-b^4}}-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) \tanh ^{-1}\left (\frac {\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {a^4 x^4-b^4}}{\sqrt {3-2 \sqrt {2}} a b}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) a x^2}{\sqrt {3-2 \sqrt {2}} b}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) b}{\sqrt {3-2 \sqrt {2}} a}}{x}\right )}{a^5 b^5}+\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) \tanh ^{-1}\left (\frac {\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {a^4 x^4-b^4}}{\sqrt {3+2 \sqrt {2}} a b}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) a x^2}{\sqrt {3+2 \sqrt {2}} b}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) b}{\sqrt {3+2 \sqrt {2}} a}}{x}\right )}{a^5 b^5}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \tan ^{-1}\left (\frac {(1+i) a b x}{\sqrt {a^4 x^4-b^4}+a^2 x^2+i b^2}\right )}{a^5 b^5} \]

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Rubi [A]  time = 0.08, antiderivative size = 120, normalized size of antiderivative = 0.38, number of steps used = 4, number of rules used = 4, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1479, 471, 21, 405} \begin {gather*} -\frac {x}{4 a^4 b^4 \sqrt {a^4 x^4-b^4}}+\frac {\tan ^{-1}\left (\frac {a x \left (b^2-a^2 x^2\right )}{b \sqrt {a^4 x^4-b^4}}\right )}{8 a^5 b^5}+\frac {\tanh ^{-1}\left (\frac {a x \left (a^2 x^2+b^2\right )}{b \sqrt {a^4 x^4-b^4}}\right )}{8 a^5 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(Sqrt[-b^4 + a^4*x^4]*(-b^8 + a^8*x^8)),x]

[Out]

-1/4*x/(a^4*b^4*Sqrt[-b^4 + a^4*x^4]) + ArcTan[(a*x*(b^2 - a^2*x^2))/(b*Sqrt[-b^4 + a^4*x^4])]/(8*a^5*b^5) + A
rcTanh[(a*x*(b^2 + a^2*x^2))/(b*Sqrt[-b^4 + a^4*x^4])]/(8*a^5*b^5)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 405

Int[Sqrt[(a_) + (b_.)*(x_)^4]/((c_) + (d_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-(a*b), 4]}, Simp[(a*ArcTan[(q*
x*(a + q^2*x^2))/(a*Sqrt[a + b*x^4])])/(2*c*q), x] + Simp[(a*ArcTanh[(q*x*(a - q^2*x^2))/(a*Sqrt[a + b*x^4])])
/(2*c*q), x]] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && NegQ[a*b]

Rule 471

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(n -
1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(n*(b*c - a*d)*(p + 1)), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 1479

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (c_.)*(x_)^(n2_))^(p_.), x_Symbol] :> Int[(f*x)^
m*(d + e*x^n)^(q + p)*(a/d + (c*x^n)/e)^p, x] /; FreeQ[{a, c, d, e, f, q, m, n, q}, x] && EqQ[n2, 2*n] && EqQ[
c*d^2 + a*e^2, 0] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {x^4}{\sqrt {-b^4+a^4 x^4} \left (-b^8+a^8 x^8\right )} \, dx &=\int \frac {x^4}{\left (-b^4+a^4 x^4\right )^{3/2} \left (b^4+a^4 x^4\right )} \, dx\\ &=-\frac {x}{4 a^4 b^4 \sqrt {-b^4+a^4 x^4}}+\frac {\int \frac {b^4-a^4 x^4}{\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )} \, dx}{4 a^4 b^4}\\ &=-\frac {x}{4 a^4 b^4 \sqrt {-b^4+a^4 x^4}}-\frac {\int \frac {\sqrt {-b^4+a^4 x^4}}{b^4+a^4 x^4} \, dx}{4 a^4 b^4}\\ &=-\frac {x}{4 a^4 b^4 \sqrt {-b^4+a^4 x^4}}+\frac {\tan ^{-1}\left (\frac {a x \left (b^2-a^2 x^2\right )}{b \sqrt {-b^4+a^4 x^4}}\right )}{8 a^5 b^5}+\frac {\tanh ^{-1}\left (\frac {a x \left (b^2+a^2 x^2\right )}{b \sqrt {-b^4+a^4 x^4}}\right )}{8 a^5 b^5}\\ \end {align*}

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Mathematica [C]  time = 0.47, size = 205, normalized size = 0.64 \begin {gather*} \frac {x \left (\frac {5 \left (b^4-a^4 x^4\right ) 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 (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 )+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 )\right )\right )}-\frac {1}{b^4}\right )}{4 a^4 \sqrt {a^4 x^4-b^4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4/(Sqrt[-b^4 + a^4*x^4]*(-b^8 + a^8*x^8)),x]

[Out]

(x*(-b^(-4) + (5*(b^4 - a^4*x^4)*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, 1/2, 1, 9/4, (a^4*x^4)/b^4, -((a^4*x^4)/b^4)])))))/(4*a^4*
Sqrt[-b^4 + a^4*x^4])

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IntegrateAlgebraic [A]  time = 0.92, size = 253, normalized size = 0.79 \begin {gather*} -\frac {x}{4 a^4 b^4 \sqrt {-b^4+a^4 x^4}}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \tan ^{-1}\left (\frac {(1+i) a b x}{i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}}\right )}{a^5 b^5}-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) \tanh ^{-1}\left (\frac {b^4-(1+i) a b^3 x-(1-i) a^3 b x^3-a^4 x^4+\left (-i b^2-(1-i) a b x-a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}{i b^4-(1-i) a b^3 x+(1+i) a^3 b x^3-i a^4 x^4+\left (b^2+(1+i) a b x-i a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}\right )}{a^5 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^4/(Sqrt[-b^4 + a^4*x^4]*(-b^8 + a^8*x^8)),x]

[Out]

-1/4*x/(a^4*b^4*Sqrt[-b^4 + a^4*x^4]) + ((1/8 - I/8)*ArcTan[((1 + I)*a*b*x)/(I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4
*x^4])])/(a^5*b^5) - ((1/16 - I/16)*ArcTanh[(b^4 - (1 + I)*a*b^3*x - (1 - I)*a^3*b*x^3 - a^4*x^4 + ((-I)*b^2 -
(1 - I)*a*b*x - a^2*x^2)*Sqrt[-b^4 + a^4*x^4])/(I*b^4 - (1 - I)*a*b^3*x + (1 + I)*a^3*b*x^3 - I*a^4*x^4 + (b^
2 + (1 + I)*a*b*x - I*a^2*x^2)*Sqrt[-b^4 + a^4*x^4])])/(a^5*b^5)

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fricas [A]  time = 1.43, size = 166, normalized size = 0.52 \begin {gather*} -\frac {4 \, \sqrt {a^{4} x^{4} - b^{4}} a b x + 2 \, {\left (a^{4} x^{4} - b^{4}\right )} \arctan \left (\frac {\sqrt {a^{4} x^{4} - b^{4}} a x}{a^{2} b x^{2} + b^{3}}\right ) - {\left (a^{4} x^{4} - b^{4}\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 )}{16 \, {\left (a^{9} b^{5} x^{4} - a^{5} b^{9}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a^4*x^4-b^4)^(1/2)/(a^8*x^8-b^8),x, algorithm="fricas")

[Out]

-1/16*(4*sqrt(a^4*x^4 - b^4)*a*b*x + 2*(a^4*x^4 - b^4)*arctan(sqrt(a^4*x^4 - b^4)*a*x/(a^2*b*x^2 + b^3)) - (a^
4*x^4 - b^4)*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^9*b^5*x^4
- a^5*b^9)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{{\left (a^{8} x^{8} - b^{8}\right )} \sqrt {a^{4} x^{4} - b^{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a^4*x^4-b^4)^(1/2)/(a^8*x^8-b^8),x, algorithm="giac")

[Out]

integrate(x^4/((a^8*x^8 - b^8)*sqrt(a^4*x^4 - b^4)), x)

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maple [A]  time = 0.25, size = 273, normalized size = 0.86

method	result	size
elliptic	\(\frac {\left (-\frac {\sqrt {2}\, \ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}\right )}{32 a^{4} b^{4} \left (a^{4} b^{4}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}+1\right )}{16 a^{4} b^{4} \left (a^{4} b^{4}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}-1\right )}{16 a^{4} b^{4} \left (a^{4} b^{4}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, x}{4 a^{4} b^{4} \sqrt {a^{4} x^{4}-b^{4}}}\right ) \sqrt {2}}{2}\)	\(273\)
default	\(-\frac {\frac {a^{4} x^{3}-a^{3} b \,x^{2}+a^{2} b^{2} x -a \,b^{3}}{2 a^{2} b^{3} \sqrt {\left (x +\frac {b}{a}\right ) \left (a^{4} x^{3}-a^{3} b \,x^{2}+a^{2} b^{2} x -a \,b^{3}\right )}}+\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 )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}-\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \left (\EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )\right )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}}{8 b^{3} a^{4}}+\frac {-\frac {a^{4} x^{3}+a^{3} b \,x^{2}+a^{2} b^{2} x +a \,b^{3}}{2 a^{2} b^{3} \sqrt {\left (x -\frac {b}{a}\right ) \left (a^{4} x^{3}+a^{3} b \,x^{2}+a^{2} b^{2} x +a \,b^{3}\right )}}-\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 )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \left (\EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )\right )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}}{8 b^{3} a^{4}}+\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} a^{4}+b^{4}\right )}{\sum }\frac {-\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}}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{32 a^{8}}-\frac {-\frac {\left (a^{4} x^{2}-a^{2} b^{2}\right ) x}{2 b^{4} a^{2} \sqrt {\left (x^{2}+\frac {b^{2}}{a^{2}}\right ) \left (a^{4} x^{2}-a^{2} b^{2}\right )}}+\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 )}{2 b^{2} \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \left (\EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )\right )}{2 b^{2} \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}}{4 a^{4} b^{2}}\)	\(921\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a^4*x^4-b^4)^(1/2)/(a^8*x^8-b^8),x,method=_RETURNVERBOSE)

[Out]

1/2*(-1/32/a^4/b^4/(a^4*b^4)^(1/4)*2^(1/2)*ln((1/2*(a^4*x^4-b^4)/x^2-(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x+(a^
4*b^4)^(1/2))/(1/2*(a^4*x^4-b^4)/x^2+(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x+(a^4*b^4)^(1/2)))-1/16/a^4/b^4/(a^4
*b^4)^(1/4)*2^(1/2)*arctan(1/(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x+1)-1/16/a^4/b^4/(a^4*b^4)^(1/4)*2^(1/2)*arc
tan(1/(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x-1)-1/4/a^4/b^4/(a^4*x^4-b^4)^(1/2)*2^(1/2)*x)*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{{\left (a^{8} x^{8} - b^{8}\right )} \sqrt {a^{4} x^{4} - b^{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a^4*x^4-b^4)^(1/2)/(a^8*x^8-b^8),x, algorithm="maxima")

[Out]

integrate(x^4/((a^8*x^8 - b^8)*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 {x^4}{\sqrt {a^4\,x^4-b^4}\,\left (b^8-a^8\,x^8\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x^4/((a^4*x^4 - b^4)^(1/2)*(b^8 - a^8*x^8)),x)

[Out]

-int(x^4/((a^4*x^4 - b^4)^(1/2)*(b^8 - a^8*x^8)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{\sqrt {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right )} \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(x**4/(a**4*x**4-b**4)**(1/2)/(a**8*x**8-b**8),x)

[Out]

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

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