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

Optimal. Leaf size=415 \[ \frac {i \left (\sqrt {2} c-(1-i) \sqrt {2 \sqrt {a} \sqrt {b}+i c^2}\right ) \sqrt {-(-1)^{3/4} c \sqrt {2 \sqrt {a} \sqrt {b}+i c^2}-i \sqrt {a} \sqrt {b}+c^2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {-(-1)^{3/4} c \sqrt {2 \sqrt {a} \sqrt {b}+i c^2}-i \sqrt {a} \sqrt {b}+c^2}}{\sqrt {a x^4-b}+\sqrt {a} x^2+i \sqrt {b}}\right )}{2 \sqrt {a} \sqrt {b} c}-\frac {i \left (\sqrt {2} c+(1-i) \sqrt {2 \sqrt {a} \sqrt {b}+i c^2}\right ) \sqrt {-(-1)^{3/4} c \sqrt {2 \sqrt {a} \sqrt {b}+i c^2}+i \sqrt {a} \sqrt {b}-c^2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {-(-1)^{3/4} c \sqrt {2 \sqrt {a} \sqrt {b}+i c^2}+i \sqrt {a} \sqrt {b}-c^2}}{\sqrt {a x^4-b}+\sqrt {a} x^2+i \sqrt {b}}\right )}{2 \sqrt {a} \sqrt {b} c} \]

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Rubi [A]  time = 0.15, antiderivative size = 22, normalized size of antiderivative = 0.05, number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {2112, 205} \begin {gather*} -\frac {\tan ^{-1}\left (\frac {c x}{\sqrt {a x^4-b}}\right )}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcTan[(c*x)/Sqrt[-b + a*x^4]]/c)

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 2112

Int[((u_)*((A_) + (B_.)*(x_)^4))/Sqrt[v_], x_Symbol] :> With[{a = Coeff[v, x, 0], b = Coeff[v, x, 2], c = Coef
f[v, x, 4], d = Coeff[1/u, x, 0], e = Coeff[1/u, x, 2], f = Coeff[1/u, x, 4]}, Dist[A, Subst[Int[1/(d - (b*d -
a*e)*x^2), x], x, x/Sqrt[v]], x] /; EqQ[a*B + A*c, 0] && EqQ[c*d - a*f, 0]] /; FreeQ[{A, B}, x] && PolyQ[v, x
^2, 2] && PolyQ[1/u, x^2, 2]

Rubi steps

\begin {align*} \int \frac {b+a x^4}{\sqrt {-b+a x^4} \left (-b+c^2 x^2+a x^4\right )} \, dx &=b \operatorname {Subst}\left (\int \frac {1}{-b-b c^2 x^2} \, dx,x,\frac {x}{\sqrt {-b+a x^4}}\right )\\ &=-\frac {\tan ^{-1}\left (\frac {c x}{\sqrt {-b+a x^4}}\right )}{c}\\ \end {align*}

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Mathematica [C]  time = 1.08, size = 190, normalized size = 0.46 \begin {gather*} -\frac {i \sqrt {1-\frac {a x^4}{b}} \left (-\Pi \left (\frac {2 \sqrt {a} \sqrt {b}}{\sqrt {c^4+4 a b}-c^2};\left .i \sinh ^{-1}\left (\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} x\right )\right |-1\right )-\Pi \left (-\frac {2 \sqrt {a} \sqrt {b}}{c^2+\sqrt {c^4+4 a b}};\left .i \sinh ^{-1}\left (\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} x\right )\right |-1\right )+F\left (\left .i \sinh ^{-1}\left (\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} x\right )\right |-1\right )\right )}{\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} \sqrt {a x^4-b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*Sqrt[1 - (a*x^4)/b]*(EllipticF[I*ArcSinh[Sqrt[-(Sqrt[a]/Sqrt[b])]*x], -1] - EllipticPi[(2*Sqrt[a]*Sqrt[b
])/(-c^2 + Sqrt[4*a*b + c^4]), I*ArcSinh[Sqrt[-(Sqrt[a]/Sqrt[b])]*x], -1] - EllipticPi[(-2*Sqrt[a]*Sqrt[b])/(c
^2 + Sqrt[4*a*b + c^4]), I*ArcSinh[Sqrt[-(Sqrt[a]/Sqrt[b])]*x], -1]))/(Sqrt[-(Sqrt[a]/Sqrt[b])]*Sqrt[-b + a*x^
4])

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IntegrateAlgebraic [A]  time = 1.18, size = 31, normalized size = 0.07 \begin {gather*} \frac {\tan ^{-1}\left (\frac {c x \sqrt {-b+a x^4}}{b-a x^4}\right )}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(c*x*Sqrt[-b + a*x^4])/(b - a*x^4)]/c

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fricas [A]  time = 1.55, size = 40, normalized size = 0.10 \begin {gather*} -\frac {\arctan \left (\frac {2 \, \sqrt {a x^{4} - b} c x}{a x^{4} - c^{2} x^{2} - b}\right )}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arctan(2*sqrt(a*x^4 - b)*c*x/(a*x^4 - c^2*x^2 - b))/c

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.21, size = 24, normalized size = 0.06

method	result	size
elliptic	\(\frac {\arctan \left (\frac {\sqrt {a \,x^{4}-b}}{x c}\right )}{c}\)	\(24\)
default	\(\frac {\sqrt {1+\frac {\sqrt {a}\, x^{2}}{\sqrt {b}}}\, \sqrt {1-\frac {\sqrt {a}\, x^{2}}{\sqrt {b}}}\, \EllipticF \left (x \sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}, i\right )}{\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}\, \sqrt {a \,x^{4}-b}}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{4}+c^{2} \textit {\_Z}^{2}-b \right )}{\sum }\frac {\left (c^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}-2 b \right ) \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2} a +a \,x^{2}-c^{2}\right )}{\sqrt {-c^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {a \,x^{4}-b}}\right )}{\sqrt {-c^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a +c^{2}\right ) \sqrt {1+\frac {\sqrt {a}\, x^{2}}{\sqrt {b}}}\, \sqrt {1-\frac {\sqrt {a}\, x^{2}}{\sqrt {b}}}\, \EllipticPi \left (x \sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}, -\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} a +c^{2}}{\sqrt {b}\, \sqrt {a}}, \frac {\sqrt {\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}\right )}{\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}\, b \sqrt {a \,x^{4}-b}}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a +c^{2}\right )}\right )}{4}\)	\(288\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a\,x^4+b}{\sqrt {a\,x^4-b}\,\left (c^2\,x^2+a\,x^4-b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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