3.2.91 \(\int \frac {\sqrt {a+b x^4}}{a c-b c x^4} \, dx\) [191]

Optimal. Leaf size=103 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a+b x^4}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} c}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a+b x^4}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} c} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {413, 218, 212, 209} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a+b x^4}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} c}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a+b x^4}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*x)/Sqrt[a + b*x^4]]/(2*Sqrt[2]*a^(1/4)*b^(1/4)*c) + ArcTanh[(Sqrt[2]*a^(1/4)*b
^(1/4)*x)/Sqrt[a + b*x^4]]/(2*Sqrt[2]*a^(1/4)*b^(1/4)*c)

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 413

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

Rubi steps

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

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Mathematica [A]
time = 0.31, size = 81, normalized size = 0.79 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a+b x^4}}\right )+\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a+b x^4}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(ArcTan[(Sqrt[2]*a^(1/4)*b^(1/4)*x)/Sqrt[a + b*x^4]] + ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*x)/Sqrt[a + b*x^4]])/(
2*Sqrt[2]*a^(1/4)*b^(1/4)*c)

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Maple [A]
time = 0.29, size = 99, normalized size = 0.96

method	result	size
default	\(\frac {\left (-\frac {\arctan \left (\frac {\sqrt {b \,x^{4}+a}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right )}{2 \left (a b \right )^{\frac {1}{4}}}+\frac {\ln \left (\frac {\frac {\sqrt {b \,x^{4}+a}\, \sqrt {2}}{2 x}+\left (a b \right )^{\frac {1}{4}}}{\frac {\sqrt {b \,x^{4}+a}\, \sqrt {2}}{2 x}-\left (a b \right )^{\frac {1}{4}}}\right )}{4 \left (a b \right )^{\frac {1}{4}}}\right ) \sqrt {2}}{2 c}\)	\(99\)
elliptic	\(\frac {\left (-\frac {\arctan \left (\frac {\sqrt {b \,x^{4}+a}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right )}{2 c \left (a b \right )^{\frac {1}{4}}}+\frac {\ln \left (\frac {\frac {\sqrt {b \,x^{4}+a}\, \sqrt {2}}{2 x}+\left (a b \right )^{\frac {1}{4}}}{\frac {\sqrt {b \,x^{4}+a}\, \sqrt {2}}{2 x}-\left (a b \right )^{\frac {1}{4}}}\right )}{4 c \left (a b \right )^{\frac {1}{4}}}\right ) \sqrt {2}}{2}\)	\(102\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (71) = 142\).
time = 4.38, size = 315, normalized size = 3.06 \begin {gather*} -\left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} \arctan \left (\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} \sqrt {b x^{4} + a} c \left (\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} - \frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b c^{3} \left (\frac {1}{a b c^{4}}\right )^{\frac {3}{4}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} b c x^{2} \left (\frac {1}{a b c^{4}}\right )^{\frac {1}{4}}}{\sqrt {b}}}{x}\right ) + \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b c^{3} x^{3} \left (\frac {1}{a b c^{4}}\right )^{\frac {3}{4}} + 2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a c x \left (\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} + \sqrt {b x^{4} + a} {\left (a c^{2} \sqrt {\frac {1}{a b c^{4}}} + x^{2}\right )}}{b x^{4} - a}\right ) - \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b c^{3} x^{3} \left (\frac {1}{a b c^{4}}\right )^{\frac {3}{4}} + 2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a c x \left (\frac {1}{a b c^{4}}\right )^{\frac {1}{4}} - \sqrt {b x^{4} + a} {\left (a c^{2} \sqrt {\frac {1}{a b c^{4}}} + x^{2}\right )}}{b x^{4} - a}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {b\,x^4+a}}{a\,c-b\,c\,x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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