3.30.15 \(\int \frac {-b+a x^4}{\sqrt {b+a x^4} (b-c^2 x^2+a x^4)} \, dx\) [2915]

Optimal. Leaf size=329 \[ -\frac {\left (c+\sqrt {2 \sqrt {a} \sqrt {b}+c^2}\right ) \sqrt {-\sqrt {a} \sqrt {b}-c^2+c \sqrt {2 \sqrt {a} \sqrt {b}+c^2}} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {-\sqrt {a} \sqrt {b}-c^2+c \sqrt {2 \sqrt {a} \sqrt {b}+c^2}} x}{\sqrt {b}+\sqrt {a} x^2+\sqrt {b+a x^4}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b} c}-\frac {\left (-c+\sqrt {2 \sqrt {a} \sqrt {b}+c^2}\right ) \sqrt {\sqrt {a} \sqrt {b}+c^2+c \sqrt {2 \sqrt {a} \sqrt {b}+c^2}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {a} \sqrt {b}+c^2+c \sqrt {2 \sqrt {a} \sqrt {b}+c^2}} x}{\sqrt {b}+\sqrt {a} x^2+\sqrt {b+a x^4}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b} c} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 20, normalized size of antiderivative = 0.06, number of steps used = 2, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2137, 214} \begin {gather*} -\frac {\tanh ^{-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]

-(ArcTanh[(c*x)/Sqrt[b + a*x^4]]/c)

Rule 214

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

Rule 2137

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

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Mathematica [A]
time = 0.89, size = 52, normalized size = 0.16 \begin {gather*} -\frac {\log \left (c x+\sqrt {b+a x^4}\right )}{2 c}+\frac {\log \left (c^2 x-c \sqrt {b+a x^4}\right )}{2 c} \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]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.14, size = 297, normalized size = 0.90

method	result	size
elliptic	\(-\frac {\arctanh \left (\frac {\sqrt {a \,x^{4}+b}}{x c}\right )}{c}\)	\(23\)
default	\(\frac {\sqrt {1-\frac {i \sqrt {a}\, x^{2}}{\sqrt {b}}}\, \sqrt {1+\frac {i \sqrt {a}\, x^{2}}{\sqrt {b}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}, i\right )}{\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, \sqrt {a \,x^{4}+b}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{4}-c^{2} \textit {\_Z}^{2}+b \right )}{\sum }\frac {\left (c^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}-2 b \right ) \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2} a +a \,x^{2}+c^{2}\right )}{\sqrt {c^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}}\, \sqrt {a \,x^{4}+b}}\right )}{\sqrt {c^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a -c^{2}\right ) \sqrt {1-\frac {i \sqrt {a}\, x^{2}}{\sqrt {b}}}\, \sqrt {1+\frac {i \sqrt {a}\, x^{2}}{\sqrt {b}}}\, \EllipticPi \left (x \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}, \frac {i \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a -c^{2}\right )}{\sqrt {a}\, \sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {a}}{\sqrt {b}}}}{\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}}\right )}{\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, b \sqrt {a \,x^{4}+b}}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a -c^{2}\right )}\right )}{4}\)	\(297\)

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/(I*a^(1/2)/b^(1/2))^(1/2)*(1-I*a^(1/2)/b^(1/2)*x^2)^(1/2)*(1+I*a^(1/2)/b^(1/2)*x^2)^(1/2)/(a*x^4+b)^(1/2)*El
lipticF(x*(I*a^(1/2)/b^(1/2))^(1/2),I)+1/4*sum((_alpha^2*c^2-2*b)/_alpha/(2*_alpha^2*a-c^2)*(-1/(c^2*_alpha^2)
^(1/2)*arctanh(_alpha^2*(-_alpha^2*a+a*x^2+c^2)/(c^2*_alpha^2)^(1/2)/(a*x^4+b)^(1/2))+2/(I*a^(1/2)/b^(1/2))^(1
/2)*_alpha*(_alpha^2*a-c^2)/b*(1-I*a^(1/2)/b^(1/2)*x^2)^(1/2)*(1+I*a^(1/2)/b^(1/2)*x^2)^(1/2)/(a*x^4+b)^(1/2)*
EllipticPi(x*(I*a^(1/2)/b^(1/2))^(1/2),I/a^(1/2)/b^(1/2)*(_alpha^2*a-c^2),(-I*a^(1/2)/b^(1/2))^(1/2)/(I*a^(1/2
)/b^(1/2))^(1/2))),_alpha=RootOf(_Z^4*a-_Z^2*c^2+b))

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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((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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Fricas [A]
time = 0.45, size = 51, normalized size = 0.16 \begin {gather*} \frac {\log \left (\frac {a x^{4} + c^{2} x^{2} - 2 \, \sqrt {a x^{4} + b} c x + b}{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*log((a*x^4 + c^2*x^2 - 2*sqrt(a*x^4 + b)*c*x + b)/(a*x^4 - c^2*x^2 + b))/c

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

[Out]

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

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