3.9.58 \(\int \frac {-2 b+c x^2}{(-b+c x^2) \sqrt [4]{-b+c x^2+a x^4}} \, dx\) [858]

Optimal. Leaf size=65 \[ \frac {\text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+c x^2+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+c x^2+a x^4}}\right )}{\sqrt [4]{a}} \]

[Out]

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

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Rubi [F]
time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-2 b+c x^2}{\left (-b+c x^2\right ) \sqrt [4]{-b+c x^2+a x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[Int][(-2*b + c*x^2)/((-b + c*x^2)*(-b + c*x^2 + a*x^4)^(1/4)), x]

Rubi steps

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

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Mathematica [A]
time = 0.36, size = 59, normalized size = 0.91 \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+c x^2+a x^4}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+c x^2+a x^4}}\right )}{\sqrt [4]{a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {c \,x^{2}-2 b}{\left (c \,x^{2}-b \right ) \left (a \,x^{4}+c \,x^{2}-b \right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (53) = 106\).
time = 0.44, size = 135, normalized size = 2.08 \begin {gather*} \frac {2 \, \arctan \left (\frac {\frac {x \sqrt {\frac {\sqrt {a} x^{2} + \sqrt {a x^{4} + c x^{2} - b}}{x^{2}}}}{a^{\frac {1}{4}}} - \frac {{\left (a x^{4} + c x^{2} - b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}}{x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (\frac {a^{\frac {1}{4}} x + {\left (a x^{4} + c x^{2} - b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} - \frac {\log \left (-\frac {a^{\frac {1}{4}} x - {\left (a x^{4} + c x^{2} - b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {2\,b-c\,x^2}{\left (b-c\,x^2\right )\,{\left (a\,x^4+c\,x^2-b\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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