3.28.47 \(\int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\)

Optimal. Leaf size=255 \[ -\frac {4 \sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt {3} \sqrt [6]{c}}\right )}{a \sqrt [6]{c}}+\frac {4 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [6]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt {3} \sqrt [6]{c}}+\frac {1}{\sqrt {3}}\right )}{a \sqrt [6]{c}}-\frac {8 \tanh ^{-1}\left (\frac {\sqrt [6]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [6]{c}}\right )}{a \sqrt [6]{c}}-\frac {4 \tanh ^{-1}\left (\frac {\frac {\sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [6]{c}}+\sqrt [6]{c}}{\sqrt [6]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}\right )}{a \sqrt [6]{c}} \]

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Rubi [F]  time = 0.35, 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 {1}{\sqrt {-b+a^2 x^2} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/(Sqrt[-b + a^2*x^2]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)),x]

[Out]

Defer[Int][1/(Sqrt[-b + a^2*x^2]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)), x]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx &=\int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.52, size = 298, normalized size = 1.17 \begin {gather*} -\frac {2 \left (-\log \left (-\sqrt [6]{c} \sqrt [6]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}+\sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}+\sqrt [3]{c}\right )+\log \left (\sqrt [6]{c} \sqrt [6]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}+\sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}+\sqrt [3]{c}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [6]{c}}}{\sqrt {3}}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [6]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [6]{c}}+1}{\sqrt {3}}\right )+4 \tanh ^{-1}\left (\frac {\sqrt [6]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [6]{c}}\right )\right )}{a \sqrt [6]{c}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[-b + a^2*x^2]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)),x]

[Out]

(-2*(2*Sqrt[3]*ArcTan[(1 - (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6))/c^(1/6))/Sqrt[3]] - 2*Sqrt[3]*ArcT
an[(1 + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6))/c^(1/6))/Sqrt[3]] + 4*ArcTanh[(c + (a*x + Sqrt[-b + a
^2*x^2])^(1/4))^(1/6)/c^(1/6)] - Log[c^(1/3) - c^(1/6)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6) + (c + (a*
x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)] + Log[c^(1/3) + c^(1/6)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6) + (
c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)]))/(a*c^(1/6))

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IntegrateAlgebraic [A]  time = 1.26, size = 255, normalized size = 1.00 \begin {gather*} -\frac {4 \sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [6]{c}}\right )}{a \sqrt [6]{c}}+\frac {4 \sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [6]{c}}\right )}{a \sqrt [6]{c}}-\frac {8 \tanh ^{-1}\left (\frac {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}\right )}{a \sqrt [6]{c}}-\frac {4 \tanh ^{-1}\left (\frac {\sqrt [6]{c}+\frac {\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}}{\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}\right )}{a \sqrt [6]{c}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(Sqrt[-b + a^2*x^2]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)),x]

[Out]

(-4*Sqrt[3]*ArcTan[1/Sqrt[3] - (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6))/(Sqrt[3]*c^(1/6))])/(a*c^(1/6)
) + (4*Sqrt[3]*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6))/(Sqrt[3]*c^(1/6))])/(a*c^(1
/6)) - (8*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)/c^(1/6)])/(a*c^(1/6)) - (4*ArcTanh[(c^(1/6) + (
c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)/c^(1/6))/(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)])/(a*c^(1/6)
)

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fricas [B]  time = 0.82, size = 590, normalized size = 2.31 \begin {gather*} -8 \, \sqrt {3} \left (\frac {1}{a^{6} c}\right )^{\frac {1}{6}} \arctan \left (\frac {2}{3} \, \sqrt {3} \sqrt {a^{5} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}} c \left (\frac {1}{a^{6} c}\right )^{\frac {5}{6}} + a^{4} c \left (\frac {1}{a^{6} c}\right )^{\frac {2}{3}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}} a \left (\frac {1}{a^{6} c}\right )^{\frac {1}{6}} - \frac {2}{3} \, \sqrt {3} a {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}} \left (\frac {1}{a^{6} c}\right )^{\frac {1}{6}} - \frac {1}{3} \, \sqrt {3}\right ) - 8 \, \sqrt {3} \left (\frac {1}{a^{6} c}\right )^{\frac {1}{6}} \arctan \left (\frac {1}{3} \, \sqrt {3} \sqrt {-4 \, a^{5} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}} c \left (\frac {1}{a^{6} c}\right )^{\frac {5}{6}} + 4 \, a^{4} c \left (\frac {1}{a^{6} c}\right )^{\frac {2}{3}} + 4 \, {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}} a \left (\frac {1}{a^{6} c}\right )^{\frac {1}{6}} - \frac {2}{3} \, \sqrt {3} a {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}} \left (\frac {1}{a^{6} c}\right )^{\frac {1}{6}} + \frac {1}{3} \, \sqrt {3}\right ) - 2 \, \left (\frac {1}{a^{6} c}\right )^{\frac {1}{6}} \log \left (4 \, a^{5} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}} c \left (\frac {1}{a^{6} c}\right )^{\frac {5}{6}} + 4 \, a^{4} c \left (\frac {1}{a^{6} c}\right )^{\frac {2}{3}} + 4 \, {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}\right ) + 2 \, \left (\frac {1}{a^{6} c}\right )^{\frac {1}{6}} \log \left (-4 \, a^{5} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}} c \left (\frac {1}{a^{6} c}\right )^{\frac {5}{6}} + 4 \, a^{4} c \left (\frac {1}{a^{6} c}\right )^{\frac {2}{3}} + 4 \, {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}\right ) - 4 \, \left (\frac {1}{a^{6} c}\right )^{\frac {1}{6}} \log \left (a^{5} c \left (\frac {1}{a^{6} c}\right )^{\frac {5}{6}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) + 4 \, \left (\frac {1}{a^{6} c}\right )^{\frac {1}{6}} \log \left (-a^{5} c \left (\frac {1}{a^{6} c}\right )^{\frac {5}{6}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^2*x^2-b)^(1/2)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6),x, algorithm="fricas")

[Out]

-8*sqrt(3)*(1/(a^6*c))^(1/6)*arctan(2/3*sqrt(3)*sqrt(a^5*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)*c*(1/(a^6
*c))^(5/6) + a^4*c*(1/(a^6*c))^(2/3) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3))*a*(1/(a^6*c))^(1/6) - 2/3*
sqrt(3)*a*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)*(1/(a^6*c))^(1/6) - 1/3*sqrt(3)) - 8*sqrt(3)*(1/(a^6*c))
^(1/6)*arctan(1/3*sqrt(3)*sqrt(-4*a^5*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)*c*(1/(a^6*c))^(5/6) + 4*a^4*
c*(1/(a^6*c))^(2/3) + 4*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3))*a*(1/(a^6*c))^(1/6) - 2/3*sqrt(3)*a*(c +
(a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)*(1/(a^6*c))^(1/6) + 1/3*sqrt(3)) - 2*(1/(a^6*c))^(1/6)*log(4*a^5*(c + (
a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)*c*(1/(a^6*c))^(5/6) + 4*a^4*c*(1/(a^6*c))^(2/3) + 4*(c + (a*x + sqrt(a^2
*x^2 - b))^(1/4))^(1/3)) + 2*(1/(a^6*c))^(1/6)*log(-4*a^5*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)*c*(1/(a^
6*c))^(5/6) + 4*a^4*c*(1/(a^6*c))^(2/3) + 4*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)) - 4*(1/(a^6*c))^(1/6)
*log(a^5*c*(1/(a^6*c))^(5/6) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)) + 4*(1/(a^6*c))^(1/6)*log(-a^5*c*(
1/(a^6*c))^(5/6) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6))

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^2*x^2-b)^(1/2)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {a^{2} x^{2}-b}\, \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {1}{6}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^2*x^2-b)^(1/2)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6),x)

[Out]

int(1/(a^2*x^2-b)^(1/2)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a^{2} x^{2} - b} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^2*x^2-b)^(1/2)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(a^2*x^2 - b)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\right )}^{1/6}\,\sqrt {a^2\,x^2-b}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((c + (a*x + (a^2*x^2 - b)^(1/2))^(1/4))^(1/6)*(a^2*x^2 - b)^(1/2)),x)

[Out]

int(1/((c + (a*x + (a^2*x^2 - b)^(1/2))^(1/4))^(1/6)*(a^2*x^2 - b)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [6]{c + \sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}}} \sqrt {a^{2} x^{2} - b}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a**2*x**2-b)**(1/2)/(c+(a*x+(a**2*x**2-b)**(1/2))**(1/4))**(1/6),x)

[Out]

Integral(1/((c + (a*x + sqrt(a**2*x**2 - b))**(1/4))**(1/6)*sqrt(a**2*x**2 - b)), x)

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