\(\int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\) [2747]

Optimal result

Integrand size = 45, antiderivative size = 255 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=-\frac {4 \sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [6]{c}}\right )}{a \sqrt [6]{c}}+\frac {4 \sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [6]{c}}\right )}{a \sqrt [6]{c}}-\frac {8 \text {arctanh}\left (\frac {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}\right )}{a \sqrt [6]{c}}-\frac {4 \text {arctanh}\left (\frac {\sqrt [6]{c}+\frac {\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}}{\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}\right )}{a \sqrt [6]{c}} \]

[Out]

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

Rubi [F]

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

[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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=-\frac {4 \left (\sqrt {3} \left (\arctan \left (\frac {1-\frac {2 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}}{\sqrt {3}}\right )-\arctan \left (\frac {1+\frac {2 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}}{\sqrt {3}}\right )\right )+2 \text {arctanh}\left (\frac {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}\right )+\text {arctanh}\left (\frac {\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}\right )\right )}{a \sqrt [6]{c}} \]

[In]

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

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.13

method	result	size
derivativedivides	\(\frac {-\frac {2 \ln \left ({\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}+c^{\frac {1}{6}} {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}+c^{\frac {1}{3}}\right )}{c^{\frac {1}{6}}}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}}{3}+\frac {2 {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}} \sqrt {3}}{3 c^{\frac {1}{6}}}\right )}{c^{\frac {1}{6}}}-\frac {4 \ln \left ({\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}+c^{\frac {1}{6}}\right )}{c^{\frac {1}{6}}}+\frac {4 \ln \left (-{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}+c^{\frac {1}{6}}\right )}{c^{\frac {1}{6}}}+\frac {2 \ln \left (c^{\frac {1}{6}} {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}-{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}-c^{\frac {1}{3}}\right )}{c^{\frac {1}{6}}}+\frac {4 \sqrt {3}\, \arctan \left (-\frac {\sqrt {3}}{3}+\frac {2 {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}} \sqrt {3}}{3 c^{\frac {1}{6}}}\right )}{c^{\frac {1}{6}}}}{a}\)	\(288\)
default	\(\frac {-\frac {2 \ln \left ({\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}+c^{\frac {1}{6}} {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}+c^{\frac {1}{3}}\right )}{c^{\frac {1}{6}}}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}}{3}+\frac {2 {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}} \sqrt {3}}{3 c^{\frac {1}{6}}}\right )}{c^{\frac {1}{6}}}-\frac {4 \ln \left ({\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}+c^{\frac {1}{6}}\right )}{c^{\frac {1}{6}}}+\frac {4 \ln \left (-{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}+c^{\frac {1}{6}}\right )}{c^{\frac {1}{6}}}+\frac {2 \ln \left (c^{\frac {1}{6}} {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}}-{\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{3}}-c^{\frac {1}{3}}\right )}{c^{\frac {1}{6}}}+\frac {4 \sqrt {3}\, \arctan \left (-\frac {\sqrt {3}}{3}+\frac {2 {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )}^{\frac {1}{6}} \sqrt {3}}{3 c^{\frac {1}{6}}}\right )}{c^{\frac {1}{6}}}}{a}\)	\(288\)

[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,method=_RETURNVERBOSE)

[Out]

4/a*(-1/2/c^(1/6)*ln((c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)+c^(1/6)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6)+c
^(1/3))+1/c^(1/6)*3^(1/2)*arctan(1/3*3^(1/2)+2/3*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6)*3^(1/2)/c^(1/6))-1/c^
(1/6)*ln((c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6)+c^(1/6))+1/c^(1/6)*ln(-(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6
)+c^(1/6))+1/2/c^(1/6)*ln(c^(1/6)*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6)-(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1
/3)-c^(1/3))+1/c^(1/6)*3^(1/2)*arctan(-1/3*3^(1/2)+2/3*(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6)*3^(1/2)/c^(1/6)
))

Fricas [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.47 \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=2 \, {\left (\sqrt {-3} - 1\right )} \left (\frac {1}{a^{6} c}\right )^{\frac {1}{6}} \log \left ({\left (\sqrt {-3} a^{5} c + a^{5} c\right )} \left (\frac {1}{a^{6} c}\right )^{\frac {5}{6}} + 2 \, {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) - 2 \, {\left (\sqrt {-3} - 1\right )} \left (\frac {1}{a^{6} c}\right )^{\frac {1}{6}} \log \left (-{\left (\sqrt {-3} a^{5} c + a^{5} c\right )} \left (\frac {1}{a^{6} c}\right )^{\frac {5}{6}} + 2 \, {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) + 2 \, {\left (\sqrt {-3} + 1\right )} \left (\frac {1}{a^{6} c}\right )^{\frac {1}{6}} \log \left ({\left (\sqrt {-3} a^{5} c - a^{5} c\right )} \left (\frac {1}{a^{6} c}\right )^{\frac {5}{6}} + 2 \, {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) - 2 \, {\left (\sqrt {-3} + 1\right )} \left (\frac {1}{a^{6} c}\right )^{\frac {1}{6}} \log \left (-{\left (\sqrt {-3} a^{5} c - a^{5} c\right )} \left (\frac {1}{a^{6} c}\right )^{\frac {5}{6}} + 2 \, {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) - 4 \, \left (\frac {1}{a^{6} c}\right )^{\frac {1}{6}} \log \left (a^{5} c \left (\frac {1}{a^{6} c}\right )^{\frac {5}{6}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) + 4 \, \left (\frac {1}{a^{6} c}\right )^{\frac {1}{6}} \log \left (-a^{5} c \left (\frac {1}{a^{6} c}\right )^{\frac {5}{6}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) \]

[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]

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

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\text {Timed out} \]

[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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {1}{{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\right )}^{1/6}\,\sqrt {a^2\,x^2-b}} \,d x \]

[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)