\(\int \frac {-1+2 x}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}} \, dx\) [822]

Optimal result

Integrand size = 29, antiderivative size = 62 \[ \int \frac {-1+2 x}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}} \, dx=-\frac {2 \arctan \left (\frac {a x}{1+3 x+3 x^2+x^3+\sqrt {1+6 x+\left (15-a^2\right ) x^2+20 x^3+15 x^4+6 x^5+x^6}}\right )}{a} \]

[Out]

-2*arctan(a*x/(1+3*x+3*x^2+x^3+(1+6*x+(-a^2+15)*x^2+20*x^3+15*x^4+6*x^5+x^6)^(1/2)))/a

Rubi [F]

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

[In]

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

[Out]

2*Defer[Int][1/Sqrt[-(a^2*x^2) + (1 + x)^6], x] - 3*Defer[Int][1/((1 + x)*Sqrt[-(a^2*x^2) + (1 + x)^6]), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2}{\sqrt {-a^2 x^2+(1+x)^6}}-\frac {3}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}}\right ) \, dx \\ & = 2 \int \frac {1}{\sqrt {-a^2 x^2+(1+x)^6}} \, dx-3 \int \frac {1}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \frac {-1+2 x}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}} \, dx=-\frac {2 \arctan \left (\frac {a x}{1+3 x+3 x^2+x^3+\sqrt {1+6 x+\left (15-a^2\right ) x^2+20 x^3+15 x^4+6 x^5+x^6}}\right )}{a} \]

[In]

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

[Out]

(-2*ArcTan[(a*x)/(1 + 3*x + 3*x^2 + x^3 + Sqrt[1 + 6*x + (15 - a^2)*x^2 + 20*x^3 + 15*x^4 + 6*x^5 + x^6])])/a

Maple [A] (verified)

Time = 1.56 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.15

method	result	size
pseudoelliptic	\(-\frac {\ln \left (2\right )+\ln \left (\frac {-a^{2} x +\sqrt {-a^{2}}\, \sqrt {-\left (-x^{3}-3 x^{2}+\left (-3+a \right ) x -1\right ) \left (x^{3}+3 x^{2}+\left (3+a \right ) x +1\right )}}{\left (1+x \right )^{3}}\right )}{\sqrt {-a^{2}}}\)	\(71\)

[In]

int((-1+2*x)/(1+x)/(-a^2*x^2+(1+x)^6)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-(ln(2)+ln((-a^2*x+(-a^2)^(1/2)*(-(-x^3-3*x^2+(-3+a)*x-1)*(x^3+3*x^2+(3+a)*x+1))^(1/2))/(1+x)^3))/(-a^2)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.76 \[ \int \frac {-1+2 x}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}} \, dx=\frac {\arctan \left (\frac {\sqrt {x^{6} + 6 \, x^{5} + 15 \, x^{4} - {\left (a^{2} - 15\right )} x^{2} + 20 \, x^{3} + 6 \, x + 1}}{a x}\right )}{a} \]

[In]

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

[Out]

arctan(sqrt(x^6 + 6*x^5 + 15*x^4 - (a^2 - 15)*x^2 + 20*x^3 + 6*x + 1)/(a*x))/a

Sympy [F]

\[ \int \frac {-1+2 x}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}} \, dx=\int \frac {2 x - 1}{\sqrt {\left (- a x + x^{3} + 3 x^{2} + 3 x + 1\right ) \left (a x + x^{3} + 3 x^{2} + 3 x + 1\right )} \left (x + 1\right )}\, dx \]

[In]

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

[Out]

Integral((2*x - 1)/(sqrt((-a*x + x**3 + 3*x**2 + 3*x + 1)*(a*x + x**3 + 3*x**2 + 3*x + 1))*(x + 1)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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