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

Optimal. Leaf size=62 \[ -\frac {2 \tan ^{-1}\left (\frac {a x}{\sqrt {\left (15-a^2\right ) x^2+x^6+6 x^5+15 x^4+20 x^3+6 x+1}+x^3+3 x^2+3 x+1}\right )}{a} \]

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Rubi [F]  time = 0.50, 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+2 x}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}} \, dx \end {gather*}

Verification is not applicable to the result.

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

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Mathematica [F]  time = 0.31, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+2 x}{(1+x) \sqrt {-a^2 x^2+(1+x)^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.43, size = 62, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-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

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fricas [A]  time = 0.54, size = 47, normalized size = 0.76 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {-1+2 x}{\left (1+x \right ) \sqrt {-a^{2} x^{2}+\left (1+x \right )^{6}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {2\,x-1}{\sqrt {{\left (x+1\right )}^6-a^2\,x^2}\,\left (x+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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