3.352 \(\int \frac{a+b x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\)

Optimal. Leaf size=47 \[ \frac{\left (2 a c^2+b\right ) \cosh ^{-1}(c x)}{2 c^3}+\frac{b x \sqrt{c x-1} \sqrt{c x+1}}{2 c^2} \]

[Out]

(b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*c^2) + ((b + 2*a*c^2)*ArcCosh[c*x])/(2*c^3)

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Rubi [A]  time = 0.0195731, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {389, 52} \[ \frac{\left (2 a c^2+b\right ) \cosh ^{-1}(c x)}{2 c^3}+\frac{b x \sqrt{c x-1} \sqrt{c x+1}}{2 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*c^2) + ((b + 2*a*c^2)*ArcCosh[c*x])/(2*c^3)

Rule 389

Int[((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symb
ol] :> Simp[(d*x*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1))/(b1*b2*(n*(p + 1) + 1)), x] - Dist[(a1*a
2*d - b1*b2*c*(n*(p + 1) + 1))/(b1*b2*(n*(p + 1) + 1)), Int[(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /;
FreeQ[{a1, b1, a2, b2, c, d, n, p}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \frac{a+b x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx &=\frac{b x \sqrt{-1+c x} \sqrt{1+c x}}{2 c^2}-\frac{\left (-b-2 a c^2\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 c^2}\\ &=\frac{b x \sqrt{-1+c x} \sqrt{1+c x}}{2 c^2}+\frac{\left (b+2 a c^2\right ) \cosh ^{-1}(c x)}{2 c^3}\\ \end{align*}

Mathematica [B]  time = 0.152255, size = 101, normalized size = 2.15 \[ \frac{4 \left (a c^2+b\right ) \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )+\frac{b \left (c x \sqrt{-(c x-1)^2} \sqrt{c x+1}-2 \sqrt{c x-1} \sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )\right )}{\sqrt{1-c x}}}{2 c^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [C]  time = 0.013, size = 103, normalized size = 2.2 \begin{align*}{\frac{{\it csgn} \left ( c \right ) }{2\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1} \left ( \sqrt{{c}^{2}{x}^{2}-1}{\it csgn} \left ( c \right ) cxb+2\,\ln \left ( \left ( \sqrt{{c}^{2}{x}^{2}-1}{\it csgn} \left ( c \right ) +cx \right ){\it csgn} \left ( c \right ) \right ) a{c}^{2}+\ln \left ( \left ( \sqrt{{c}^{2}{x}^{2}-1}{\it csgn} \left ( c \right ) +cx \right ){\it csgn} \left ( c \right ) \right ) b \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.977853, size = 120, normalized size = 2.55 \begin{align*} \frac{a \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}}} + \frac{\sqrt{c^{2} x^{2} - 1} b x}{2 \, c^{2}} + \frac{b \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{2 \, \sqrt{c^{2}} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.45222, size = 136, normalized size = 2.89 \begin{align*} \frac{\sqrt{c x + 1} \sqrt{c x - 1} b c x -{\left (2 \, a c^{2} + b\right )} \log \left (-c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{2 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 14.6635, size = 182, normalized size = 3.87 \begin{align*} \frac{a{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c} - \frac{i a{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c} + \frac{b{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & - \frac{1}{2}, - \frac{1}{2}, 0, 1 \\-1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{3}} - \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, -1, -1, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*meijerg(((1/4, 3/4), (1/2, 1/2, 1, 1)), ((0, 1/4, 1/2, 3/4, 1, 0), ()), 1/(c**2*x**2))/(4*pi**(3/2)*c) - I*a
*meijerg(((-1/2, -1/4, 0, 1/4, 1/2, 1), ()), ((-1/4, 1/4), (-1/2, 0, 0, 0)), exp_polar(2*I*pi)/(c**2*x**2))/(4
*pi**(3/2)*c) + b*meijerg(((-3/4, -1/4), (-1/2, -1/2, 0, 1)), ((-1, -3/4, -1/2, -1/4, 0, 0), ()), 1/(c**2*x**2
))/(4*pi**(3/2)*c**3) - I*b*meijerg(((-3/2, -5/4, -1, -3/4, -1/2, 1), ()), ((-5/4, -3/4), (-3/2, -1, -1, 0)),
exp_polar(2*I*pi)/(c**2*x**2))/(4*pi**(3/2)*c**3)

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Giac [A]  time = 1.22789, size = 96, normalized size = 2.04 \begin{align*} \frac{{\left ({\left (c x + 1\right )} b c^{4} - b c^{4}\right )} \sqrt{c x + 1} \sqrt{c x - 1} - 2 \,{\left (2 \, a c^{6} + b c^{4}\right )} \log \left ({\left | -\sqrt{c x + 1} + \sqrt{c x - 1} \right |}\right )}{384 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(((c*x + 1)*b*c^4 - b*c^4)*sqrt(c*x + 1)*sqrt(c*x - 1) - 2*(2*a*c^6 + b*c^4)*log(abs(-sqrt(c*x + 1) + sq
rt(c*x - 1))))/c