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

Optimal. Leaf size=16 \[ \frac{\log \left (a+b \sin ^{-1}(c x)\right )}{b c} \]

[Out]

Log[a + b*ArcSin[c*x]]/(b*c)

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Rubi [A]  time = 0.0480692, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {4639} \[ \frac{\log \left (a+b \sin ^{-1}(c x)\right )}{b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[a + b*ArcSin[c*x]]/(b*c)

Rule 4639

Int[1/(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Simp[Log[a + b*ArcSin[c*x]]
/(b*c*Sqrt[d]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]

Rubi steps

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

Mathematica [A]  time = 0.047529, size = 16, normalized size = 1. \[ \frac{\log \left (a+b \sin ^{-1}(c x)\right )}{b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[a + b*ArcSin[c*x]]/(b*c)

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Maple [A]  time = 0.006, size = 17, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( a+b\arcsin \left ( cx \right ) \right ) }{bc}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(a+b*arcsin(c*x))/b/c

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Maxima [A]  time = 1.50592, size = 22, normalized size = 1.38 \begin{align*} \frac{\log \left (b \arcsin \left (c x\right ) + a\right )}{b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*arcsin(c*x) + a)/(b*c)

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Fricas [A]  time = 1.93635, size = 42, normalized size = 2.62 \begin{align*} \frac{\log \left (-b \arcsin \left (c x\right ) - a\right )}{b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(-b*arcsin(c*x) - a)/(b*c)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [C]  time = 1.3367, size = 43, normalized size = 2.69 \begin{align*} \frac{\log \left (b^{2} \Im \left ( \arcsin \left (c x\right ) \right )^{2} +{\left (b \Re \left ( \arcsin \left (c x\right ) \right ) + a\right )}^{2}\right )}{2 \, b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(b^2*imag_part(arcsin(c*x))^2 + (b*real_part(arcsin(c*x)) + a)^2)/(b*c)