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

Optimal. Leaf size=120 \[ \frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{c}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{c}+x \sqrt{a+b \sin ^{-1}(c x)} \]

[Out]

x*Sqrt[a + b*ArcSin[c*x]] - (Sqrt[b]*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]
])/c + (Sqrt[b]*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/c

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Rubi [A]  time = 0.270951, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {4619, 4723, 3306, 3305, 3351, 3304, 3352} \[ \frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{c}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{c}+x \sqrt{a+b \sin ^{-1}(c x)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*ArcSin[c*x]],x]

[Out]

x*Sqrt[a + b*ArcSin[c*x]] - (Sqrt[b]*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]
])/c + (Sqrt[b]*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c*x]])/Sqrt[b]]*Sin[a/b])/c

Rule 4619

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[b*c*n, Int[
(x*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c^(
m + 1), Subst[Int[(a + b*x)^n*Sin[x]^m*Cos[x]^(2*p + 1), x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 3306

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d +
f*x]/Sqrt[c + d*x], x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c
, d, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]

Rule 3305

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[(f*x^2)/d], x], x,
Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[(f*x^2)/d],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rubi steps

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

Mathematica [C]  time = 0.0926495, size = 119, normalized size = 0.99 \[ \frac{b e^{-\frac{i a}{b}} \left (\sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+e^{\frac{2 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{3}{2},\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )}{2 c \sqrt{a+b \sin ^{-1}(c x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[a + b*ArcSin[c*x]],x]

[Out]

(b*(Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcSin[c*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I*(a +
b*ArcSin[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c*x]))/b]))/(2*c*E^((I*a)/b)*Sqrt[a + b*ArcSin[c*x]])

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Maple [A]  time = 0.001, size = 178, normalized size = 1.5 \begin{align*}{\frac{1}{2\,c} \left ( -\sqrt{2}\sqrt{\pi }\sqrt{{b}^{-1}}\sqrt{a+b\arcsin \left ( cx \right ) }\cos \left ({\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi }b}\sqrt{a+b\arcsin \left ( cx \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) b+\sqrt{2}\sqrt{\pi }\sqrt{{b}^{-1}}\sqrt{a+b\arcsin \left ( cx \right ) }\sin \left ({\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }b}\sqrt{a+b\arcsin \left ( cx \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) b+2\,\arcsin \left ( cx \right ) \sin \left ({\frac{a+b\arcsin \left ( cx \right ) }{b}}-{\frac{a}{b}} \right ) b+2\,\sin \left ({\frac{a+b\arcsin \left ( cx \right ) }{b}}-{\frac{a}{b}} \right ) a \right ){\frac{1}{\sqrt{a+b\arcsin \left ( cx \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/c/(a+b*arcsin(c*x))^(1/2)*(-2^(1/2)*Pi^(1/2)*(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*cos(a/b)*FresnelS(2^(1/2)
/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b+2^(1/2)*Pi^(1/2)*(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)*sin(a/
b)*FresnelC(2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(c*x))^(1/2)/b)*b+2*arcsin(c*x)*sin((a+b*arcsin(c*x))/b-a/
b)*b+2*sin((a+b*arcsin(c*x))/b-a/b)*a)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \arcsin \left (c x\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*arcsin(c*x) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \operatorname{asin}{\left (c x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*asin(c*x)), x)

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Giac [C]  time = 1.50831, size = 266, normalized size = 2.22 \begin{align*} \frac{i \, \sqrt{2} \sqrt{\pi } b \operatorname{erf}\left (-\frac{i \, \sqrt{2} \sqrt{b \arcsin \left (c x\right ) + a}}{2 \, \sqrt{{\left | b \right |}}} - \frac{\sqrt{2} \sqrt{b \arcsin \left (c x\right ) + a} \sqrt{{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac{i \, a}{b}\right )}}{4 \, c{\left (\frac{i \, b}{\sqrt{{\left | b \right |}}} + \sqrt{{\left | b \right |}}\right )}} - \frac{i \, \sqrt{2} \sqrt{\pi } b \operatorname{erf}\left (\frac{i \, \sqrt{2} \sqrt{b \arcsin \left (c x\right ) + a}}{2 \, \sqrt{{\left | b \right |}}} - \frac{\sqrt{2} \sqrt{b \arcsin \left (c x\right ) + a} \sqrt{{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac{i \, a}{b}\right )}}{4 \, c{\left (-\frac{i \, b}{\sqrt{{\left | b \right |}}} + \sqrt{{\left | b \right |}}\right )}} - \frac{i \, \sqrt{b \arcsin \left (c x\right ) + a} e^{\left (i \, \arcsin \left (c x\right )\right )}}{2 \, c} + \frac{i \, \sqrt{b \arcsin \left (c x\right ) + a} e^{\left (-i \, \arcsin \left (c x\right )\right )}}{2 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*I*sqrt(2)*sqrt(pi)*b*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c
*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(c*(I*b/sqrt(abs(b)) + sqrt(abs(b)))) - 1/4*I*sqrt(2)*sqrt(pi)*b*erf(1/2*I*
sqrt(2)*sqrt(b*arcsin(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/
(c*(-I*b/sqrt(abs(b)) + sqrt(abs(b)))) - 1/2*I*sqrt(b*arcsin(c*x) + a)*e^(I*arcsin(c*x))/c + 1/2*I*sqrt(b*arcs
in(c*x) + a)*e^(-I*arcsin(c*x))/c