3.194 \(\int \frac {x}{(a+b \sin ^{-1}(c x))^{3/2}} \, dx\)

Optimal. Leaf size=130 \[ \frac {2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2} c^2}+\frac {2 \sqrt {\pi } \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2} c^2}-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \sin ^{-1}(c x)}} \]

[Out]

2*cos(2*a/b)*FresnelC(2*(a+b*arcsin(c*x))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/2)/b^(3/2)/c^2+2*FresnelS(2*(a+b*arcsi
n(c*x))^(1/2)/b^(1/2)/Pi^(1/2))*sin(2*a/b)*Pi^(1/2)/b^(3/2)/c^2-2*x*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arcsin(c*x))^(
1/2)

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Rubi [A]  time = 0.16, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4631, 3306, 3305, 3351, 3304, 3352} \[ \frac {2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {\pi } \sqrt {b}}\right )}{b^{3/2} c^2}+\frac {2 \sqrt {\pi } \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2} c^2}-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \sin ^{-1}(c x)}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*ArcSin[c*x])^(3/2),x]

[Out]

(-2*x*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[a + b*ArcSin[c*x]]) + (2*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSi
n[c*x]])/(Sqrt[b]*Sqrt[Pi])])/(b^(3/2)*c^2) + (2*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt[P
i])]*Sin[(2*a)/b])/(b^(3/2)*c^2)

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 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 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 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 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]

Rule 4631

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[1 - c^2*x^2]*(a + b*ArcSin
[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a + b*x)^(n + 1)
, Sin[x]^(m - 1)*(m - (m + 1)*Sin[x]^2), x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && G
eQ[n, -2] && LtQ[n, -1]

Rubi steps

\begin {align*} \int \frac {x}{\left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}+\frac {2 \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}+\frac {\left (2 \cos \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}+\frac {\left (2 \sin \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}+\frac {\left (4 \cos \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{b^2 c^2}+\frac {\left (4 \sin \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{b^2 c^2}\\ &=-\frac {2 x \sqrt {1-c^2 x^2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}+\frac {2 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2} c^2}+\frac {2 \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{b^{3/2} c^2}\\ \end {align*}

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Mathematica [C]  time = 0.15, size = 155, normalized size = 1.19 \[ \frac {i e^{-\frac {2 i a}{b}} \left (2 i e^{\frac {2 i a}{b}} \sin \left (2 \sin ^{-1}(c x)\right )-\sqrt {2} \sqrt {-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {1}{2},-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+\sqrt {2} e^{\frac {4 i a}{b}} \sqrt {\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {1}{2},\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )}{2 b c^2 \sqrt {a+b \sin ^{-1}(c x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x/(a + b*ArcSin[c*x])^(3/2),x]

[Out]

((I/2)*(-(Sqrt[2]*Sqrt[((-I)*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c*x]))/b]) + Sqrt[2]*E^(
((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c*x]))/b]*Gamma[1/2, ((2*I)*(a + b*ArcSin[c*x]))/b] + (2*I)*E^(((2*I)*a)/b)
*Sin[2*ArcSin[c*x]]))/(b*c^2*E^(((2*I)*a)/b)*Sqrt[a + b*ArcSin[c*x]])

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*arcsin(c*x))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*arcsin(c*x))^(3/2),x, algorithm="giac")

[Out]

integrate(x/(b*arcsin(c*x) + a)^(3/2), x)

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maple [A]  time = 0.07, size = 143, normalized size = 1.10 \[ -\frac {-2 \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right )-2 \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right )+\sin \left (\frac {2 a +2 b \arcsin \left (c x \right )}{b}-\frac {2 a}{b}\right )}{c^{2} b \sqrt {a +b \arcsin \left (c x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a+b*arcsin(c*x))^(3/2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*arcsin(c*x))^(3/2),x, algorithm="maxima")

[Out]

integrate(x/(b*arcsin(c*x) + a)^(3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*asin(c*x))^(3/2),x)

[Out]

int(x/(a + b*asin(c*x))^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*asin(c*x))**(3/2),x)

[Out]

Integral(x/(a + b*asin(c*x))**(3/2), x)

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