3.260 \(\int \frac{(c e+d e x)^3}{\sqrt{a+b \sin ^{-1}(c+d x)}} \, dx\)

Optimal. Leaf size=233 \[ -\frac{\sqrt{\pi } e^3 \sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{4 \sqrt{b} d}+\frac{\sqrt{\frac{\pi }{2}} e^3 \sin \left (\frac{4 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 \sqrt{b} d}-\frac{\sqrt{\frac{\pi }{2}} e^3 \cos \left (\frac{4 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 \sqrt{b} d}+\frac{\sqrt{\pi } e^3 \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{4 \sqrt{b} d} \]

[Out]

-(e^3*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(8*Sqrt[b]*d) + (e
^3*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(4*Sqrt[b]*d) - (e^3*Sq
rt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(4*Sqrt[b]*d) + (e^3*Sqrt[Pi
/2]*FresnelC[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(4*a)/b])/(8*Sqrt[b]*d)

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Rubi [A]  time = 0.51703, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {4805, 12, 4635, 4406, 3306, 3305, 3351, 3304, 3352} \[ -\frac{\sqrt{\pi } e^3 \sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{\pi } \sqrt{b}}\right )}{4 \sqrt{b} d}+\frac{\sqrt{\frac{\pi }{2}} e^3 \sin \left (\frac{4 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 \sqrt{b} d}-\frac{\sqrt{\frac{\pi }{2}} e^3 \cos \left (\frac{4 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 \sqrt{b} d}+\frac{\sqrt{\pi } e^3 \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{4 \sqrt{b} d} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^3/Sqrt[a + b*ArcSin[c + d*x]],x]

[Out]

-(e^3*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(8*Sqrt[b]*d) + (e
^3*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(4*Sqrt[b]*d) - (e^3*Sq
rt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(4*Sqrt[b]*d) + (e^3*Sqrt[Pi
/2]*FresnelC[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(4*a)/b])/(8*Sqrt[b]*d)

Rule 4805

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 4635

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*S
in[x]^m*Cos[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 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 \frac{(c e+d e x)^3}{\sqrt{a+b \sin ^{-1}(c+d x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^3 x^3}{\sqrt{a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 \operatorname{Subst}\left (\int \frac{\cos (x) \sin ^3(x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{d}\\ &=\frac{e^3 \operatorname{Subst}\left (\int \left (\frac{\sin (2 x)}{4 \sqrt{a+b x}}-\frac{\sin (4 x)}{8 \sqrt{a+b x}}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{d}\\ &=-\frac{e^3 \operatorname{Subst}\left (\int \frac{\sin (4 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 d}+\frac{e^3 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{4 d}\\ &=\frac{\left (e^3 \cos \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+d x)\right )}{4 d}-\frac{\left (e^3 \cos \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{4 a}{b}+4 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 d}-\frac{\left (e^3 \sin \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+d x)\right )}{4 d}+\frac{\left (e^3 \sin \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{4 a}{b}+4 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 d}\\ &=\frac{\left (e^3 \cos \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+d x)}\right )}{2 b d}-\frac{\left (e^3 \cos \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{4 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{4 b d}-\frac{\left (e^3 \sin \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+d x)}\right )}{2 b d}+\frac{\left (e^3 \sin \left (\frac{4 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{4 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{4 b d}\\ &=-\frac{e^3 \sqrt{\frac{\pi }{2}} \cos \left (\frac{4 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 \sqrt{b} d}+\frac{e^3 \sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right )}{4 \sqrt{b} d}-\frac{e^3 \sqrt{\pi } C\left (\frac{2 \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b} \sqrt{\pi }}\right ) \sin \left (\frac{2 a}{b}\right )}{4 \sqrt{b} d}+\frac{e^3 \sqrt{\frac{\pi }{2}} C\left (\frac{2 \sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{4 a}{b}\right )}{8 \sqrt{b} d}\\ \end{align*}

Mathematica [C]  time = 0.137109, size = 249, normalized size = 1.07 \[ \frac{e^3 e^{-\frac{4 i a}{b}} \left (-2 \sqrt{2} e^{\frac{2 i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-2 \sqrt{2} e^{\frac{6 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+\sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{4 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac{8 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{4 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )}{32 d \sqrt{a+b \sin ^{-1}(c+d x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(e^3*(-2*Sqrt[2]*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c + d
*x]))/b] - 2*Sqrt[2]*E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((2*I)*(a + b*ArcSin[c + d
*x]))/b] + Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-4*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((8*I)*a
)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((4*I)*(a + b*ArcSin[c + d*x]))/b]))/(32*d*E^(((4*I)*a)/b)
*Sqrt[a + b*ArcSin[c + d*x]])

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Maple [A]  time = 0.066, size = 169, normalized size = 0.7 \begin{align*}{\frac{{e}^{3}\sqrt{\pi }}{16\,d}\sqrt{{b}^{-1}} \left ( -\cos \left ( 4\,{\frac{a}{b}} \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) \sqrt{2}+\sin \left ( 4\,{\frac{a}{b}} \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) \sqrt{2}+4\,\cos \left ( 2\,{\frac{a}{b}} \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) -4\,\sin \left ( 2\,{\frac{a}{b}} \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^3/(a+b*arcsin(d*x+c))^(1/2),x)

[Out]

1/16/d*e^3*(1/b)^(1/2)*Pi^(1/2)*(-cos(4*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)
/b)*2^(1/2)+sin(4*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*2^(1/2)+4*cos(2*a/
b)*FresnelS(2/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)-4*sin(2*a/b)*FresnelC(2/Pi^(1/2)/(1/b)^(1/2)*(
a+b*arcsin(d*x+c))^(1/2)/b))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d e x + c e\right )}^{3}}{\sqrt{b \arcsin \left (d x + c\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^3/(a+b*arcsin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate((d*e*x + c*e)^3/sqrt(b*arcsin(d*x + c) + 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((d*e*x+c*e)^3/(a+b*arcsin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} e^{3} \left (\int \frac{c^{3}}{\sqrt{a + b \operatorname{asin}{\left (c + d x \right )}}}\, dx + \int \frac{d^{3} x^{3}}{\sqrt{a + b \operatorname{asin}{\left (c + d x \right )}}}\, dx + \int \frac{3 c d^{2} x^{2}}{\sqrt{a + b \operatorname{asin}{\left (c + d x \right )}}}\, dx + \int \frac{3 c^{2} d x}{\sqrt{a + b \operatorname{asin}{\left (c + d x \right )}}}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**3/(a+b*asin(d*x+c))**(1/2),x)

[Out]

e**3*(Integral(c**3/sqrt(a + b*asin(c + d*x)), x) + Integral(d**3*x**3/sqrt(a + b*asin(c + d*x)), x) + Integra
l(3*c*d**2*x**2/sqrt(a + b*asin(c + d*x)), x) + Integral(3*c**2*d*x/sqrt(a + b*asin(c + d*x)), x))

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Giac [A]  time = 2.27816, size = 441, normalized size = 1.89 \begin{align*} \frac{\sqrt{\pi } i \operatorname{erf}\left (-\frac{\sqrt{2} \sqrt{b \arcsin \left (d x + c\right ) + a} \sqrt{b} i}{{\left | b \right |}} - \frac{\sqrt{2} \sqrt{b \arcsin \left (d x + c\right ) + a}}{\sqrt{b}}\right ) e^{\left (\frac{4 \, a i}{b} + 3\right )}}{16 \,{\left (\frac{\sqrt{2} b^{\frac{3}{2}} i}{{\left | b \right |}} + \sqrt{2} \sqrt{b}\right )} d} - \frac{\sqrt{\pi } i \operatorname{erf}\left (\frac{\sqrt{b \arcsin \left (d x + c\right ) + a} \sqrt{b} i}{{\left | b \right |}} - \frac{\sqrt{b \arcsin \left (d x + c\right ) + a}}{\sqrt{b}}\right ) e^{\left (-\frac{2 \, a i}{b} + 3\right )}}{8 \,{\left (\frac{b^{\frac{3}{2}} i}{{\left | b \right |}} - \sqrt{b}\right )} d} + \frac{\sqrt{\pi } i \operatorname{erf}\left (\frac{\sqrt{2} \sqrt{b \arcsin \left (d x + c\right ) + a} \sqrt{b} i}{{\left | b \right |}} - \frac{\sqrt{2} \sqrt{b \arcsin \left (d x + c\right ) + a}}{\sqrt{b}}\right ) e^{\left (-\frac{4 \, a i}{b} + 3\right )}}{16 \,{\left (\frac{\sqrt{2} b^{\frac{3}{2}} i}{{\left | b \right |}} - \sqrt{2} \sqrt{b}\right )} d} - \frac{\sqrt{\pi } i \operatorname{erf}\left (-\frac{\sqrt{b \arcsin \left (d x + c\right ) + a} \sqrt{b} i}{{\left | b \right |}} - \frac{\sqrt{b \arcsin \left (d x + c\right ) + a}}{\sqrt{b}}\right ) e^{\left (\frac{2 \, a i}{b} + 3\right )}}{8 \, \sqrt{b} d{\left (\frac{b i}{{\left | b \right |}} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^3/(a+b*arcsin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

1/16*sqrt(pi)*i*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(2)*sqrt(b*arcsin(d*x + c) + a
)/sqrt(b))*e^(4*a*i/b + 3)/((sqrt(2)*b^(3/2)*i/abs(b) + sqrt(2)*sqrt(b))*d) - 1/8*sqrt(pi)*i*erf(sqrt(b*arcsin
(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-2*a*i/b + 3)/((b^(3/2)*i/abs(b) - s
qrt(b))*d) + 1/16*sqrt(pi)*i*erf(sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(2)*sqrt(b*arcsin(
d*x + c) + a)/sqrt(b))*e^(-4*a*i/b + 3)/((sqrt(2)*b^(3/2)*i/abs(b) - sqrt(2)*sqrt(b))*d) - 1/8*sqrt(pi)*i*erf(
-sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(2*a*i/b + 3)/(sqrt(b)*
d*(b*i/abs(b) + 1))