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

Optimal. Leaf size=105 \[ \frac {\sqrt {\pi } e \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{2 \sqrt {b} d}-\frac {\sqrt {\pi } e \sin \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{2 \sqrt {b} d} \]

[Out]

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

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Rubi [A]  time = 0.23, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {4805, 12, 4635, 4406, 3306, 3305, 3351, 3304, 3352} \[ \frac {\sqrt {\pi } e \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{2 \sqrt {b} d}-\frac {\sqrt {\pi } e \sin \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {\pi } \sqrt {b}}\right )}{2 \sqrt {b} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

Rubi steps

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

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Mathematica [C]  time = 0.08, size = 134, normalized size = 1.28 \[ -\frac {e e^{-\frac {2 i a}{b}} \left (\sqrt {-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {1}{2},-\frac {2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac {4 i a}{b}} \sqrt {\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {1}{2},\frac {2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )}{4 \sqrt {2} d \sqrt {a+b \sin ^{-1}(c+d x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/4*(e*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((4*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[2]*d*E^(((2*I)*a)
/b)*Sqrt[a + b*ArcSin[c + d*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((d*e*x+c*e)/(a+b*arcsin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 2.55, size = 151, normalized size = 1.44 \[ -\frac {\sqrt {\pi } i \operatorname {erf}\left (\frac {\sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b} i}{{\left | b \right |}} - \frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}}\right ) e^{\left (-\frac {2 \, a i}{b} + 1\right )}}{4 \, {\left (\frac {b^{\frac {3}{2}} i}{{\left | b \right |}} - \sqrt {b}\right )} d} - \frac {\sqrt {\pi } i \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b} i}{{\left | b \right |}} - \frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}}\right ) e^{\left (\frac {2 \, a i}{b} + 1\right )}}{4 \, \sqrt {b} d {\left (\frac {b i}{{\left | b \right |}} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*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 + 1)/((b^(3/2)*i/abs(b) - sqrt(b))*d) - 1/4*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 + 1)/(sqrt(b)*d*(b*i/abs(b) + 1))

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maple [A]  time = 0.09, size = 85, normalized size = 0.81 \[ \frac {\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, e \left (\cos \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right )-\sin \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right )\right )}{2 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*e*x + c*e)/sqrt(b*arcsin(d*x + c) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e*(Integral(c/sqrt(a + b*asin(c + d*x)), x) + Integral(d*x/sqrt(a + b*asin(c + d*x)), x))

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