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

Optimal. Leaf size=69 \[ \frac{e \cos \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 b d}-\frac{e \sin \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{2 b d} \]

[Out]

-(e*CosIntegral[(2*(a + b*ArcSin[c + d*x]))/b]*Sin[(2*a)/b])/(2*b*d) + (e*Cos[(2*a)/b]*SinIntegral[(2*(a + b*A
rcSin[c + d*x]))/b])/(2*b*d)

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Rubi [A]  time = 0.145213, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4805, 12, 4635, 4406, 3303, 3299, 3302} \[ \frac{e \cos \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \sin ^{-1}(c+d x)\right )}{2 b d}-\frac{e \sin \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 a}{b}+2 \sin ^{-1}(c+d x)\right )}{2 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e*CosIntegral[(2*a)/b + 2*ArcSin[c + d*x]]*Sin[(2*a)/b])/(2*b*d) + (e*Cos[(2*a)/b]*SinIntegral[(2*a)/b + 2*A
rcSin[c + d*x]])/(2*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 3303

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

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3302

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

Rubi steps

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

Mathematica [A]  time = 0.0852078, size = 61, normalized size = 0.88 \[ \frac{e \left (\cos \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \sin ^{-1}(c+d x)\right )-\sin \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 a}{b}+2 \sin ^{-1}(c+d x)\right )\right )}{2 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*(-(CosIntegral[(2*a)/b + 2*ArcSin[c + d*x]]*Sin[(2*a)/b]) + Cos[(2*a)/b]*SinIntegral[(2*a)/b + 2*ArcSin[c +
d*x]]))/(2*b*d)

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Maple [A]  time = 0.03, size = 60, normalized size = 0.9 \begin{align*}{\frac{e}{2\,bd} \left ({\it Si} \left ( 2\,\arcsin \left ( dx+c \right ) +2\,{\frac{a}{b}} \right ) \cos \left ( 2\,{\frac{a}{b}} \right ) -{\it Ci} \left ( 2\,\arcsin \left ( dx+c \right ) +2\,{\frac{a}{b}} \right ) \sin \left ( 2\,{\frac{a}{b}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/d*e*(Si(2*arcsin(d*x+c)+2*a/b)*cos(2*a/b)-Ci(2*arcsin(d*x+c)+2*a/b)*sin(2*a/b))/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d e x + c e}{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)/(a+b*arcsin(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{d e x + c e}{b \arcsin \left (d x + c\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)/(a+b*arcsin(d*x+c)),x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [A]  time = 1.22327, size = 132, normalized size = 1.91 \begin{align*} -\frac{\cos \left (\frac{a}{b}\right ) \operatorname{Ci}\left (\frac{2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right ) e \sin \left (\frac{a}{b}\right )}{b d} + \frac{\cos \left (\frac{a}{b}\right )^{2} e \operatorname{Si}\left (\frac{2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right )}{b d} - \frac{e \operatorname{Si}\left (\frac{2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right )}{2 \, b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(a/b)*cos_integral(2*a/b + 2*arcsin(d*x + c))*e*sin(a/b)/(b*d) + cos(a/b)^2*e*sin_integral(2*a/b + 2*arcsi
n(d*x + c))/(b*d) - 1/2*e*sin_integral(2*a/b + 2*arcsin(d*x + c))/(b*d)