3.239 \(\int \frac{\sin (a+b (c+d x)^{2/3})}{(c e+d e x)^{2/3}} \, dx\)

Optimal. Leaf size=133 \[ \frac{3 \sqrt{\frac{\pi }{2}} \sin (a) (c+d x)^{2/3} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [3]{c+d x}\right )}{\sqrt{b} d (e (c+d x))^{2/3}}+\frac{3 \sqrt{\frac{\pi }{2}} \cos (a) (c+d x)^{2/3} S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [3]{c+d x}\right )}{\sqrt{b} d (e (c+d x))^{2/3}} \]

[Out]

(3*Sqrt[Pi/2]*(c + d*x)^(2/3)*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)])/(Sqrt[b]*d*(e*(c + d*x))^(2
/3)) + (3*Sqrt[Pi/2]*(c + d*x)^(2/3)*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)]*Sin[a])/(Sqrt[b]*d*(e*(c + d
*x))^(2/3))

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Rubi [A]  time = 0.124505, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3435, 3417, 3383, 3353, 3352, 3351} \[ \frac{3 \sqrt{\frac{\pi }{2}} \sin (a) (c+d x)^{2/3} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [3]{c+d x}\right )}{\sqrt{b} d (e (c+d x))^{2/3}}+\frac{3 \sqrt{\frac{\pi }{2}} \cos (a) (c+d x)^{2/3} S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [3]{c+d x}\right )}{\sqrt{b} d (e (c+d x))^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*(c + d*x)^(2/3)]/(c*e + d*e*x)^(2/3),x]

[Out]

(3*Sqrt[Pi/2]*(c + d*x)^(2/3)*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)])/(Sqrt[b]*d*(e*(c + d*x))^(2
/3)) + (3*Sqrt[Pi/2]*(c + d*x)^(2/3)*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)]*Sin[a])/(Sqrt[b]*d*(e*(c + d
*x))^(2/3))

Rule 3435

Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_Symbol] :
> Dist[1/f, Subst[Int[((h*x)/f)^m*(a + b*Sin[c + d*x^n])^p, x], x, e + f*x], x] /; FreeQ[{a, b, c, d, e, f, g,
h, m}, x] && IGtQ[p, 0] && EqQ[f*g - e*h, 0]

Rule 3417

Int[((e_)*(x_))^(m_)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[(e^IntPart[m]*(e*x)
^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && Integ
erQ[p] && FractionQ[n]

Rule 3383

Int[(x_)^(m_.)*Sin[(a_.) + (b_.)*(x_)^(n_)], x_Symbol] :> Dist[2/n, Subst[Int[Sin[a + b*x^2], x], x, x^(n/2)],
x] /; FreeQ[{a, b, m, n}, x] && EqQ[m, n/2 - 1]

Rule 3353

Int[Sin[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Dist[Sin[c], Int[Cos[d*(e + f*x)^2], x], x] + Dist[
Cos[c], Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, 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 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]

Rubi steps

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

Mathematica [A]  time = 0.147337, size = 96, normalized size = 0.72 \[ \frac{3 \sqrt{\frac{\pi }{2}} (c+d x)^{2/3} \left (\sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [3]{c+d x}\right )+\cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [3]{c+d x}\right )\right )}{\sqrt{b} d (e (c+d x))^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Sqrt[Pi/2]*(c + d*x)^(2/3)*(Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)] + FresnelC[Sqrt[b]*Sqrt[2/P
i]*(c + d*x)^(1/3)]*Sin[a]))/(Sqrt[b]*d*(e*(c + d*x))^(2/3))

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \begin{align*} \int{\sin \left ( a+b \left ( dx+c \right ) ^{{\frac{2}{3}}} \right ) \left ( dex+ce \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: IndexError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sin \left ({\left (d x + c\right )}^{\frac{2}{3}} b + a\right )}{{\left (d e x + c e\right )}^{\frac{2}{3}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sin((d*x + c)^(2/3)*b + a)/(d*e*x + c*e)^(2/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(a + b*(c + d*x)**(2/3))/(e*(c + d*x))**(2/3), x)

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Giac [C]  time = 1.13899, size = 113, normalized size = 0.85 \begin{align*} -\frac{3 \,{\left (-\frac{i \, \sqrt{\pi } \operatorname{erf}\left (-{\left (d x e + c e\right )}^{\frac{1}{3}} \sqrt{-i \, b e^{\left (-\frac{2}{3}\right )}}\right ) e^{\left (i \, a\right )}}{\sqrt{-i \, b e^{\left (-\frac{2}{3}\right )}}} + \frac{i \, \sqrt{\pi } \operatorname{erf}\left (-{\left (d x e + c e\right )}^{\frac{1}{3}} \sqrt{i \, b e^{\left (-\frac{2}{3}\right )}}\right ) e^{\left (-i \, a\right )}}{\sqrt{i \, b e^{\left (-\frac{2}{3}\right )}}}\right )} e^{\left (-1\right )}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/4*(-I*sqrt(pi)*erf(-(d*x*e + c*e)^(1/3)*sqrt(-I*b*e^(-2/3)))*e^(I*a)/sqrt(-I*b*e^(-2/3)) + I*sqrt(pi)*erf(-
(d*x*e + c*e)^(1/3)*sqrt(I*b*e^(-2/3)))*e^(-I*a)/sqrt(I*b*e^(-2/3)))*e^(-1)/d