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

Optimal. Leaf size=227 \[ -\frac{9 \sqrt{\pi } \sin (a) (e (c+d x))^{2/3} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [3]{c+d x}\right )}{4 \sqrt{2} b^{5/2} d (c+d x)^{2/3}}-\frac{9 \sqrt{\pi } \cos (a) (e (c+d x))^{2/3} S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [3]{c+d x}\right )}{4 \sqrt{2} b^{5/2} d (c+d x)^{2/3}}+\frac{9 (e (c+d x))^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d \sqrt [3]{c+d x}}-\frac{3 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d} \]

[Out]

(-3*(c + d*x)^(1/3)*(e*(c + d*x))^(2/3)*Cos[a + b*(c + d*x)^(2/3)])/(2*b*d) - (9*Sqrt[Pi]*(e*(c + d*x))^(2/3)*
Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)])/(4*Sqrt[2]*b^(5/2)*d*(c + d*x)^(2/3)) - (9*Sqrt[Pi]*(e*(c
+ d*x))^(2/3)*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)]*Sin[a])/(4*Sqrt[2]*b^(5/2)*d*(c + d*x)^(2/3)) + (9
*(e*(c + d*x))^(2/3)*Sin[a + b*(c + d*x)^(2/3)])/(4*b^2*d*(c + d*x)^(1/3))

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Rubi [A]  time = 0.197811, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {3435, 3417, 3415, 3385, 3386, 3353, 3352, 3351} \[ -\frac{9 \sqrt{\pi } \sin (a) (e (c+d x))^{2/3} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [3]{c+d x}\right )}{4 \sqrt{2} b^{5/2} d (c+d x)^{2/3}}-\frac{9 \sqrt{\pi } \cos (a) (e (c+d x))^{2/3} S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [3]{c+d x}\right )}{4 \sqrt{2} b^{5/2} d (c+d x)^{2/3}}+\frac{9 (e (c+d x))^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d \sqrt [3]{c+d x}}-\frac{3 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(c + d*x)^(1/3)*(e*(c + d*x))^(2/3)*Cos[a + b*(c + d*x)^(2/3)])/(2*b*d) - (9*Sqrt[Pi]*(e*(c + d*x))^(2/3)*
Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)])/(4*Sqrt[2]*b^(5/2)*d*(c + d*x)^(2/3)) - (9*Sqrt[Pi]*(e*(c
+ d*x))^(2/3)*FresnelC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)]*Sin[a])/(4*Sqrt[2]*b^(5/2)*d*(c + d*x)^(2/3)) + (9
*(e*(c + d*x))^(2/3)*Sin[a + b*(c + d*x)^(2/3)])/(4*b^2*d*(c + d*x)^(1/3))

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 3415

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Module[{k = Denominator[n]}, D
ist[k, Subst[Int[x^(k*(m + 1) - 1)*(a + b*Sin[c + d*x^(k*n)])^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, m}
, x] && IntegerQ[p] && FractionQ[n]

Rule 3385

Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> -Simp[(e^(n - 1)*(e*x)^(m - n + 1)*Cos[c + d
*x^n])/(d*n), x] + Dist[(e^n*(m - n + 1))/(d*n), Int[(e*x)^(m - n)*Cos[c + d*x^n], x], x] /; FreeQ[{c, d, e},
x] && IGtQ[n, 0] && LtQ[n, m + 1]

Rule 3386

Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[(e^(n - 1)*(e*x)^(m - n + 1)*Sin[c + d*
x^n])/(d*n), x] - Dist[(e^n*(m - n + 1))/(d*n), Int[(e*x)^(m - n)*Sin[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x
] && IGtQ[n, 0] && LtQ[n, m + 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 (c e+d e x)^{2/3} \sin \left (a+b (c+d x)^{2/3}\right ) \, dx &=\frac{\operatorname{Subst}\left (\int (e x)^{2/3} \sin \left (a+b x^{2/3}\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{(e (c+d x))^{2/3} \operatorname{Subst}\left (\int x^{2/3} \sin \left (a+b x^{2/3}\right ) \, dx,x,c+d x\right )}{d (c+d x)^{2/3}}\\ &=\frac{\left (3 (e (c+d x))^{2/3}\right ) \operatorname{Subst}\left (\int x^4 \sin \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d (c+d x)^{2/3}}\\ &=-\frac{3 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac{\left (9 (e (c+d x))^{2/3}\right ) \operatorname{Subst}\left (\int x^2 \cos \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{2 b d (c+d x)^{2/3}}\\ &=-\frac{3 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac{9 (e (c+d x))^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d \sqrt [3]{c+d x}}-\frac{\left (9 (e (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 )}{4 b^2 d (c+d x)^{2/3}}\\ &=-\frac{3 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}+\frac{9 (e (c+d x))^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d \sqrt [3]{c+d x}}-\frac{\left (9 (e (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 )}{4 b^2 d (c+d x)^{2/3}}-\frac{\left (9 (e (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 )}{4 b^2 d (c+d x)^{2/3}}\\ &=-\frac{3 \sqrt [3]{c+d x} (e (c+d x))^{2/3} \cos \left (a+b (c+d x)^{2/3}\right )}{2 b d}-\frac{9 \sqrt{\pi } (e (c+d x))^{2/3} \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [3]{c+d x}\right )}{4 \sqrt{2} b^{5/2} d (c+d x)^{2/3}}-\frac{9 \sqrt{\pi } (e (c+d x))^{2/3} C\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{4 \sqrt{2} b^{5/2} d (c+d x)^{2/3}}+\frac{9 (e (c+d x))^{2/3} \sin \left (a+b (c+d x)^{2/3}\right )}{4 b^2 d \sqrt [3]{c+d x}}\\ \end{align*}

Mathematica [A]  time = 0.45467, size = 160, normalized size = 0.7 \[ -\frac{3 (e (c+d x))^{2/3} \left (3 \sqrt{2 \pi } \sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} \sqrt [3]{c+d x}\right )+3 \sqrt{2 \pi } \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} \sqrt [3]{c+d x}\right )+2 \sqrt{b} \left (2 b (c+d x) \cos \left (a+b (c+d x)^{2/3}\right )-3 \sqrt [3]{c+d x} \sin \left (a+b (c+d x)^{2/3}\right )\right )\right )}{8 b^{5/2} d (c+d x)^{2/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(e*(c + d*x))^(2/3)*(3*Sqrt[2*Pi]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)] + 3*Sqrt[2*Pi]*Fresn
elC[Sqrt[b]*Sqrt[2/Pi]*(c + d*x)^(1/3)]*Sin[a] + 2*Sqrt[b]*(2*b*(c + d*x)*Cos[a + b*(c + d*x)^(2/3)] - 3*(c +
d*x)^(1/3)*Sin[a + b*(c + d*x)^(2/3)])))/(8*b^(5/2)*d*(c + d*x)^(2/3))

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) ^{{\frac{2}{3}}}\sin \left ( a+b \left ( dx+c \right ) ^{{\frac{2}{3}}} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((d*e*x+c*e)^(2/3)*sin(a+b*(d*x+c)^(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((d*e*x+c*e)^(2/3)*sin(a+b*(d*x+c)^(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 ({\left (d e x + c e\right )}^{\frac{2}{3}} \sin \left ({\left (d x + c\right )}^{\frac{2}{3}} b + a\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [C]  time = 1.2228, size = 284, normalized size = 1.25 \begin{align*} -\frac{3 \,{\left (-\frac{2 i \,{\left (2 i \,{\left (d x e + c e\right )} b e^{\left (-\frac{2}{3}\right )} - 3 \,{\left (d x e + c e\right )}^{\frac{1}{3}}\right )} e^{\left (i \,{\left (d x e + c e\right )}^{\frac{2}{3}} b e^{\left (-\frac{2}{3}\right )} + i \, a + \frac{4}{3}\right )}}{b^{2}} - \frac{2 i \,{\left (2 i \,{\left (d x e + c e\right )} b e^{\left (-\frac{2}{3}\right )} + 3 \,{\left (d x e + c e\right )}^{\frac{1}{3}}\right )} e^{\left (-i \,{\left (d x e + c e\right )}^{\frac{2}{3}} b e^{\left (-\frac{2}{3}\right )} - i \, a + \frac{4}{3}\right )}}{b^{2}} + \frac{3 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 + \frac{4}{3}\right )}}{\sqrt{-i \, b e^{\left (-\frac{2}{3}\right )}} b^{2}} - \frac{3 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 + \frac{4}{3}\right )}}{\sqrt{i \, b e^{\left (-\frac{2}{3}\right )}} b^{2}}\right )} e^{\left (-1\right )}}{16 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/16*(-2*I*(2*I*(d*x*e + c*e)*b*e^(-2/3) - 3*(d*x*e + c*e)^(1/3))*e^(I*(d*x*e + c*e)^(2/3)*b*e^(-2/3) + I*a +
4/3)/b^2 - 2*I*(2*I*(d*x*e + c*e)*b*e^(-2/3) + 3*(d*x*e + c*e)^(1/3))*e^(-I*(d*x*e + c*e)^(2/3)*b*e^(-2/3) -
I*a + 4/3)/b^2 + 3*I*sqrt(pi)*erf(-(d*x*e + c*e)^(1/3)*sqrt(-I*b*e^(-2/3)))*e^(I*a + 4/3)/(sqrt(-I*b*e^(-2/3))
*b^2) - 3*I*sqrt(pi)*erf(-(d*x*e + c*e)^(1/3)*sqrt(I*b*e^(-2/3)))*e^(-I*a + 4/3)/(sqrt(I*b*e^(-2/3))*b^2))*e^(
-1)/d