3.240 \(\int \frac {\sin (a+b (c+d x)^{2/3})}{(c e+d e x)^{4/3}} \, dx\)

Optimal. Leaf size=168 \[ \frac {3 \sqrt {2 \pi } \sqrt {b} \cos (a) \sqrt [3]{c+d x} C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac {3 \sqrt {2 \pi } \sqrt {b} \sin (a) \sqrt [3]{c+d x} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac {3 \sin \left (a+b (c+d x)^{2/3}\right )}{d e \sqrt [3]{e (c+d x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.16, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3435, 3417, 3415, 3387, 3354, 3352, 3351} \[ \frac {3 \sqrt {2 \pi } \sqrt {b} \cos (a) \sqrt [3]{c+d x} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac {3 \sqrt {2 \pi } \sqrt {b} \sin (a) \sqrt [3]{c+d x} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac {3 \sin \left (a+b (c+d x)^{2/3}\right )}{d e \sqrt [3]{e (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*(c + d*x)^(2/3)]/(c*e + d*e*x)^(4/3),x]

[Out]

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

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 3354

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

Rule 3387

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

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 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 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]

Rubi steps

\begin {align*} \int \frac {\sin \left (a+b (c+d x)^{2/3}\right )}{(c e+d e x)^{4/3}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sin \left (a+b x^{2/3}\right )}{(e x)^{4/3}} \, dx,x,c+d x\right )}{d}\\ &=\frac {\sqrt [3]{c+d x} \operatorname {Subst}\left (\int \frac {\sin \left (a+b x^{2/3}\right )}{x^{4/3}} \, dx,x,c+d x\right )}{d e \sqrt [3]{e (c+d x)}}\\ &=\frac {\left (3 \sqrt [3]{c+d x}\right ) \operatorname {Subst}\left (\int \frac {\sin \left (a+b x^2\right )}{x^2} \, dx,x,\sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}\\ &=-\frac {3 \sin \left (a+b (c+d x)^{2/3}\right )}{d e \sqrt [3]{e (c+d x)}}+\frac {\left (6 b \sqrt [3]{c+d x}\right ) \operatorname {Subst}\left (\int \cos \left (a+b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}\\ &=-\frac {3 \sin \left (a+b (c+d x)^{2/3}\right )}{d e \sqrt [3]{e (c+d x)}}+\frac {\left (6 b \sqrt [3]{c+d x} \cos (a)\right ) \operatorname {Subst}\left (\int \cos \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac {\left (6 b \sqrt [3]{c+d x} \sin (a)\right ) \operatorname {Subst}\left (\int \sin \left (b x^2\right ) \, dx,x,\sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}\\ &=\frac {3 \sqrt {b} \sqrt {2 \pi } \sqrt [3]{c+d x} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )}{d e \sqrt [3]{e (c+d x)}}-\frac {3 \sqrt {b} \sqrt {2 \pi } \sqrt [3]{c+d x} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)}{d e \sqrt [3]{e (c+d x)}}-\frac {3 \sin \left (a+b (c+d x)^{2/3}\right )}{d e \sqrt [3]{e (c+d x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.24, size = 133, normalized size = 0.79 \[ -\frac {3 \left (\sqrt {2 \pi } \left (-\sqrt {b}\right ) \cos (a) \sqrt [3]{c+d x} C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )+\sqrt {2 \pi } \sqrt {b} \sin (a) \sqrt [3]{c+d x} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )+\sin \left (a+b (c+d x)^{2/3}\right )\right )}{d e \sqrt [3]{e (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*(c + d*x)^(2/3)]/(c*e + d*e*x)^(4/3),x]

[Out]

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

________________________________________________________________________________________

fricas [F]  time = 1.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d e x + c e\right )}^{\frac {2}{3}} \sin \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )}{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}, x\right ) \]

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)^(4/3),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left ({\left (d x + c\right )}^{\frac {2}{3}} b + a\right )}{{\left (d e x + c e\right )}^{\frac {4}{3}}}\,{d x} \]

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)^(4/3),x, algorithm="giac")

[Out]

integrate(sin((d*x + c)^(2/3)*b + a)/(d*e*x + c*e)^(4/3), x)

________________________________________________________________________________________

maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{\left (d e x +c e \right )^{\frac {4}{3}}}\, dx \]

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)^(4/3),x)

[Out]

int(sin(a+b*(d*x+c)^(2/3))/(d*e*x+c*e)^(4/3),x)

________________________________________________________________________________________

maxima [C]  time = 1.34, size = 386, normalized size = 2.30 \[ \frac {{\left ({\left ({\left (3 i \, \Gamma \left (-\frac {1}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - 3 i \, \Gamma \left (-\frac {1}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (-3 i \, \Gamma \left (-\frac {1}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + 3 i \, \Gamma \left (-\frac {1}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + 3 \, {\left (\Gamma \left (-\frac {1}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (-\frac {1}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) - 3 \, {\left (\Gamma \left (-\frac {1}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (-\frac {1}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right )\right )} \cos \relax (a) - {\left (3 \, {\left (\Gamma \left (-\frac {1}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (-\frac {1}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + 3 \, {\left (\Gamma \left (-\frac {1}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) + \Gamma \left (-\frac {1}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) - {\left (3 i \, \Gamma \left (-\frac {1}{2}, -i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - 3 i \, \Gamma \left (-\frac {1}{2}, i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) - {\left (3 i \, \Gamma \left (-\frac {1}{2}, i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}\right ) - 3 i \, \Gamma \left (-\frac {1}{2}, -i \, {\left (d x + c\right )}^{\frac {2}{3}} b\right )\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right )\right )} \sin \relax (a)\right )} \sqrt {{\left (d x + c\right )}^{\frac {2}{3}} b}}{8 \, {\left (d x + c\right )}^{\frac {1}{3}} d e^{\frac {4}{3}}} \]

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)^(4/3),x, algorithm="maxima")

[Out]

1/8*(((3*I*gamma(-1/2, -I*b*conjugate((d*x + c)^(2/3))) - 3*I*gamma(-1/2, I*(d*x + c)^(2/3)*b))*cos(1/4*pi + 1
/3*arctan2(0, d*x + c)) + (-3*I*gamma(-1/2, I*b*conjugate((d*x + c)^(2/3))) + 3*I*gamma(-1/2, -I*(d*x + c)^(2/
3)*b))*cos(-1/4*pi + 1/3*arctan2(0, d*x + c)) + 3*(gamma(-1/2, -I*b*conjugate((d*x + c)^(2/3))) + gamma(-1/2,
I*(d*x + c)^(2/3)*b))*sin(1/4*pi + 1/3*arctan2(0, d*x + c)) - 3*(gamma(-1/2, I*b*conjugate((d*x + c)^(2/3))) +
gamma(-1/2, -I*(d*x + c)^(2/3)*b))*sin(-1/4*pi + 1/3*arctan2(0, d*x + c)))*cos(a) - (3*(gamma(-1/2, -I*b*conj
ugate((d*x + c)^(2/3))) + gamma(-1/2, I*(d*x + c)^(2/3)*b))*cos(1/4*pi + 1/3*arctan2(0, d*x + c)) + 3*(gamma(-
1/2, I*b*conjugate((d*x + c)^(2/3))) + gamma(-1/2, -I*(d*x + c)^(2/3)*b))*cos(-1/4*pi + 1/3*arctan2(0, d*x + c
)) - (3*I*gamma(-1/2, -I*b*conjugate((d*x + c)^(2/3))) - 3*I*gamma(-1/2, I*(d*x + c)^(2/3)*b))*sin(1/4*pi + 1/
3*arctan2(0, d*x + c)) - (3*I*gamma(-1/2, I*b*conjugate((d*x + c)^(2/3))) - 3*I*gamma(-1/2, -I*(d*x + c)^(2/3)
*b))*sin(-1/4*pi + 1/3*arctan2(0, d*x + c)))*sin(a))*sqrt((d*x + c)^(2/3)*b)/((d*x + c)^(1/3)*d*e^(4/3))

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sin \left (a+b\,{\left (c+d\,x\right )}^{2/3}\right )}{{\left (c\,e+d\,e\,x\right )}^{4/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*(c + d*x)^(2/3))/(c*e + d*e*x)^(4/3),x)

[Out]

int(sin(a + b*(c + d*x)^(2/3))/(c*e + d*e*x)^(4/3), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin {\left (a + b \left (c + d x\right )^{\frac {2}{3}} \right )}}{\left (e \left (c + d x\right )\right )^{\frac {4}{3}}}\, dx \]

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)**(4/3),x)

[Out]

Integral(sin(a + b*(c + d*x)**(2/3))/(e*(c + d*x))**(4/3), x)

________________________________________________________________________________________