3.3.39 \(\int \frac {\sin (a+b (c+d x)^{2/3})}{(c e+d e x)^{2/3}} \, dx\) [239]

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

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 133, normalized size of antiderivative = 1.00, 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 = {3516, 3498, 3464, 3434, 3433, 3432} \begin {gather*} \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}} \end {gather*}

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 3432

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 3433

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 3434

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 3464

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 3498

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 3516

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)^{2/3}} \, dx &=\frac {\text {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} \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.10, size = 96, normalized size = 0.72 \begin {gather*} \frac {3 \sqrt {\frac {\pi }{2}} (c+d x)^{2/3} \left (\cos (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right )+C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt [3]{c+d x}\right ) \sin (a)\right )}{\sqrt {b} d (e (c+d x))^{2/3}} \end {gather*}

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 = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sin \left (a +b \left (d x +c \right )^{\frac {2}{3}}\right )}{\left (d e x +c e \right )^{\frac {2}{3}}}\, dx\]

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)^(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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.58, size = 486, normalized size = 3.65 \begin {gather*} \frac {3 \, {\left ({\left ({\left (-i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} + i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\right )}\right )} \cos \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} - i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\right )}\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} + \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\right )}\right )} \sin \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) - {\left (\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} + \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\right )}\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right )\right )} \cos \left (a\right ) + {\left ({\left (\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} + \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\right )}\right )} \cos \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (\sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} + \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\right )}\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} - i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\right )}\right )} \sin \left (\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right ) + {\left (i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {i \, b \overline {{\left (d x + c\right )}^{\frac {2}{3}}}}\right ) - 1\right )} - i \, \sqrt {\pi } {\left (\operatorname {erf}\left (\sqrt {-i \, {\left (d x + c\right )}^{\frac {2}{3}} b}\right ) - 1\right )}\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{3} \, \arctan \left (0, d x + c\right )\right )\right )} \sin \left (a\right )\right )} \sqrt {{\left (d x + c\right )}^{\frac {2}{3}} b} e^{\left (-\frac {2}{3}\right )}}{8 \, {\left (d x + c\right )}^{\frac {1}{3}} b d} \end {gather*}

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]

3/8*(((-I*sqrt(pi)*(erf(sqrt(-I*b*conjugate((d*x + c)^(2/3)))) - 1) + I*sqrt(pi)*(erf(sqrt(I*(d*x + c)^(2/3)*b
)) - 1))*cos(1/4*pi + 1/3*arctan2(0, d*x + c)) + (I*sqrt(pi)*(erf(sqrt(I*b*conjugate((d*x + c)^(2/3)))) - 1) -
I*sqrt(pi)*(erf(sqrt(-I*(d*x + c)^(2/3)*b)) - 1))*cos(-1/4*pi + 1/3*arctan2(0, d*x + c)) + (sqrt(pi)*(erf(sqr
t(-I*b*conjugate((d*x + c)^(2/3)))) - 1) + sqrt(pi)*(erf(sqrt(I*(d*x + c)^(2/3)*b)) - 1))*sin(1/4*pi + 1/3*arc
tan2(0, d*x + c)) - (sqrt(pi)*(erf(sqrt(I*b*conjugate((d*x + c)^(2/3)))) - 1) + sqrt(pi)*(erf(sqrt(-I*(d*x + c
)^(2/3)*b)) - 1))*sin(-1/4*pi + 1/3*arctan2(0, d*x + c)))*cos(a) + ((sqrt(pi)*(erf(sqrt(-I*b*conjugate((d*x +
c)^(2/3)))) - 1) + sqrt(pi)*(erf(sqrt(I*(d*x + c)^(2/3)*b)) - 1))*cos(1/4*pi + 1/3*arctan2(0, d*x + c)) + (sqr
t(pi)*(erf(sqrt(I*b*conjugate((d*x + c)^(2/3)))) - 1) + sqrt(pi)*(erf(sqrt(-I*(d*x + c)^(2/3)*b)) - 1))*cos(-1
/4*pi + 1/3*arctan2(0, d*x + c)) + (I*sqrt(pi)*(erf(sqrt(-I*b*conjugate((d*x + c)^(2/3)))) - 1) - I*sqrt(pi)*(
erf(sqrt(I*(d*x + c)^(2/3)*b)) - 1))*sin(1/4*pi + 1/3*arctan2(0, d*x + c)) + (I*sqrt(pi)*(erf(sqrt(I*b*conjuga
te((d*x + c)^(2/3)))) - 1) - I*sqrt(pi)*(erf(sqrt(-I*(d*x + c)^(2/3)*b)) - 1))*sin(-1/4*pi + 1/3*arctan2(0, d*
x + c)))*sin(a))*sqrt((d*x + c)^(2/3)*b)*e^(-2/3)/((d*x + c)^(1/3)*b*d)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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] Result contains complex when optimal does not.
time = 6.30, size = 84, normalized size = 0.63 \begin {gather*} -\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 {gather*}

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

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

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)^(2/3),x)

[Out]

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

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