3.248 \(\int \frac {\sin (a+\frac {b}{\sqrt [3]{c+d x}})}{(c e+d e x)^{8/3}} \, dx\)

Optimal. Leaf size=217 \[ \frac {72 (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^5 d e^2 (e (c+d x))^{2/3}}+\frac {72 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^4 d e^2 (e (c+d x))^{2/3}}-\frac {36 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^3 d e^2 (e (c+d x))^{2/3}}-\frac {12 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{c+d x} (e (c+d x))^{2/3}}+\frac {3 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} (e (c+d x))^{2/3}} \]

[Out]

-36*cos(a+b/(d*x+c)^(1/3))/b^3/d/e^2/(e*(d*x+c))^(2/3)+3*cos(a+b/(d*x+c)^(1/3))/b/d/e^2/(d*x+c)^(2/3)/(e*(d*x+
c))^(2/3)+72*(d*x+c)^(2/3)*cos(a+b/(d*x+c)^(1/3))/b^5/d/e^2/(e*(d*x+c))^(2/3)-12*sin(a+b/(d*x+c)^(1/3))/b^2/d/
e^2/(d*x+c)^(1/3)/(e*(d*x+c))^(2/3)+72*(d*x+c)^(1/3)*sin(a+b/(d*x+c)^(1/3))/b^4/d/e^2/(e*(d*x+c))^(2/3)

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Rubi [A]  time = 0.19, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3431, 15, 3296, 2638} \[ \frac {72 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^4 d e^2 (e (c+d x))^{2/3}}-\frac {12 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{c+d x} (e (c+d x))^{2/3}}+\frac {72 (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^5 d e^2 (e (c+d x))^{2/3}}-\frac {36 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^3 d e^2 (e (c+d x))^{2/3}}+\frac {3 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} (e (c+d x))^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b/(c + d*x)^(1/3)]/(c*e + d*e*x)^(8/3),x]

[Out]

(-36*Cos[a + b/(c + d*x)^(1/3)])/(b^3*d*e^2*(e*(c + d*x))^(2/3)) + (3*Cos[a + b/(c + d*x)^(1/3)])/(b*d*e^2*(c
+ d*x)^(2/3)*(e*(c + d*x))^(2/3)) + (72*(c + d*x)^(2/3)*Cos[a + b/(c + d*x)^(1/3)])/(b^5*d*e^2*(e*(c + d*x))^(
2/3)) - (12*Sin[a + b/(c + d*x)^(1/3)])/(b^2*d*e^2*(c + d*x)^(1/3)*(e*(c + d*x))^(2/3)) + (72*(c + d*x)^(1/3)*
Sin[a + b/(c + d*x)^(1/3)])/(b^4*d*e^2*(e*(c + d*x))^(2/3))

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3431

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

Rubi steps

\begin {align*} \int \frac {\sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{(c e+d e x)^{8/3}} \, dx &=-\frac {3 \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{\left (\frac {e}{x^3}\right )^{8/3} x^4} \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d}\\ &=-\frac {\left (3 (c+d x)^{2/3}\right ) \operatorname {Subst}\left (\int x^4 \sin (a+b x) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{d e^2 (e (c+d x))^{2/3}}\\ &=\frac {3 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} (e (c+d x))^{2/3}}-\frac {\left (12 (c+d x)^{2/3}\right ) \operatorname {Subst}\left (\int x^3 \cos (a+b x) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{b d e^2 (e (c+d x))^{2/3}}\\ &=\frac {3 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} (e (c+d x))^{2/3}}-\frac {12 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{c+d x} (e (c+d x))^{2/3}}+\frac {\left (36 (c+d x)^{2/3}\right ) \operatorname {Subst}\left (\int x^2 \sin (a+b x) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 (e (c+d x))^{2/3}}\\ &=-\frac {36 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^3 d e^2 (e (c+d x))^{2/3}}+\frac {3 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} (e (c+d x))^{2/3}}-\frac {12 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{c+d x} (e (c+d x))^{2/3}}+\frac {\left (72 (c+d x)^{2/3}\right ) \operatorname {Subst}\left (\int x \cos (a+b x) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{b^3 d e^2 (e (c+d x))^{2/3}}\\ &=-\frac {36 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^3 d e^2 (e (c+d x))^{2/3}}+\frac {3 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} (e (c+d x))^{2/3}}-\frac {12 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{c+d x} (e (c+d x))^{2/3}}+\frac {72 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^4 d e^2 (e (c+d x))^{2/3}}-\frac {\left (72 (c+d x)^{2/3}\right ) \operatorname {Subst}\left (\int \sin (a+b x) \, dx,x,\frac {1}{\sqrt [3]{c+d x}}\right )}{b^4 d e^2 (e (c+d x))^{2/3}}\\ &=-\frac {36 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^3 d e^2 (e (c+d x))^{2/3}}+\frac {3 \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b d e^2 (c+d x)^{2/3} (e (c+d x))^{2/3}}+\frac {72 (c+d x)^{2/3} \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^5 d e^2 (e (c+d x))^{2/3}}-\frac {12 \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^2 d e^2 \sqrt [3]{c+d x} (e (c+d x))^{2/3}}+\frac {72 \sqrt [3]{c+d x} \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )}{b^4 d e^2 (e (c+d x))^{2/3}}\\ \end {align*}

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Mathematica [A]  time = 0.30, size = 112, normalized size = 0.52 \[ \frac {(c+d x)^{4/3} \left (12 b \left (b^2 \left (-\sqrt [3]{c+d x}\right )+6 c+6 d x\right ) \sin \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )+3 \left (b^4-12 b^2 (c+d x)^{2/3}+24 (c+d x)^{4/3}\right ) \cos \left (a+\frac {b}{\sqrt [3]{c+d x}}\right )\right )}{b^5 d (e (c+d x))^{8/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b/(c + d*x)^(1/3)]/(c*e + d*e*x)^(8/3),x]

[Out]

((c + d*x)^(4/3)*(3*(b^4 - 12*b^2*(c + d*x)^(2/3) + 24*(c + d*x)^(4/3))*Cos[a + b/(c + d*x)^(1/3)] + 12*b*(6*c
+ 6*d*x - b^2*(c + d*x)^(1/3))*Sin[a + b/(c + d*x)^(1/3)]))/(b^5*d*(e*(c + d*x))^(8/3))

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fricas [A]  time = 2.12, size = 181, normalized size = 0.83 \[ \frac {3 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b^{4} - 12 \, b^{2} d x - 12 \, b^{2} c + 24 \, {\left (d x + c\right )}^{\frac {5}{3}}\right )} {\left (d e x + c e\right )}^{\frac {1}{3}} \cos \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {2}{3}} b}{d x + c}\right ) - 4 \, {\left ({\left (d x + c\right )}^{\frac {2}{3}} b^{3} - 6 \, {\left (b d x + b c\right )} {\left (d x + c\right )}^{\frac {1}{3}}\right )} {\left (d e x + c e\right )}^{\frac {1}{3}} \sin \left (\frac {a d x + a c + {\left (d x + c\right )}^{\frac {2}{3}} b}{d x + c}\right )\right )}}{b^{5} d^{3} e^{3} x^{2} + 2 \, b^{5} c d^{2} e^{3} x + b^{5} c^{2} d e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)^(1/3))/(d*e*x+c*e)^(8/3),x, algorithm="fricas")

[Out]

3*(((d*x + c)^(1/3)*b^4 - 12*b^2*d*x - 12*b^2*c + 24*(d*x + c)^(5/3))*(d*e*x + c*e)^(1/3)*cos((a*d*x + a*c + (
d*x + c)^(2/3)*b)/(d*x + c)) - 4*((d*x + c)^(2/3)*b^3 - 6*(b*d*x + b*c)*(d*x + c)^(1/3))*(d*e*x + c*e)^(1/3)*s
in((a*d*x + a*c + (d*x + c)^(2/3)*b)/(d*x + c)))/(b^5*d^3*e^3*x^2 + 2*b^5*c*d^2*e^3*x + b^5*c^2*d*e^3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (a + \frac {b}{{\left (d x + c\right )}^{\frac {1}{3}}}\right )}{{\left (d e x + c e\right )}^{\frac {8}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)^(1/3))/(d*e*x+c*e)^(8/3),x, algorithm="giac")

[Out]

integrate(sin(a + b/(d*x + c)^(1/3))/(d*e*x + c*e)^(8/3), x)

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maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {1}{3}}}\right )}{\left (d e x +c e \right )^{\frac {8}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a+b/(d*x+c)^(1/3))/(d*e*x+c*e)^(8/3),x)

[Out]

int(sin(a+b/(d*x+c)^(1/3))/(d*e*x+c*e)^(8/3),x)

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maxima [C]  time = 3.57, size = 1965, normalized size = 9.06 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)^(1/3))/(d*e*x+c*e)^(8/3),x, algorithm="maxima")

[Out]

-1/2000*(600*((cos(a)^2 + sin(a)^2)*b^5*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)) - (b^5*cos(((d*x + c)^(1/
3)*a + b)/(d*x + c)^(1/3))^2*sin(a) + b^5*sin(a)*sin(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))^2)*cos((2*(d*x +
c)^(1/3)*a + b)/(d*x + c)^(1/3)) + (b^5*cos(a)*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))^2 + b^5*cos(a)*si
n(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))^2)*sin((2*(d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3)))*(d*x + c)^(1/3)*
e^(1/3) - (((300*(gamma(6, I*b*conjugate((d*x + c)^(-1/3))) + gamma(6, -I*b*conjugate((d*x + c)^(-1/3))) + gam
ma(6, I*b/(d*x + c)^(1/3)) + gamma(6, -I*b/(d*x + c)^(1/3)))*cos(a)^3 - (300*I*gamma(6, I*b*conjugate((d*x + c
)^(-1/3))) - 300*I*gamma(6, -I*b*conjugate((d*x + c)^(-1/3))) + 300*I*gamma(6, I*b/(d*x + c)^(1/3)) - 300*I*ga
mma(6, -I*b/(d*x + c)^(1/3)))*cos(a)^2*sin(a) + 300*(gamma(6, I*b*conjugate((d*x + c)^(-1/3))) + gamma(6, -I*b
*conjugate((d*x + c)^(-1/3))) + gamma(6, I*b/(d*x + c)^(1/3)) + gamma(6, -I*b/(d*x + c)^(1/3)))*cos(a)*sin(a)^
2 - (300*I*gamma(6, I*b*conjugate((d*x + c)^(-1/3))) - 300*I*gamma(6, -I*b*conjugate((d*x + c)^(-1/3))) + 300*
I*gamma(6, I*b/(d*x + c)^(1/3)) - 300*I*gamma(6, -I*b/(d*x + c)^(1/3)))*sin(a)^3)*d^2*x^2 + (600*(gamma(6, I*b
*conjugate((d*x + c)^(-1/3))) + gamma(6, -I*b*conjugate((d*x + c)^(-1/3))) + gamma(6, I*b/(d*x + c)^(1/3)) + g
amma(6, -I*b/(d*x + c)^(1/3)))*cos(a)^3 - (600*I*gamma(6, I*b*conjugate((d*x + c)^(-1/3))) - 600*I*gamma(6, -I
*b*conjugate((d*x + c)^(-1/3))) + 600*I*gamma(6, I*b/(d*x + c)^(1/3)) - 600*I*gamma(6, -I*b/(d*x + c)^(1/3)))*
cos(a)^2*sin(a) + 600*(gamma(6, I*b*conjugate((d*x + c)^(-1/3))) + gamma(6, -I*b*conjugate((d*x + c)^(-1/3)))
+ gamma(6, I*b/(d*x + c)^(1/3)) + gamma(6, -I*b/(d*x + c)^(1/3)))*cos(a)*sin(a)^2 - (600*I*gamma(6, I*b*conjug
ate((d*x + c)^(-1/3))) - 600*I*gamma(6, -I*b*conjugate((d*x + c)^(-1/3))) + 600*I*gamma(6, I*b/(d*x + c)^(1/3)
) - 600*I*gamma(6, -I*b/(d*x + c)^(1/3)))*sin(a)^3)*c*d*x + (300*(gamma(6, I*b*conjugate((d*x + c)^(-1/3))) +
gamma(6, -I*b*conjugate((d*x + c)^(-1/3))) + gamma(6, I*b/(d*x + c)^(1/3)) + gamma(6, -I*b/(d*x + c)^(1/3)))*c
os(a)^3 - (300*I*gamma(6, I*b*conjugate((d*x + c)^(-1/3))) - 300*I*gamma(6, -I*b*conjugate((d*x + c)^(-1/3)))
+ 300*I*gamma(6, I*b/(d*x + c)^(1/3)) - 300*I*gamma(6, -I*b/(d*x + c)^(1/3)))*cos(a)^2*sin(a) + 300*(gamma(6,
I*b*conjugate((d*x + c)^(-1/3))) + gamma(6, -I*b*conjugate((d*x + c)^(-1/3))) + gamma(6, I*b/(d*x + c)^(1/3))
+ gamma(6, -I*b/(d*x + c)^(1/3)))*cos(a)*sin(a)^2 - (300*I*gamma(6, I*b*conjugate((d*x + c)^(-1/3))) - 300*I*g
amma(6, -I*b*conjugate((d*x + c)^(-1/3))) + 300*I*gamma(6, I*b/(d*x + c)^(1/3)) - 300*I*gamma(6, -I*b/(d*x + c
)^(1/3)))*sin(a)^3)*c^2)*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))^2 + ((300*(gamma(6, I*b*conjugate((d*x +
c)^(-1/3))) + gamma(6, -I*b*conjugate((d*x + c)^(-1/3))) + gamma(6, I*b/(d*x + c)^(1/3)) + gamma(6, -I*b/(d*x
+ c)^(1/3)))*cos(a)^3 - (300*I*gamma(6, I*b*conjugate((d*x + c)^(-1/3))) - 300*I*gamma(6, -I*b*conjugate((d*x
+ c)^(-1/3))) + 300*I*gamma(6, I*b/(d*x + c)^(1/3)) - 300*I*gamma(6, -I*b/(d*x + c)^(1/3)))*cos(a)^2*sin(a) +
300*(gamma(6, I*b*conjugate((d*x + c)^(-1/3))) + gamma(6, -I*b*conjugate((d*x + c)^(-1/3))) + gamma(6, I*b/(d
*x + c)^(1/3)) + gamma(6, -I*b/(d*x + c)^(1/3)))*cos(a)*sin(a)^2 - (300*I*gamma(6, I*b*conjugate((d*x + c)^(-1
/3))) - 300*I*gamma(6, -I*b*conjugate((d*x + c)^(-1/3))) + 300*I*gamma(6, I*b/(d*x + c)^(1/3)) - 300*I*gamma(6
, -I*b/(d*x + c)^(1/3)))*sin(a)^3)*d^2*x^2 + (600*(gamma(6, I*b*conjugate((d*x + c)^(-1/3))) + gamma(6, -I*b*c
onjugate((d*x + c)^(-1/3))) + gamma(6, I*b/(d*x + c)^(1/3)) + gamma(6, -I*b/(d*x + c)^(1/3)))*cos(a)^3 - (600*
I*gamma(6, I*b*conjugate((d*x + c)^(-1/3))) - 600*I*gamma(6, -I*b*conjugate((d*x + c)^(-1/3))) + 600*I*gamma(6
, I*b/(d*x + c)^(1/3)) - 600*I*gamma(6, -I*b/(d*x + c)^(1/3)))*cos(a)^2*sin(a) + 600*(gamma(6, I*b*conjugate((
d*x + c)^(-1/3))) + gamma(6, -I*b*conjugate((d*x + c)^(-1/3))) + gamma(6, I*b/(d*x + c)^(1/3)) + gamma(6, -I*b
/(d*x + c)^(1/3)))*cos(a)*sin(a)^2 - (600*I*gamma(6, I*b*conjugate((d*x + c)^(-1/3))) - 600*I*gamma(6, -I*b*co
njugate((d*x + c)^(-1/3))) + 600*I*gamma(6, I*b/(d*x + c)^(1/3)) - 600*I*gamma(6, -I*b/(d*x + c)^(1/3)))*sin(a
)^3)*c*d*x + (300*(gamma(6, I*b*conjugate((d*x + c)^(-1/3))) + gamma(6, -I*b*conjugate((d*x + c)^(-1/3))) + ga
mma(6, I*b/(d*x + c)^(1/3)) + gamma(6, -I*b/(d*x + c)^(1/3)))*cos(a)^3 - (300*I*gamma(6, I*b*conjugate((d*x +
c)^(-1/3))) - 300*I*gamma(6, -I*b*conjugate((d*x + c)^(-1/3))) + 300*I*gamma(6, I*b/(d*x + c)^(1/3)) - 300*I*g
amma(6, -I*b/(d*x + c)^(1/3)))*cos(a)^2*sin(a) + 300*(gamma(6, I*b*conjugate((d*x + c)^(-1/3))) + gamma(6, -I*
b*conjugate((d*x + c)^(-1/3))) + gamma(6, I*b/(d*x + c)^(1/3)) + gamma(6, -I*b/(d*x + c)^(1/3)))*cos(a)*sin(a)
^2 - (300*I*gamma(6, I*b*conjugate((d*x + c)^(-1/3))) - 300*I*gamma(6, -I*b*conjugate((d*x + c)^(-1/3))) + 300
*I*gamma(6, I*b/(d*x + c)^(1/3)) - 300*I*gamma(6, -I*b/(d*x + c)^(1/3)))*sin(a)^3)*c^2)*sin(((d*x + c)^(1/3)*a
+ b)/(d*x + c)^(1/3))^2)*e^(1/3))/(((cos(a)^2 + sin(a)^2)*b^5*d^3*e^3*x^2 + 2*(cos(a)^2 + sin(a)^2)*b^5*c*d^2
*e^3*x + (cos(a)^2 + sin(a)^2)*b^5*c^2*d*e^3)*cos(((d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))^2 + ((cos(a)^2 + si
n(a)^2)*b^5*d^3*e^3*x^2 + 2*(cos(a)^2 + sin(a)^2)*b^5*c*d^2*e^3*x + (cos(a)^2 + sin(a)^2)*b^5*c^2*d*e^3)*sin((
(d*x + c)^(1/3)*a + b)/(d*x + c)^(1/3))^2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sin \left (a+\frac {b}{{\left (c+d\,x\right )}^{1/3}}\right )}{{\left (c\,e+d\,e\,x\right )}^{8/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b/(c + d*x)^(1/3))/(c*e + d*e*x)^(8/3),x)

[Out]

int(sin(a + b/(c + d*x)^(1/3))/(c*e + d*e*x)^(8/3), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)**(1/3))/(d*e*x+c*e)**(8/3),x)

[Out]

Timed out

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