3.3.27 \(\int (c e+d e x)^{4/3} \sin (a+b \sqrt [3]{c+d x}) \, dx\) [227]

Optimal. Leaf size=289 \[ \frac {2160 e \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d \sqrt [3]{c+d x}}-\frac {1080 e \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d}+\frac {90 e (c+d x) \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {2160 e \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d}-\frac {360 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d}+\frac {18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d} \]

[Out]

2160*e*(e*(d*x+c))^(1/3)*cos(a+b*(d*x+c)^(1/3))/b^7/d/(d*x+c)^(1/3)-1080*e*(d*x+c)^(1/3)*(e*(d*x+c))^(1/3)*cos
(a+b*(d*x+c)^(1/3))/b^5/d+90*e*(d*x+c)*(e*(d*x+c))^(1/3)*cos(a+b*(d*x+c)^(1/3))/b^3/d-3*e*(d*x+c)^(5/3)*(e*(d*
x+c))^(1/3)*cos(a+b*(d*x+c)^(1/3))/b/d+2160*e*(e*(d*x+c))^(1/3)*sin(a+b*(d*x+c)^(1/3))/b^6/d-360*e*(d*x+c)^(2/
3)*(e*(d*x+c))^(1/3)*sin(a+b*(d*x+c)^(1/3))/b^4/d+18*e*(d*x+c)^(4/3)*(e*(d*x+c))^(1/3)*sin(a+b*(d*x+c)^(1/3))/
b^2/d

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Rubi [A]
time = 0.18, antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3512, 15, 3377, 2718} \begin {gather*} \frac {2160 e \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d \sqrt [3]{c+d x}}+\frac {2160 e \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d}-\frac {1080 e \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d}-\frac {360 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d}+\frac {90 e (c+d x) \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}+\frac {18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac {3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^(4/3)*Sin[a + b*(c + d*x)^(1/3)],x]

[Out]

(2160*e*(e*(c + d*x))^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^7*d*(c + d*x)^(1/3)) - (1080*e*(c + d*x)^(1/3)*(e*(
c + d*x))^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b^5*d) + (90*e*(c + d*x)*(e*(c + d*x))^(1/3)*Cos[a + b*(c + d*x)^
(1/3)])/(b^3*d) - (3*e*(c + d*x)^(5/3)*(e*(c + d*x))^(1/3)*Cos[a + b*(c + d*x)^(1/3)])/(b*d) + (2160*e*(e*(c +
d*x))^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^6*d) - (360*e*(c + d*x)^(2/3)*(e*(c + d*x))^(1/3)*Sin[a + b*(c + d
*x)^(1/3)])/(b^4*d) + (18*e*(c + d*x)^(4/3)*(e*(c + d*x))^(1/3)*Sin[a + b*(c + d*x)^(1/3)])/(b^2*d)

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 2718

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

Rule 3377

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 3512

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 (c e+d e x)^{4/3} \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx &=\frac {3 \text {Subst}\left (\int x^2 \left (e x^3\right )^{4/3} \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=\frac {\left (3 e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x^6 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d \sqrt [3]{c+d x}}\\ &=-\frac {3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {\left (18 e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x^5 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d \sqrt [3]{c+d x}}\\ &=-\frac {3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac {\left (90 e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x^4 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d \sqrt [3]{c+d x}}\\ &=\frac {90 e (c+d x) \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac {\left (360 e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x^3 \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^3 d \sqrt [3]{c+d x}}\\ &=\frac {90 e (c+d x) \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac {360 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d}+\frac {18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}+\frac {\left (1080 e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x^2 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^4 d \sqrt [3]{c+d x}}\\ &=-\frac {1080 e \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d}+\frac {90 e (c+d x) \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}-\frac {360 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d}+\frac {18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}+\frac {\left (2160 e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int x \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^5 d \sqrt [3]{c+d x}}\\ &=-\frac {1080 e \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d}+\frac {90 e (c+d x) \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {2160 e \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d}-\frac {360 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d}+\frac {18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac {\left (2160 e \sqrt [3]{e (c+d x)}\right ) \text {Subst}\left (\int \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^6 d \sqrt [3]{c+d x}}\\ &=\frac {2160 e \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^7 d \sqrt [3]{c+d x}}-\frac {1080 e \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^5 d}+\frac {90 e (c+d x) \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {3 e (c+d x)^{5/3} \sqrt [3]{e (c+d x)} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {2160 e \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^6 d}-\frac {360 e (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^4 d}+\frac {18 e (c+d x)^{4/3} \sqrt [3]{e (c+d x)} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}\\ \end {align*}

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Mathematica [A]
time = 0.38, size = 226, normalized size = 0.78 \begin {gather*} \frac {3 (e (c+d x))^{4/3} \left (-\cos \left (b \sqrt [3]{c+d x}\right ) \left (\left (-720+360 b^2 (c+d x)^{2/3}-30 b^4 (c+d x)^{4/3}+b^6 (c+d x)^2\right ) \cos (a)-6 b \left (120 \sqrt [3]{c+d x}-20 b^2 (c+d x)+b^4 (c+d x)^{5/3}\right ) \sin (a)\right )+\left (6 b \left (120 \sqrt [3]{c+d x}-20 b^2 (c+d x)+b^4 (c+d x)^{5/3}\right ) \cos (a)+\left (-720+360 b^2 (c+d x)^{2/3}-30 b^4 (c+d x)^{4/3}+b^6 (c+d x)^2\right ) \sin (a)\right ) \sin \left (b \sqrt [3]{c+d x}\right )\right )}{b^7 d (c+d x)^{4/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^(4/3)*Sin[a + b*(c + d*x)^(1/3)],x]

[Out]

(3*(e*(c + d*x))^(4/3)*(-(Cos[b*(c + d*x)^(1/3)]*((-720 + 360*b^2*(c + d*x)^(2/3) - 30*b^4*(c + d*x)^(4/3) + b
^6*(c + d*x)^2)*Cos[a] - 6*b*(120*(c + d*x)^(1/3) - 20*b^2*(c + d*x) + b^4*(c + d*x)^(5/3))*Sin[a])) + (6*b*(1
20*(c + d*x)^(1/3) - 20*b^2*(c + d*x) + b^4*(c + d*x)^(5/3))*Cos[a] + (-720 + 360*b^2*(c + d*x)^(2/3) - 30*b^4
*(c + d*x)^(4/3) + b^6*(c + d*x)^2)*Sin[a])*Sin[b*(c + d*x)^(1/3)]))/(b^7*d*(c + d*x)^(4/3))

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (d e x +c e \right )^{\frac {4}{3}} \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^(4/3)*sin(a+b*(d*x+c)^(1/3)),x)

[Out]

int((d*e*x+c*e)^(4/3)*sin(a+b*(d*x+c)^(1/3)),x)

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Maxima [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.49, size = 195, normalized size = 0.67 \begin {gather*} -\frac {3 \, {\left (2 \, {\left (b^{6} d^{2} x^{2} e + 2 \, b^{6} c d x e + b^{6} c^{2} e\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - 3 \, {\left (e \Gamma \left (6, i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + e \Gamma \left (6, -i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + e \Gamma \left (6, i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right ) + e \Gamma \left (6, -i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )\right )} \cos \left (a\right ) - 3 \, {\left (-i \, e \Gamma \left (6, i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) + i \, e \Gamma \left (6, -i \, b \overline {{\left (d x + c\right )}^{\frac {1}{3}}}\right ) - i \, e \Gamma \left (6, i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right ) + i \, e \Gamma \left (6, -i \, {\left (d x + c\right )}^{\frac {1}{3}} b\right )\right )} \sin \left (a\right )\right )} e^{\frac {1}{3}}}{2 \, b^{7} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(4/3)*sin(a+b*(d*x+c)^(1/3)),x, algorithm="maxima")

[Out]

-3/2*(2*(b^6*d^2*x^2*e + 2*b^6*c*d*x*e + b^6*c^2*e)*cos((d*x + c)^(1/3)*b + a) - 3*(e*gamma(6, I*b*conjugate((
d*x + c)^(1/3))) + e*gamma(6, -I*b*conjugate((d*x + c)^(1/3))) + e*gamma(6, I*(d*x + c)^(1/3)*b) + e*gamma(6,
-I*(d*x + c)^(1/3)*b))*cos(a) - 3*(-I*e*gamma(6, I*b*conjugate((d*x + c)^(1/3))) + I*e*gamma(6, -I*b*conjugate
((d*x + c)^(1/3))) - I*e*gamma(6, I*(d*x + c)^(1/3)*b) + I*e*gamma(6, -I*(d*x + c)^(1/3)*b))*sin(a))*e^(1/3)/(
b^7*d)

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Fricas [A]
time = 0.76, size = 229, normalized size = 0.79 \begin {gather*} -\frac {3 \, {\left ({\left ({\left (b^{6} d^{2} x^{2} + 2 \, b^{6} c d x + b^{6} c^{2} - 720\right )} {\left (d x + c\right )}^{\frac {2}{3}} e + 360 \, {\left (b^{2} d x + b^{2} c\right )} {\left (d x + c\right )}^{\frac {1}{3}} e - 30 \, {\left (b^{4} d^{2} x^{2} + 2 \, b^{4} c d x + b^{4} c^{2}\right )} e\right )} {\left (d x + c\right )}^{\frac {1}{3}} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) e^{\frac {1}{3}} + 6 \, {\left (20 \, {\left (b^{3} d x + b^{3} c\right )} {\left (d x + c\right )}^{\frac {2}{3}} e - {\left (b^{5} d^{2} x^{2} + 2 \, b^{5} c d x + b^{5} c^{2}\right )} {\left (d x + c\right )}^{\frac {1}{3}} e - 120 \, {\left (b d x + b c\right )} e\right )} {\left (d x + c\right )}^{\frac {1}{3}} e^{\frac {1}{3}} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{7} d^{2} x + b^{7} c d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(4/3)*sin(a+b*(d*x+c)^(1/3)),x, algorithm="fricas")

[Out]

-3*(((b^6*d^2*x^2 + 2*b^6*c*d*x + b^6*c^2 - 720)*(d*x + c)^(2/3)*e + 360*(b^2*d*x + b^2*c)*(d*x + c)^(1/3)*e -
30*(b^4*d^2*x^2 + 2*b^4*c*d*x + b^4*c^2)*e)*(d*x + c)^(1/3)*cos((d*x + c)^(1/3)*b + a)*e^(1/3) + 6*(20*(b^3*d
*x + b^3*c)*(d*x + c)^(2/3)*e - (b^5*d^2*x^2 + 2*b^5*c*d*x + b^5*c^2)*(d*x + c)^(1/3)*e - 120*(b*d*x + b*c)*e)
*(d*x + c)^(1/3)*e^(1/3)*sin((d*x + c)^(1/3)*b + a))/(b^7*d^2*x + b^7*c*d)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**(4/3)*sin(a+b*(d*x+c)**(1/3)),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3003 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 566 vs. \(2 (195) = 390\).
time = 4.27, size = 566, normalized size = 1.96 \begin {gather*} \frac {3 \, {\left ({\left (2 \, {\left (\frac {{\left ({\left (d x e + c e\right )} b^{3} c e^{3} - 6 \, {\left (d x e + c e\right )}^{\frac {1}{3}} b c e^{\frac {11}{3}}\right )} \cos \left ({\left ({\left (d x e + c e\right )}^{\frac {1}{3}} b e^{\frac {2}{3}} + a e\right )} e^{\left (-1\right )}\right ) e^{\left (-\frac {8}{3}\right )}}{b^{4} d^{2}} - \frac {3 \, {\left ({\left (d x e + c e\right )}^{\frac {2}{3}} b^{2} c e^{\frac {10}{3}} - 2 \, c e^{4}\right )} e^{\left (-\frac {8}{3}\right )} \sin \left ({\left ({\left (d x e + c e\right )}^{\frac {1}{3}} b e^{\frac {2}{3}} + a e\right )} e^{\left (-1\right )}\right )}{b^{4} d^{2}}\right )} e^{\left (-1\right )} - {\left (\frac {{\left ({\left (d x e + c e\right )}^{2} b^{6} e^{5} - 30 \, {\left (d x e + c e\right )}^{\frac {4}{3}} b^{4} e^{\frac {17}{3}} + 360 \, {\left (d x e + c e\right )}^{\frac {2}{3}} b^{2} e^{\frac {19}{3}} - 720 \, e^{7}\right )} \cos \left ({\left ({\left (d x e + c e\right )}^{\frac {1}{3}} b e^{\frac {2}{3}} + a e\right )} e^{\left (-1\right )}\right ) e^{\left (-\frac {14}{3}\right )}}{b^{7} d^{2}} - \frac {6 \, {\left ({\left (d x e + c e\right )}^{\frac {5}{3}} b^{5} e^{\frac {16}{3}} - 20 \, {\left (d x e + c e\right )} b^{3} e^{6} + 120 \, {\left (d x e + c e\right )}^{\frac {1}{3}} b e^{\frac {20}{3}}\right )} e^{\left (-\frac {14}{3}\right )} \sin \left ({\left ({\left (d x e + c e\right )}^{\frac {1}{3}} b e^{\frac {2}{3}} + a e\right )} e^{\left (-1\right )}\right )}{b^{7} d^{2}}\right )} e^{\left (-2\right )} - \frac {c^{2} \cos \left ({\left ({\left (d x e + c e\right )}^{\frac {1}{3}} b e^{\frac {2}{3}} + a e\right )} e^{\left (-1\right )}\right ) e^{\frac {1}{3}}}{b d^{2}}\right )} d^{2} e - \frac {c^{2} \cos \left ({\left ({\left (d x e + c e\right )}^{\frac {1}{3}} b e^{\frac {2}{3}} + a e\right )} e^{\left (-1\right )}\right ) e^{\frac {4}{3}}}{b} + 2 \, {\left (\frac {c \cos \left ({\left ({\left (d x e + c e\right )}^{\frac {1}{3}} b e^{\frac {2}{3}} + a e\right )} e^{\left (-1\right )}\right ) e^{\frac {4}{3}}}{b} - \frac {{\left ({\left (d x e + c e\right )} b^{3} e^{3} - 6 \, {\left (d x e + c e\right )}^{\frac {1}{3}} b e^{\frac {11}{3}}\right )} \cos \left ({\left ({\left (d x e + c e\right )}^{\frac {1}{3}} b e^{\frac {2}{3}} + a e\right )} e^{\left (-1\right )}\right ) e^{\left (-\frac {8}{3}\right )}}{b^{4}} + \frac {3 \, {\left ({\left (d x e + c e\right )}^{\frac {2}{3}} b^{2} e^{\frac {10}{3}} - 2 \, e^{4}\right )} e^{\left (-\frac {8}{3}\right )} \sin \left ({\left ({\left (d x e + c e\right )}^{\frac {1}{3}} b e^{\frac {2}{3}} + a e\right )} e^{\left (-1\right )}\right )}{b^{4}}\right )} c\right )}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^(4/3)*sin(a+b*(d*x+c)^(1/3)),x, algorithm="giac")

[Out]

3*((2*(((d*x*e + c*e)*b^3*c*e^3 - 6*(d*x*e + c*e)^(1/3)*b*c*e^(11/3))*cos(((d*x*e + c*e)^(1/3)*b*e^(2/3) + a*e
)*e^(-1))*e^(-8/3)/(b^4*d^2) - 3*((d*x*e + c*e)^(2/3)*b^2*c*e^(10/3) - 2*c*e^4)*e^(-8/3)*sin(((d*x*e + c*e)^(1
/3)*b*e^(2/3) + a*e)*e^(-1))/(b^4*d^2))*e^(-1) - (((d*x*e + c*e)^2*b^6*e^5 - 30*(d*x*e + c*e)^(4/3)*b^4*e^(17/
3) + 360*(d*x*e + c*e)^(2/3)*b^2*e^(19/3) - 720*e^7)*cos(((d*x*e + c*e)^(1/3)*b*e^(2/3) + a*e)*e^(-1))*e^(-14/
3)/(b^7*d^2) - 6*((d*x*e + c*e)^(5/3)*b^5*e^(16/3) - 20*(d*x*e + c*e)*b^3*e^6 + 120*(d*x*e + c*e)^(1/3)*b*e^(2
0/3))*e^(-14/3)*sin(((d*x*e + c*e)^(1/3)*b*e^(2/3) + a*e)*e^(-1))/(b^7*d^2))*e^(-2) - c^2*cos(((d*x*e + c*e)^(
1/3)*b*e^(2/3) + a*e)*e^(-1))*e^(1/3)/(b*d^2))*d^2*e - c^2*cos(((d*x*e + c*e)^(1/3)*b*e^(2/3) + a*e)*e^(-1))*e
^(4/3)/b + 2*(c*cos(((d*x*e + c*e)^(1/3)*b*e^(2/3) + a*e)*e^(-1))*e^(4/3)/b - ((d*x*e + c*e)*b^3*e^3 - 6*(d*x*
e + c*e)^(1/3)*b*e^(11/3))*cos(((d*x*e + c*e)^(1/3)*b*e^(2/3) + a*e)*e^(-1))*e^(-8/3)/b^4 + 3*((d*x*e + c*e)^(
2/3)*b^2*e^(10/3) - 2*e^4)*e^(-8/3)*sin(((d*x*e + c*e)^(1/3)*b*e^(2/3) + a*e)*e^(-1))/b^4)*c)/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sin \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )\,{\left (c\,e+d\,e\,x\right )}^{4/3} \,d x \end {gather*}

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)^(4/3),x)

[Out]

int(sin(a + b*(c + d*x)^(1/3))*(c*e + d*e*x)^(4/3), x)

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