3.2.71 \(\int (e+f x)^3 \sin (a+b (c+d x)^3) \, dx\) [171]

3.2.71.1 Optimal result

Integrand size = 20, antiderivative size = 434 \[ \int (e+f x)^3 \sin \left (a+b (c+d x)^3\right ) \, dx=-\frac {f^2 (d e-c f) \cos \left (a+b (c+d x)^3\right )}{b d^4}-\frac {f^3 (c+d x) \cos \left (a+b (c+d x)^3\right )}{3 b d^4}-\frac {e^{i a} f^3 (c+d x) \Gamma \left (\frac {1}{3},-i b (c+d x)^3\right )}{18 b d^4 \sqrt [3]{-i b (c+d x)^3}}+\frac {i e^{i a} (d e-c f)^3 (c+d x) \Gamma \left (\frac {1}{3},-i b (c+d x)^3\right )}{6 d^4 \sqrt [3]{-i b (c+d x)^3}}-\frac {e^{-i a} f^3 (c+d x) \Gamma \left (\frac {1}{3},i b (c+d x)^3\right )}{18 b d^4 \sqrt [3]{i b (c+d x)^3}}-\frac {i e^{-i a} (d e-c f)^3 (c+d x) \Gamma \left (\frac {1}{3},i b (c+d x)^3\right )}{6 d^4 \sqrt [3]{i b (c+d x)^3}}+\frac {i e^{i a} f (d e-c f)^2 (c+d x)^2 \Gamma \left (\frac {2}{3},-i b (c+d x)^3\right )}{2 d^4 \left (-i b (c+d x)^3\right )^{2/3}}-\frac {i e^{-i a} f (d e-c f)^2 (c+d x)^2 \Gamma \left (\frac {2}{3},i b (c+d x)^3\right )}{2 d^4 \left (i b (c+d x)^3\right )^{2/3}} \]

output

-f^2*(-c*f+d*e)*cos(a+b*(d*x+c)^3)/b/d^4-1/3*f^3*(d*x+c)*cos(a+b*(d*x+c)^3 
)/b/d^4-1/18*exp(I*a)*f^3*(d*x+c)*GAMMA(1/3,-I*b*(d*x+c)^3)/b/d^4/(-I*b*(d 
*x+c)^3)^(1/3)+1/6*I*exp(I*a)*(-c*f+d*e)^3*(d*x+c)*GAMMA(1/3,-I*b*(d*x+c)^ 
3)/d^4/(-I*b*(d*x+c)^3)^(1/3)-1/18*f^3*(d*x+c)*GAMMA(1/3,I*b*(d*x+c)^3)/b/ 
d^4/exp(I*a)/(I*b*(d*x+c)^3)^(1/3)-1/6*I*(-c*f+d*e)^3*(d*x+c)*GAMMA(1/3,I* 
b*(d*x+c)^3)/d^4/exp(I*a)/(I*b*(d*x+c)^3)^(1/3)+1/2*I*exp(I*a)*f*(-c*f+d*e 
)^2*(d*x+c)^2*GAMMA(2/3,-I*b*(d*x+c)^3)/d^4/(-I*b*(d*x+c)^3)^(2/3)-1/2*I*f 
*(-c*f+d*e)^2*(d*x+c)^2*GAMMA(2/3,I*b*(d*x+c)^3)/d^4/exp(I*a)/(I*b*(d*x+c) 
^3)^(2/3)
 

3.2.71.2 Mathematica [A] (verified)

Time = 13.94 (sec) , antiderivative size = 353, normalized size of antiderivative = 0.81 \[ \int (e+f x)^3 \sin \left (a+b (c+d x)^3\right ) \, dx=\frac {-6 f^2 (3 d e-2 c f+d f x) \cos \left (a+b c^3\right ) \cos \left (b d x \left (3 c^2+3 c d x+d^2 x^2\right )\right )+\frac {(c+d x) \left (-\left (\left (f^3+3 i b (d e-c f)^3\right ) \sqrt [3]{i b (c+d x)^3} \Gamma \left (\frac {1}{3},i b (c+d x)^3\right )\right )-9 i b f (d e-c f)^2 (c+d x) \Gamma \left (\frac {2}{3},i b (c+d x)^3\right )\right ) (\cos (a)-i \sin (a))}{\left (i b (c+d x)^3\right )^{2/3}}+\frac {(c+d x) \left (-\left (\left (f^3-3 i b (d e-c f)^3\right ) \sqrt [3]{-i b (c+d x)^3} \Gamma \left (\frac {1}{3},-i b (c+d x)^3\right )\right )+9 i b f (d e-c f)^2 (c+d x) \Gamma \left (\frac {2}{3},-i b (c+d x)^3\right )\right ) (\cos (a)+i \sin (a))}{\left (-i b (c+d x)^3\right )^{2/3}}+6 f^2 (3 d e-2 c f+d f x) \sin \left (a+b c^3\right ) \sin \left (b d x \left (3 c^2+3 c d x+d^2 x^2\right )\right )}{18 b d^4} \]

input

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

output

(-6*f^2*(3*d*e - 2*c*f + d*f*x)*Cos[a + b*c^3]*Cos[b*d*x*(3*c^2 + 3*c*d*x 
+ d^2*x^2)] + ((c + d*x)*(-((f^3 + (3*I)*b*(d*e - c*f)^3)*(I*b*(c + d*x)^3 
)^(1/3)*Gamma[1/3, I*b*(c + d*x)^3]) - (9*I)*b*f*(d*e - c*f)^2*(c + d*x)*G 
amma[2/3, I*b*(c + d*x)^3])*(Cos[a] - I*Sin[a]))/(I*b*(c + d*x)^3)^(2/3) + 
 ((c + d*x)*(-((f^3 - (3*I)*b*(d*e - c*f)^3)*((-I)*b*(c + d*x)^3)^(1/3)*Ga 
mma[1/3, (-I)*b*(c + d*x)^3]) + (9*I)*b*f*(d*e - c*f)^2*(c + d*x)*Gamma[2/ 
3, (-I)*b*(c + d*x)^3])*(Cos[a] + I*Sin[a]))/((-I)*b*(c + d*x)^3)^(2/3) + 
6*f^2*(3*d*e - 2*c*f + d*f*x)*Sin[a + b*c^3]*Sin[b*d*x*(3*c^2 + 3*c*d*x + 
d^2*x^2)])/(18*b*d^4)
 

3.2.71.3 Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 414, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3914, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (e+f x)^3 \sin \left (a+b (c+d x)^3\right ) \, dx\)

\(\Big \downarrow \) 3914

\(\displaystyle \frac {\int \left (\sin \left (b (c+d x)^3+a\right ) (d e-c f)^3+3 f (c+d x) \sin \left (b (c+d x)^3+a\right ) (d e-c f)^2+3 f^2 (c+d x)^2 \sin \left (b (c+d x)^3+a\right ) (d e-c f)+f^3 (c+d x)^3 \sin \left (b (c+d x)^3+a\right )\right )d(c+d x)}{d^4}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {f^2 (d e-c f) \cos \left (a+b (c+d x)^3\right )}{b}+\frac {i e^{i a} f (c+d x)^2 (d e-c f)^2 \Gamma \left (\frac {2}{3},-i b (c+d x)^3\right )}{2 \left (-i b (c+d x)^3\right )^{2/3}}-\frac {i e^{-i a} f (c+d x)^2 (d e-c f)^2 \Gamma \left (\frac {2}{3},i b (c+d x)^3\right )}{2 \left (i b (c+d x)^3\right )^{2/3}}+\frac {i e^{i a} (c+d x) (d e-c f)^3 \Gamma \left (\frac {1}{3},-i b (c+d x)^3\right )}{6 \sqrt [3]{-i b (c+d x)^3}}-\frac {i e^{-i a} (c+d x) (d e-c f)^3 \Gamma \left (\frac {1}{3},i b (c+d x)^3\right )}{6 \sqrt [3]{i b (c+d x)^3}}-\frac {f^3 (c+d x) \cos \left (a+b (c+d x)^3\right )}{3 b}-\frac {e^{i a} f^3 (c+d x) \Gamma \left (\frac {1}{3},-i b (c+d x)^3\right )}{18 b \sqrt [3]{-i b (c+d x)^3}}-\frac {e^{-i a} f^3 (c+d x) \Gamma \left (\frac {1}{3},i b (c+d x)^3\right )}{18 b \sqrt [3]{i b (c+d x)^3}}}{d^4}\)

input

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

output

(-((f^2*(d*e - c*f)*Cos[a + b*(c + d*x)^3])/b) - (f^3*(c + d*x)*Cos[a + b* 
(c + d*x)^3])/(3*b) - (E^(I*a)*f^3*(c + d*x)*Gamma[1/3, (-I)*b*(c + d*x)^3 
])/(18*b*((-I)*b*(c + d*x)^3)^(1/3)) + ((I/6)*E^(I*a)*(d*e - c*f)^3*(c + d 
*x)*Gamma[1/3, (-I)*b*(c + d*x)^3])/((-I)*b*(c + d*x)^3)^(1/3) - (f^3*(c + 
 d*x)*Gamma[1/3, I*b*(c + d*x)^3])/(18*b*E^(I*a)*(I*b*(c + d*x)^3)^(1/3)) 
- ((I/6)*(d*e - c*f)^3*(c + d*x)*Gamma[1/3, I*b*(c + d*x)^3])/(E^(I*a)*(I* 
b*(c + d*x)^3)^(1/3)) + ((I/2)*E^(I*a)*f*(d*e - c*f)^2*(c + d*x)^2*Gamma[2 
/3, (-I)*b*(c + d*x)^3])/((-I)*b*(c + d*x)^3)^(2/3) - ((I/2)*f*(d*e - c*f) 
^2*(c + d*x)^2*Gamma[2/3, I*b*(c + d*x)^3])/(E^(I*a)*(I*b*(c + d*x)^3)^(2/ 
3)))/d^4
 

3.2.71.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3914

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

3.2.71.4 Maple [F]

\[\int \left (f x +e \right )^{3} \sin \left (a +b \left (d x +c \right )^{3}\right )d x\]

input

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

output

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

3.2.71.5 Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 591, normalized size of antiderivative = 1.36 \[ \int (e+f x)^3 \sin \left (a+b (c+d x)^3\right ) \, dx=-\frac {\left (i \, b d^{3}\right )^{\frac {2}{3}} {\left ({\left (3 \, b d^{3} e^{3} - 9 \, b c d^{2} e^{2} f + 9 \, b c^{2} d e f^{2} - 3 \, b c^{3} f^{3} - i \, f^{3}\right )} \cos \left (a\right ) - {\left (3 i \, b d^{3} e^{3} - 9 i \, b c d^{2} e^{2} f + 9 i \, b c^{2} d e f^{2} - 3 i \, b c^{3} f^{3} + f^{3}\right )} \sin \left (a\right )\right )} \Gamma \left (\frac {1}{3}, i \, b d^{3} x^{3} + 3 i \, b c d^{2} x^{2} + 3 i \, b c^{2} d x + i \, b c^{3}\right ) + \left (-i \, b d^{3}\right )^{\frac {2}{3}} {\left ({\left (3 \, b d^{3} e^{3} - 9 \, b c d^{2} e^{2} f + 9 \, b c^{2} d e f^{2} - 3 \, b c^{3} f^{3} + i \, f^{3}\right )} \cos \left (a\right ) - {\left (-3 i \, b d^{3} e^{3} + 9 i \, b c d^{2} e^{2} f - 9 i \, b c^{2} d e f^{2} + 3 i \, b c^{3} f^{3} + f^{3}\right )} \sin \left (a\right )\right )} \Gamma \left (\frac {1}{3}, -i \, b d^{3} x^{3} - 3 i \, b c d^{2} x^{2} - 3 i \, b c^{2} d x - i \, b c^{3}\right ) + 9 \, \left (i \, b d^{3}\right )^{\frac {1}{3}} {\left ({\left (b d^{3} e^{2} f - 2 \, b c d^{2} e f^{2} + b c^{2} d f^{3}\right )} \cos \left (a\right ) + {\left (-i \, b d^{3} e^{2} f + 2 i \, b c d^{2} e f^{2} - i \, b c^{2} d f^{3}\right )} \sin \left (a\right )\right )} \Gamma \left (\frac {2}{3}, i \, b d^{3} x^{3} + 3 i \, b c d^{2} x^{2} + 3 i \, b c^{2} d x + i \, b c^{3}\right ) + 9 \, \left (-i \, b d^{3}\right )^{\frac {1}{3}} {\left ({\left (b d^{3} e^{2} f - 2 \, b c d^{2} e f^{2} + b c^{2} d f^{3}\right )} \cos \left (a\right ) + {\left (i \, b d^{3} e^{2} f - 2 i \, b c d^{2} e f^{2} + i \, b c^{2} d f^{3}\right )} \sin \left (a\right )\right )} \Gamma \left (\frac {2}{3}, -i \, b d^{3} x^{3} - 3 i \, b c d^{2} x^{2} - 3 i \, b c^{2} d x - i \, b c^{3}\right ) + 6 \, {\left (b d^{3} f^{3} x + 3 \, b d^{3} e f^{2} - 2 \, b c d^{2} f^{3}\right )} \cos \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{18 \, b^{2} d^{6}} \]

input

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

output

-1/18*((I*b*d^3)^(2/3)*((3*b*d^3*e^3 - 9*b*c*d^2*e^2*f + 9*b*c^2*d*e*f^2 - 
 3*b*c^3*f^3 - I*f^3)*cos(a) - (3*I*b*d^3*e^3 - 9*I*b*c*d^2*e^2*f + 9*I*b* 
c^2*d*e*f^2 - 3*I*b*c^3*f^3 + f^3)*sin(a))*gamma(1/3, I*b*d^3*x^3 + 3*I*b* 
c*d^2*x^2 + 3*I*b*c^2*d*x + I*b*c^3) + (-I*b*d^3)^(2/3)*((3*b*d^3*e^3 - 9* 
b*c*d^2*e^2*f + 9*b*c^2*d*e*f^2 - 3*b*c^3*f^3 + I*f^3)*cos(a) - (-3*I*b*d^ 
3*e^3 + 9*I*b*c*d^2*e^2*f - 9*I*b*c^2*d*e*f^2 + 3*I*b*c^3*f^3 + f^3)*sin(a 
))*gamma(1/3, -I*b*d^3*x^3 - 3*I*b*c*d^2*x^2 - 3*I*b*c^2*d*x - I*b*c^3) + 
9*(I*b*d^3)^(1/3)*((b*d^3*e^2*f - 2*b*c*d^2*e*f^2 + b*c^2*d*f^3)*cos(a) + 
(-I*b*d^3*e^2*f + 2*I*b*c*d^2*e*f^2 - I*b*c^2*d*f^3)*sin(a))*gamma(2/3, I* 
b*d^3*x^3 + 3*I*b*c*d^2*x^2 + 3*I*b*c^2*d*x + I*b*c^3) + 9*(-I*b*d^3)^(1/3 
)*((b*d^3*e^2*f - 2*b*c*d^2*e*f^2 + b*c^2*d*f^3)*cos(a) + (I*b*d^3*e^2*f - 
 2*I*b*c*d^2*e*f^2 + I*b*c^2*d*f^3)*sin(a))*gamma(2/3, -I*b*d^3*x^3 - 3*I* 
b*c*d^2*x^2 - 3*I*b*c^2*d*x - I*b*c^3) + 6*(b*d^3*f^3*x + 3*b*d^3*e*f^2 - 
2*b*c*d^2*f^3)*cos(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a))/( 
b^2*d^6)
 

3.2.71.6 Sympy [F]

\[ \int (e+f x)^3 \sin \left (a+b (c+d x)^3\right ) \, dx=\int \left (e + f x\right )^{3} \sin {\left (a + b c^{3} + 3 b c^{2} d x + 3 b c d^{2} x^{2} + b d^{3} x^{3} \right )}\, dx \]

input

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

output

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

3.2.71.7 Maxima [F]

\[ \int (e+f x)^3 \sin \left (a+b (c+d x)^3\right ) \, dx=\int { {\left (f x + e\right )}^{3} \sin \left ({\left (d x + c\right )}^{3} b + a\right ) \,d x } \]

input

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

output

integrate((f*x + e)^3*sin((d*x + c)^3*b + a), x)
 

3.2.71.8 Giac [F]

\[ \int (e+f x)^3 \sin \left (a+b (c+d x)^3\right ) \, dx=\int { {\left (f x + e\right )}^{3} \sin \left ({\left (d x + c\right )}^{3} b + a\right ) \,d x } \]

input

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

output

integrate((f*x + e)^3*sin((d*x + c)^3*b + a), x)
 

3.2.71.9 Mupad [F(-1)]

Timed out. \[ \int (e+f x)^3 \sin \left (a+b (c+d x)^3\right ) \, dx=\int \sin \left (a+b\,{\left (c+d\,x\right )}^3\right )\,{\left (e+f\,x\right )}^3 \,d x \]

input

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

output

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