Integrand size = 12, antiderivative size = 107 \[ \int \sin \left (a+b (c+d x)^3\right ) \, dx=\frac {i e^{i a} (c+d x) \Gamma \left (\frac {1}{3},-i b (c+d x)^3\right )}{6 d \sqrt [3]{-i b (c+d x)^3}}-\frac {i e^{-i a} (c+d x) \Gamma \left (\frac {1}{3},i b (c+d x)^3\right )}{6 d \sqrt [3]{i b (c+d x)^3}} \] Output:
1/6*I*exp(I*a)*(d*x+c)*GAMMA(1/3,-I*b*(d*x+c)^3)/d/(-I*b*(d*x+c)^3)^(1/3)- 1/6*I*(d*x+c)*GAMMA(1/3,I*b*(d*x+c)^3)/d/exp(I*a)/(I*b*(d*x+c)^3)^(1/3)
Time = 0.15 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.07 \[ \int \sin \left (a+b (c+d x)^3\right ) \, dx=\frac {i (c+d x) \left (-\sqrt [3]{-i b (c+d x)^3} \Gamma \left (\frac {1}{3},i b (c+d x)^3\right ) (\cos (a)-i \sin (a))+\sqrt [3]{i b (c+d x)^3} \Gamma \left (\frac {1}{3},-i b (c+d x)^3\right ) (\cos (a)+i \sin (a))\right )}{6 d \sqrt [3]{b^2 (c+d x)^6}} \] Input:
Integrate[Sin[a + b*(c + d*x)^3],x]
Output:
((I/6)*(c + d*x)*(-(((-I)*b*(c + d*x)^3)^(1/3)*Gamma[1/3, I*b*(c + d*x)^3] *(Cos[a] - I*Sin[a])) + (I*b*(c + d*x)^3)^(1/3)*Gamma[1/3, (-I)*b*(c + d*x )^3]*(Cos[a] + I*Sin[a])))/(d*(b^2*(c + d*x)^6)^(1/3))
Time = 0.23 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3836, 2637}
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.
\(\displaystyle \int \sin \left (a+b (c+d x)^3\right ) \, dx\) |
\(\Big \downarrow \) 3836 |
\(\displaystyle \frac {1}{2} i \int e^{-i b (c+d x)^3-i a}dx-\frac {1}{2} i \int e^{i b (c+d x)^3+i a}dx\) |
\(\Big \downarrow \) 2637 |
\(\displaystyle \frac {i e^{i a} (c+d x) \Gamma \left (\frac {1}{3},-i b (c+d x)^3\right )}{6 d \sqrt [3]{-i b (c+d x)^3}}-\frac {i e^{-i a} (c+d x) \Gamma \left (\frac {1}{3},i b (c+d x)^3\right )}{6 d \sqrt [3]{i b (c+d x)^3}}\) |
Input:
Int[Sin[a + b*(c + d*x)^3],x]
Output:
((I/6)*E^(I*a)*(c + d*x)*Gamma[1/3, (-I)*b*(c + d*x)^3])/(d*((-I)*b*(c + d *x)^3)^(1/3)) - ((I/6)*(c + d*x)*Gamma[1/3, I*b*(c + d*x)^3])/(d*E^(I*a)*( I*b*(c + d*x)^3)^(1/3))
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[(-F^a )*(c + d*x)*(Gamma[1/n, (-b)*(c + d*x)^n*Log[F]]/(d*n*((-b)*(c + d*x)^n*Log [F])^(1/n))), x] /; FreeQ[{F, a, b, c, d, n}, x] && !IntegerQ[2/n]
Int[Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)], x_Symbol] :> Simp[I/2 I nt[E^((-c)*I - d*I*(e + f*x)^n), x], x] - Simp[I/2 Int[E^(c*I + d*I*(e + f*x)^n), x], x] /; FreeQ[{c, d, e, f}, x] && IGtQ[n, 2]
\[\int \sin \left (a +b \left (d x +c \right )^{3}\right )d x\]
Input:
int(sin(a+b*(d*x+c)^3),x)
Output:
int(sin(a+b*(d*x+c)^3),x)
Time = 0.08 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.06 \[ \int \sin \left (a+b (c+d x)^3\right ) \, dx=-\frac {\left (i \, b d^{3}\right )^{\frac {2}{3}} {\left (\cos \left (a\right ) - i \, \sin \left (a\right )\right )} \Gamma \left (\frac {1}{3}, i \, b d^{3} x^{3} + 3 i \, b c d^{2} x^{2} + 3 i \, b c^{2} d x + i \, b c^{3}\right ) + \left (-i \, b d^{3}\right )^{\frac {2}{3}} {\left (\cos \left (a\right ) + i \, \sin \left (a\right )\right )} \Gamma \left (\frac {1}{3}, -i \, b d^{3} x^{3} - 3 i \, b c d^{2} x^{2} - 3 i \, b c^{2} d x - i \, b c^{3}\right )}{6 \, b d^{3}} \] Input:
integrate(sin(a+b*(d*x+c)^3),x, algorithm="fricas")
Output:
-1/6*((I*b*d^3)^(2/3)*(cos(a) - I*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)*(cos(a) + I*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))/(b* d^3)
\[ \int \sin \left (a+b (c+d x)^3\right ) \, dx=\int \sin {\left (a + b \left (c + d x\right )^{3} \right )}\, dx \] Input:
integrate(sin(a+b*(d*x+c)**3),x)
Output:
Integral(sin(a + b*(c + d*x)**3), x)
\[ \int \sin \left (a+b (c+d x)^3\right ) \, dx=\int { \sin \left ({\left (d x + c\right )}^{3} b + a\right ) \,d x } \] Input:
integrate(sin(a+b*(d*x+c)^3),x, algorithm="maxima")
Output:
integrate(sin((d*x + c)^3*b + a), x)
\[ \int \sin \left (a+b (c+d x)^3\right ) \, dx=\int { \sin \left ({\left (d x + c\right )}^{3} b + a\right ) \,d x } \] Input:
integrate(sin(a+b*(d*x+c)^3),x, algorithm="giac")
Output:
integrate(sin((d*x + c)^3*b + a), x)
Timed out. \[ \int \sin \left (a+b (c+d x)^3\right ) \, dx=\int \sin \left (a+b\,{\left (c+d\,x\right )}^3\right ) \,d x \] Input:
int(sin(a + b*(c + d*x)^3),x)
Output:
int(sin(a + b*(c + d*x)^3), x)
\[ \int \sin \left (a+b (c+d x)^3\right ) \, dx=\int \sin \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )d x \] Input:
int(sin(a+b*(d*x+c)^3),x)
Output:
int(sin(a + b*c**3 + 3*b*c**2*d*x + 3*b*c*d**2*x**2 + b*d**3*x**3),x)