Integrand size = 18, antiderivative size = 235 \[ \int (e+f x) \sin \left (a+\frac {b}{(c+d x)^3}\right ) \, dx=-\frac {i e^{i a} f \left (-\frac {i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac {2}{3},-\frac {i b}{(c+d x)^3}\right )}{6 d^2}+\frac {i e^{-i a} f \left (\frac {i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac {2}{3},\frac {i b}{(c+d x)^3}\right )}{6 d^2}-\frac {i e^{i a} (d e-c f) \sqrt [3]{-\frac {i b}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},-\frac {i b}{(c+d x)^3}\right )}{6 d^2}+\frac {i e^{-i a} (d e-c f) \sqrt [3]{\frac {i b}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},\frac {i b}{(c+d x)^3}\right )}{6 d^2} \] Output:
-1/6*I*exp(I*a)*f*(-I*b/(d*x+c)^3)^(2/3)*(d*x+c)^2*GAMMA(-2/3,-I*b/(d*x+c) ^3)/d^2+1/6*I*f*(I*b/(d*x+c)^3)^(2/3)*(d*x+c)^2*GAMMA(-2/3,I*b/(d*x+c)^3)/ d^2/exp(I*a)-1/6*I*exp(I*a)*(-c*f+d*e)*(-I*b/(d*x+c)^3)^(1/3)*(d*x+c)*GAMM A(-1/3,-I*b/(d*x+c)^3)/d^2+1/6*I*(-c*f+d*e)*(I*b/(d*x+c)^3)^(1/3)*(d*x+c)* GAMMA(-1/3,I*b/(d*x+c)^3)/d^2/exp(I*a)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(705\) vs. \(2(235)=470\).
Time = 2.46 (sec) , antiderivative size = 705, normalized size of antiderivative = 3.00 \[ \int (e+f x) \sin \left (a+\frac {b}{(c+d x)^3}\right ) \, dx=\frac {3 b f \cos (a) \left (\frac {\Gamma \left (\frac {1}{3},-\frac {i b}{(c+d x)^3}\right )}{3 \sqrt [3]{-\frac {i b}{(c+d x)^3}} (c+d x)}+\frac {\Gamma \left (\frac {1}{3},\frac {i b}{(c+d x)^3}\right )}{3 \sqrt [3]{\frac {i b}{(c+d x)^3}} (c+d x)}\right )}{4 d^2}+\frac {3 b e \cos (a) \left (\frac {\Gamma \left (\frac {2}{3},-\frac {i b}{(c+d x)^3}\right )}{3 \left (-\frac {i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2}+\frac {\Gamma \left (\frac {2}{3},\frac {i b}{(c+d x)^3}\right )}{3 \left (\frac {i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2}\right )}{2 d}-\frac {3 b c f \cos (a) \left (\frac {\Gamma \left (\frac {2}{3},-\frac {i b}{(c+d x)^3}\right )}{3 \left (-\frac {i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2}+\frac {\Gamma \left (\frac {2}{3},\frac {i b}{(c+d x)^3}\right )}{3 \left (\frac {i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2}\right )}{2 d^2}+\frac {e (c+d x) \cos \left (\frac {b}{(c+d x)^3}\right ) \sin (a)}{d}+\frac {f (-c+d x) (c+d x) \cos \left (\frac {b}{(c+d x)^3}\right ) \sin (a)}{2 d^2}+\frac {3 i b f \left (\frac {\Gamma \left (\frac {1}{3},-\frac {i b}{(c+d x)^3}\right )}{3 \sqrt [3]{-\frac {i b}{(c+d x)^3}} (c+d x)}-\frac {\Gamma \left (\frac {1}{3},\frac {i b}{(c+d x)^3}\right )}{3 \sqrt [3]{\frac {i b}{(c+d x)^3}} (c+d x)}\right ) \sin (a)}{4 d^2}+\frac {3 i b e \left (\frac {\Gamma \left (\frac {2}{3},-\frac {i b}{(c+d x)^3}\right )}{3 \left (-\frac {i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2}-\frac {\Gamma \left (\frac {2}{3},\frac {i b}{(c+d x)^3}\right )}{3 \left (\frac {i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2}\right ) \sin (a)}{2 d}-\frac {3 i b c f \left (\frac {\Gamma \left (\frac {2}{3},-\frac {i b}{(c+d x)^3}\right )}{3 \left (-\frac {i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2}-\frac {\Gamma \left (\frac {2}{3},\frac {i b}{(c+d x)^3}\right )}{3 \left (\frac {i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2}\right ) \sin (a)}{2 d^2}+\frac {e (c+d x) \cos (a) \sin \left (\frac {b}{(c+d x)^3}\right )}{d}+\frac {f (-c+d x) (c+d x) \cos (a) \sin \left (\frac {b}{(c+d x)^3}\right )}{2 d^2} \] Input:
Integrate[(e + f*x)*Sin[a + b/(c + d*x)^3],x]
Output:
(3*b*f*Cos[a]*(Gamma[1/3, ((-I)*b)/(c + d*x)^3]/(3*(((-I)*b)/(c + d*x)^3)^ (1/3)*(c + d*x)) + Gamma[1/3, (I*b)/(c + d*x)^3]/(3*((I*b)/(c + d*x)^3)^(1 /3)*(c + d*x))))/(4*d^2) + (3*b*e*Cos[a]*(Gamma[2/3, ((-I)*b)/(c + d*x)^3] /(3*(((-I)*b)/(c + d*x)^3)^(2/3)*(c + d*x)^2) + Gamma[2/3, (I*b)/(c + d*x) ^3]/(3*((I*b)/(c + d*x)^3)^(2/3)*(c + d*x)^2)))/(2*d) - (3*b*c*f*Cos[a]*(G amma[2/3, ((-I)*b)/(c + d*x)^3]/(3*(((-I)*b)/(c + d*x)^3)^(2/3)*(c + d*x)^ 2) + Gamma[2/3, (I*b)/(c + d*x)^3]/(3*((I*b)/(c + d*x)^3)^(2/3)*(c + d*x)^ 2)))/(2*d^2) + (e*(c + d*x)*Cos[b/(c + d*x)^3]*Sin[a])/d + (f*(-c + d*x)*( c + d*x)*Cos[b/(c + d*x)^3]*Sin[a])/(2*d^2) + (((3*I)/4)*b*f*(Gamma[1/3, ( (-I)*b)/(c + d*x)^3]/(3*(((-I)*b)/(c + d*x)^3)^(1/3)*(c + d*x)) - Gamma[1/ 3, (I*b)/(c + d*x)^3]/(3*((I*b)/(c + d*x)^3)^(1/3)*(c + d*x)))*Sin[a])/d^2 + (((3*I)/2)*b*e*(Gamma[2/3, ((-I)*b)/(c + d*x)^3]/(3*(((-I)*b)/(c + d*x) ^3)^(2/3)*(c + d*x)^2) - Gamma[2/3, (I*b)/(c + d*x)^3]/(3*((I*b)/(c + d*x) ^3)^(2/3)*(c + d*x)^2))*Sin[a])/d - (((3*I)/2)*b*c*f*(Gamma[2/3, ((-I)*b)/ (c + d*x)^3]/(3*(((-I)*b)/(c + d*x)^3)^(2/3)*(c + d*x)^2) - Gamma[2/3, (I* b)/(c + d*x)^3]/(3*((I*b)/(c + d*x)^3)^(2/3)*(c + d*x)^2))*Sin[a])/d^2 + ( e*(c + d*x)*Cos[a]*Sin[b/(c + d*x)^3])/d + (f*(-c + d*x)*(c + d*x)*Cos[a]* Sin[b/(c + d*x)^3])/(2*d^2)
Time = 0.31 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.
| \(\displaystyle \int (e+f x) \sin \left (a+\frac {b}{(c+d x)^3}\right ) \, dx\) |
| \(\Big \downarrow \) 3914 |
| \(\displaystyle \frac {\int \left ((d e-c f) \sin \left (a+\frac {b}{(c+d x)^3}\right )+f (c+d x) \sin \left (a+\frac {b}{(c+d x)^3}\right )\right )d(c+d x)}{d^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {1}{6} i e^{i a} (c+d x) \sqrt [3]{-\frac {i b}{(c+d x)^3}} (d e-c f) \Gamma \left (-\frac {1}{3},-\frac {i b}{(c+d x)^3}\right )+\frac {1}{6} i e^{-i a} (c+d x) \sqrt [3]{\frac {i b}{(c+d x)^3}} (d e-c f) \Gamma \left (-\frac {1}{3},\frac {i b}{(c+d x)^3}\right )-\frac {1}{6} i e^{i a} f (c+d x)^2 \left (-\frac {i b}{(c+d x)^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {i b}{(c+d x)^3}\right )+\frac {1}{6} i e^{-i a} f (c+d x)^2 \left (\frac {i b}{(c+d x)^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},\frac {i b}{(c+d x)^3}\right )}{d^2}\) |
Input:
Int[(e + f*x)*Sin[a + b/(c + d*x)^3],x]
Output:
((-1/6*I)*E^(I*a)*f*(((-I)*b)/(c + d*x)^3)^(2/3)*(c + d*x)^2*Gamma[-2/3, ( (-I)*b)/(c + d*x)^3] + ((I/6)*f*((I*b)/(c + d*x)^3)^(2/3)*(c + d*x)^2*Gamm a[-2/3, (I*b)/(c + d*x)^3])/E^(I*a) - (I/6)*E^(I*a)*(d*e - c*f)*(((-I)*b)/ (c + d*x)^3)^(1/3)*(c + d*x)*Gamma[-1/3, ((-I)*b)/(c + d*x)^3] + ((I/6)*(d *e - c*f)*((I*b)/(c + d*x)^3)^(1/3)*(c + d*x)*Gamma[-1/3, (I*b)/(c + d*x)^ 3])/E^(I*a))/d^2
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]
\[\int \left (f x +e \right ) \sin \left (a +\frac {b}{\left (d x +c \right )^{3}}\right )d x\]
Input:
int((f*x+e)*sin(a+b/(d*x+c)^3),x)
Output:
int((f*x+e)*sin(a+b/(d*x+c)^3),x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (176) = 352\).
Time = 0.09 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.56 \[ \int (e+f x) \sin \left (a+\frac {b}{(c+d x)^3}\right ) \, dx=\frac {{\left (-i \, d^{2} f \cos \left (a\right ) - d^{2} f \sin \left (a\right )\right )} \left (\frac {i \, b}{d^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, \frac {i \, b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) + {\left (i \, d^{2} f \cos \left (a\right ) - d^{2} f \sin \left (a\right )\right )} \left (-\frac {i \, b}{d^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -\frac {i \, b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - 2 \, {\left ({\left (i \, d^{2} e - i \, c d f\right )} \cos \left (a\right ) + {\left (d^{2} e - c d f\right )} \sin \left (a\right )\right )} \left (\frac {i \, b}{d^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, \frac {i \, b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - 2 \, {\left ({\left (-i \, d^{2} e + i \, c d f\right )} \cos \left (a\right ) + {\left (d^{2} e - c d f\right )} \sin \left (a\right )\right )} \left (-\frac {i \, b}{d^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {i \, b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) + 2 \, {\left (d^{2} f x^{2} + 2 \, d^{2} e x + 2 \, c d e - c^{2} f\right )} \sin \left (\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}{4 \, d^{2}} \] Input:
integrate((f*x+e)*sin(a+b/(d*x+c)^3),x, algorithm="fricas")
Output:
1/4*((-I*d^2*f*cos(a) - d^2*f*sin(a))*(I*b/d^3)^(2/3)*gamma(1/3, I*b/(d^3* x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) + (I*d^2*f*cos(a) - d^2*f*sin(a))*(- I*b/d^3)^(2/3)*gamma(1/3, -I*b/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) - 2*((I*d^2*e - I*c*d*f)*cos(a) + (d^2*e - c*d*f)*sin(a))*(I*b/d^3)^(1/3)* gamma(2/3, I*b/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) - 2*((-I*d^2*e + I*c*d*f)*cos(a) + (d^2*e - c*d*f)*sin(a))*(-I*b/d^3)^(1/3)*gamma(2/3, -I* b/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) + 2*(d^2*f*x^2 + 2*d^2*e*x + 2*c*d*e - c^2*f)*sin((a*d^3*x^3 + 3*a*c*d^2*x^2 + 3*a*c^2*d*x + a*c^3 + b) /(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)))/d^2
\[ \int (e+f x) \sin \left (a+\frac {b}{(c+d x)^3}\right ) \, dx=\int \left (e + f x\right ) \sin {\left (a + \frac {b}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}} \right )}\, dx \] Input:
integrate((f*x+e)*sin(a+b/(d*x+c)**3),x)
Output:
Integral((e + f*x)*sin(a + b/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x** 3)), x)
\[ \int (e+f x) \sin \left (a+\frac {b}{(c+d x)^3}\right ) \, dx=\int { {\left (f x + e\right )} \sin \left (a + \frac {b}{{\left (d x + c\right )}^{3}}\right ) \,d x } \] Input:
integrate((f*x+e)*sin(a+b/(d*x+c)^3),x, algorithm="maxima")
Output:
1/2*(f*x^2 + 2*e*x)*sin((a*d^3*x^3 + 3*a*c*d^2*x^2 + 3*a*c^2*d*x + a*c^3 + b)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) + integrate(3/4*(b*d*f*x^2 + 2*b*d*e*x)*cos((a*d^3*x^3 + 3*a*c*d^2*x^2 + 3*a*c^2*d*x + a*c^3 + b)/(d^ 3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))/(d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2 *x^2 + 4*c^3*d*x + c^4), x) + integrate(3/4*(b*d*f*x^2 + 2*b*d*e*x)*cos((a *d^3*x^3 + 3*a*c*d^2*x^2 + 3*a*c^2*d*x + a*c^3 + b)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))/((d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x + c^4)*cos((a*d^3*x^3 + 3*a*c*d^2*x^2 + 3*a*c^2*d*x + a*c^3 + b)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))^2 + (d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^ 2 + 4*c^3*d*x + c^4)*sin((a*d^3*x^3 + 3*a*c*d^2*x^2 + 3*a*c^2*d*x + a*c^3 + b)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))^2), x)
\[ \int (e+f x) \sin \left (a+\frac {b}{(c+d x)^3}\right ) \, dx=\int { {\left (f x + e\right )} \sin \left (a + \frac {b}{{\left (d x + c\right )}^{3}}\right ) \,d x } \] Input:
integrate((f*x+e)*sin(a+b/(d*x+c)^3),x, algorithm="giac")
Output:
integrate((f*x + e)*sin(a + b/(d*x + c)^3), x)
Timed out. \[ \int (e+f x) \sin \left (a+\frac {b}{(c+d x)^3}\right ) \, dx=\int \sin \left (a+\frac {b}{{\left (c+d\,x\right )}^3}\right )\,\left (e+f\,x\right ) \,d x \] Input:
int(sin(a + b/(c + d*x)^3)*(e + f*x),x)
Output:
int(sin(a + b/(c + d*x)^3)*(e + f*x), x)
\[ \int (e+f x) \sin \left (a+\frac {b}{(c+d x)^3}\right ) \, dx=\text {too large to display} \] Input:
int((f*x+e)*sin(a+b/(d*x+c)^3),x)
Output:
(294*cos((a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*c**11*f + 210*cos((a*c**3 + 3* a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b)/(c**3 + 3*c**2*d*x + 3*c*d **2*x**2 + d**3*x**3))*c**10*d*e + 1764*cos((a*c**3 + 3*a*c**2*d*x + 3*a*c *d**2*x**2 + a*d**3*x**3 + b)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x* *3))*c**10*d*f*x + 1260*cos((a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d **3*x**3 + b)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*c**9*d**2*e *x + 4410*cos((a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b)/ (c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*c**9*d**2*f*x**2 + 3150*c os((a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b)/(c**3 + 3*c **2*d*x + 3*c*d**2*x**2 + d**3*x**3))*c**8*d**3*e*x**2 + 5880*cos((a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b)/(c**3 + 3*c**2*d*x + 3 *c*d**2*x**2 + d**3*x**3))*c**8*d**3*f*x**3 + 4200*cos((a*c**3 + 3*a*c**2* d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b)/(c**3 + 3*c**2*d*x + 3*c*d**2*x** 2 + d**3*x**3))*c**7*d**4*e*x**3 + 4410*cos((a*c**3 + 3*a*c**2*d*x + 3*a*c *d**2*x**2 + a*d**3*x**3 + b)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x* *3))*c**7*d**4*f*x**4 + 3150*cos((a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x**3 + b)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*c**6*d **5*e*x**4 + 1764*cos((a*c**3 + 3*a*c**2*d*x + 3*a*c*d**2*x**2 + a*d**3*x* *3 + b)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))*c**6*d**5*f*x*...