Integrand size = 16, antiderivative size = 121 \[ \int \frac {\sin ^3(a+b x)}{c+d x} \, dx=-\frac {\operatorname {CosIntegral}\left (\frac {3 b c}{d}+3 b x\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{4 d}+\frac {3 \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )}{4 d}+\frac {3 \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{4 d}-\frac {\cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{4 d} \] Output:
-1/4*Ci(3*b*c/d+3*b*x)*sin(3*a-3*b*c/d)/d+3/4*Ci(b*c/d+b*x)*sin(a-b*c/d)/d +3/4*cos(a-b*c/d)*Si(b*c/d+b*x)/d-1/4*cos(3*a-3*b*c/d)*Si(3*b*c/d+3*b*x)/d
Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.84 \[ \int \frac {\sin ^3(a+b x)}{c+d x} \, dx=-\frac {\operatorname {CosIntegral}\left (\frac {3 b (c+d x)}{d}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )-3 \operatorname {CosIntegral}\left (b \left (\frac {c}{d}+x\right )\right ) \sin \left (a-\frac {b c}{d}\right )-3 \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (b \left (\frac {c}{d}+x\right )\right )+\cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b (c+d x)}{d}\right )}{4 d} \] Input:
Integrate[Sin[a + b*x]^3/(c + d*x),x]
Output:
-1/4*(CosIntegral[(3*b*(c + d*x))/d]*Sin[3*a - (3*b*c)/d] - 3*CosIntegral[ b*(c/d + x)]*Sin[a - (b*c)/d] - 3*Cos[a - (b*c)/d]*SinIntegral[b*(c/d + x) ] + Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d])/d
Time = 0.47 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3042, 3793, 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.
\(\displaystyle \int \frac {\sin ^3(a+b x)}{c+d x} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (a+b x)^3}{c+d x}dx\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \int \left (\frac {3 \sin (a+b x)}{4 (c+d x)}-\frac {\sin (3 a+3 b x)}{4 (c+d x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {3 b c}{d}+3 b x\right )}{4 d}+\frac {3 \sin \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right )}{4 d}+\frac {3 \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{4 d}-\frac {\cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{4 d}\) |
Input:
Int[Sin[a + b*x]^3/(c + d*x),x]
Output:
-1/4*(CosIntegral[(3*b*c)/d + 3*b*x]*Sin[3*a - (3*b*c)/d])/d + (3*CosInteg ral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/(4*d) + (3*Cos[a - (b*c)/d]*SinIntegr al[(b*c)/d + b*x])/(4*d) - (Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3 *b*x])/(4*d)
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Time = 1.34 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.42
method | result | size |
derivativedivides | \(\frac {-\frac {b \left (-\frac {3 \,\operatorname {Si}\left (-3 b x -3 a -\frac {3 \left (-a d +b c \right )}{d}\right ) \cos \left (\frac {-3 a d +3 b c}{d}\right )}{d}-\frac {3 \,\operatorname {Ci}\left (3 b x +3 a +\frac {-3 a d +3 b c}{d}\right ) \sin \left (\frac {-3 a d +3 b c}{d}\right )}{d}\right )}{12}+\frac {3 b \left (-\frac {\operatorname {Si}\left (-b x -a -\frac {-a d +b c}{d}\right ) \cos \left (\frac {-a d +b c}{d}\right )}{d}-\frac {\operatorname {Ci}\left (b x +a +\frac {-a d +b c}{d}\right ) \sin \left (\frac {-a d +b c}{d}\right )}{d}\right )}{4}}{b}\) | \(172\) |
default | \(\frac {-\frac {b \left (-\frac {3 \,\operatorname {Si}\left (-3 b x -3 a -\frac {3 \left (-a d +b c \right )}{d}\right ) \cos \left (\frac {-3 a d +3 b c}{d}\right )}{d}-\frac {3 \,\operatorname {Ci}\left (3 b x +3 a +\frac {-3 a d +3 b c}{d}\right ) \sin \left (\frac {-3 a d +3 b c}{d}\right )}{d}\right )}{12}+\frac {3 b \left (-\frac {\operatorname {Si}\left (-b x -a -\frac {-a d +b c}{d}\right ) \cos \left (\frac {-a d +b c}{d}\right )}{d}-\frac {\operatorname {Ci}\left (b x +a +\frac {-a d +b c}{d}\right ) \sin \left (\frac {-a d +b c}{d}\right )}{d}\right )}{4}}{b}\) | \(172\) |
risch | \(-\frac {i {\mathrm e}^{\frac {3 i \left (a d -b c \right )}{d}} \operatorname {expIntegral}_{1}\left (-3 i b x -3 i a -\frac {3 \left (-i a d +i b c \right )}{d}\right )}{8 d}+\frac {i {\mathrm e}^{-\frac {3 i \left (a d -b c \right )}{d}} \operatorname {expIntegral}_{1}\left (3 i b x +3 i a -\frac {3 i \left (a d -b c \right )}{d}\right )}{8 d}-\frac {3 i {\mathrm e}^{-\frac {i \left (a d -b c \right )}{d}} \operatorname {expIntegral}_{1}\left (i b x +i a -\frac {i \left (a d -b c \right )}{d}\right )}{8 d}+\frac {3 i {\mathrm e}^{\frac {i \left (a d -b c \right )}{d}} \operatorname {expIntegral}_{1}\left (-i b x -i a -\frac {-i a d +i b c}{d}\right )}{8 d}\) | \(194\) |
Input:
int(sin(b*x+a)^3/(d*x+c),x,method=_RETURNVERBOSE)
Output:
1/b*(-1/12*b*(-3*Si(-3*b*x-3*a-3*(-a*d+b*c)/d)*cos(3*(-a*d+b*c)/d)/d-3*Ci( 3*b*x+3*a+3*(-a*d+b*c)/d)*sin(3*(-a*d+b*c)/d)/d)+3/4*b*(-Si(-b*x-a-(-a*d+b *c)/d)*cos((-a*d+b*c)/d)/d-Ci(b*x+a+(-a*d+b*c)/d)*sin((-a*d+b*c)/d)/d))
Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02 \[ \int \frac {\sin ^3(a+b x)}{c+d x} \, dx=\frac {3 \, \operatorname {Ci}\left (\frac {b d x + b c}{d}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - \operatorname {Ci}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + 3 \, \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (\frac {b d x + b c}{d}\right )}{4 \, d} \] Input:
integrate(sin(b*x+a)^3/(d*x+c),x, algorithm="fricas")
Output:
1/4*(3*cos_integral((b*d*x + b*c)/d)*sin(-(b*c - a*d)/d) - cos_integral(3* (b*d*x + b*c)/d)*sin(-3*(b*c - a*d)/d) - cos(-3*(b*c - a*d)/d)*sin_integra l(3*(b*d*x + b*c)/d) + 3*cos(-(b*c - a*d)/d)*sin_integral((b*d*x + b*c)/d) )/d
\[ \int \frac {\sin ^3(a+b x)}{c+d x} \, dx=\int \frac {\sin ^{3}{\left (a + b x \right )}}{c + d x}\, dx \] Input:
integrate(sin(b*x+a)**3/(d*x+c),x)
Output:
Integral(sin(a + b*x)**3/(c + d*x), x)
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.31 \[ \int \frac {\sin ^3(a+b x)}{c+d x} \, dx=-\frac {3 \, b {\left (i \, E_{1}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) - i \, E_{1}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) - b {\left (-i \, E_{1}\left (\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + i \, E_{1}\left (-\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 3 \, b {\left (E_{1}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + E_{1}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right ) - b {\left (E_{1}\left (\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{1}\left (-\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{8 \, b d} \] Input:
integrate(sin(b*x+a)^3/(d*x+c),x, algorithm="maxima")
Output:
-1/8*(3*b*(I*exp_integral_e(1, (I*b*c + I*(b*x + a)*d - I*a*d)/d) - I*exp_ integral_e(1, -(I*b*c + I*(b*x + a)*d - I*a*d)/d))*cos(-(b*c - a*d)/d) - b *(-I*exp_integral_e(1, 3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + I*exp_integ ral_e(1, -3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*cos(-3*(b*c - a*d)/d) + 3 *b*(exp_integral_e(1, (I*b*c + I*(b*x + a)*d - I*a*d)/d) + exp_integral_e( 1, -(I*b*c + I*(b*x + a)*d - I*a*d)/d))*sin(-(b*c - a*d)/d) - b*(exp_integ ral_e(1, 3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + exp_integral_e(1, -3*(-I* b*c - I*(b*x + a)*d + I*a*d)/d))*sin(-3*(b*c - a*d)/d))/(b*d)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.62 (sec) , antiderivative size = 6296, normalized size of antiderivative = 52.03 \[ \int \frac {\sin ^3(a+b x)}{c+d x} \, dx=\text {Too large to display} \] Input:
integrate(sin(b*x+a)^3/(d*x+c),x, algorithm="giac")
Output:
-1/8*(imag_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*a)^2*tan(1/2*a)^2*t an(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 - 3*imag_part(cos_integral(b*x + b*c/d))* tan(3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 + 3*imag_part( cos_integral(-b*x - b*c/d))*tan(3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan (1/2*b*c/d)^2 - imag_part(cos_integral(-3*b*x - 3*b*c/d))*tan(3/2*a)^2*tan (1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 + 2*sin_integral(3*(b*d*x + b* c)/d)*tan(3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 - 6*sin_ integral((b*d*x + b*c)/d)*tan(3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1 /2*b*c/d)^2 - 6*real_part(cos_integral(b*x + b*c/d))*tan(3/2*a)^2*tan(1/2* a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d) - 6*real_part(cos_integral(-b*x - b*c /d))*tan(3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d) + 2*real_pa rt(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d) *tan(1/2*b*c/d)^2 + 2*real_part(cos_integral(-3*b*x - 3*b*c/d))*tan(3/2*a) ^2*tan(1/2*a)^2*tan(3/2*b*c/d)*tan(1/2*b*c/d)^2 + 6*real_part(cos_integral (b*x + b*c/d))*tan(3/2*a)^2*tan(1/2*a)*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 + 6*real_part(cos_integral(-b*x - b*c/d))*tan(3/2*a)^2*tan(1/2*a)*tan(3/2*b *c/d)^2*tan(1/2*b*c/d)^2 - 2*real_part(cos_integral(3*b*x + 3*b*c/d))*tan( 3/2*a)*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 - 2*real_part(cos_in tegral(-3*b*x - 3*b*c/d))*tan(3/2*a)*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2 *b*c/d)^2 + imag_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*a)^2*tan(1...
Timed out. \[ \int \frac {\sin ^3(a+b x)}{c+d x} \, dx=\int \frac {{\sin \left (a+b\,x\right )}^3}{c+d\,x} \,d x \] Input:
int(sin(a + b*x)^3/(c + d*x),x)
Output:
int(sin(a + b*x)^3/(c + d*x), x)
\[ \int \frac {\sin ^3(a+b x)}{c+d x} \, dx=\int \frac {\sin \left (b x +a \right )^{3}}{d x +c}d x \] Input:
int(sin(b*x+a)^3/(d*x+c),x)
Output:
int(sin(a + b*x)**3/(c + d*x),x)