\(\int \sin ^3(\frac {a+b x}{c+d x}) \, dx\) [38]

Optimal result

Integrand size = 16, antiderivative size = 194 \[ \int \sin ^3\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {3 (b c-a d) \cos \left (\frac {b}{d}\right ) \operatorname {CosIntegral}\left (\frac {b c-a d}{d (c+d x)}\right )}{4 d^2}-\frac {3 (b c-a d) \cos \left (\frac {3 b}{d}\right ) \operatorname {CosIntegral}\left (\frac {3 (b c-a d)}{d (c+d x)}\right )}{4 d^2}+\frac {(c+d x) \sin ^3\left (\frac {a+b x}{c+d x}\right )}{d}+\frac {3 (b c-a d) \sin \left (\frac {b}{d}\right ) \text {Si}\left (\frac {b c-a d}{d (c+d x)}\right )}{4 d^2}-\frac {3 (b c-a d) \sin \left (\frac {3 b}{d}\right ) \text {Si}\left (\frac {3 (b c-a d)}{d (c+d x)}\right )}{4 d^2} \]

[Out]

3/4*(-a*d+b*c)*Ci((-a*d+b*c)/d/(d*x+c))*cos(b/d)/d^2-3/4*(-a*d+b*c)*Ci(3*(-a*d+b*c)/d/(d*x+c))*cos(3*b/d)/d^2+
3/4*(-a*d+b*c)*Si((-a*d+b*c)/d/(d*x+c))*sin(b/d)/d^2-3/4*(-a*d+b*c)*Si(3*(-a*d+b*c)/d/(d*x+c))*sin(3*b/d)/d^2+
(d*x+c)*sin((b*x+a)/(d*x+c))^3/d

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {4659, 3394, 3384, 3380, 3383} \[ \int \sin ^3\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {3 \cos \left (\frac {b}{d}\right ) (b c-a d) \operatorname {CosIntegral}\left (\frac {b c-a d}{d (c+d x)}\right )}{4 d^2}-\frac {3 \cos \left (\frac {3 b}{d}\right ) (b c-a d) \operatorname {CosIntegral}\left (\frac {3 (b c-a d)}{d (c+d x)}\right )}{4 d^2}+\frac {3 \sin \left (\frac {b}{d}\right ) (b c-a d) \text {Si}\left (\frac {b c-a d}{d (c+d x)}\right )}{4 d^2}-\frac {3 \sin \left (\frac {3 b}{d}\right ) (b c-a d) \text {Si}\left (\frac {3 (b c-a d)}{d (c+d x)}\right )}{4 d^2}+\frac {(c+d x) \sin ^3\left (\frac {a+b x}{c+d x}\right )}{d} \]

[In]

Int[Sin[(a + b*x)/(c + d*x)]^3,x]

[Out]

(3*(b*c - a*d)*Cos[b/d]*CosIntegral[(b*c - a*d)/(d*(c + d*x))])/(4*d^2) - (3*(b*c - a*d)*Cos[(3*b)/d]*CosInteg
ral[(3*(b*c - a*d))/(d*(c + d*x))])/(4*d^2) + ((c + d*x)*Sin[(a + b*x)/(c + d*x)]^3)/d + (3*(b*c - a*d)*Sin[b/
d]*SinIntegral[(b*c - a*d)/(d*(c + d*x))])/(4*d^2) - (3*(b*c - a*d)*Sin[(3*b)/d]*SinIntegral[(3*(b*c - a*d))/(
d*(c + d*x))])/(4*d^2)

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3394

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]^
n/(d*(m + 1))), x] - Dist[f*(n/(d*(m + 1))), Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]
^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] && LtQ[m, -1]

Rule 4659

Int[Sin[((e_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_))]^(n_.), x_Symbol] :> Dist[-d^(-1), Subst[Int[Sin[b*(
e/d) - e*(b*c - a*d)*(x/d)]^n/x^2, x], x, 1/(c + d*x)], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && NeQ[b*c
- a*d, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\sin ^3\left (\frac {b}{d}-\frac {(b c-a d) x}{d}\right )}{x^2} \, dx,x,\frac {1}{c+d x}\right )}{d} \\ & = \frac {(c+d x) \sin ^3\left (\frac {a+b x}{c+d x}\right )}{d}+\frac {(3 (b c-a d)) \text {Subst}\left (\int \left (-\frac {\cos \left (\frac {3 b}{d}-\frac {3 (b c-a d) x}{d}\right )}{4 x}+\frac {\cos \left (\frac {b}{d}-\frac {(b c-a d) x}{d}\right )}{4 x}\right ) \, dx,x,\frac {1}{c+d x}\right )}{d^2} \\ & = \frac {(c+d x) \sin ^3\left (\frac {a+b x}{c+d x}\right )}{d}-\frac {(3 (b c-a d)) \text {Subst}\left (\int \frac {\cos \left (\frac {3 b}{d}-\frac {3 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{4 d^2}+\frac {(3 (b c-a d)) \text {Subst}\left (\int \frac {\cos \left (\frac {b}{d}-\frac {(b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{4 d^2} \\ & = \frac {(c+d x) \sin ^3\left (\frac {a+b x}{c+d x}\right )}{d}+\frac {\left (3 (b c-a d) \cos \left (\frac {b}{d}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {(b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{4 d^2}-\frac {\left (3 (b c-a d) \cos \left (\frac {3 b}{d}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{4 d^2}+\frac {\left (3 (b c-a d) \sin \left (\frac {b}{d}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {(b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{4 d^2}-\frac {\left (3 (b c-a d) \sin \left (\frac {3 b}{d}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 (b c-a d) x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{4 d^2} \\ & = \frac {3 (b c-a d) \cos \left (\frac {b}{d}\right ) \operatorname {CosIntegral}\left (\frac {b c-a d}{d (c+d x)}\right )}{4 d^2}-\frac {3 (b c-a d) \cos \left (\frac {3 b}{d}\right ) \operatorname {CosIntegral}\left (\frac {3 (b c-a d)}{d (c+d x)}\right )}{4 d^2}+\frac {(c+d x) \sin ^3\left (\frac {a+b x}{c+d x}\right )}{d}+\frac {3 (b c-a d) \sin \left (\frac {b}{d}\right ) \text {Si}\left (\frac {b c-a d}{d (c+d x)}\right )}{4 d^2}-\frac {3 (b c-a d) \sin \left (\frac {3 b}{d}\right ) \text {Si}\left (\frac {3 (b c-a d)}{d (c+d x)}\right )}{4 d^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.78 (sec) , antiderivative size = 692, normalized size of antiderivative = 3.57 \[ \int \sin ^3\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {3 i c d e^{-\frac {i (a+b x)}{c+d x}}-3 i c d e^{\frac {i (a+b x)}{c+d x}}-i c d e^{-\frac {3 i (a+b x)}{c+d x}}+i c d e^{\frac {3 i (a+b x)}{c+d x}}+6 d^2 x \cos \left (\frac {-b c+a d}{d (c+d x)}\right ) \sin \left (\frac {b}{d}\right )-2 d^2 x \cos \left (\frac {3 (-b c+a d)}{d (c+d x)}\right ) \sin \left (\frac {3 b}{d}\right )+6 d^2 x \cos \left (\frac {b}{d}\right ) \sin \left (\frac {-b c+a d}{d (c+d x)}\right )-2 d^2 x \cos \left (\frac {3 b}{d}\right ) \sin \left (\frac {3 (-b c+a d)}{d (c+d x)}\right )+3 (b c-a d) \left (-\cos \left (\frac {3 b}{d}\right ) \operatorname {CosIntegral}\left (\frac {3 b c-3 a d}{c d+d^2 x}\right )+\cos \left (\frac {b}{d}\right ) \operatorname {CosIntegral}\left (\frac {b c-a d}{c d+d^2 x}\right )+\operatorname {CosIntegral}\left (\frac {-b c+a d}{d (c+d x)}\right ) \left (\cos \left (\frac {b}{d}\right )-i \sin \left (\frac {b}{d}\right )\right )+i \operatorname {CosIntegral}\left (\frac {b c-a d}{c d+d^2 x}\right ) \sin \left (\frac {b}{d}\right )-\operatorname {CosIntegral}\left (\frac {3 (-b c+a d)}{d (c+d x)}\right ) \left (\cos \left (\frac {3 b}{d}\right )-i \sin \left (\frac {3 b}{d}\right )\right )-i \operatorname {CosIntegral}\left (\frac {3 b c-3 a d}{c d+d^2 x}\right ) \sin \left (\frac {3 b}{d}\right )+i \cos \left (\frac {b}{d}\right ) \text {Si}\left (\frac {-b c+a d}{d (c+d x)}\right )-\sin \left (\frac {b}{d}\right ) \text {Si}\left (\frac {-b c+a d}{d (c+d x)}\right )-i \cos \left (\frac {3 b}{d}\right ) \text {Si}\left (\frac {3 (-b c+a d)}{d (c+d x)}\right )+\sin \left (\frac {3 b}{d}\right ) \text {Si}\left (\frac {3 (-b c+a d)}{d (c+d x)}\right )-i \cos \left (\frac {3 b}{d}\right ) \text {Si}\left (\frac {3 b c-3 a d}{c d+d^2 x}\right )-\sin \left (\frac {3 b}{d}\right ) \text {Si}\left (\frac {3 b c-3 a d}{c d+d^2 x}\right )+i \cos \left (\frac {b}{d}\right ) \text {Si}\left (\frac {b c-a d}{c d+d^2 x}\right )+\sin \left (\frac {b}{d}\right ) \text {Si}\left (\frac {b c-a d}{c d+d^2 x}\right )\right )}{8 d^2} \]

[In]

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

[Out]

(((3*I)*c*d)/E^((I*(a + b*x))/(c + d*x)) - (3*I)*c*d*E^((I*(a + b*x))/(c + d*x)) - (I*c*d)/E^(((3*I)*(a + b*x)
)/(c + d*x)) + I*c*d*E^(((3*I)*(a + b*x))/(c + d*x)) + 6*d^2*x*Cos[(-(b*c) + a*d)/(d*(c + d*x))]*Sin[b/d] - 2*
d^2*x*Cos[(3*(-(b*c) + a*d))/(d*(c + d*x))]*Sin[(3*b)/d] + 6*d^2*x*Cos[b/d]*Sin[(-(b*c) + a*d)/(d*(c + d*x))]
- 2*d^2*x*Cos[(3*b)/d]*Sin[(3*(-(b*c) + a*d))/(d*(c + d*x))] + 3*(b*c - a*d)*(-(Cos[(3*b)/d]*CosIntegral[(3*b*
c - 3*a*d)/(c*d + d^2*x)]) + Cos[b/d]*CosIntegral[(b*c - a*d)/(c*d + d^2*x)] + CosIntegral[(-(b*c) + a*d)/(d*(
c + d*x))]*(Cos[b/d] - I*Sin[b/d]) + I*CosIntegral[(b*c - a*d)/(c*d + d^2*x)]*Sin[b/d] - CosIntegral[(3*(-(b*c
) + a*d))/(d*(c + d*x))]*(Cos[(3*b)/d] - I*Sin[(3*b)/d]) - I*CosIntegral[(3*b*c - 3*a*d)/(c*d + d^2*x)]*Sin[(3
*b)/d] + I*Cos[b/d]*SinIntegral[(-(b*c) + a*d)/(d*(c + d*x))] - Sin[b/d]*SinIntegral[(-(b*c) + a*d)/(d*(c + d*
x))] - I*Cos[(3*b)/d]*SinIntegral[(3*(-(b*c) + a*d))/(d*(c + d*x))] + Sin[(3*b)/d]*SinIntegral[(3*(-(b*c) + a*
d))/(d*(c + d*x))] - I*Cos[(3*b)/d]*SinIntegral[(3*b*c - 3*a*d)/(c*d + d^2*x)] - Sin[(3*b)/d]*SinIntegral[(3*b
*c - 3*a*d)/(c*d + d^2*x)] + I*Cos[b/d]*SinIntegral[(b*c - a*d)/(c*d + d^2*x)] + Sin[b/d]*SinIntegral[(b*c - a
*d)/(c*d + d^2*x)]))/(8*d^2)

Maple [A] (verified)

Time = 1.73 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.52

method	result	size
derivativedivides	\(-\frac {\left (a d -c b \right ) \left (-\frac {d^{2} \left (-\frac {3 \sin \left (\frac {3 a d -3 c b}{d \left (d x +c \right )}+\frac {3 b}{d}\right )}{\left (\left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right ) d -b \right ) d}+\frac {-\frac {9 \,\operatorname {Si}\left (\frac {3 a d -3 c b}{d \left (d x +c \right )}\right ) \sin \left (\frac {3 b}{d}\right )}{d}+\frac {9 \,\operatorname {Ci}\left (\frac {3 a d -3 c b}{d \left (d x +c \right )}\right ) \cos \left (\frac {3 b}{d}\right )}{d}}{d}\right )}{12}+\frac {3 d^{2} \left (-\frac {\sin \left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right )}{\left (\left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right ) d -b \right ) d}+\frac {-\frac {\operatorname {Si}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) \sin \left (\frac {b}{d}\right )}{d}+\frac {\operatorname {Ci}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) \cos \left (\frac {b}{d}\right )}{d}}{d}\right )}{4}\right )}{d^{2}}\)	\(295\)
default	\(-\frac {\left (a d -c b \right ) \left (-\frac {d^{2} \left (-\frac {3 \sin \left (\frac {3 a d -3 c b}{d \left (d x +c \right )}+\frac {3 b}{d}\right )}{\left (\left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right ) d -b \right ) d}+\frac {-\frac {9 \,\operatorname {Si}\left (\frac {3 a d -3 c b}{d \left (d x +c \right )}\right ) \sin \left (\frac {3 b}{d}\right )}{d}+\frac {9 \,\operatorname {Ci}\left (\frac {3 a d -3 c b}{d \left (d x +c \right )}\right ) \cos \left (\frac {3 b}{d}\right )}{d}}{d}\right )}{12}+\frac {3 d^{2} \left (-\frac {\sin \left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right )}{\left (\left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right ) d -b \right ) d}+\frac {-\frac {\operatorname {Si}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) \sin \left (\frac {b}{d}\right )}{d}+\frac {\operatorname {Ci}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) \cos \left (\frac {b}{d}\right )}{d}}{d}\right )}{4}\right )}{d^{2}}\)	\(295\)
risch	\(-\frac {3 i {\mathrm e}^{-\frac {i b}{d}} \operatorname {Si}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) b c}{4 d^{2}}-\frac {3 i {\mathrm e}^{-\frac {i b}{d}} \pi \,\operatorname {csgn}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) a}{8 d}+\frac {3 i {\mathrm e}^{-\frac {i b}{d}} \operatorname {Si}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) a}{4 d}+\frac {3 i {\mathrm e}^{-\frac {i b}{d}} \pi \,\operatorname {csgn}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) b c}{8 d^{2}}-\frac {3 \,{\mathrm e}^{-\frac {3 i b}{d}} \operatorname {Ei}_{1}\left (-\frac {3 i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) a}{8 d}+\frac {3 \,{\mathrm e}^{-\frac {3 i b}{d}} \operatorname {Ei}_{1}\left (-\frac {3 i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) b c}{8 d^{2}}-\frac {3 \,\operatorname {Ei}_{1}\left (-\frac {3 i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) {\mathrm e}^{\frac {3 i b}{d}} a}{8 d}+\frac {3 \,\operatorname {Ei}_{1}\left (-\frac {3 i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) {\mathrm e}^{\frac {3 i b}{d}} c b}{8 d^{2}}+\frac {3 \,\operatorname {Ei}_{1}\left (-\frac {i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) {\mathrm e}^{\frac {i b}{d}} a}{8 d}-\frac {3 \,\operatorname {Ei}_{1}\left (-\frac {i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) {\mathrm e}^{\frac {i b}{d}} c b}{8 d^{2}}-\frac {3 i {\mathrm e}^{-\frac {3 i b}{d}} \operatorname {csgn}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) \pi b c}{8 d^{2}}-\frac {3 i {\mathrm e}^{-\frac {3 i b}{d}} \operatorname {Si}\left (\frac {3 a d -3 c b}{d \left (d x +c \right )}\right ) a}{4 d}+\frac {3 i {\mathrm e}^{-\frac {3 i b}{d}} \operatorname {csgn}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) \pi a}{8 d}+\frac {3 i {\mathrm e}^{-\frac {3 i b}{d}} \operatorname {Si}\left (\frac {3 a d -3 c b}{d \left (d x +c \right )}\right ) b c}{4 d^{2}}+\frac {3 \,{\mathrm e}^{-\frac {i b}{d}} \operatorname {Ei}_{1}\left (-\frac {i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) a}{8 d}-\frac {3 \,{\mathrm e}^{-\frac {i b}{d}} \operatorname {Ei}_{1}\left (-\frac {i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) b c}{8 d^{2}}+\frac {3 \sin \left (\frac {x b +a}{d x +c}\right ) x}{4}+\frac {3 \sin \left (\frac {x b +a}{d x +c}\right ) c}{4 d}-\frac {\sin \left (\frac {3 x b +3 a}{d x +c}\right ) x}{4}-\frac {\sin \left (\frac {3 x b +3 a}{d x +c}\right ) c}{4 d}\)	\(668\)

[In]

int(sin((b*x+a)/(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-1/d^2*(a*d-b*c)*(-1/12*d^2*(-3*sin(3*(a*d-b*c)/d/(d*x+c)+3*b/d)/((b/d+(a*d-b*c)/d/(d*x+c))*d-b)/d+3*(-3*Si(3*
(a*d-b*c)/d/(d*x+c))*sin(3*b/d)/d+3*Ci(3*(a*d-b*c)/d/(d*x+c))*cos(3*b/d)/d)/d)+3/4*d^2*(-sin(b/d+(a*d-b*c)/d/(
d*x+c))/((b/d+(a*d-b*c)/d/(d*x+c))*d-b)/d+(-Si((a*d-b*c)/d/(d*x+c))*sin(b/d)/d+Ci((a*d-b*c)/d/(d*x+c))*cos(b/d
)/d)/d))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.09 \[ \int \sin ^3\left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {3 \, {\left (b c - a d\right )} \cos \left (\frac {b}{d}\right ) \operatorname {Ci}\left (-\frac {b c - a d}{d^{2} x + c d}\right ) - 3 \, {\left (b c - a d\right )} \cos \left (\frac {3 \, b}{d}\right ) \operatorname {Ci}\left (-\frac {3 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) - 3 \, {\left (b c - a d\right )} \sin \left (\frac {b}{d}\right ) \operatorname {Si}\left (-\frac {b c - a d}{d^{2} x + c d}\right ) + 3 \, {\left (b c - a d\right )} \sin \left (\frac {3 \, b}{d}\right ) \operatorname {Si}\left (-\frac {3 \, {\left (b c - a d\right )}}{d^{2} x + c d}\right ) + 4 \, {\left (d^{2} x - {\left (d^{2} x + c d\right )} \cos \left (\frac {b x + a}{d x + c}\right )^{2} + c d\right )} \sin \left (\frac {b x + a}{d x + c}\right )}{4 \, d^{2}} \]

[In]

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

[Out]

1/4*(3*(b*c - a*d)*cos(b/d)*cos_integral(-(b*c - a*d)/(d^2*x + c*d)) - 3*(b*c - a*d)*cos(3*b/d)*cos_integral(-
3*(b*c - a*d)/(d^2*x + c*d)) - 3*(b*c - a*d)*sin(b/d)*sin_integral(-(b*c - a*d)/(d^2*x + c*d)) + 3*(b*c - a*d)
*sin(3*b/d)*sin_integral(-3*(b*c - a*d)/(d^2*x + c*d)) + 4*(d^2*x - (d^2*x + c*d)*cos((b*x + a)/(d*x + c))^2 +
c*d)*sin((b*x + a)/(d*x + c)))/d^2

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1239 vs. \(2 (186) = 372\).

Time = 66.42 (sec) , antiderivative size = 1239, normalized size of antiderivative = 6.39 \[ \int \sin ^3\left (\frac {a+b x}{c+d x}\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/4*(3*b^3*c^2*cos(b/d)*cos_integral(-(b - (b*x + a)*d/(d*x + c))/d) - 6*a*b^2*c*d*cos(b/d)*cos_integral(-(b -
(b*x + a)*d/(d*x + c))/d) - 3*(b*x + a)*b^2*c^2*d*cos(b/d)*cos_integral(-(b - (b*x + a)*d/(d*x + c))/d)/(d*x
+ c) + 3*a^2*b*d^2*cos(b/d)*cos_integral(-(b - (b*x + a)*d/(d*x + c))/d) + 6*(b*x + a)*a*b*c*d^2*cos(b/d)*cos_
integral(-(b - (b*x + a)*d/(d*x + c))/d)/(d*x + c) - 3*(b*x + a)*a^2*d^3*cos(b/d)*cos_integral(-(b - (b*x + a)
*d/(d*x + c))/d)/(d*x + c) - 3*b^3*c^2*cos(3*b/d)*cos_integral(-3*(b - (b*x + a)*d/(d*x + c))/d) + 6*a*b^2*c*d
*cos(3*b/d)*cos_integral(-3*(b - (b*x + a)*d/(d*x + c))/d) + 3*(b*x + a)*b^2*c^2*d*cos(3*b/d)*cos_integral(-3*
(b - (b*x + a)*d/(d*x + c))/d)/(d*x + c) - 3*a^2*b*d^2*cos(3*b/d)*cos_integral(-3*(b - (b*x + a)*d/(d*x + c))/
d) - 6*(b*x + a)*a*b*c*d^2*cos(3*b/d)*cos_integral(-3*(b - (b*x + a)*d/(d*x + c))/d)/(d*x + c) + 3*(b*x + a)*a
^2*d^3*cos(3*b/d)*cos_integral(-3*(b - (b*x + a)*d/(d*x + c))/d)/(d*x + c) - 3*b^3*c^2*sin(3*b/d)*sin_integral
(3*(b - (b*x + a)*d/(d*x + c))/d) + 6*a*b^2*c*d*sin(3*b/d)*sin_integral(3*(b - (b*x + a)*d/(d*x + c))/d) + 3*(
b*x + a)*b^2*c^2*d*sin(3*b/d)*sin_integral(3*(b - (b*x + a)*d/(d*x + c))/d)/(d*x + c) - 3*a^2*b*d^2*sin(3*b/d)
*sin_integral(3*(b - (b*x + a)*d/(d*x + c))/d) - 6*(b*x + a)*a*b*c*d^2*sin(3*b/d)*sin_integral(3*(b - (b*x + a
)*d/(d*x + c))/d)/(d*x + c) + 3*(b*x + a)*a^2*d^3*sin(3*b/d)*sin_integral(3*(b - (b*x + a)*d/(d*x + c))/d)/(d*
x + c) + 3*b^3*c^2*sin(b/d)*sin_integral((b - (b*x + a)*d/(d*x + c))/d) - 6*a*b^2*c*d*sin(b/d)*sin_integral((b
- (b*x + a)*d/(d*x + c))/d) - 3*(b*x + a)*b^2*c^2*d*sin(b/d)*sin_integral((b - (b*x + a)*d/(d*x + c))/d)/(d*x
+ c) + 3*a^2*b*d^2*sin(b/d)*sin_integral((b - (b*x + a)*d/(d*x + c))/d) + 6*(b*x + a)*a*b*c*d^2*sin(b/d)*sin_
integral((b - (b*x + a)*d/(d*x + c))/d)/(d*x + c) - 3*(b*x + a)*a^2*d^3*sin(b/d)*sin_integral((b - (b*x + a)*d
/(d*x + c))/d)/(d*x + c) - b^2*c^2*d*sin(3*(b*x + a)/(d*x + c)) + 2*a*b*c*d^2*sin(3*(b*x + a)/(d*x + c)) - a^2
*d^3*sin(3*(b*x + a)/(d*x + c)) + 3*b^2*c^2*d*sin((b*x + a)/(d*x + c)) - 6*a*b*c*d^2*sin((b*x + a)/(d*x + c))
+ 3*a^2*d^3*sin((b*x + a)/(d*x + c)))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)/(b*d^2 - (b*x + a)*d^3/(d*x + c)
)

Mupad [F(-1)]

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

[In]

int(sin((a + b*x)/(c + d*x))^3,x)

[Out]

int(sin((a + b*x)/(c + d*x))^3, x)