\(\int \frac {\sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx\) [194]

Optimal result

Integrand size = 21, antiderivative size = 75 \[ \int \frac {\sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 x}{2 a}+\frac {2 \cos (c+d x)}{a d}-\frac {3 \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {\cos (c+d x) \sin ^2(c+d x)}{d (a+a \sin (c+d x))} \]

[Out]

3/2*x/a+2*cos(d*x+c)/a/d-3/2*cos(d*x+c)*sin(d*x+c)/a/d+cos(d*x+c)*sin(d*x+c)^2/d/(a+a*sin(d*x+c))

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2846, 2813} \[ \int \frac {\sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2 \cos (c+d x)}{a d}+\frac {\sin ^2(c+d x) \cos (c+d x)}{d (a \sin (c+d x)+a)}-\frac {3 \sin (c+d x) \cos (c+d x)}{2 a d}+\frac {3 x}{2 a} \]

[In]

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

[Out]

(3*x)/(2*a) + (2*Cos[c + d*x])/(a*d) - (3*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) + (Cos[c + d*x]*Sin[c + d*x]^2)/(
d*(a + a*Sin[c + d*x]))

Rule 2813

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*a*c +
b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Cos[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rule 2846

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(
b*c - a*d))*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n - 1)/(a*f*(a + b*Sin[e + f*x]))), x] - Dist[d/(a*b), Int[(c
+ d*Sin[e + f*x])^(n - 2)*Simp[b*d*(n - 1) - a*c*n + (b*c*(n - 1) - a*d*n)*Sin[e + f*x], x], x], x] /; FreeQ[{
a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 1] && (IntegerQ
[2*n] || EqQ[c, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {\cos (c+d x) \sin ^2(c+d x)}{d (a+a \sin (c+d x))}-\frac {\int \sin (c+d x) (2 a-3 a \sin (c+d x)) \, dx}{a^2} \\ & = \frac {3 x}{2 a}+\frac {2 \cos (c+d x)}{a d}-\frac {3 \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {\cos (c+d x) \sin ^2(c+d x)}{d (a+a \sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.56 \[ \int \frac {\sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right ) (-8+6 c+6 d x+4 \cos (c+d x)-\sin (2 (c+d x)))+\cos \left (\frac {1}{2} (c+d x)\right ) (6 c+6 d x+4 \cos (c+d x)-\sin (2 (c+d x)))\right )}{4 a d (1+\sin (c+d x))} \]

[In]

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

[Out]

((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(Sin[(c + d*x)/2]*(-8 + 6*c + 6*d*x + 4*Cos[c + d*x] - Sin[2*(c + d*x)]
) + Cos[(c + d*x)/2]*(6*c + 6*d*x + 4*Cos[c + d*x] - Sin[2*(c + d*x)])))/(4*a*d*(1 + Sin[c + d*x]))

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.19 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.08

method	result	size
risch	\(\frac {3 x}{2 a}+\frac {{\mathrm e}^{i \left (d x +c \right )}}{2 a d}+\frac {{\mathrm e}^{-i \left (d x +c \right )}}{2 a d}+\frac {2}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}-\frac {\sin \left (2 d x +2 c \right )}{4 d a}\)	\(81\)
derivativedivides	\(\frac {\frac {16}{8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+8}+\frac {2 \left (\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+1\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\)	\(91\)
default	\(\frac {\frac {16}{8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+8}+\frac {2 \left (\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+1\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}\)	\(91\)
parallelrisch	\(\frac {\left (12 d x +4\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+12 d x \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-\sin \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+3 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+\cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )-20 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+3 \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )}{8 a d \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\)	\(113\)
norman	\(\frac {-\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {3 x}{2 a}+\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}+\frac {9 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {9 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {9 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {9 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {3 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {3 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {8}{3 a d}-\frac {4 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 d a}+\frac {5 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\)	\(274\)

[In]

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

[Out]

3/2*x/a+1/2/a/d*exp(I*(d*x+c))+1/2/a/d*exp(-I*(d*x+c))+2/d/a/(exp(I*(d*x+c))+I)-1/4/d/a*sin(2*d*x+2*c)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.23 \[ \int \frac {\sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\cos \left (d x + c\right )^{3} + 3 \, d x + 3 \, {\left (d x + 1\right )} \cos \left (d x + c\right ) + 2 \, \cos \left (d x + c\right )^{2} + {\left (3 \, d x - \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + 2}{2 \, {\left (a d \cos \left (d x + c\right ) + a d \sin \left (d x + c\right ) + a d\right )}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1127 vs. \(2 (65) = 130\).

Time = 1.89 (sec) , antiderivative size = 1127, normalized size of antiderivative = 15.03 \[ \int \frac {\sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((3*d*x*tan(c/2 + d*x/2)**5/(2*a*d*tan(c/2 + d*x/2)**5 + 2*a*d*tan(c/2 + d*x/2)**4 + 4*a*d*tan(c/2 +
d*x/2)**3 + 4*a*d*tan(c/2 + d*x/2)**2 + 2*a*d*tan(c/2 + d*x/2) + 2*a*d) + 3*d*x*tan(c/2 + d*x/2)**4/(2*a*d*tan
(c/2 + d*x/2)**5 + 2*a*d*tan(c/2 + d*x/2)**4 + 4*a*d*tan(c/2 + d*x/2)**3 + 4*a*d*tan(c/2 + d*x/2)**2 + 2*a*d*t
an(c/2 + d*x/2) + 2*a*d) + 6*d*x*tan(c/2 + d*x/2)**3/(2*a*d*tan(c/2 + d*x/2)**5 + 2*a*d*tan(c/2 + d*x/2)**4 +
4*a*d*tan(c/2 + d*x/2)**3 + 4*a*d*tan(c/2 + d*x/2)**2 + 2*a*d*tan(c/2 + d*x/2) + 2*a*d) + 6*d*x*tan(c/2 + d*x/
2)**2/(2*a*d*tan(c/2 + d*x/2)**5 + 2*a*d*tan(c/2 + d*x/2)**4 + 4*a*d*tan(c/2 + d*x/2)**3 + 4*a*d*tan(c/2 + d*x
/2)**2 + 2*a*d*tan(c/2 + d*x/2) + 2*a*d) + 3*d*x*tan(c/2 + d*x/2)/(2*a*d*tan(c/2 + d*x/2)**5 + 2*a*d*tan(c/2 +
d*x/2)**4 + 4*a*d*tan(c/2 + d*x/2)**3 + 4*a*d*tan(c/2 + d*x/2)**2 + 2*a*d*tan(c/2 + d*x/2) + 2*a*d) + 3*d*x/(
2*a*d*tan(c/2 + d*x/2)**5 + 2*a*d*tan(c/2 + d*x/2)**4 + 4*a*d*tan(c/2 + d*x/2)**3 + 4*a*d*tan(c/2 + d*x/2)**2
+ 2*a*d*tan(c/2 + d*x/2) + 2*a*d) + 6*tan(c/2 + d*x/2)**4/(2*a*d*tan(c/2 + d*x/2)**5 + 2*a*d*tan(c/2 + d*x/2)*
*4 + 4*a*d*tan(c/2 + d*x/2)**3 + 4*a*d*tan(c/2 + d*x/2)**2 + 2*a*d*tan(c/2 + d*x/2) + 2*a*d) + 6*tan(c/2 + d*x
/2)**3/(2*a*d*tan(c/2 + d*x/2)**5 + 2*a*d*tan(c/2 + d*x/2)**4 + 4*a*d*tan(c/2 + d*x/2)**3 + 4*a*d*tan(c/2 + d*
x/2)**2 + 2*a*d*tan(c/2 + d*x/2) + 2*a*d) + 10*tan(c/2 + d*x/2)**2/(2*a*d*tan(c/2 + d*x/2)**5 + 2*a*d*tan(c/2
+ d*x/2)**4 + 4*a*d*tan(c/2 + d*x/2)**3 + 4*a*d*tan(c/2 + d*x/2)**2 + 2*a*d*tan(c/2 + d*x/2) + 2*a*d) + 2*tan(
c/2 + d*x/2)/(2*a*d*tan(c/2 + d*x/2)**5 + 2*a*d*tan(c/2 + d*x/2)**4 + 4*a*d*tan(c/2 + d*x/2)**3 + 4*a*d*tan(c/
2 + d*x/2)**2 + 2*a*d*tan(c/2 + d*x/2) + 2*a*d) + 8/(2*a*d*tan(c/2 + d*x/2)**5 + 2*a*d*tan(c/2 + d*x/2)**4 + 4
*a*d*tan(c/2 + d*x/2)**3 + 4*a*d*tan(c/2 + d*x/2)**2 + 2*a*d*tan(c/2 + d*x/2) + 2*a*d), Ne(d, 0)), (x*sin(c)**
3/(a*sin(c) + a), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (71) = 142\).

Time = 0.28 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.83 \[ \int \frac {\sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 4}{a + \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}} + \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{d} \]

[In]

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

[Out]

((sin(d*x + c)/(cos(d*x + c) + 1) + 5*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*sin(d*x + c)^3/(cos(d*x + c) + 1
)^3 + 3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4)/(a + a*sin(d*x + c)/(cos(d*x + c) + 1) + 2*a*sin(d*x + c)^2/(
cos(d*x + c) + 1)^2 + 2*a*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + a*sin(
d*x + c)^5/(cos(d*x + c) + 1)^5) + 3*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.21 \[ \int \frac {\sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {3 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a} + \frac {4}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}}}{2 \, d} \]

[In]

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

[Out]

1/2*(3*(d*x + c)/a + 2*(tan(1/2*d*x + 1/2*c)^3 + 2*tan(1/2*d*x + 1/2*c)^2 - tan(1/2*d*x + 1/2*c) + 2)/((tan(1/
2*d*x + 1/2*c)^2 + 1)^2*a) + 4/(a*(tan(1/2*d*x + 1/2*c) + 1)))/d

Mupad [B] (verification not implemented)

Time = 3.29 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.23 \[ \int \frac {\sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3\,x}{2\,a}+\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+4}{a\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2} \]

[In]

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

[Out]

(3*x)/(2*a) + (tan(c/2 + (d*x)/2) + 5*tan(c/2 + (d*x)/2)^2 + 3*tan(c/2 + (d*x)/2)^3 + 3*tan(c/2 + (d*x)/2)^4 +
4)/(a*d*(tan(c/2 + (d*x)/2) + 1)*(tan(c/2 + (d*x)/2)^2 + 1)^2)