\(\int \frac {(c-c \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx\) [271]

Optimal result

Integrand size = 26, antiderivative size = 90 \[ \int \frac {(c-c \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx=\frac {5 c^3 x}{a^2}+\frac {5 c^3 \cos (e+f x)}{a^2 f}-\frac {2 a^2 c^3 \cos ^5(e+f x)}{3 f (a+a \sin (e+f x))^4}+\frac {10 c^3 \cos ^3(e+f x)}{3 f (a+a \sin (e+f x))^2} \]

[Out]

5*c^3*x/a^2+5*c^3*cos(f*x+e)/a^2/f-2/3*a^2*c^3*cos(f*x+e)^5/f/(a+a*sin(f*x+e))^4+10/3*c^3*cos(f*x+e)^3/f/(a+a*
sin(f*x+e))^2

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2815, 2759, 2761, 8} \[ \int \frac {(c-c \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx=\frac {5 c^3 \cos (e+f x)}{a^2 f}-\frac {2 a^2 c^3 \cos ^5(e+f x)}{3 f (a \sin (e+f x)+a)^4}+\frac {5 c^3 x}{a^2}+\frac {10 c^3 \cos ^3(e+f x)}{3 f (a \sin (e+f x)+a)^2} \]

[In]

Int[(c - c*Sin[e + f*x])^3/(a + a*Sin[e + f*x])^2,x]

[Out]

(5*c^3*x)/a^2 + (5*c^3*Cos[e + f*x])/(a^2*f) - (2*a^2*c^3*Cos[e + f*x]^5)/(3*f*(a + a*Sin[e + f*x])^4) + (10*c
^3*Cos[e + f*x]^3)/(3*f*(a + a*Sin[e + f*x])^2)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2759

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[2*g*(g
*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(2*m + p + 1))), x] + Dist[g^2*((p - 1)/(b^2*(2*m +
p + 1))), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && Eq
Q[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] &&  !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*
p]

Rule 2761

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g*((g*Cos[e
+ f*x])^(p - 1)/(b*f*(p - 1))), x] + Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g
}, x] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]

Rule 2815

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

Rubi steps \begin{align*} \text {integral}& = \left (a^3 c^3\right ) \int \frac {\cos ^6(e+f x)}{(a+a \sin (e+f x))^5} \, dx \\ & = -\frac {2 a^2 c^3 \cos ^5(e+f x)}{3 f (a+a \sin (e+f x))^4}-\frac {1}{3} \left (5 a c^3\right ) \int \frac {\cos ^4(e+f x)}{(a+a \sin (e+f x))^3} \, dx \\ & = -\frac {2 a^2 c^3 \cos ^5(e+f x)}{3 f (a+a \sin (e+f x))^4}+\frac {10 c^3 \cos ^3(e+f x)}{3 f (a+a \sin (e+f x))^2}+\frac {\left (5 c^3\right ) \int \frac {\cos ^2(e+f x)}{a+a \sin (e+f x)} \, dx}{a} \\ & = \frac {5 c^3 \cos (e+f x)}{a^2 f}-\frac {2 a^2 c^3 \cos ^5(e+f x)}{3 f (a+a \sin (e+f x))^4}+\frac {10 c^3 \cos ^3(e+f x)}{3 f (a+a \sin (e+f x))^2}+\frac {\left (5 c^3\right ) \int 1 \, dx}{a^2} \\ & = \frac {5 c^3 x}{a^2}+\frac {5 c^3 \cos (e+f x)}{a^2 f}-\frac {2 a^2 c^3 \cos ^5(e+f x)}{3 f (a+a \sin (e+f x))^4}+\frac {10 c^3 \cos ^3(e+f x)}{3 f (a+a \sin (e+f x))^2} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(210\) vs. \(2(90)=180\).

Time = 11.96 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.33 \[ \int \frac {(c-c \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (16 \sin \left (\frac {1}{2} (e+f x)\right )-8 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-56 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+15 (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+3 \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3\right ) (c-c \sin (e+f x))^3}{3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (a+a \sin (e+f x))^2} \]

[In]

Integrate[(c - c*Sin[e + f*x])^3/(a + a*Sin[e + f*x])^2,x]

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(16*Sin[(e + f*x)/2] - 8*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 56*Sin
[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 15*(e + f*x)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 +
3*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)*(c - c*Sin[e + f*x])^3)/(3*f*(Cos[(e + f*x)/2] - Sin[
(e + f*x)/2])^6*(a + a*Sin[e + f*x])^2)

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.94

method	result	size
derivativedivides	\(\frac {2 c^{3} \left (\frac {1}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+5 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {16}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {8}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {4}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f \,a^{2}}\)	\(85\)
default	\(\frac {2 c^{3} \left (\frac {1}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+5 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {16}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {8}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {4}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f \,a^{2}}\)	\(85\)
risch	\(\frac {5 c^{3} x}{a^{2}}+\frac {c^{3} {\mathrm e}^{i \left (f x +e \right )}}{2 a^{2} f}+\frac {c^{3} {\mathrm e}^{-i \left (f x +e \right )}}{2 a^{2} f}+\frac {32 i c^{3} {\mathrm e}^{i \left (f x +e \right )}+24 c^{3} {\mathrm e}^{2 i \left (f x +e \right )}-\frac {56 c^{3}}{3}}{f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3}}\)	\(108\)
parallelrisch	\(\frac {c^{3} \left (30 f x \cos \left (3 f x +3 e \right )+90 \cos \left (f x +e \right ) f x +46 \cos \left (3 f x +3 e \right )+156 \cos \left (2 f x +2 e \right )+72 \sin \left (f x +e \right )+138 \cos \left (f x +e \right )+3 \cos \left (4 f x +4 e \right )-56 \sin \left (3 f x +3 e \right )+25\right )}{6 f \,a^{2} \left (\cos \left (3 f x +3 e \right )+3 \cos \left (f x +e \right )\right )}\)	\(117\)
norman	\(\frac {\frac {8 c^{3} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}+\frac {38 c^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f a}+\frac {5 c^{3} x}{a}+\frac {46 c^{3}}{3 f a}+\frac {15 c^{3} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a}+\frac {30 c^{3} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {50 c^{3} x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {60 c^{3} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {60 c^{3} x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {50 c^{3} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {30 c^{3} x \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {15 c^{3} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {5 c^{3} x \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {110 c^{3} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}+\frac {78 c^{3} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}+\frac {130 c^{3} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}+\frac {34 c^{3} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}+\frac {58 c^{3} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}+\frac {106 c^{3} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3} a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\)	\(406\)

[In]

int((c-c*sin(f*x+e))^3/(a+a*sin(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

2/f*c^3/a^2*(1/(1+tan(1/2*f*x+1/2*e)^2)+5*arctan(tan(1/2*f*x+1/2*e))-16/3/(tan(1/2*f*x+1/2*e)+1)^3+8/(tan(1/2*
f*x+1/2*e)+1)^2+4/(tan(1/2*f*x+1/2*e)+1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (86) = 172\).

Time = 0.25 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.06 \[ \int \frac {(c-c \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx=\frac {3 \, c^{3} \cos \left (f x + e\right )^{3} - 30 \, c^{3} f x + 8 \, c^{3} + {\left (15 \, c^{3} f x - 31 \, c^{3}\right )} \cos \left (f x + e\right )^{2} - {\left (15 \, c^{3} f x + 26 \, c^{3}\right )} \cos \left (f x + e\right ) - {\left (30 \, c^{3} f x + 3 \, c^{3} \cos \left (f x + e\right )^{2} + 8 \, c^{3} + {\left (15 \, c^{3} f x + 34 \, c^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{3 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \]

[In]

integrate((c-c*sin(f*x+e))^3/(a+a*sin(f*x+e))^2,x, algorithm="fricas")

[Out]

1/3*(3*c^3*cos(f*x + e)^3 - 30*c^3*f*x + 8*c^3 + (15*c^3*f*x - 31*c^3)*cos(f*x + e)^2 - (15*c^3*f*x + 26*c^3)*
cos(f*x + e) - (30*c^3*f*x + 3*c^3*cos(f*x + e)^2 + 8*c^3 + (15*c^3*f*x + 34*c^3)*cos(f*x + e))*sin(f*x + e))/
(a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1282 vs. \(2 (87) = 174\).

Time = 3.91 (sec) , antiderivative size = 1282, normalized size of antiderivative = 14.24 \[ \int \frac {(c-c \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

integrate((c-c*sin(f*x+e))**3/(a+a*sin(f*x+e))**2,x)

[Out]

Piecewise((15*c**3*f*x*tan(e/2 + f*x/2)**5/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a
**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) + 45*c**3*f*
x*tan(e/2 + f*x/2)**4/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2
)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) + 60*c**3*f*x*tan(e/2 + f*x/2)**3
/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(
e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) + 60*c**3*f*x*tan(e/2 + f*x/2)**2/(3*a**2*f*tan(e/2 +
f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a
**2*f*tan(e/2 + f*x/2) + 3*a**2*f) + 45*c**3*f*x*tan(e/2 + f*x/2)/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan
(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) +
3*a**2*f) + 15*c**3*f*x/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*
x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) + 24*c**3*tan(e/2 + f*x/2)**4/
(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e
/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) + 102*c**3*tan(e/2 + f*x/2)**3/(3*a**2*f*tan(e/2 + f*x/
2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*
f*tan(e/2 + f*x/2) + 3*a**2*f) + 82*c**3*tan(e/2 + f*x/2)**2/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2
+ f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a*
*2*f) + 114*c**3*tan(e/2 + f*x/2)/(3*a**2*f*tan(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan
(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f) + 46*c**3/(3*a**2*f*t
an(e/2 + f*x/2)**5 + 9*a**2*f*tan(e/2 + f*x/2)**4 + 12*a**2*f*tan(e/2 + f*x/2)**3 + 12*a**2*f*tan(e/2 + f*x/2)
**2 + 9*a**2*f*tan(e/2 + f*x/2) + 3*a**2*f), Ne(f, 0)), (x*(-c*sin(e) + c)**3/(a*sin(e) + a)**2, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (86) = 172\).

Time = 0.30 (sec) , antiderivative size = 590, normalized size of antiderivative = 6.56 \[ \int \frac {(c-c \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx=\frac {2 \, {\left (2 \, c^{3} {\left (\frac {\frac {12 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {11 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {9 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 5}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {4 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {4 \, a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{2} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} + \frac {3 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{2}}\right )} + 3 \, c^{3} {\left (\frac {\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 4}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {3 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{2}}\right )} - \frac {c^{3} {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 2\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {3 \, c^{3} {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}\right )}}{3 \, f} \]

[In]

integrate((c-c*sin(f*x+e))^3/(a+a*sin(f*x+e))^2,x, algorithm="maxima")

[Out]

2/3*(2*c^3*((12*sin(f*x + e)/(cos(f*x + e) + 1) + 11*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 9*sin(f*x + e)^3/(c
os(f*x + e) + 1)^3 + 3*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 5)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) +
4*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 4*a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*a^2*sin(f*x + e)^4/
(cos(f*x + e) + 1)^4 + a^2*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 3*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^
2) + 3*c^3*((9*sin(f*x + e)/(cos(f*x + e) + 1) + 3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 4)/(a^2 + 3*a^2*sin(f
*x + e)/(cos(f*x + e) + 1) + 3*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)
^3) + 3*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^2) - c^3*(3*sin(f*x + e)/(cos(f*x + e) + 1) + 3*sin(f*x + e)
^2/(cos(f*x + e) + 1)^2 + 2)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*sin(f*x + e)^2/(cos(f*x + e)
+ 1)^2 + a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3) + 3*c^3*(3*sin(f*x + e)/(cos(f*x + e) + 1) + 1)/(a^2 + 3*a^
2*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*x + e)^3/(cos(f*x +
e) + 1)^3))/f

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.07 \[ \int \frac {(c-c \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx=\frac {\frac {15 \, {\left (f x + e\right )} c^{3}}{a^{2}} + \frac {6 \, c^{3}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )} a^{2}} + \frac {8 \, {\left (3 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 5 \, c^{3}\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}}}{3 \, f} \]

[In]

integrate((c-c*sin(f*x+e))^3/(a+a*sin(f*x+e))^2,x, algorithm="giac")

[Out]

1/3*(15*(f*x + e)*c^3/a^2 + 6*c^3/((tan(1/2*f*x + 1/2*e)^2 + 1)*a^2) + 8*(3*c^3*tan(1/2*f*x + 1/2*e)^2 + 12*c^
3*tan(1/2*f*x + 1/2*e) + 5*c^3)/(a^2*(tan(1/2*f*x + 1/2*e) + 1)^3))/f

Mupad [B] (verification not implemented)

Time = 9.30 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.41 \[ \int \frac {(c-c \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx=\frac {5\,c^3\,x}{a^2}-\frac {5\,c^3\,\left (e+f\,x\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (15\,c^3\,\left (e+f\,x\right )-\frac {c^3\,\left (45\,e+45\,f\,x+114\right )}{3}\right )-\frac {c^3\,\left (15\,e+15\,f\,x+46\right )}{3}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (15\,c^3\,\left (e+f\,x\right )-\frac {c^3\,\left (45\,e+45\,f\,x+24\right )}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (20\,c^3\,\left (e+f\,x\right )-\frac {c^3\,\left (60\,e+60\,f\,x+82\right )}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (20\,c^3\,\left (e+f\,x\right )-\frac {c^3\,\left (60\,e+60\,f\,x+102\right )}{3}\right )}{a^2\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^3\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]

[In]

int((c - c*sin(e + f*x))^3/(a + a*sin(e + f*x))^2,x)

[Out]

(5*c^3*x)/a^2 - (5*c^3*(e + f*x) + tan(e/2 + (f*x)/2)*(15*c^3*(e + f*x) - (c^3*(45*e + 45*f*x + 114))/3) - (c^
3*(15*e + 15*f*x + 46))/3 + tan(e/2 + (f*x)/2)^4*(15*c^3*(e + f*x) - (c^3*(45*e + 45*f*x + 24))/3) + tan(e/2 +
(f*x)/2)^2*(20*c^3*(e + f*x) - (c^3*(60*e + 60*f*x + 82))/3) + tan(e/2 + (f*x)/2)^3*(20*c^3*(e + f*x) - (c^3*
(60*e + 60*f*x + 102))/3))/(a^2*f*(tan(e/2 + (f*x)/2) + 1)^3*(tan(e/2 + (f*x)/2)^2 + 1))