\(\int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2} \, dx\) [309]

Optimal result

Integrand size = 28, antiderivative size = 65 \[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2} \, dx=\frac {24 c^5 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}+\frac {6 c^4 \cos ^7(e+f x)}{f (c-c \sin (e+f x))^{5/2}} \]

[Out]

8/63*a^3*c^5*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^(7/2)+2/9*a^3*c^4*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^(5/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2815, 2753, 2752} \[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2} \, dx=\frac {8 a^3 c^5 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^{7/2}}+\frac {2 a^3 c^4 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}} \]

[In]

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

[Out]

(8*a^3*c^5*Cos[e + f*x]^7)/(63*f*(c - c*Sin[e + f*x])^(7/2)) + (2*a^3*c^4*Cos[e + f*x]^7)/(9*f*(c - c*Sin[e +
f*x])^(5/2))

Rule 2752

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

Rule 2753

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

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)}{(c-c \sin (e+f x))^{3/2}} \, dx \\ & = \frac {2 a^3 c^4 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}}+\frac {1}{9} \left (4 a^3 c^4\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx \\ & = \frac {8 a^3 c^5 \cos ^7(e+f x)}{63 f (c-c \sin (e+f x))^{7/2}}+\frac {2 a^3 c^4 \cos ^7(e+f x)}{9 f (c-c \sin (e+f x))^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.75 \[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2} \, dx=-\frac {6 c \sec (e+f x) (1+\sin (e+f x))^4 (-11+7 \sin (e+f x)) \sqrt {c-c \sin (e+f x)}}{7 f} \]

[In]

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

[Out]

(-6*c*Sec[e + f*x]*(1 + Sin[e + f*x])^4*(-11 + 7*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(7*f)

Maple [A] (verified)

Time = 2.18 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {2 \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (\sin \left (f x +e \right )+1\right )^{4} a^{3} \left (7 \sin \left (f x +e \right )-11\right )}{63 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(61\)
parts	\(\frac {2 a^{3} \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (\sin \left (f x +e \right )+1\right ) \left (\sin \left (f x +e \right )-5\right )}{3 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 a^{3} \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (\sin \left (f x +e \right )+1\right ) \left (35 \left (\sin ^{4}\left (f x +e \right )\right )-85 \left (\sin ^{3}\left (f x +e \right )\right )+102 \left (\sin ^{2}\left (f x +e \right )\right )-136 \sin \left (f x +e \right )+272\right )}{315 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {6 a^{3} \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (\sin \left (f x +e \right )+1\right ) \left (\sin ^{2}\left (f x +e \right )-3 \sin \left (f x +e \right )+6\right )}{5 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 a^{3} \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (\sin \left (f x +e \right )+1\right ) \left (15 \left (\sin ^{3}\left (f x +e \right )\right )-39 \left (\sin ^{2}\left (f x +e \right )\right )+52 \sin \left (f x +e \right )-104\right )}{35 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(290\)

[In]

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

[Out]

2/63*(sin(f*x+e)-1)*c^2*(sin(f*x+e)+1)^4*a^3*(7*sin(f*x+e)-11)/cos(f*x+e)/(c-c*sin(f*x+e))^(1/2)/f

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (65) = 130\).

Time = 0.27 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.75 \[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2} \, dx=-\frac {2 \, {\left (7 \, a^{3} c \cos \left (f x + e\right )^{5} + 17 \, a^{3} c \cos \left (f x + e\right )^{4} - 2 \, a^{3} c \cos \left (f x + e\right )^{3} + 4 \, a^{3} c \cos \left (f x + e\right )^{2} - 16 \, a^{3} c \cos \left (f x + e\right ) - 32 \, a^{3} c + {\left (7 \, a^{3} c \cos \left (f x + e\right )^{4} - 10 \, a^{3} c \cos \left (f x + e\right )^{3} - 12 \, a^{3} c \cos \left (f x + e\right )^{2} - 16 \, a^{3} c \cos \left (f x + e\right ) - 32 \, a^{3} c\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{63 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \]

[In]

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

[Out]

-2/63*(7*a^3*c*cos(f*x + e)^5 + 17*a^3*c*cos(f*x + e)^4 - 2*a^3*c*cos(f*x + e)^3 + 4*a^3*c*cos(f*x + e)^2 - 16
*a^3*c*cos(f*x + e) - 32*a^3*c + (7*a^3*c*cos(f*x + e)^4 - 10*a^3*c*cos(f*x + e)^3 - 12*a^3*c*cos(f*x + e)^2 -
16*a^3*c*cos(f*x + e) - 32*a^3*c)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e) - f*sin(f*x + e) +
f)

Sympy [F]

\[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2} \, dx=a^{3} \left (\int c \sqrt {- c \sin {\left (e + f x \right )} + c}\, dx + \int 2 c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )}\, dx + \int \left (- 2 c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{3}{\left (e + f x \right )}\right )\, dx + \int \left (- c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{4}{\left (e + f x \right )}\right )\, dx\right ) \]

[In]

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

[Out]

a**3*(Integral(c*sqrt(-c*sin(e + f*x) + c), x) + Integral(2*c*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x), x) + Int
egral(-2*c*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x)**3, x) + Integral(-c*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x)*
*4, x))

Maxima [F]

\[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (65) = 130\).

Time = 0.38 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.09 \[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2} \, dx=-\frac {\sqrt {2} {\left (378 \, a^{3} c \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 168 \, a^{3} c \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 27 \, a^{3} c \cos \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 7 \, a^{3} c \cos \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, f x + \frac {9}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {c}}{504 \, f} \]

[In]

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

[Out]

-1/504*sqrt(2)*(378*a^3*c*cos(-1/4*pi + 1/2*f*x + 1/2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) + 168*a^3*c*cos(-
3/4*pi + 3/2*f*x + 3/2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)) - 27*a^3*c*cos(-7/4*pi + 7/2*f*x + 7/2*e)*sgn(si
n(-1/4*pi + 1/2*f*x + 1/2*e)) - 7*a^3*c*cos(-9/4*pi + 9/2*f*x + 9/2*e)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*sq
rt(c)/f

Mupad [F(-1)]

Timed out. \[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2} \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \]

[In]

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

[Out]

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