\(\int (3+3 \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx\) [373]

Optimal result

Integrand size = 30, antiderivative size = 43 \[ \int (3+3 \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\frac {c \cos (e+f x) (3+3 \sin (e+f x))^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2817} \[ \int (3+3 \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\frac {c \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}} \]

[In]

Int[(a + a*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]],x]

[Out]

(c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(4*f*Sqrt[c - c*Sin[e + f*x]])

Rule 2817

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

Rubi steps \begin{align*} \text {integral}& = \frac {c \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 f \sqrt {c-c \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.63 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.91 \[ \int (3+3 \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\frac {27 \sqrt {3} \sec (e+f x) \sqrt {1+\sin (e+f x)} \sqrt {c-c \sin (e+f x)} (-28 \cos (2 (e+f x))+\cos (4 (e+f x))+56 \sin (e+f x)-8 \sin (3 (e+f x)))}{32 f} \]

[In]

Integrate[(3 + 3*Sin[e + f*x])^(7/2)*Sqrt[c - c*Sin[e + f*x]],x]

[Out]

(27*Sqrt[3]*Sec[e + f*x]*Sqrt[1 + Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]*(-28*Cos[2*(e + f*x)] + Cos[4*(e + f*
x)] + 56*Sin[e + f*x] - 8*Sin[3*(e + f*x)]))/(32*f)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(80\) vs. \(2(37)=74\).

Time = 3.10 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.88

method	result	size
default	\(\frac {\sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, a^{3} \left (\cos ^{3}\left (f x +e \right )-4 \sin \left (f x +e \right ) \cos \left (f x +e \right )-8 \cos \left (f x +e \right )+8 \tan \left (f x +e \right )+7 \sec \left (f x +e \right )\right )}{4 f}\)	\(81\)

[In]

int((a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/4/f*(a*(sin(f*x+e)+1))^(1/2)*(-c*(sin(f*x+e)-1))^(1/2)*a^3*(cos(f*x+e)^3-4*sin(f*x+e)*cos(f*x+e)-8*cos(f*x+e
)+8*tan(f*x+e)+7*sec(f*x+e))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (37) = 74\).

Time = 0.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.21 \[ \int (3+3 \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\frac {{\left (a^{3} \cos \left (f x + e\right )^{4} - 8 \, a^{3} \cos \left (f x + e\right )^{2} + 7 \, a^{3} - 4 \, {\left (a^{3} \cos \left (f x + e\right )^{2} - 2 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{4 \, f \cos \left (f x + e\right )} \]

[In]

integrate((a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

1/4*(a^3*cos(f*x + e)^4 - 8*a^3*cos(f*x + e)^2 + 7*a^3 - 4*(a^3*cos(f*x + e)^2 - 2*a^3)*sin(f*x + e))*sqrt(a*s
in(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e))

Sympy [F(-1)]

Timed out. \[ \int (3+3 \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(f*x+e))**(7/2)*(c-c*sin(f*x+e))**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int (3+3 \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} \sqrt {-c \sin \left (f x + e\right ) + c} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^(7/2)*sqrt(-c*sin(f*x + e) + c), x)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.19 \[ \int (3+3 \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=-\frac {4 \, a^{\frac {7}{2}} \sqrt {c} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{f} \]

[In]

integrate((a+a*sin(f*x+e))^(7/2)*(c-c*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

-4*a^(7/2)*sqrt(c)*cos(-1/4*pi + 1/2*f*x + 1/2*e)^8*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*
f*x + 1/2*e))/f

Mupad [B] (verification not implemented)

Time = 8.62 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.30 \[ \int (3+3 \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=-\frac {a^3\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (28\,\cos \left (e+f\,x\right )+27\,\cos \left (3\,e+3\,f\,x\right )-\cos \left (5\,e+5\,f\,x\right )-48\,\sin \left (2\,e+2\,f\,x\right )+8\,\sin \left (4\,e+4\,f\,x\right )\right )}{32\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]

[In]

int((a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(1/2),x)

[Out]

-(a^3*(a*(sin(e + f*x) + 1))^(1/2)*(-c*(sin(e + f*x) - 1))^(1/2)*(28*cos(e + f*x) + 27*cos(3*e + 3*f*x) - cos(
5*e + 5*f*x) - 48*sin(2*e + 2*f*x) + 8*sin(4*e + 4*f*x)))/(32*f*(cos(2*e + 2*f*x) + 1))