\(\int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx\) [347]

Optimal result

Integrand size = 30, antiderivative size = 39 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {\cos (e+f x)}{f \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{7/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.10, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2817} \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {a \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}} \]

[In]

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

[Out]

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

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 {a \cos (e+f x)}{3 f \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(87\) vs. \(2(39)=78\).

Time = 1.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.23 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {\sqrt {1+\sin (e+f x)} \sqrt {c-c \sin (e+f x)}}{\sqrt {3} c^4 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]

[In]

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

[Out]

(Sqrt[1 + Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])/(Sqrt[3]*c^4*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[
(e + f*x)/2] + Sin[(e + f*x)/2]))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(91\) vs. \(2(37)=74\).

Time = 3.64 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.36

method	result	size
default	\(\frac {\sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, \left (\sin \left (f x +e \right ) \cos \left (f x +e \right )-3 \cos \left (f x +e \right )-4 \tan \left (f x +e \right )+3 \sec \left (f x +e \right )\right )}{3 f \left (2 \sin \left (f x +e \right )+\cos ^{2}\left (f x +e \right )-2\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{3}}\)	\(92\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (37) = 74\).

Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.31 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=-\frac {\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{3 \, {\left (3 \, c^{4} f \cos \left (f x + e\right )^{3} - 4 \, c^{4} f \cos \left (f x + e\right ) - {\left (c^{4} f \cos \left (f x + e\right )^{3} - 4 \, c^{4} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \]

[In]

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

[Out]

-1/3*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(3*c^4*f*cos(f*x + e)^3 - 4*c^4*f*cos(f*x + e) - (c^4*
f*cos(f*x + e)^3 - 4*c^4*f*cos(f*x + e))*sin(f*x + e))

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.36 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=-\frac {\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{24 \, c^{\frac {7}{2}} f \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6}} \]

[In]

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

[Out]

-1/24*sqrt(a)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))/(c^(7/2)*f*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi +
1/2*f*x + 1/2*e)^6)

Mupad [B] (verification not implemented)

Time = 11.07 (sec) , antiderivative size = 190, normalized size of antiderivative = 4.87 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=-\frac {{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sqrt {a+a\,\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}\,\sqrt {c-c\,\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}\,16{}\mathrm {i}}{3\,c^4\,f\,\left (1+14\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}-{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}-14\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,6{}\mathrm {i}-{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,14{}\mathrm {i}-{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,14{}\mathrm {i}+{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,6{}\mathrm {i}\right )} \]

[In]

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

[Out]

-(exp(e*4i + f*x*4i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*(c - c*((exp(- e*
1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*16i)/(3*c^4*f*(exp(e*1i + f*x*1i)*6i - 14*exp(e*2i + f*
x*2i) - exp(e*3i + f*x*3i)*14i - exp(e*5i + f*x*5i)*14i + 14*exp(e*6i + f*x*6i) + exp(e*7i + f*x*7i)*6i - exp(
e*8i + f*x*8i) + 1))