\(\int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx\) [568]

Optimal result

Integrand size = 29, antiderivative size = 44 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {6 \cos (e+f x)}{(c+d) f \sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.02, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2850} \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {2 a \cos (e+f x)}{f (c+d) \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}} \]

[In]

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

[Out]

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

Rule 2850

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(3/2), x_Symbol] :> Sim
p[-2*b^2*(Cos[e + f*x]/(f*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*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]

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

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.98 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {2 \sqrt {3} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {1+\sin (e+f x)}}{(c+d) f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(83\) vs. \(2(41)=82\).

Time = 3.36 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.91

method	result	size
default	\(-\frac {2 \sec \left (f x +e \right ) \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {c +d \sin \left (f x +e \right )}\, \left (\sin \left (f x +e \right )-1\right ) \left (d \sin \left (f x +e \right )-c \right )}{f \left (\left (\cos ^{2}\left (f x +e \right )\right ) d^{2}+c^{2}-d^{2}\right ) \left (c +d \right )}\)	\(84\)

[In]

int((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (41) = 82\).

Time = 0.27 (sec) , antiderivative size = 129, normalized size of antiderivative = 2.93 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 \, \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )}}{{\left (c d + d^{2}\right )} f \cos \left (f x + e\right )^{2} - {\left (c^{2} + c d\right )} f \cos \left (f x + e\right ) - {\left (c^{2} + 2 \, c d + d^{2}\right )} f - {\left ({\left (c d + d^{2}\right )} f \cos \left (f x + e\right ) + {\left (c^{2} + 2 \, c d + d^{2}\right )} f\right )} \sin \left (f x + e\right )} \]

[In]

integrate((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}{\left (c + d \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((a+a*sin(f*x+e))**(1/2)/(c+d*sin(f*x+e))**(3/2),x)

[Out]

Integral(sqrt(a*(sin(e + f*x) + 1))/(c + d*sin(e + f*x))**(3/2), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (41) = 82\).

Time = 0.32 (sec) , antiderivative size = 179, normalized size of antiderivative = 4.07 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {2 \, {\left (\sqrt {a} c - \frac {\sqrt {a} {\left (c - 2 \, d\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sqrt {a} {\left (c - 2 \, d\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {\sqrt {a} c \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}}{{\left (c + d + \frac {{\left (c + d\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (c + \frac {2 \, d \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {c \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}^{\frac {3}{2}} f} \]

[In]

integrate((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (41) = 82\).

Time = 0.56 (sec) , antiderivative size = 151, normalized size of antiderivative = 3.43 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {4 \, \sqrt {2} {\left (c^{2} d^{2} - 2 \, c d^{3} + d^{4}\right )} \sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )}{{\left (c^{3} d^{2} - c^{2} d^{3} - c d^{4} + d^{5}\right )} \sqrt {c \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{4} + d \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{4} + 2 \, c \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{2} - 6 \, d \tan \left (-\frac {1}{8} \, \pi + \frac {1}{4} \, f x + \frac {1}{4} \, e\right )^{2} + c + d} f} \]

[In]

integrate((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))^(3/2),x, algorithm="giac")

[Out]

4*sqrt(2)*(c^2*d^2 - 2*c*d^3 + d^4)*sqrt(a)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(-1/8*pi + 1/4*f*x + 1/4*e)
/((c^3*d^2 - c^2*d^3 - c*d^4 + d^5)*sqrt(c*tan(-1/8*pi + 1/4*f*x + 1/4*e)^4 + d*tan(-1/8*pi + 1/4*f*x + 1/4*e)
^4 + 2*c*tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 - 6*d*tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 + c + d)*f)

Mupad [B] (verification not implemented)

Time = 9.51 (sec) , antiderivative size = 145, normalized size of antiderivative = 3.30 \[ \int \frac {\sqrt {3+3 \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {4\,\left (2\,c\,\cos \left (e+f\,x\right )+d\,\sin \left (2\,e+2\,f\,x\right )\right )\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {c+d\,\sin \left (e+f\,x\right )}}{f\,\left (c+d\right )\,\left (4\,c\,d+4\,c^2\,\sin \left (e+f\,x\right )+3\,d^2\,\sin \left (e+f\,x\right )+4\,c^2+2\,d^2-2\,d^2\,\cos \left (2\,e+2\,f\,x\right )-d^2\,\sin \left (3\,e+3\,f\,x\right )+8\,c\,d\,\sin \left (e+f\,x\right )-4\,c\,d\,\cos \left (2\,e+2\,f\,x\right )\right )} \]

[In]

int((a + a*sin(e + f*x))^(1/2)/(c + d*sin(e + f*x))^(3/2),x)

[Out]

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