\(\int \frac {\sqrt {c+d \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx\) [506]

Optimal result

Integrand size = 27, antiderivative size = 169 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (3+3 \sin (e+f x))}-\frac {E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(c+d) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 f \sqrt {c+d \sin (e+f x)}} \]

[Out]

-cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/f/(a+a*sin(f*x+e))+(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2
*f*x)*EllipticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f*x+e))^(1/2)/a/f/((c+d*sin(f*x+e)
)/(c+d))^(1/2)-(c+d)*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticF(cos(1/2*e+1/4*Pi+
1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/a/f/(c+d*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2848, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a \sin (e+f x)+a)}+\frac {(c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{a f \sqrt {c+d \sin (e+f x)}}-\frac {\sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{a f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \]

[In]

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

[Out]

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

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2831

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

Rule 2848

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a+a \sin (e+f x))}+\frac {d \int \frac {a-a \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{2 a^2} \\ & = -\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a+a \sin (e+f x))}-\frac {\int \sqrt {c+d \sin (e+f x)} \, dx}{2 a}+\frac {(c+d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{2 a} \\ & = -\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a+a \sin (e+f x))}-\frac {\sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{2 a \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left ((c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{2 a \sqrt {c+d \sin (e+f x)}} \\ & = -\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f (a+a \sin (e+f x))}-\frac {E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{a f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(c+d) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{a f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.87 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (2 \sin \left (\frac {1}{2} (e+f x)\right ) (c+d \sin (e+f x))-\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (c+d \sin (e+f x)-(c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+(c+d) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right )\right )}{3 f (1+\sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \]

[In]

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

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(2*Sin[(e + f*x)/2]*(c + d*Sin[e + f*x]) - (Cos[(e + f*x)/2] + Sin[(e +
f*x)/2])*(c + d*Sin[e + f*x] - (c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*
x])/(c + d)] + (c + d)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])))/(
3*f*(1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]])

Maple [A] (verified)

Time = 1.33 (sec) , antiderivative size = 382, normalized size of antiderivative = 2.26

method	result	size
default	\(\frac {\sqrt {\left (\cos ^{2}\left (f x +e \right )\right ) d \sin \left (f x +e \right )+c \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, E\left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2}-\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, E\left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{2}-\left (\cos ^{2}\left (f x +e \right )\right ) d^{2}+c d \sin \left (f x +e \right )-\sin \left (f x +e \right ) d^{2}-c d +d^{2}\right )}{d \sqrt {-\left (c +d \sin \left (f x +e \right )\right ) \left (\sin \left (f x +e \right )-1\right ) \left (\sin \left (f x +e \right )+1\right )}\, a \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\)	\(382\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.12 (sec) , antiderivative size = 555, normalized size of antiderivative = 3.28 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=\frac {{\left (\sqrt {2} {\left (2 \, c + 3 \, d\right )} \cos \left (f x + e\right ) + \sqrt {2} {\left (2 \, c + 3 \, d\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (2 \, c + 3 \, d\right )}\right )} \sqrt {i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) + {\left (\sqrt {2} {\left (2 \, c + 3 \, d\right )} \cos \left (f x + e\right ) + \sqrt {2} {\left (2 \, c + 3 \, d\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (2 \, c + 3 \, d\right )}\right )} \sqrt {-i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right ) - 3 \, {\left (-i \, \sqrt {2} d \cos \left (f x + e\right ) - i \, \sqrt {2} d \sin \left (f x + e\right ) - i \, \sqrt {2} d\right )} \sqrt {i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) - 3 \, {\left (i \, \sqrt {2} d \cos \left (f x + e\right ) + i \, \sqrt {2} d \sin \left (f x + e\right ) + i \, \sqrt {2} d\right )} \sqrt {-i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right ) - 6 \, {\left (d \cos \left (f x + e\right ) - d \sin \left (f x + e\right ) + d\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{6 \, {\left (a d f \cos \left (f x + e\right ) + a d f \sin \left (f x + e\right ) + a d f\right )}} \]

[In]

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

[Out]

1/6*((sqrt(2)*(2*c + 3*d)*cos(f*x + e) + sqrt(2)*(2*c + 3*d)*sin(f*x + e) + sqrt(2)*(2*c + 3*d))*sqrt(I*d)*wei
erstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(
f*x + e) - 2*I*c)/d) + (sqrt(2)*(2*c + 3*d)*cos(f*x + e) + sqrt(2)*(2*c + 3*d)*sin(f*x + e) + sqrt(2)*(2*c + 3
*d))*sqrt(-I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f
*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d) - 3*(-I*sqrt(2)*d*cos(f*x + e) - I*sqrt(2)*d*sin(f*x + e) - I*sqrt(2)
*d)*sqrt(I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, weierstrassPInverse(-
4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/
d)) - 3*(I*sqrt(2)*d*cos(f*x + e) + I*sqrt(2)*d*sin(f*x + e) + I*sqrt(2)*d)*sqrt(-I*d)*weierstrassZeta(-4/3*(4
*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I
*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d)) - 6*(d*cos(f*x + e) - d*sin(f*x
+ e) + d)*sqrt(d*sin(f*x + e) + c))/(a*d*f*cos(f*x + e) + a*d*f*sin(f*x + e) + a*d*f)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate(sqrt(d*sin(f*x + e) + c)/(a*sin(f*x + e) + a), x)

Giac [F]

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

[In]

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

[Out]

integrate(sqrt(d*sin(f*x + e) + c)/(a*sin(f*x + e) + a), x)

Mupad [F(-1)]

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

[In]

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

[Out]

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