\(\int \frac {3+b \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx\) [726]

Optimal result

Integrand size = 25, antiderivative size = 139 \[ \int \frac {3+b \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx=\frac {2 b E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 (b c-3 d) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{d f \sqrt {c+d \sin (e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2831, 2742, 2740, 2734, 2732} \[ \int \frac {3+b \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx=\frac {2 b \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 3.21 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.73 \[ \int \frac {3+b \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx=-\frac {2 \left (b (c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )+(-b c+3 d) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{d f \sqrt {c+d \sin (e+f x)}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 5.11 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.75

method	result	size
default	\(\frac {2 \left (c -d \right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, \left (a F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d +F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) b d -E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) b c -E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) b d \right )}{d^{2} \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\)	\(243\)
parts	\(\frac {2 a \left (c -d \right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{d \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}-\frac {2 b \left (c -d \right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, \left (E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c +E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d -F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d \right )}{d^{2} \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\)	\(330\)
risch	\(\text {Expression too large to display}\)	\(1429\)

[In]

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

[Out]

2*(c-d)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*(a*Elli
pticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*d+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c
+d))^(1/2))*b*d-EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*b*c-EllipticE(((c+d*sin(f*x+e))/
(c-d))^(1/2),((c-d)/(c+d))^(1/2))*b*d)/d^2/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 = 0.10 (sec) , antiderivative size = 383, normalized size of antiderivative = 2.76 \[ \int \frac {3+b \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx=\frac {-3 i \, \sqrt {2} b \sqrt {i \, d} d {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) + 3 i \, \sqrt {2} b \sqrt {-i \, d} d {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right ) - \sqrt {2} {\left (2 \, b c - 3 \, a d\right )} \sqrt {i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) - \sqrt {2} {\left (2 \, b c - 3 \, a d\right )} \sqrt {-i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )}{3 \, d^{2} f} \]

[In]

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

[Out]

1/3*(-3*I*sqrt(2)*b*sqrt(I*d)*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, 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)) + 3*I*sqrt(2)*b*sqrt(-I*d)*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)) - sqrt(2)*(2*b*c - 3*a*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) - sqrt(
2)*(2*b*c - 3*a*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))/(d^2*f)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.08 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.27 \[ \int \frac {3+b \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx=\frac {b\,\left (2\,c\,\mathrm {F}\left (\mathrm {asin}\left (\frac {\sqrt {2}\,\sqrt {1-\sin \left (e+f\,x\right )}}{2}\right )\middle |\frac {2\,d}{c+d}\right )-2\,\left (c+d\right )\,\mathrm {E}\left (\mathrm {asin}\left (\frac {\sqrt {2}\,\sqrt {1-\sin \left (e+f\,x\right )}}{2}\right )\middle |\frac {2\,d}{c+d}\right )\right )\,\sqrt {{\cos \left (e+f\,x\right )}^2}\,\sqrt {\frac {c+d\,\sin \left (e+f\,x\right )}{c+d}}}{d\,f\,\cos \left (e+f\,x\right )\,\sqrt {c+d\,\sin \left (e+f\,x\right )}}-\frac {2\,a\,\mathrm {F}\left (\frac {\pi }{4}-\frac {e}{2}-\frac {f\,x}{2}\middle |\frac {2\,d}{c+d}\right )\,\sqrt {\frac {c+d\,\sin \left (e+f\,x\right )}{c+d}}}{f\,\sqrt {c+d\,\sin \left (e+f\,x\right )}} \]

[In]

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

[Out]

(b*(2*c*ellipticF(asin((2^(1/2)*(1 - sin(e + f*x))^(1/2))/2), (2*d)/(c + d)) - 2*(c + d)*ellipticE(asin((2^(1/
2)*(1 - sin(e + f*x))^(1/2))/2), (2*d)/(c + d)))*(cos(e + f*x)^2)^(1/2)*((c + d*sin(e + f*x))/(c + d))^(1/2))/
(d*f*cos(e + f*x)*(c + d*sin(e + f*x))^(1/2)) - (2*a*ellipticF(pi/4 - e/2 - (f*x)/2, (2*d)/(c + d))*((c + d*si
n(e + f*x))/(c + d))^(1/2))/(f*(c + d*sin(e + f*x))^(1/2))