\(\int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx\) [25]

Optimal result

Integrand size = 39, antiderivative size = 149 \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx=-\frac {2 \sqrt {a} \sqrt {g} \arctan \left (\frac {\sqrt {a} \sqrt {g} \cos (e+f x)}{\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\right )}{d f}+\frac {2 \sqrt {a} \sqrt {c} \sqrt {g} \arctan \left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \cos (e+f x)}{\sqrt {c+d} \sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\right )}{d \sqrt {c+d} f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3007, 2854, 211, 3009} \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx=\frac {2 \sqrt {a} \sqrt {c} \sqrt {g} \arctan \left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a} \sqrt {g \sin (e+f x)}}\right )}{d f \sqrt {c+d}}-\frac {2 \sqrt {a} \sqrt {g} \arctan \left (\frac {\sqrt {a} \sqrt {g} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {g \sin (e+f x)}}\right )}{d f} \]

[In]

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

[Out]

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

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2854

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[
-2*(b/f), Subst[Int[1/(b + d*x^2), x], x, b*(Cos[e + f*x]/(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]

Rule 3007

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

Rule 3009

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.40 (sec) , antiderivative size = 662, normalized size of antiderivative = 4.44 \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx=\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) e^{-\frac {5}{2} i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )^{5/2} \left (-2 i \sqrt {-1+e^{2 i (e+f x)}}+\left (i+\frac {c-d}{\sqrt {-c^2+d^2}}\right ) \sqrt {-1+e^{2 i (e+f x)}}+\left (i+\frac {-c+d}{\sqrt {-c^2+d^2}}\right ) \sqrt {-1+e^{2 i (e+f x)}}+2 i \arctan \left (\sqrt {-1+e^{2 i (e+f x)}}\right )+\frac {\left (i+\frac {-c+d}{\sqrt {-c^2+d^2}}\right ) \left (\sqrt {2} \sqrt {c} \sqrt {c+i \sqrt {-c^2+d^2}} \arctan \left (\frac {d-\left (-i c+\sqrt {-c^2+d^2}\right ) e^{i (e+f x)}}{\sqrt {2} \sqrt {c} \sqrt {c+i \sqrt {-c^2+d^2}} \sqrt {-1+e^{2 i (e+f x)}}}\right )+\left (-i c+\sqrt {-c^2+d^2}\right ) \text {arctanh}\left (\frac {e^{i (e+f x)}}{\sqrt {-1+e^{2 i (e+f x)}}}\right )\right )}{d}+\frac {\left (i+\frac {c-d}{\sqrt {-c^2+d^2}}\right ) \left (\sqrt {2} \sqrt {c} \sqrt {c-i \sqrt {-c^2+d^2}} \arctan \left (\frac {d+\left (i c+\sqrt {-c^2+d^2}\right ) e^{i (e+f x)}}{\sqrt {2} \sqrt {c} \sqrt {c-i \sqrt {-c^2+d^2}} \sqrt {-1+e^{2 i (e+f x)}}}\right )-\left (i c+\sqrt {-c^2+d^2}\right ) \text {arctanh}\left (\frac {e^{i (e+f x)}}{\sqrt {-1+e^{2 i (e+f x)}}}\right )\right )}{d}\right ) \sqrt {g \sin (e+f x)} \sqrt {a (1+\sin (e+f x))}}{\sqrt {2} d \left (-i e^{-i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )\right )^{5/2} f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {\sin (e+f x)}} \]

[In]

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

[Out]

((1/2 + I/2)*(-1 + E^((2*I)*(e + f*x)))^(5/2)*((-2*I)*Sqrt[-1 + E^((2*I)*(e + f*x))] + (I + (c - d)/Sqrt[-c^2
+ d^2])*Sqrt[-1 + E^((2*I)*(e + f*x))] + (I + (-c + d)/Sqrt[-c^2 + d^2])*Sqrt[-1 + E^((2*I)*(e + f*x))] + (2*I
)*ArcTan[Sqrt[-1 + E^((2*I)*(e + f*x))]] + ((I + (-c + d)/Sqrt[-c^2 + d^2])*(Sqrt[2]*Sqrt[c]*Sqrt[c + I*Sqrt[-
c^2 + d^2]]*ArcTan[(d - ((-I)*c + Sqrt[-c^2 + d^2])*E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[c]*Sqrt[c + I*Sqrt[-c^2 + d
^2]]*Sqrt[-1 + E^((2*I)*(e + f*x))])] + ((-I)*c + Sqrt[-c^2 + d^2])*ArcTanh[E^(I*(e + f*x))/Sqrt[-1 + E^((2*I)
*(e + f*x))]]))/d + ((I + (c - d)/Sqrt[-c^2 + d^2])*(Sqrt[2]*Sqrt[c]*Sqrt[c - I*Sqrt[-c^2 + d^2]]*ArcTan[(d +
(I*c + Sqrt[-c^2 + d^2])*E^(I*(e + f*x)))/(Sqrt[2]*Sqrt[c]*Sqrt[c - I*Sqrt[-c^2 + d^2]]*Sqrt[-1 + E^((2*I)*(e
+ f*x))])] - (I*c + Sqrt[-c^2 + d^2])*ArcTanh[E^(I*(e + f*x))/Sqrt[-1 + E^((2*I)*(e + f*x))]]))/d)*Sqrt[g*Sin[
e + f*x]]*Sqrt[a*(1 + Sin[e + f*x])])/(Sqrt[2]*d*E^(((5*I)/2)*(e + f*x))*(((-I)*(-1 + E^((2*I)*(e + f*x))))/E^
(I*(e + f*x)))^(5/2)*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[Sin[e + f*x]])

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(934\) vs. \(2(117)=234\).

Time = 3.64 (sec) , antiderivative size = 935, normalized size of antiderivative = 6.28

method	result	size
default	\(\text {Expression too large to display}\)	\(935\)

[In]

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

[Out]

1/2/f*(g*sin(f*x+e))^(1/2)*(a*(1+sin(f*x+e)))^(1/2)*(2^(1/2)*(-(c-d)*(c+d))^(1/2)*(((-(c-d)*(c+d))^(1/2)+d)*c)
^(1/2)*(((-(c-d)*(c+d))^(1/2)-d)*c)^(1/2)*ln(-(csc(f*x+e)-cot(f*x+e)+(csc(f*x+e)-cot(f*x+e))^(1/2)*2^(1/2)+1)/
((csc(f*x+e)-cot(f*x+e))^(1/2)*2^(1/2)-csc(f*x+e)+cot(f*x+e)-1))+4*2^(1/2)*(-(c-d)*(c+d))^(1/2)*(((-(c-d)*(c+d
))^(1/2)+d)*c)^(1/2)*(((-(c-d)*(c+d))^(1/2)-d)*c)^(1/2)*arctan((csc(f*x+e)-cot(f*x+e))^(1/2)*2^(1/2)+1)+4*2^(1
/2)*(-(c-d)*(c+d))^(1/2)*(((-(c-d)*(c+d))^(1/2)+d)*c)^(1/2)*(((-(c-d)*(c+d))^(1/2)-d)*c)^(1/2)*arctan((csc(f*x
+e)-cot(f*x+e))^(1/2)*2^(1/2)-1)+2^(1/2)*(-(c-d)*(c+d))^(1/2)*(((-(c-d)*(c+d))^(1/2)+d)*c)^(1/2)*(((-(c-d)*(c+
d))^(1/2)-d)*c)^(1/2)*ln(-((csc(f*x+e)-cot(f*x+e))^(1/2)*2^(1/2)-csc(f*x+e)+cot(f*x+e)-1)/(csc(f*x+e)-cot(f*x+
e)+(csc(f*x+e)-cot(f*x+e))^(1/2)*2^(1/2)+1))+4*(-(c-d)*(c+d))^(1/2)*(((-(c-d)*(c+d))^(1/2)+d)*c)^(1/2)*arctanh
((csc(f*x+e)-cot(f*x+e))^(1/2)*c/(((-(c-d)*(c+d))^(1/2)-d)*c)^(1/2))*c-4*(-(c-d)*(c+d))^(1/2)*(((-(c-d)*(c+d))
^(1/2)-d)*c)^(1/2)*arctan((csc(f*x+e)-cot(f*x+e))^(1/2)*c/(((-(c-d)*(c+d))^(1/2)+d)*c)^(1/2))*c+4*(((-(c-d)*(c
+d))^(1/2)+d)*c)^(1/2)*arctanh((csc(f*x+e)-cot(f*x+e))^(1/2)*c/(((-(c-d)*(c+d))^(1/2)-d)*c)^(1/2))*c^2-4*(((-(
c-d)*(c+d))^(1/2)+d)*c)^(1/2)*arctanh((csc(f*x+e)-cot(f*x+e))^(1/2)*c/(((-(c-d)*(c+d))^(1/2)-d)*c)^(1/2))*c*d+
4*(((-(c-d)*(c+d))^(1/2)-d)*c)^(1/2)*arctan((csc(f*x+e)-cot(f*x+e))^(1/2)*c/(((-(c-d)*(c+d))^(1/2)+d)*c)^(1/2)
)*c^2-4*(((-(c-d)*(c+d))^(1/2)-d)*c)^(1/2)*arctan((csc(f*x+e)-cot(f*x+e))^(1/2)*c/(((-(c-d)*(c+d))^(1/2)+d)*c)
^(1/2))*c*d)/(cos(f*x+e)+sin(f*x+e)+1)/(csc(f*x+e)-cot(f*x+e))^(1/2)/d/(-(c-d)*(c+d))^(1/2)/(((-(c-d)*(c+d))^(
1/2)+d)*c)^(1/2)/(((-(c-d)*(c+d))^(1/2)-d)*c)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (117) = 234\).

Time = 1.47 (sec) , antiderivative size = 3273, normalized size of antiderivative = 21.97 \[ \int \frac {\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/4*(sqrt(-a*c*g/(c + d))*log(((128*a*c^4 + 256*a*c^3*d + 160*a*c^2*d^2 + 32*a*c*d^3 + a*d^4)*g*cos(f*x + e)^
5 - (128*a*c^4 + 192*a*c^3*d + 64*a*c^2*d^2 - 4*a*c*d^3 - a*d^4)*g*cos(f*x + e)^4 - 2*(208*a*c^4 + 368*a*c^3*d
+ 195*a*c^2*d^2 + 32*a*c*d^3 + a*d^4)*g*cos(f*x + e)^3 + 2*(64*a*c^4 + 94*a*c^3*d + 29*a*c^2*d^2 - 4*a*c*d^3
- a*d^4)*g*cos(f*x + e)^2 + (289*a*c^4 + 480*a*c^3*d + 230*a*c^2*d^2 + 32*a*c*d^3 + a*d^4)*g*cos(f*x + e) + 8*
((16*c^4 + 40*c^3*d + 34*c^2*d^2 + 11*c*d^3 + d^4)*cos(f*x + e)^4 + 51*c^4 + 110*c^3*d + 76*c^2*d^2 + 18*c*d^3
+ d^4 - (24*c^4 + 52*c^3*d + 35*c^2*d^2 + 7*c*d^3)*cos(f*x + e)^3 - (66*c^4 + 149*c^3*d + 110*c^2*d^2 + 29*c*
d^3 + 2*d^4)*cos(f*x + e)^2 + (25*c^4 + 53*c^3*d + 35*c^2*d^2 + 7*c*d^3)*cos(f*x + e) - (51*c^4 + 110*c^3*d +
76*c^2*d^2 + 18*c*d^3 + d^4 - (16*c^4 + 40*c^3*d + 34*c^2*d^2 + 11*c*d^3 + d^4)*cos(f*x + e)^3 - (40*c^4 + 92*
c^3*d + 69*c^2*d^2 + 18*c*d^3 + d^4)*cos(f*x + e)^2 + (26*c^4 + 57*c^3*d + 41*c^2*d^2 + 11*c*d^3 + d^4)*cos(f*
x + e))*sin(f*x + e))*sqrt(-a*c*g/(c + d))*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e)) + (a*c^4 + 4*a*c^3*d
+ 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4)*g + ((128*a*c^4 + 256*a*c^3*d + 160*a*c^2*d^2 + 32*a*c*d^3 + a*d^4)*g*cos(f
*x + e)^4 + 4*(64*a*c^4 + 112*a*c^3*d + 56*a*c^2*d^2 + 7*a*c*d^3)*g*cos(f*x + e)^3 - 2*(80*a*c^4 + 144*a*c^3*d
+ 83*a*c^2*d^2 + 18*a*c*d^3 + a*d^4)*g*cos(f*x + e)^2 - 4*(72*a*c^4 + 119*a*c^3*d + 56*a*c^2*d^2 + 7*a*c*d^3)
*g*cos(f*x + e) + (a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4)*g)*sin(f*x + e))/(d^4*cos(f*x + e)^5 +
(4*c*d^3 + d^4)*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 2*(3*c^2*d^2 + d^4)*cos(f*x + e)
^3 - 2*(2*c^3*d + 3*c^2*d^2 + 4*c*d^3 + d^4)*cos(f*x + e)^2 + (c^4 + 6*c^2*d^2 + d^4)*cos(f*x + e) + (d^4*cos(
f*x + e)^4 - 4*c*d^3*cos(f*x + e)^3 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 2*(3*c^2*d^2 + 2*c*d^3 + d^4
)*cos(f*x + e)^2 + 4*(c^3*d + c*d^3)*cos(f*x + e))*sin(f*x + e))) + sqrt(-a*g)*log((128*a*g*cos(f*x + e)^5 - 1
28*a*g*cos(f*x + e)^4 - 416*a*g*cos(f*x + e)^3 + 128*a*g*cos(f*x + e)^2 + 289*a*g*cos(f*x + e) - 8*(16*cos(f*x
+ e)^4 - 24*cos(f*x + e)^3 - 66*cos(f*x + e)^2 + (16*cos(f*x + e)^3 + 40*cos(f*x + e)^2 - 26*cos(f*x + e) - 5
1)*sin(f*x + e) + 25*cos(f*x + e) + 51)*sqrt(-a*g)*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e)) + a*g + (128*
a*g*cos(f*x + e)^4 + 256*a*g*cos(f*x + e)^3 - 160*a*g*cos(f*x + e)^2 - 288*a*g*cos(f*x + e) + a*g)*sin(f*x + e
))/(cos(f*x + e) + sin(f*x + e) + 1)))/(d*f), -1/4*(2*sqrt(a*c*g/(c + d))*arctan(1/4*((8*c^2 + 8*c*d + d^2)*co
s(f*x + e)^2 - 9*c^2 - 8*c*d - d^2 + 2*(4*c^2 + 3*c*d)*sin(f*x + e))*sqrt(a*c*g/(c + d))*sqrt(a*sin(f*x + e) +
a)*sqrt(g*sin(f*x + e))/(a*c^2*g*cos(f*x + e)*sin(f*x + e) + (2*a*c^2 + a*c*d)*g*cos(f*x + e)^3 - (2*a*c^2 +
a*c*d)*g*cos(f*x + e))) - sqrt(-a*g)*log((128*a*g*cos(f*x + e)^5 - 128*a*g*cos(f*x + e)^4 - 416*a*g*cos(f*x +
e)^3 + 128*a*g*cos(f*x + e)^2 + 289*a*g*cos(f*x + e) - 8*(16*cos(f*x + e)^4 - 24*cos(f*x + e)^3 - 66*cos(f*x +
e)^2 + (16*cos(f*x + e)^3 + 40*cos(f*x + e)^2 - 26*cos(f*x + e) - 51)*sin(f*x + e) + 25*cos(f*x + e) + 51)*sq
rt(-a*g)*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e)) + a*g + (128*a*g*cos(f*x + e)^4 + 256*a*g*cos(f*x + e)^
3 - 160*a*g*cos(f*x + e)^2 - 288*a*g*cos(f*x + e) + a*g)*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)))/(d*
f), 1/4*(2*sqrt(a*g)*arctan(1/4*sqrt(a*g)*(8*cos(f*x + e)^2 + 8*sin(f*x + e) - 9)*sqrt(a*sin(f*x + e) + a)*sqr
t(g*sin(f*x + e))/(2*a*g*cos(f*x + e)^3 + a*g*cos(f*x + e)*sin(f*x + e) - 2*a*g*cos(f*x + e))) + sqrt(-a*c*g/(
c + d))*log(((128*a*c^4 + 256*a*c^3*d + 160*a*c^2*d^2 + 32*a*c*d^3 + a*d^4)*g*cos(f*x + e)^5 - (128*a*c^4 + 19
2*a*c^3*d + 64*a*c^2*d^2 - 4*a*c*d^3 - a*d^4)*g*cos(f*x + e)^4 - 2*(208*a*c^4 + 368*a*c^3*d + 195*a*c^2*d^2 +
32*a*c*d^3 + a*d^4)*g*cos(f*x + e)^3 + 2*(64*a*c^4 + 94*a*c^3*d + 29*a*c^2*d^2 - 4*a*c*d^3 - a*d^4)*g*cos(f*x
+ e)^2 + (289*a*c^4 + 480*a*c^3*d + 230*a*c^2*d^2 + 32*a*c*d^3 + a*d^4)*g*cos(f*x + e) + 8*((16*c^4 + 40*c^3*d
+ 34*c^2*d^2 + 11*c*d^3 + d^4)*cos(f*x + e)^4 + 51*c^4 + 110*c^3*d + 76*c^2*d^2 + 18*c*d^3 + d^4 - (24*c^4 +
52*c^3*d + 35*c^2*d^2 + 7*c*d^3)*cos(f*x + e)^3 - (66*c^4 + 149*c^3*d + 110*c^2*d^2 + 29*c*d^3 + 2*d^4)*cos(f*
x + e)^2 + (25*c^4 + 53*c^3*d + 35*c^2*d^2 + 7*c*d^3)*cos(f*x + e) - (51*c^4 + 110*c^3*d + 76*c^2*d^2 + 18*c*d
^3 + d^4 - (16*c^4 + 40*c^3*d + 34*c^2*d^2 + 11*c*d^3 + d^4)*cos(f*x + e)^3 - (40*c^4 + 92*c^3*d + 69*c^2*d^2
+ 18*c*d^3 + d^4)*cos(f*x + e)^2 + (26*c^4 + 57*c^3*d + 41*c^2*d^2 + 11*c*d^3 + d^4)*cos(f*x + e))*sin(f*x + e
))*sqrt(-a*c*g/(c + d))*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e)) + (a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a
*c*d^3 + a*d^4)*g + ((128*a*c^4 + 256*a*c^3*d + 160*a*c^2*d^2 + 32*a*c*d^3 + a*d^4)*g*cos(f*x + e)^4 + 4*(64*a
*c^4 + 112*a*c^3*d + 56*a*c^2*d^2 + 7*a*c*d^3)*g*cos(f*x + e)^3 - 2*(80*a*c^4 + 144*a*c^3*d + 83*a*c^2*d^2 + 1
8*a*c*d^3 + a*d^4)*g*cos(f*x + e)^2 - 4*(72*a*c^4 + 119*a*c^3*d + 56*a*c^2*d^2 + 7*a*c*d^3)*g*cos(f*x + e) + (
a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4)*g)*sin(f*x + e))/(d^4*cos(f*x + e)^5 + (4*c*d^3 + d^4)*co
s(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 2*(3*c^2*d^2 + d^4)*cos(f*x + e)^3 - 2*(2*c^3*d + 3
*c^2*d^2 + 4*c*d^3 + d^4)*cos(f*x + e)^2 + (c^4 + 6*c^2*d^2 + d^4)*cos(f*x + e) + (d^4*cos(f*x + e)^4 - 4*c*d^
3*cos(f*x + e)^3 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 2*(3*c^2*d^2 + 2*c*d^3 + d^4)*cos(f*x + e)^2 +
4*(c^3*d + c*d^3)*cos(f*x + e))*sin(f*x + e))))/(d*f), -1/2*(sqrt(a*c*g/(c + d))*arctan(1/4*((8*c^2 + 8*c*d +
d^2)*cos(f*x + e)^2 - 9*c^2 - 8*c*d - d^2 + 2*(4*c^2 + 3*c*d)*sin(f*x + e))*sqrt(a*c*g/(c + d))*sqrt(a*sin(f*x
+ e) + a)*sqrt(g*sin(f*x + e))/(a*c^2*g*cos(f*x + e)*sin(f*x + e) + (2*a*c^2 + a*c*d)*g*cos(f*x + e)^3 - (2*a
*c^2 + a*c*d)*g*cos(f*x + e))) - sqrt(a*g)*arctan(1/4*sqrt(a*g)*(8*cos(f*x + e)^2 + 8*sin(f*x + e) - 9)*sqrt(a
*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))/(2*a*g*cos(f*x + e)^3 + a*g*cos(f*x + e)*sin(f*x + e) - 2*a*g*cos(f*x
+ e))))/(d*f)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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