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

Optimal. Leaf size=149 \[ \frac{2 \sqrt{a} \sqrt{c} \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{c} \sqrt{g} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right )}{d f \sqrt{c+d}}-\frac{2 \sqrt{a} \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{g} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right )}{d f} \]

[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)

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Rubi [A]  time = 0.508214, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2928, 2775, 205, 2930} \[ \frac{2 \sqrt{a} \sqrt{c} \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{c} \sqrt{g} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right )}{d f \sqrt{c+d}}-\frac{2 \sqrt{a} \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{g} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right )}{d f} \]

Antiderivative was successfully verified.

[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 2928

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 2775

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 205

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

Rule 2930

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*
Sin[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*} \int \frac{\sqrt{g \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx &=\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) \operatorname{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) \operatorname{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} \tan ^{-1}\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} \tan ^{-1}\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]  time = 55.3841, size = 661, normalized size = 4.44 \[ \frac{\left (\frac{1}{2}+\frac{i}{2}\right ) e^{-\frac{5}{2} i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )^{5/2} \sqrt{a (\sin (e+f x)+1)} \sqrt{g \sin (e+f x)} \left (\left (\frac{c-d}{\sqrt{d^2-c^2}}+i\right ) \sqrt{-1+e^{2 i (e+f x)}}+\left (\frac{d-c}{\sqrt{d^2-c^2}}+i\right ) \sqrt{-1+e^{2 i (e+f x)}}+\frac{\left (\frac{c-d}{\sqrt{d^2-c^2}}+i\right ) \left (\sqrt{2} \sqrt{c} \sqrt{c-i \sqrt{d^2-c^2}} \tan ^{-1}\left (\frac{d+\left (\sqrt{d^2-c^2}+i c\right ) e^{i (e+f x)}}{\sqrt{2} \sqrt{c} \sqrt{c-i \sqrt{d^2-c^2}} \sqrt{-1+e^{2 i (e+f x)}}}\right )-\left (\sqrt{d^2-c^2}+i c\right ) \tanh ^{-1}\left (\frac{e^{i (e+f x)}}{\sqrt{-1+e^{2 i (e+f x)}}}\right )\right )}{d}+\frac{\left (\frac{d-c}{\sqrt{d^2-c^2}}+i\right ) \left (\sqrt{2} \sqrt{c} \sqrt{c+i \sqrt{d^2-c^2}} \tan ^{-1}\left (\frac{d-\left (\sqrt{d^2-c^2}-i c\right ) e^{i (e+f x)}}{\sqrt{2} \sqrt{c} \sqrt{c+i \sqrt{d^2-c^2}} \sqrt{-1+e^{2 i (e+f x)}}}\right )+\left (\sqrt{d^2-c^2}-i c\right ) \tanh ^{-1}\left (\frac{e^{i (e+f x)}}{\sqrt{-1+e^{2 i (e+f x)}}}\right )\right )}{d}-2 i \left (\sqrt{-1+e^{2 i (e+f x)}}-\tan ^{-1}\left (\sqrt{-1+e^{2 i (e+f x)}}\right )\right )\right )}{\sqrt{2} d f \left (-i e^{-i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )\right )^{5/2} \sqrt{\sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]

Warning: Unable to verify antiderivative.

[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)*((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)*(Sqrt[-1 + E^((2*I)*(e + f*x))] - ArcT
an[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]])

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Maple [B]  time = 0.522, size = 1025, normalized size = 6.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 15.711, size = 8049, normalized size = 54.02 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

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