3.26 \(\int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{g \sin (e+f x)} (c+d \sin (e+f x))} \, dx\)
Optimal. Leaf size=83 \[ -\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{c} \sqrt{g} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right )}{\sqrt{c} f \sqrt{g} \sqrt{c+d}} \]
[Out]
(-2*Sqrt[a]*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]])])/(Sqrt[c]*Sqrt[c + d]*f*Sqrt[g])
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Rubi [A] time = 0.228218, antiderivative size = 83, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used =
{2930, 205} \[ -\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{c} \sqrt{g} \cos (e+f x)}{\sqrt{c+d} \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right )}{\sqrt{c} f \sqrt{g} \sqrt{c+d}} \]
Antiderivative was successfully verified.
[In]
Int[Sqrt[a + a*Sin[e + f*x]]/(Sqrt[g*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]
[Out]
(-2*Sqrt[a]*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]])])/(Sqrt[c]*Sqrt[c + d]*f*Sqrt[g])
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]
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]
Rubi steps
\begin{align*} \int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{g \sin (e+f x)} (c+d \sin (e+f x))} \, dx &=-\frac{(2 a) \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 )}{f}\\ &=-\frac{2 \sqrt{a} \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 )}{\sqrt{c} \sqrt{c+d} f \sqrt{g}}\\ \end{align*}
Mathematica [C] time = 54.3944, size = 436, normalized size = 5.25 \[ \frac{\left (\frac{1}{4}+\frac{i}{4}\right ) g \sqrt{a (\sin (e+f x)+1)} \left (\cos \left (\frac{3}{2} (e+f x)\right )-i \sin \left (\frac{3}{2} (e+f x)\right )\right ) (i \sin (2 (e+f x))+\cos (2 (e+f x))-1)^{3/2} \left (\sqrt{c+i \sqrt{d^2-c^2}} \left (\sqrt{d^2-c^2}+i c-i d\right ) \tan ^{-1}\left (\frac{d-\left (\sqrt{d^2-c^2}-i c\right ) (\cos (e+f x)+i \sin (e+f x))}{\sqrt{2} \sqrt{c} \sqrt{c+i \sqrt{d^2-c^2}} \sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))-1}}\right )+\sqrt{c-i \sqrt{d^2-c^2}} \left (\sqrt{d^2-c^2}-i c+i d\right ) \tan ^{-1}\left (\frac{d+\left (\sqrt{d^2-c^2}+i c\right ) (\cos (e+f x)+i \sin (e+f x))}{\sqrt{2} \sqrt{c} \sqrt{c-i \sqrt{d^2-c^2}} \sqrt{i \sin (2 (e+f x))+\cos (2 (e+f x))-1}}\right )\right )}{\sqrt{2} \sqrt{c} d f \sqrt{d^2-c^2} (g \sin (e+f x))^{3/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Sqrt[a + a*Sin[e + f*x]]/(Sqrt[g*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]
[Out]
((1/4 + I/4)*g*(Sqrt[c + I*Sqrt[-c^2 + d^2]]*(I*c - I*d + Sqrt[-c^2 + d^2])*ArcTan[(d - ((-I)*c + Sqrt[-c^2 +
d^2])*(Cos[e + f*x] + I*Sin[e + f*x]))/(Sqrt[2]*Sqrt[c]*Sqrt[c + I*Sqrt[-c^2 + d^2]]*Sqrt[-1 + Cos[2*(e + f*x)
] + I*Sin[2*(e + f*x)]])] + Sqrt[c - I*Sqrt[-c^2 + d^2]]*((-I)*c + I*d + Sqrt[-c^2 + d^2])*ArcTan[(d + (I*c +
Sqrt[-c^2 + d^2])*(Cos[e + f*x] + I*Sin[e + f*x]))/(Sqrt[2]*Sqrt[c]*Sqrt[c - I*Sqrt[-c^2 + d^2]]*Sqrt[-1 + Cos
[2*(e + f*x)] + I*Sin[2*(e + f*x)]])])*Sqrt[a*(1 + Sin[e + f*x])]*(Cos[(3*(e + f*x))/2] - I*Sin[(3*(e + f*x))/
2])*(-1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)])^(3/2))/(Sqrt[2]*Sqrt[c]*d*Sqrt[-c^2 + d^2]*f*(Cos[(e + f*x)/2
] + Sin[(e + f*x)/2])*(g*Sin[e + f*x])^(3/2))
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Maple [B] time = 0.316, size = 526, normalized size = 6.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))/(g*sin(f*x+e))^(1/2),x)
[Out]
2/f/(-(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)*(-(-1+cos(f*x+
e))/sin(f*x+e))^(1/2)*(a*(1+sin(f*x+e)))^(1/2)*sin(f*x+e)*(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)*(-(c-d)*(c+d))^(1/2)-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)*(-(c-d)*(c+d))^
(1/2)+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)*c-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)*d+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)*c-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)*d)/(g*sin(f*x+e))^(1/2)/(-1+cos(f*x+e)-sin(f*x+e))
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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}}{{\left (d \sin \left (f x + e\right ) + c\right )} \sqrt{g \sin \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))/(g*sin(f*x+e))^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(a*sin(f*x + e) + a)/((d*sin(f*x + e) + c)*sqrt(g*sin(f*x + e))), x)
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Fricas [B] time = 8.67625, size = 3043, normalized size = 36.66 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))/(g*sin(f*x+e))^(1/2),x, algorithm="fricas")
[Out]
[1/4*sqrt(-a/((c^2 + c*d)*g))*log(((128*a*c^4 + 256*a*c^3*d + 160*a*c^2*d^2 + 32*a*c*d^3 + a*d^4)*cos(f*x + e)
^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - (128*a*c^4 + 192*a*c^3*d + 64*a*c^2*d^2 - 4*a*c*d^3
- a*d^4)*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)*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)*cos(f*x + e)^2 - 8*(51*c^5 + 110*c^4*d + 76*c^3*d^
2 + 18*c^2*d^3 + c*d^4 + (16*c^5 + 40*c^4*d + 34*c^3*d^2 + 11*c^2*d^3 + c*d^4)*cos(f*x + e)^4 - (24*c^5 + 52*c
^4*d + 35*c^3*d^2 + 7*c^2*d^3)*cos(f*x + e)^3 - (66*c^5 + 149*c^4*d + 110*c^3*d^2 + 29*c^2*d^3 + 2*c*d^4)*cos(
f*x + e)^2 + (25*c^5 + 53*c^4*d + 35*c^3*d^2 + 7*c^2*d^3)*cos(f*x + e) - (51*c^5 + 110*c^4*d + 76*c^3*d^2 + 18
*c^2*d^3 + c*d^4 - (16*c^5 + 40*c^4*d + 34*c^3*d^2 + 11*c^2*d^3 + c*d^4)*cos(f*x + e)^3 - (40*c^5 + 92*c^4*d +
69*c^3*d^2 + 18*c^2*d^3 + c*d^4)*cos(f*x + e)^2 + (26*c^5 + 57*c^4*d + 41*c^3*d^2 + 11*c^2*d^3 + c*d^4)*cos(f
*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))*sqrt(-a/((c^2 + c*d)*g)) + (289*a*c^4 + 4
80*a*c^3*d + 230*a*c^2*d^2 + 32*a*c*d^3 + a*d^4)*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 + (128*a*c^4 + 256*a*c^3*d + 160*a*c^2*d^2 + 32*a*c*d^3 + a*d^4)*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)*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)*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)*cos(f*x + e))*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)))/f, 1/2*sqrt(a/((c^2 + c*d)*g))*
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*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))*sqrt(a/((c^2 + c*d)*g))/((2*a*c + a*d)*cos(f*x + e)^3 + a*c*cos(f*x +
e)*sin(f*x + e) - (2*a*c + a*d)*cos(f*x + e)))/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 )}} \left (c + d \sin{\left (e + f x \right )}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))**(1/2)/(c+d*sin(f*x+e))/(g*sin(f*x+e))**(1/2),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}}{{\left (d \sin \left (f x + e\right ) + c\right )} \sqrt{g \sin \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))/(g*sin(f*x+e))^(1/2),x, algorithm="giac")
[Out]
integrate(sqrt(a*sin(f*x + e) + a)/((d*sin(f*x + e) + c)*sqrt(g*sin(f*x + e))), x)