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

Optimal. Leaf size=141 \[ -\frac{2 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right )}{\sqrt{a} f}-\frac{\sqrt{2} \sqrt{c-d} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right )}{\sqrt{a} f} \]

[Out]

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

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Rubi [A]  time = 0.296327, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {2777, 2775, 205, 2782, 208} \[ -\frac{2 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right )}{\sqrt{a} f}-\frac{\sqrt{2} \sqrt{c-d} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right )}{\sqrt{a} f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2777

Int[Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[
d/b, Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] + Dist[(b*c - a*d)/b, Int[1/(Sqrt[a + b*Sin
[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), 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 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 2782

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> D
ist[(-2*a)/f, Subst[Int[1/(2*b^2 - (a*c - 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 208

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

Rubi steps

\begin{align*} \int \frac{\sqrt{c+d \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx &=(c-d) \int \frac{1}{\sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)}} \, dx+\frac{d \int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{c+d \sin (e+f x)}} \, dx}{a}\\ &=-\frac{(2 a (c-d)) \operatorname{Subst}\left (\int \frac{1}{2 a^2-(a c-a d) x^2} \, dx,x,\frac{a \cos (e+f x)}{\sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}\right )}{f}-\frac{(2 d) \operatorname{Subst}\left (\int \frac{1}{a+d x^2} \, dx,x,\frac{a \cos (e+f x)}{\sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}\right )}{f}\\ &=-\frac{2 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}\right )}{\sqrt{a} f}-\frac{\sqrt{2} \sqrt{c-d} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{c-d} \cos (e+f x)}{\sqrt{2} \sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}\right )}{\sqrt{a} f}\\ \end{align*}

Mathematica [C]  time = 15.016, size = 1251, normalized size = 8.87 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((Sqrt[2]*Sqrt[c - d]*Log[1 + Tan[(e + f*x)/2]] - Sqrt[2]*Sqrt[c - d]*Log[c - d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[
e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]] - I*Sqrt[d]*(Log[(2*(c - I*d + (1 - I)*S
qrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + ((-I)*c + d)*Tan[(e + f*x)/2]))/(d^(3/
2)*(I + Tan[(e + f*x)/2]))] - Log[(2*(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c +
d*Sin[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2]))/(d^(3/2)*(-I + Tan[(e + f*x)/2]))]))*Sqrt[c + d*Sin[e + f*x]])
/(f*Sqrt[a*(1 + Sin[e + f*x])]*((Sqrt[c - d]*Sec[(e + f*x)/2]^2)/(Sqrt[2]*(1 + Tan[(e + f*x)/2])) - (Sqrt[2]*S
qrt[c - d]*(((-c + d)*Sec[(e + f*x)/2]^2)/2 + (Sqrt[c - d]*d*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/Sqrt[
c + d*Sin[e + f*x]] + Sqrt[c - d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]]))/(c -
d + 2*Sqrt[c - d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (-c + d)*Tan[(e + f*x)/2]) - I*Sqr
t[d]*((d^(3/2)*(I + Tan[(e + f*x)/2])*((2*((((-I)*c + d)*Sec[(e + f*x)/2]^2)/2 + ((1 - I)*d^(3/2)*Cos[e + f*x]
*Sqrt[(1 + Cos[e + f*x])^(-1)])/(Sqrt[2]*Sqrt[c + d*Sin[e + f*x]]) + ((1 - I)*Sqrt[d]*((1 + Cos[e + f*x])^(-1)
)^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/Sqrt[2]))/(d^(3/2)*(I + Tan[(e + f*x)/2])) - (Sec[(e + f*x)/2]^
2*(c - I*d + (1 - I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + ((-I)*c + d)*Tan
[(e + f*x)/2]))/(d^(3/2)*(I + Tan[(e + f*x)/2])^2)))/(2*(c - I*d + (1 - I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f
*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + ((-I)*c + d)*Tan[(e + f*x)/2])) - (d^(3/2)*(-I + Tan[(e + f*x)/2])*((2*(
((I*c + d)*Sec[(e + f*x)/2]^2)/2 + ((1 + I)*d^(3/2)*Cos[e + f*x]*Sqrt[(1 + Cos[e + f*x])^(-1)])/(Sqrt[2]*Sqrt[
c + d*Sin[e + f*x]]) + ((1 + I)*Sqrt[d]*((1 + Cos[e + f*x])^(-1))^(3/2)*Sin[e + f*x]*Sqrt[c + d*Sin[e + f*x]])
/Sqrt[2]))/(d^(3/2)*(-I + Tan[(e + f*x)/2])) - (Sec[(e + f*x)/2]^2*(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1
+ Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)*Tan[(e + f*x)/2]))/(d^(3/2)*(-I + Tan[(e + f*x)/2])
^2)))/(2*(c + I*d + (1 + I)*Sqrt[2]*Sqrt[d]*Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[c + d*Sin[e + f*x]] + (I*c + d)
*Tan[(e + f*x)/2])))))

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Maple [B]  time = 0.273, size = 3359, normalized size = 23.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/f/d/(c^2-2*c*d+d^2)/(-(d^2/c^2)^(1/2)*c)^(1/2)*(-2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*(-(d^2/c^
2)^(1/2)*c)^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e
)-d*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*(2*c-2*d)^(1/2)*c^2*d*sin(f*x+e)+2*2^
(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*s
in(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*
x+e)+sin(f*x+e)))*(2*c-2*d)^(1/2)*c*d^2*sin(f*x+e)-2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*(-(d^2/c^2)
^(1/2)*c)^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-
d*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*(2*c-2*d)^(1/2)*d^3*sin(f*x+e)-cos(f*x+
e)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+
c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*(-(d^2/c^2)^(1/2)*c)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))
/(cos(f*x+e)+1))^(1/2)*(2*c-2*d)^(1/2)*c^2*d+2*cos(f*x+e)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos
(f*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)
))*(-(d^2/c^2)^(1/2)*c)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*(2*c-2*d)^(1/2)*c*d^2-cos(f*x+e)
*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*
cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*(-(d^2/c^2)^(1/2)*c)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(
cos(f*x+e)+1))^(1/2)*(2*c-2*d)^(1/2)*d^3-2*(d^2/c^2)^(1/2)*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d
))^(1/2)*arctan(1/(-(d^2/c^2)^(1/2)*c)^(1/2)*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2))*c^3*
d*cos(f*x+e)+4*(d^2/c^2)^(1/2)*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*arctan(1/(-(d^2/c^2
)^(1/2)*c)^(1/2)*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2))*c^2*d^2*cos(f*x+e)-2*(d^2/c^2)^(
1/2)*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*arctan(1/(-(d^2/c^2)^(1/2)*c)^(1/2)*(d*(c+d*s
in(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2))*c*d^3*cos(f*x+e)-ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f
*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)
+sin(f*x+e)))*(-(d^2/c^2)^(1/2)*c)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*(2*c-2*d)^(1/2)*c^2*d
+2*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+
c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*(-(d^2/c^2)^(1/2)*c)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))
/(cos(f*x+e)+1))^(1/2)*(2*c-2*d)^(1/2)*c*d^2-ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(
1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*(-(d^2/c^2
)^(1/2)*c)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*(2*c-2*d)^(1/2)*d^3+2*(d^2/c^2)^(1/2)*(-(d^2/
c^2)^(1/2)*c)^(1/2)*(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*d^2*c^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1
/2)*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*arctan(((d^2/c^2)^(1/2)*c^2-d^2)*c*((d^2/c^2)^
(1/2)-1)/(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*d^2*c^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2)*((d^2/c
^2)^(1/2)*c*sin(f*x+e)+d*cos(f*x+e)-d)/(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/((d^2/c^2)^
(1/2)*c*sin(f*x+e)-d*cos(f*x+e)+d))*c*sin(f*x+e)+2*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)
*arctan(1/(-(d^2/c^2)^(1/2)*c)^(1/2)*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2))*c^2*d^2*cos(
f*x+e)-4*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*arctan(1/(-(d^2/c^2)^(1/2)*c)^(1/2)*(d*(c
+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2))*c*d^3*cos(f*x+e)+2*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2
)*c*sin(f*x+e)+d))^(1/2)*arctan(1/(-(d^2/c^2)^(1/2)*c)^(1/2)*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)
+d))^(1/2))*d^4*cos(f*x+e)+2*(-(d^2/c^2)^(1/2)*c)^(1/2)*(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*d^2*c^2+d^4*(d
^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2)*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*arctan(((d
^2/c^2)^(1/2)*c^2-d^2)*c*((d^2/c^2)^(1/2)-1)/(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*d^2*c^2+d^4*(d^2/c^2)^(1/
2)-4*c^2*d^2-4*d^4)*c)^(1/2)*((d^2/c^2)^(1/2)*c*sin(f*x+e)+d*cos(f*x+e)-d)/(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2
)*c*sin(f*x+e)+d))^(1/2)/((d^2/c^2)^(1/2)*c*sin(f*x+e)-d*cos(f*x+e)+d))*d*sin(f*x+e)+2*(d^2/c^2)^(1/2)*(-(d^2/
c^2)^(1/2)*c)^(1/2)*(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*d^2*c^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1
/2)*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*arctan(((d^2/c^2)^(1/2)*c^2-d^2)*c*((d^2/c^2)^
(1/2)-1)/(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*d^2*c^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2)*((d^2/c
^2)^(1/2)*c*sin(f*x+e)+d*cos(f*x+e)-d)/(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/((d^2/c^2)^
(1/2)*c*sin(f*x+e)-d*cos(f*x+e)+d))*c+2*(-(d^2/c^2)^(1/2)*c)^(1/2)*(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*d^2
*c^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2)*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)
*arctan(((d^2/c^2)^(1/2)*c^2-d^2)*c*((d^2/c^2)^(1/2)-1)/(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*d^2*c^2+d^4*(d
^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2)*((d^2/c^2)^(1/2)*c*sin(f*x+e)+d*cos(f*x+e)-d)/(d*(c+d*sin(f*x+e))/((d^
2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/((d^2/c^2)^(1/2)*c*sin(f*x+e)-d*cos(f*x+e)+d))*d)/(c+d*sin(f*x+e))^(1/2)/(
a*(1+sin(f*x+e)))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d \sin \left (f x + e\right ) + c}}{\sqrt{a \sin \left (f x + e\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 4.08283, size = 4674, normalized size = 33.15 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(2*sqrt(2)*sqrt((c - d)/a)*log((2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt((c - d)/
a)*(cos(f*x + e) - sin(f*x + e) + 1) - (c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f*x
+ e) - 2*c - 2*d)*sin(f*x + e) - 2*c - 2*d)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) -
2)) + sqrt(-d/a)*log((128*d^4*cos(f*x + e)^5 + 128*(2*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 - 32*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e)^3 - 32*(c^3*d - 2*c^2*d^2 + 9*c*d^3 - 4*d^4
)*cos(f*x + e)^2 - 8*(16*d^3*cos(f*x + e)^4 + 24*(c*d^2 - d^3)*cos(f*x + e)^3 - c^3 + 17*c^2*d - 59*c*d^2 + 51
*d^3 - 2*(5*c^2*d - 26*c*d^2 + 33*d^3)*cos(f*x + e)^2 - (c^3 - 7*c^2*d + 31*c*d^2 - 25*d^3)*cos(f*x + e) + (16
*d^3*cos(f*x + e)^3 + c^3 - 17*c^2*d + 59*c*d^2 - 51*d^3 - 8*(3*c*d^2 - 5*d^3)*cos(f*x + e)^2 - 2*(5*c^2*d - 1
4*c*d^2 + 13*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-d/a) + (
c^4 - 28*c^3*d + 230*c^2*d^2 - 476*c*d^3 + 289*d^4)*cos(f*x + e) + (128*d^4*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6
*c^2*d^2 + 4*c*d^3 + d^4 - 256*(c*d^3 - d^4)*cos(f*x + e)^3 - 32*(5*c^2*d^2 - 6*c*d^3 + 5*d^4)*cos(f*x + e)^2
+ 32*(c^3*d - 7*c^2*d^2 + 15*c*d^3 - 9*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)))/f,
1/2*(sqrt(2)*sqrt((c - d)/a)*log((2*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt((c - d)/a)
*(cos(f*x + e) - sin(f*x + e) + 1) - (c - 3*d)*cos(f*x + e)^2 - (3*c - d)*cos(f*x + e) + ((c - 3*d)*cos(f*x +
e) - 2*c - 2*d)*sin(f*x + e) - 2*c - 2*d)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2
)) + sqrt(d/a)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin
(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(d/a)/(2*d^3*cos(f*x + e)^3 - (3*c*d^2 - d^3)*cos(f*x + e)*sin(f*x
+ e) - (c^2*d - c*d^2 + 2*d^3)*cos(f*x + e))))/f, -1/4*(4*sqrt(2)*sqrt(-(c - d)/a)*arctan(-sqrt(2)*sqrt(a*sin
(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-(c - d)/a)/((c - d)*cos(f*x + e))) - sqrt(-d/a)*log((128*d^4*cos
(f*x + e)^5 + 128*(2*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 - 32*(5*c^2*d^2 -
14*c*d^3 + 13*d^4)*cos(f*x + e)^3 - 32*(c^3*d - 2*c^2*d^2 + 9*c*d^3 - 4*d^4)*cos(f*x + e)^2 - 8*(16*d^3*cos(f
*x + e)^4 + 24*(c*d^2 - d^3)*cos(f*x + e)^3 - c^3 + 17*c^2*d - 59*c*d^2 + 51*d^3 - 2*(5*c^2*d - 26*c*d^2 + 33*
d^3)*cos(f*x + e)^2 - (c^3 - 7*c^2*d + 31*c*d^2 - 25*d^3)*cos(f*x + e) + (16*d^3*cos(f*x + e)^3 + c^3 - 17*c^2
*d + 59*c*d^2 - 51*d^3 - 8*(3*c*d^2 - 5*d^3)*cos(f*x + e)^2 - 2*(5*c^2*d - 14*c*d^2 + 13*d^3)*cos(f*x + e))*si
n(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-d/a) + (c^4 - 28*c^3*d + 230*c^2*d^2 - 476
*c*d^3 + 289*d^4)*cos(f*x + e) + (128*d^4*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 256*(c*
d^3 - d^4)*cos(f*x + e)^3 - 32*(5*c^2*d^2 - 6*c*d^3 + 5*d^4)*cos(f*x + e)^2 + 32*(c^3*d - 7*c^2*d^2 + 15*c*d^3
- 9*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)))/f, -1/2*(2*sqrt(2)*sqrt(-(c - d)/a)*
arctan(-sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-(c - d)/a)/((c - d)*cos(f*x + e))) - s
qrt(d/a)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x +
e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(d/a)/(2*d^3*cos(f*x + e)^3 - (3*c*d^2 - d^3)*cos(f*x + e)*sin(f*x + e)
- (c^2*d - c*d^2 + 2*d^3)*cos(f*x + e))))/f]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c + d \sin{\left (e + f x \right )}}}{\sqrt{a \left (\sin{\left (e + f x \right )} + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d \sin \left (f x + e\right ) + c}}{\sqrt{a \sin \left (f x + e\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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