\(\int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {3+3 \sin (e+f x)}} \, dx\) [590]

Optimal result

Integrand size = 29, antiderivative size = 135 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {3+3 \sin (e+f x)}} \, dx=-\frac {2 \sqrt {d} \arctan \left (\frac {\sqrt {3} \sqrt {d} \cos (e+f x)}{\sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {3} f}-\frac {\sqrt {\frac {2}{3}} \sqrt {c-d} \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {c-d} \cos (e+f x)}{\sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2856, 2854, 211, 2861, 214} \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {3+3 \sin (e+f x)}} \, dx=-\frac {2 \sqrt {d} \arctan \left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {a} f}-\frac {\sqrt {2} \sqrt {c-d} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {a} f} \]

[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 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 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[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 2856

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 2861

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]

Rubi steps \begin{align*} \text {integral}& = (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)) \text {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) \text {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} \arctan \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} \text {arctanh}\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] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 14.20 (sec) , antiderivative size = 1251, normalized size of antiderivative = 9.27 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {3+3 \sin (e+f x)}} \, dx=\frac {\left (\sqrt {2} \sqrt {c-d} \log \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )-\sqrt {2} \sqrt {c-d} \log \left (c-d+2 \sqrt {c-d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )-i \sqrt {d} \left (\log \left (\frac {2 \left (c-i d+(1-i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-i c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}{d^{3/2} \left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right )}\right )-\log \left (\frac {2 \left (c+i d+(1+i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(i c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}{d^{3/2} \left (-i+\tan \left (\frac {1}{2} (e+f x)\right )\right )}\right )\right )\right ) \sqrt {c+d \sin (e+f x)}}{f \sqrt {3+3 \sin (e+f x)} \left (\frac {\sqrt {c-d} \sec ^2\left (\frac {1}{2} (e+f x)\right )}{\sqrt {2} \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )}-\frac {\sqrt {2} \sqrt {c-d} \left (\frac {1}{2} (-c+d) \sec ^2\left (\frac {1}{2} (e+f x)\right )+\frac {\sqrt {c-d} d \cos (e+f x) \sqrt {\frac {1}{1+\cos (e+f x)}}}{\sqrt {c+d \sin (e+f x)}}+\sqrt {c-d} \left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} \sin (e+f x) \sqrt {c+d \sin (e+f x)}\right )}{c-d+2 \sqrt {c-d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )}-i \sqrt {d} \left (\frac {d^{3/2} \left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right ) \left (\frac {2 \left (\frac {1}{2} (-i c+d) \sec ^2\left (\frac {1}{2} (e+f x)\right )+\frac {(1-i) d^{3/2} \cos (e+f x) \sqrt {\frac {1}{1+\cos (e+f x)}}}{\sqrt {2} \sqrt {c+d \sin (e+f x)}}+\frac {(1-i) \sqrt {d} \left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} \sin (e+f x) \sqrt {c+d \sin (e+f x)}}{\sqrt {2}}\right )}{d^{3/2} \left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right )}-\frac {\sec ^2\left (\frac {1}{2} (e+f x)\right ) \left (c-i d+(1-i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-i c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}{d^{3/2} \left (i+\tan \left (\frac {1}{2} (e+f x)\right )\right )^2}\right )}{2 \left (c-i d+(1-i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-i c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}-\frac {d^{3/2} \left (-i+\tan \left (\frac {1}{2} (e+f x)\right )\right ) \left (\frac {2 \left (\frac {1}{2} (i c+d) \sec ^2\left (\frac {1}{2} (e+f x)\right )+\frac {(1+i) d^{3/2} \cos (e+f x) \sqrt {\frac {1}{1+\cos (e+f x)}}}{\sqrt {2} \sqrt {c+d \sin (e+f x)}}+\frac {(1+i) \sqrt {d} \left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} \sin (e+f x) \sqrt {c+d \sin (e+f x)}}{\sqrt {2}}\right )}{d^{3/2} \left (-i+\tan \left (\frac {1}{2} (e+f x)\right )\right )}-\frac {\sec ^2\left (\frac {1}{2} (e+f x)\right ) \left (c+i d+(1+i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(i c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}{d^{3/2} \left (-i+\tan \left (\frac {1}{2} (e+f x)\right )\right )^2}\right )}{2 \left (c+i d+(1+i) \sqrt {2} \sqrt {d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(i c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}\right )\right )} \]

[In]

Integrate[Sqrt[c + d*Sin[e + f*x]]/Sqrt[3 + 3*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[3 + 3*Sin[e + f*x]]*((Sqrt[c - d]*Sec[(e + f*x)/2]^2)/(Sqrt[2]*(1 + Tan[(e + f*x)/2])) - (Sqrt[2]*Sqr
t[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*Sqrt[
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]*S
qrt[(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]])/S
qrt[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)*T
an[(e + f*x)/2])))))

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(3360\) vs. \(2(114)=228\).

Time = 1.48 (sec) , antiderivative size = 3361, normalized size of antiderivative = 24.90

\[\text {output too large to display}\]

[In]

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

[Out]

-1/2/f*(c^2*(2*c-2*d)^(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)/(-cos(f*x+e)+1+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)*d*sin(f*x+e)-2*c*(2*c-2*d)^(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)/
(-cos(f*x+e)+1+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)*d^2*sin
(f*x+e)+(2*c-2*d)^(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)/(-cos(f*x+e)+1+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)*d^3*sin(f*x+e)+((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*(2*c-2*d)^
(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*2^(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)/(-cos(f*x+e)+1+sin(f*x+e)))*c^2*cos(f*x+e
)*d-2*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*(2*c-2*d)^(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*2^(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*c
os(f*x+e)-c+d)/(-cos(f*x+e)+1+sin(f*x+e)))*c*cos(f*x+e)*d^2+((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*(2*c-2*d)^
(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*2^(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)/(-cos(f*x+e)+1+sin(f*x+e)))*cos(f*x+e)*d^
3+((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*(2*c-2*d)^(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*2^(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)/(-cos(f*x+e)+1+sin(f*x+e)))*c^2*d-2*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*(2*c-2*d)^(1/2)*(-(d^2/
c^2)^(1/2)*c)^(1/2)*2^(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)/(-cos(f*x+e)+1+sin(f*x+e)))*c*d^2+((c+d*sin(f*x+e))/(c
os(f*x+e)+1))^(1/2)*(2*c-2*d)^(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*2^(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)/(-cos(f*x+e
)+1+sin(f*x+e)))*d^3+2*c^3*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*arctan(((c+d*sin(f*x+e)
)*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*(d^2/c^2)^(1/2)*d*cos(f*x+e)-4*c^2*((c
+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*arctan(((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x
+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*(d^2/c^2)^(1/2)*d^2*cos(f*x+e)+2*c*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1
/2)*c*sin(f*x+e)+d))^(1/2)*arctan(((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2
)*c)^(1/2))*(d^2/c^2)^(1/2)*d^3*cos(f*x+e)+2*c*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*arc
tan(((d^2/c^2)^(1/2)*c*sin(f*x+e)+d*cos(f*x+e)-d)/((c+d*sin(f*x+e))*d/((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^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)*c^2*d^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)*(-(d^2/c^2)^(1
/2)*c)^(1/2)*(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*c^2*d^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2)*sin
(f*x+e)-2*c^2*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*arctan(((c+d*sin(f*x+e))*d/((d^2/c^2
)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*d^2*cos(f*x+e)+4*c*((c+d*sin(f*x+e))*d/((d^2/c^2)^(
1/2)*c*sin(f*x+e)+d))^(1/2)*arctan(((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/
2)*c)^(1/2))*d^3*cos(f*x+e)-2*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*arctan(((c+d*sin(f*x
+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*d^4*cos(f*x+e)+2*((c+d*sin(f*x+e))*
d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*arctan(((d^2/c^2)^(1/2)*c*sin(f*x+e)+d*cos(f*x+e)-d)/((c+d*sin(f*x+e
))*d/((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^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)*c^2*d^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)^(1/2)*(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*c^2*d^2+d^4*(d^2/c^2)^(1/2)-
4*c^2*d^2-4*d^4)*c)^(1/2)*d*sin(f*x+e)+2*c*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*arctan(
((d^2/c^2)^(1/2)*c*sin(f*x+e)+d*cos(f*x+e)-d)/((c+d*sin(f*x+e))*d/((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^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)*c^2*d^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)*(-(d^2/c^2)^(1/2)*
c)^(1/2)*(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*c^2*d^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2)+2*((c+d
*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*arctan(((d^2/c^2)^(1/2)*c*sin(f*x+e)+d*cos(f*x+e)-d)/((
c+d*sin(f*x+e))*d/((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^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)*c^2*d^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)^(1/2)*(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*c^2*d^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))^(1/2)/(a*(sin(f*x+e)+1))^(1/2)/(-(d^2/c^2)^(1/2)*c)
^(1/2)/d/(c^2-2*c*d+d^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (114) = 228\).

Time = 0.62 (sec) , antiderivative size = 1944, normalized size of antiderivative = 14.40 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {3+3 \sin (e+f x)}} \, dx=\text {Too large to display} \]

[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]

Sympy [F]

\[ \int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {3+3 \sin (e+f x)}} \, dx=\int \frac {\sqrt {c + d \sin {\left (e + f x \right )}}}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \]

[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)

Maxima [F]

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

[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)

Giac [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {3+3 \sin (e+f x)}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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