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

Optimal. Leaf size=123 \[ -\frac{2 \sqrt{a} \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right )}{f}-\frac{2 \sqrt{a} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{c} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right )}{f} \]

[Out]

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

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Rubi [A]  time = 0.459175, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2949, 2775, 205, 2943, 206} \[ -\frac{2 \sqrt{a} \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right )}{f}-\frac{2 \sqrt{a} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{c} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right )}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2949

Int[(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]])/sin[(e_.) + (f_.)*
(x_)], x_Symbol] :> Dist[d, Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] + Dist[c, Int[Sqrt[a
+ b*Sin[e + f*x]]/(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] && (NeQ[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 2943

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/(sin[(e_.) + (f_.)*(x_)]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x
_)]]), x_Symbol] :> Dist[(-2*a)/f, Subst[Int[1/(1 - a*c*x^2), x], x, Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sq
rt[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[
b*c + a*d, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [C]  time = 3.10763, size = 567, normalized size = 4.61 \[ -\frac{\sqrt{a (\sin (e+f x)+1)} \left (\cos \left (\frac{1}{2} (e+f x)\right )+i \sin \left (\frac{1}{2} (e+f x)\right )\right ) \sqrt{c+d \sin (e+f x)} \left (\sqrt{c} \log \left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) e^{-\frac{i e}{2}} f \left (2 i \sqrt{c} \sqrt{2 c e^{i (e+f x)}-i d \left (-1+e^{2 i (e+f x)}\right )}-\sqrt{2} c \left (-1+e^{i (e+f x)}\right )-i \sqrt{2} d \left (1+e^{i (e+f x)}\right )\right )}{c^{3/2} \left (1+e^{i (e+f x)}\right )}\right )+\sqrt{c} \log \left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) e^{-\frac{i e}{2}} f \left (2 \sqrt{c} \sqrt{2 c e^{i (e+f x)}-i d \left (-1+e^{2 i (e+f x)}\right )}+\sqrt{2} c \left (1+e^{i (e+f x)}\right )-i \sqrt{2} d \left (-1+e^{i (e+f x)}\right )\right )}{c^{3/2} \left (-1+e^{i (e+f x)}\right )}\right )-i \sqrt{d} \left (\log \left (\frac{2 f e^{-\frac{1}{2} i (e+2 f x)} \left (i \sqrt{d} \sqrt{2 c e^{i (e+f x)}-i d \left (-1+e^{2 i (e+f x)}\right )}+\sqrt [4]{-1} c e^{i (e+f x)}+(-1)^{3/4} d\right )}{d^{3/2}}\right )-\log \left (\frac{(1+i) \sqrt{2} \left ((1-i) \sqrt{d} \sqrt{(\cos (e+f x)+i \sin (e+f x)) (c+d \sin (e+f x))}+c+d \sin (e+f x)-i d \cos (e+f x)\right )}{\sqrt{d}}\right )\right )\right )}{f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \sqrt{(\cos (e+f x)+i \sin (e+f x)) (c+d \sin (e+f x))}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(((Sqrt[c]*Log[((1/2 + I/2)*(-(Sqrt[2]*c*(-1 + E^(I*(e + f*x)))) - I*Sqrt[2]*d*(1 + E^(I*(e + f*x))) + (2*I)*
Sqrt[c]*Sqrt[2*c*E^(I*(e + f*x)) - I*d*(-1 + E^((2*I)*(e + f*x)))])*f)/(c^(3/2)*E^((I/2)*e)*(1 + E^(I*(e + f*x
))))] + Sqrt[c]*Log[((1/2 + I/2)*((-I)*Sqrt[2]*d*(-1 + E^(I*(e + f*x))) + Sqrt[2]*c*(1 + E^(I*(e + f*x))) + 2*
Sqrt[c]*Sqrt[2*c*E^(I*(e + f*x)) - I*d*(-1 + E^((2*I)*(e + f*x)))])*f)/(c^(3/2)*E^((I/2)*e)*(-1 + E^(I*(e + f*
x))))] - I*Sqrt[d]*(Log[(2*((-1)^(3/4)*d + (-1)^(1/4)*c*E^(I*(e + f*x)) + I*Sqrt[d]*Sqrt[2*c*E^(I*(e + f*x)) -
I*d*(-1 + E^((2*I)*(e + f*x)))])*f)/(d^(3/2)*E^((I/2)*(e + 2*f*x)))] - Log[((1 + I)*Sqrt[2]*(c - I*d*Cos[e +
f*x] + d*Sin[e + f*x] + (1 - I)*Sqrt[d]*Sqrt[(Cos[e + f*x] + I*Sin[e + f*x])*(c + d*Sin[e + f*x])]))/Sqrt[d]])
)*(Cos[(e + f*x)/2] + I*Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c + d*Sin[e + f*x]])/(f*(Cos[(e + f*
x)/2] + Sin[(e + f*x)/2])*Sqrt[(Cos[e + f*x] + I*Sin[e + f*x])*(c + d*Sin[e + f*x])]))

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Maple [B]  time = 0.46, size = 4568, normalized size = 37.1 \begin{align*} \text{output 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))^(1/2)/sin(f*x+e),x)

[Out]

1/f/d/(c^2-2*c*d+d^2)/(-(d^2/c^2)^(1/2)*c)^(1/2)/c^(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^(7/2)*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)+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^(5/2)*d^2*(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)+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*(((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)*(-(d^2/c^2)^(1/2)*c)^(1/2)*c^(1/2)*cos(f*x+e)-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*(((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)*(-(d^2/c^2)^(1/2)*c)^(1/2)*c^(1/2)*sin(
f*x+e)-ln(-1/c^(1/2)*(-c^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+c*cos(f*x+e)-d*sin(f
*x+e)-c)/sin(f*x+e))*c^3*d*(-(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)*sin(f*x+
e)+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^(3/2)*(((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)*(-(d^2/c^2)^(1/2)*c)^(1/2)*(d^2/c^2)^(1/2)*cos(f*x+e)-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^(3/2)*(((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)*(-(d^
2/c^2)^(1/2)*c)^(1/2)*(d^2/c^2)^(1/2)*sin(f*x+e)-ln(-1/c^(1/2)*(-c^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)
+1))^(1/2)*sin(f*x+e)+c*cos(f*x+e)-d*sin(f*x+e)-c)/sin(f*x+e))*c*d^3*(-(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)*sin(f*x+e)+ln(2*(-c^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*si
n(f*x+e)+d*cos(f*x+e)-c*sin(f*x+e)-d)/(-1+cos(f*x+e)))*c^3*d*(-(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)*sin(f*x+e)-2*ln(2*(-c^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+
e)+d*cos(f*x+e)-c*sin(f*x+e)-d)/(-1+cos(f*x+e)))*c^2*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)*sin(f*x+e)+ln(2*(-c^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+d*c
os(f*x+e)-c*sin(f*x+e)-d)/(-1+cos(f*x+e)))*c*d^3*(-(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)*sin(f*x+e)+2*ln(-1/c^(1/2)*(-c^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+
c*cos(f*x+e)-d*sin(f*x+e)-c)/sin(f*x+e))*c^2*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)*sin(f*x+e)-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^(
3/2)*(((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)*(-(d^2/c^2)^(1/2)*c)^(1/2)*(d^2/c^2)^(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^(7/2)*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)*sin(f*x+e)+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^(3/2)*d^3*(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)*cos(f*x+e)+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^(3/2)*d^3*(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)*sin(f*x+e)-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*(((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)*(-(d^2/c^2)^(1/2)*c)^(1/2)*c^(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^(7/2)*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)*cos(f*x+e)-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^(3/2)*d^3*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)+arct
an(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*(d*(c+d*sin(f
*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*c^(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^(5/2)*d^2*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1
/2)-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^(5/2)
*d^2*(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)*cos(f*x+e)-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^(5/2)*d^2*(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)*sin(f*x+e)-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^(5/2)*d^2*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*si
n(f*x+e)+d))^(1/2)*cos(f*x+e)-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^(5/2)*d^2*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*sin(f*x+e)+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^(3/2)*d^3*(d*(c+d*s
in(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*cos(f*x+e)+2*arctan(1/(-(d^2/c^2)^(1/2)*c)^(1/2)*(d*(c+d*si
n(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2))*c^(3/2)*d^3*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+
e)+d))^(1/2)*sin(f*x+e)-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*(d*(c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*c^(1/2)*cos(f*x+e)-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*(d*(c+d*sin(f*x+e))/((d^2
/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*c^(1/2)*sin(f*x+e)-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^(3/2)*d^3*(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))*(a*(1+sin(f*x+e)))^(1/2)*(c+d*sin(f*x+e))^(1/2)/(cos(f*x+e)^2*d+d*sin(f*x+e)*cos(f*x+e)+c*
cos(f*x+e)-c*sin(f*x+e)-d*sin(f*x+e)-c-d)

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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{d \sin \left (f x + e\right ) + c}}{\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))^(1/2)/sin(f*x+e),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 8.71986, size = 8421, normalized size = 68.46 \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))^(1/2)/sin(f*x+e),x, algorithm="fricas")

[Out]

[1/4*(sqrt(a*c)*log(((a*c^4 - 28*a*c^3*d + 70*a*c^2*d^2 - 28*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 - (31*a*c^4 - 196*a*c^3*d + 154*a*c^2*d^2 - 4*a*c*d^3 - a*d^4)*cos(f*x +
e)^4 - 2*(81*a*c^4 - 252*a*c^3*d + 150*a*c^2*d^2 - 28*a*c*d^3 + a*d^4)*cos(f*x + e)^3 + 2*(79*a*c^4 - 100*a*c^
3*d + 74*a*c^2*d^2 - 4*a*c*d^3 - a*d^4)*cos(f*x + e)^2 - 8*((c^3 - 7*c^2*d + 7*c*d^2 - d^3)*cos(f*x + e)^4 - 2
*(5*c^3 - 14*c^2*d + 5*c*d^2)*cos(f*x + e)^3 + 51*c^3 - 59*c^2*d + 17*c*d^2 - d^3 - 2*(18*c^3 - 33*c^2*d + 12*
c*d^2 - d^3)*cos(f*x + e)^2 + 2*(13*c^3 - 14*c^2*d + 5*c*d^2)*cos(f*x + e) + ((c^3 - 7*c^2*d + 7*c*d^2 - d^3)*
cos(f*x + e)^3 - 51*c^3 + 59*c^2*d - 17*c*d^2 + d^3 + (11*c^3 - 35*c^2*d + 17*c*d^2 - d^3)*cos(f*x + e)^2 - (2
5*c^3 - 31*c^2*d + 7*c*d^2 - d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*
x + e) + c) + (289*a*c^4 - 476*a*c^3*d + 230*a*c^2*d^2 - 28*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 + (a*c^4 - 28*a*c^3*d + 70*a*c^2*d^2 - 28*a*c*d^3 + a*d^4)*cos(f*x + e)^4 +
32*(a*c^4 - 7*a*c^3*d + 7*a*c^2*d^2 - a*c*d^3)*cos(f*x + e)^3 - 2*(65*a*c^4 - 140*a*c^3*d + 38*a*c^2*d^2 - 12
*a*c*d^3 + a*d^4)*cos(f*x + e)^2 - 32*(9*a*c^4 - 15*a*c^3*d + 7*a*c^2*d^2 - a*c*d^3)*cos(f*x + e))*sin(f*x + e
))/(cos(f*x + e)^5 + cos(f*x + e)^4 - 2*cos(f*x + e)^3 - 2*cos(f*x + e)^2 + (cos(f*x + e)^4 - 2*cos(f*x + e)^2
+ 1)*sin(f*x + e) + cos(f*x + e) + 1)) + sqrt(-a*d)*log((128*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*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)
*cos(f*x + e)^3 - 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*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))*sin(f*x + e))
*sqrt(-a*d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c) + (a*c^4 - 28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c*
d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^
4 - 256*(a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d
- 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)))/f, 1/4*
(2*sqrt(-a*c)*arctan(-1/4*((c^2 - 6*c*d + d^2)*cos(f*x + e)^2 - 9*c^2 + 6*c*d - d^2 + 8*(c^2 - c*d)*sin(f*x +
e))*sqrt(-a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/((a*c^2*d - a*c*d^2)*cos(f*x + e)^3 - (a*c^3
- 3*a*c^2*d)*cos(f*x + e)*sin(f*x + e) + (2*a*c^3 - a*c^2*d + a*c*d^2)*cos(f*x + e))) + sqrt(-a*d)*log((128*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*(2*a*c*d^3 - a*d^4)*cos(f*x + e
)^4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*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 + 5
1*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) + (1
6*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))*sin(f*x + e))*sqrt(-a*d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c) +
(a*c^4 - 28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^
4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a
*c*d^3 + 5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e
))/(cos(f*x + e) + sin(f*x + e) + 1)))/f, 1/4*(2*sqrt(a*d)*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*d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(2*a*d^3*cos(f*x
+ e)^3 - (3*a*c*d^2 - a*d^3)*cos(f*x + e)*sin(f*x + e) - (a*c^2*d - a*c*d^2 + 2*a*d^3)*cos(f*x + e))) + sqrt(
a*c)*log(((a*c^4 - 28*a*c^3*d + 70*a*c^2*d^2 - 28*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 - (31*a*c^4 - 196*a*c^3*d + 154*a*c^2*d^2 - 4*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 2*(8
1*a*c^4 - 252*a*c^3*d + 150*a*c^2*d^2 - 28*a*c*d^3 + a*d^4)*cos(f*x + e)^3 + 2*(79*a*c^4 - 100*a*c^3*d + 74*a*
c^2*d^2 - 4*a*c*d^3 - a*d^4)*cos(f*x + e)^2 - 8*((c^3 - 7*c^2*d + 7*c*d^2 - d^3)*cos(f*x + e)^4 - 2*(5*c^3 - 1
4*c^2*d + 5*c*d^2)*cos(f*x + e)^3 + 51*c^3 - 59*c^2*d + 17*c*d^2 - d^3 - 2*(18*c^3 - 33*c^2*d + 12*c*d^2 - d^3
)*cos(f*x + e)^2 + 2*(13*c^3 - 14*c^2*d + 5*c*d^2)*cos(f*x + e) + ((c^3 - 7*c^2*d + 7*c*d^2 - d^3)*cos(f*x + e
)^3 - 51*c^3 + 59*c^2*d - 17*c*d^2 + d^3 + (11*c^3 - 35*c^2*d + 17*c*d^2 - d^3)*cos(f*x + e)^2 - (25*c^3 - 31*
c^2*d + 7*c*d^2 - d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)
+ (289*a*c^4 - 476*a*c^3*d + 230*a*c^2*d^2 - 28*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 + (a*c^4 - 28*a*c^3*d + 70*a*c^2*d^2 - 28*a*c*d^3 + a*d^4)*cos(f*x + e)^4 + 32*(a*c^4
- 7*a*c^3*d + 7*a*c^2*d^2 - a*c*d^3)*cos(f*x + e)^3 - 2*(65*a*c^4 - 140*a*c^3*d + 38*a*c^2*d^2 - 12*a*c*d^3 +
a*d^4)*cos(f*x + e)^2 - 32*(9*a*c^4 - 15*a*c^3*d + 7*a*c^2*d^2 - a*c*d^3)*cos(f*x + e))*sin(f*x + e))/(cos(f*x
+ e)^5 + cos(f*x + e)^4 - 2*cos(f*x + e)^3 - 2*cos(f*x + e)^2 + (cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*sin(f
*x + e) + cos(f*x + e) + 1)))/f, 1/2*(sqrt(-a*c)*arctan(-1/4*((c^2 - 6*c*d + d^2)*cos(f*x + e)^2 - 9*c^2 + 6*c
*d - d^2 + 8*(c^2 - c*d)*sin(f*x + e))*sqrt(-a*c)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/((a*c^2*d
- a*c*d^2)*cos(f*x + e)^3 - (a*c^3 - 3*a*c^2*d)*cos(f*x + e)*sin(f*x + e) + (2*a*c^3 - a*c^2*d + a*c*d^2)*cos(
f*x + e))) + sqrt(a*d)*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))*sq
rt(a*d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(2*a*d^3*cos(f*x + e)^3 - (3*a*c*d^2 - a*d^3)*cos(f*
x + e)*sin(f*x + e) - (a*c^2*d - a*c*d^2 + 2*a*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{a \left (\sin{\left (e + f x \right )} + 1\right )} \sqrt{c + d \sin{\left (e + f x \right )}}}{\sin{\left (e + f x \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))**(1/2)/sin(f*x+e),x)

[Out]

Integral(sqrt(a*(sin(e + f*x) + 1))*sqrt(c + d*sin(e + f*x))/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{d \sin \left (f x + e\right ) + c}}{\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))^(1/2)/sin(f*x+e),x, algorithm="giac")

[Out]

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