\(\int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx\) [36]

Optimal result

Integrand size = 35, antiderivative size = 61 \[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx=-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {c} f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {3022, 212} \[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx=-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {c} f} \]

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

Rule 212

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

Rule 3022

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {(2 a) \text {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 \sqrt {a} \text {arctanh}\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 )}{\sqrt {c} f} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.84 (sec) , antiderivative size = 367, normalized size of antiderivative = 6.02 \[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx=-\frac {\left (\log \left (-\frac {(1+i) e^{\frac {i e}{2}} \left (\sqrt {2} c \left (-1+e^{i (e+f x)}\right )+i \sqrt {2} d \left (1+e^{i (e+f x)}\right )-2 i \sqrt {c} \sqrt {2 c e^{i (e+f x)}-i d \left (-1+e^{2 i (e+f x)}\right )}\right ) f}{\sqrt {c} \left (1+e^{i (e+f x)}\right )}\right )+\log \left (\frac {(1+i) e^{\frac {i e}{2}} \left (-i \sqrt {2} d \left (-1+e^{i (e+f x)}\right )+\sqrt {2} c \left (1+e^{i (e+f x)}\right )+2 \sqrt {c} \sqrt {2 c e^{i (e+f x)}-i d \left (-1+e^{2 i (e+f x)}\right )}\right ) f}{\sqrt {c} \left (-1+e^{i (e+f x)}\right )}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-i \sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} \sqrt {(\cos (e+f x)+i \sin (e+f x)) (c+d \sin (e+f x))}}{\sqrt {c} f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(202\) vs. \(2(49)=98\).

Time = 2.04 (sec) , antiderivative size = 203, normalized size of antiderivative = 3.33

method	result	size
default	\(\frac {\sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {c +d \sin \left (f x +e \right )}\, \left (\ln \left (-\frac {-\sqrt {c}\, \sqrt {2}\, \sqrt {\frac {c +d \sin \left (f x +e \right )}{1+\cos \left (f x +e \right )}}+c \cot \left (f x +e \right )-c \csc \left (f x +e \right )-d}{\sqrt {c}}\right )-\ln \left (-\frac {2 \left (\sqrt {c}\, \sqrt {2}\, \sqrt {\frac {c +d \sin \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sin \left (f x +e \right )+c \sin \left (f x +e \right )-\cos \left (f x +e \right ) d +d \right )}{\cos \left (f x +e \right )-1}\right )\right ) \sqrt {2}}{f \left (\cos \left (f x +e \right )+\sin \left (f x +e \right )+1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {c}}\)	\(203\)

[In]

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

[Out]

1/f*(a*(1+sin(f*x+e)))^(1/2)*(c+d*sin(f*x+e))^(1/2)*(ln(-(-c^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(1+cos(f*x+e)))^(
1/2)+c*cot(f*x+e)-c*csc(f*x+e)-d)/c^(1/2))-ln(-2*(c^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(1+cos(f*x+e)))^(1/2)*sin(
f*x+e)+c*sin(f*x+e)-cos(f*x+e)*d+d)/(cos(f*x+e)-1)))*2^(1/2)/(cos(f*x+e)+sin(f*x+e)+1)/((c+d*sin(f*x+e))/(1+co
s(f*x+e)))^(1/2)/c^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (49) = 98\).

Time = 0.46 (sec) , antiderivative size = 1044, normalized size of antiderivative = 17.11 \[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sin(f*x+e))^(1/2)/sin(f*x+e)/(c+d*sin(f*x+e))^(1/2),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^4 - 7*c^3*d + 7*c^2*d^2 - c*d^3)*cos(f*x + e)^4
+ 51*c^4 - 59*c^3*d + 17*c^2*d^2 - c*d^3 - 2*(5*c^4 - 14*c^3*d + 5*c^2*d^2)*cos(f*x + e)^3 - 2*(18*c^4 - 33*c^
3*d + 12*c^2*d^2 - c*d^3)*cos(f*x + e)^2 + 2*(13*c^4 - 14*c^3*d + 5*c^2*d^2)*cos(f*x + e) - (51*c^4 - 59*c^3*d
+ 17*c^2*d^2 - c*d^3 - (c^4 - 7*c^3*d + 7*c^2*d^2 - c*d^3)*cos(f*x + e)^3 - (11*c^4 - 35*c^3*d + 17*c^2*d^2 -
c*d^3)*cos(f*x + e)^2 + (25*c^4 - 31*c^3*d + 7*c^2*d^2 - c*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(a/c) + (289*a*c^4 - 476*a*c^3*d + 230*a*c^2*d^2 - 28*a*c*d^3 + a*d^4)*co
s(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 + (co
s(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*sin(f*x + e) + a)*sqrt(d
*sin(f*x + e) + c)*sqrt(-a/c)/((a*c*d - a*d^2)*cos(f*x + e)^3 - (a*c^2 - 3*a*c*d)*cos(f*x + e)*sin(f*x + e) +
(2*a*c^2 - a*c*d + a*d^2)*cos(f*x + e)))/f]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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