\(\int \frac {1}{\sqrt {a+c \cot (d+e x)+b \csc (d+e x)} \sqrt {\sin (d+e x)}} \, dx\) [469]

Optimal result

Integrand size = 33, antiderivative size = 118 \[ \int \frac {1}{\sqrt {a+c \cot (d+e x)+b \csc (d+e x)} \sqrt {\sin (d+e x)}} \, dx=\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right ),\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) \sqrt {\frac {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}}}{e \sqrt {a+c \cot (d+e x)+b \csc (d+e x)} \sqrt {\sin (d+e x)}} \]

[Out]

2*(cos(1/2*d+1/2*e*x-1/2*arctan(c,a))^2)^(1/2)/cos(1/2*d+1/2*e*x-1/2*arctan(c,a))*EllipticF(sin(1/2*d+1/2*e*x-
1/2*arctan(c,a)),2^(1/2)*((a^2+c^2)^(1/2)/(b+(a^2+c^2)^(1/2)))^(1/2))*((b+c*cos(e*x+d)+a*sin(e*x+d))/(b+(a^2+c
^2)^(1/2)))^(1/2)/e/(a+c*cot(e*x+d)+b*csc(e*x+d))^(1/2)/sin(e*x+d)^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3243, 3206, 2740} \[ \int \frac {1}{\sqrt {a+c \cot (d+e x)+b \csc (d+e x)} \sqrt {\sin (d+e x)}} \, dx=\frac {2 \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right ),\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {\sin (d+e x)} \sqrt {a+b \csc (d+e x)+c \cot (d+e x)}} \]

[In]

Int[1/(Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]*Sqrt[Sin[d + e*x]]),x]

[Out]

(2*EllipticF[(d + e*x - ArcTan[c, a])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[(b + c*Cos[d + e*x] +
a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])/(e*Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]*Sqrt[Sin[d + e*x]])

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 3206

Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a +
b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], Int[1/Sqrt[
a/(a + Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - ArcTan[b, c]]], x], x] /; Free
Q[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] &&  !GtQ[a + Sqrt[b^2 + c^2], 0]

Rule 3243

Int[((a_.) + csc[(d_.) + (e_.)*(x_)]*(b_.) + cot[(d_.) + (e_.)*(x_)]*(c_.))^(n_)*sin[(d_.) + (e_.)*(x_)]^(n_),
x_Symbol] :> Dist[Sin[d + e*x]^n*((a + b*Csc[d + e*x] + c*Cot[d + e*x])^n/(b + a*Sin[d + e*x] + c*Cos[d + e*x
])^n), Int[(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] &&  !IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {b+c \cos (d+e x)+a \sin (d+e x)} \int \frac {1}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}} \, dx}{\sqrt {a+c \cot (d+e x)+b \csc (d+e x)} \sqrt {\sin (d+e x)}} \\ & = \frac {\sqrt {\frac {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}} \int \frac {1}{\sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \cos \left (d+e x-\tan ^{-1}(c,a)\right )}{b+\sqrt {a^2+c^2}}}} \, dx}{\sqrt {a+c \cot (d+e x)+b \csc (d+e x)} \sqrt {\sin (d+e x)}} \\ & = \frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right ),\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) \sqrt {\frac {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}}}{e \sqrt {a+c \cot (d+e x)+b \csc (d+e x)} \sqrt {\sin (d+e x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 18.77 (sec) , antiderivative size = 519, normalized size of antiderivative = 4.40 \[ \int \frac {1}{\sqrt {a+c \cot (d+e x)+b \csc (d+e x)} \sqrt {\sin (d+e x)}} \, dx=\frac {4 \left (-i a-b+c+i \sqrt {a^2-b^2+c^2}\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\left (-a-i b+i c+\sqrt {a^2-b^2+c^2}\right ) (-\cos (d+e x)+i \sin (d+e x))}{-a+i b-i c+\sqrt {a^2-b^2+c^2}}}\right ),\frac {i b+\sqrt {a^2-b^2+c^2}}{i b-\sqrt {a^2-b^2+c^2}}\right ) \sqrt {\frac {\left (-a-i b+i c+\sqrt {a^2-b^2+c^2}\right ) (-\cos (d+e x)+i \sin (d+e x))}{-a+i b-i c+\sqrt {a^2-b^2+c^2}}} (\cos (d+e x)+i \sin (d+e x)) \sqrt {-\frac {i \left (a+\sqrt {a^2-b^2+c^2}+(b-c) \tan \left (\frac {1}{2} (d+e x)\right )\right )}{\left (a-i b+i c+\sqrt {a^2-b^2+c^2}\right ) \left (-i+\tan \left (\frac {1}{2} (d+e x)\right )\right )}} \sqrt {-\frac {i \left (-a+\sqrt {a^2-b^2+c^2}+(-b+c) \tan \left (\frac {1}{2} (d+e x)\right )\right )}{\left (-a+i b-i c+\sqrt {a^2-b^2+c^2}\right ) \left (-i+\tan \left (\frac {1}{2} (d+e x)\right )\right )}}}{\left (a+i b-i c-\sqrt {a^2-b^2+c^2}\right ) e \sqrt {a+c \cot (d+e x)+b \csc (d+e x)} \sqrt {\sin (d+e x)}} \]

[In]

Integrate[1/(Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]*Sqrt[Sin[d + e*x]]),x]

[Out]

(4*((-I)*a - b + c + I*Sqrt[a^2 - b^2 + c^2])*EllipticF[ArcSin[Sqrt[((-a - I*b + I*c + Sqrt[a^2 - b^2 + c^2])*
(-Cos[d + e*x] + I*Sin[d + e*x]))/(-a + I*b - I*c + Sqrt[a^2 - b^2 + c^2])]], (I*b + Sqrt[a^2 - b^2 + c^2])/(I
*b - Sqrt[a^2 - b^2 + c^2])]*Sqrt[((-a - I*b + I*c + Sqrt[a^2 - b^2 + c^2])*(-Cos[d + e*x] + I*Sin[d + e*x]))/
(-a + I*b - I*c + Sqrt[a^2 - b^2 + c^2])]*(Cos[d + e*x] + I*Sin[d + e*x])*Sqrt[((-I)*(a + Sqrt[a^2 - b^2 + c^2
] + (b - c)*Tan[(d + e*x)/2]))/((a - I*b + I*c + Sqrt[a^2 - b^2 + c^2])*(-I + Tan[(d + e*x)/2]))]*Sqrt[((-I)*(
-a + Sqrt[a^2 - b^2 + c^2] + (-b + c)*Tan[(d + e*x)/2]))/((-a + I*b - I*c + Sqrt[a^2 - b^2 + c^2])*(-I + Tan[(
d + e*x)/2]))])/((a + I*b - I*c - Sqrt[a^2 - b^2 + c^2])*e*Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]*Sqrt[Sin[
d + e*x]])

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 28.57 (sec) , antiderivative size = 716, normalized size of antiderivative = 6.07

method	result	size
default	\(-\frac {2 i \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {-\frac {\left (i \cos \left (e x +d \right )+i-\sin \left (e x +d \right )\right )^{2} \left (i b -i c -\sqrt {a^{2}-b^{2}+c^{2}}-a \right )}{\left (1+\cos \left (e x +d \right )\right ) \left (i b -i c +\sqrt {a^{2}-b^{2}+c^{2}}+a \right )}}}{2}, \sqrt {\frac {\left (i b -i c +\sqrt {a^{2}-b^{2}+c^{2}}+a \right ) \left (i b -i c +\sqrt {a^{2}-b^{2}+c^{2}}-a \right )}{\left (i b -i c -\sqrt {a^{2}-b^{2}+c^{2}}-a \right ) \left (i b -i c -\sqrt {a^{2}-b^{2}+c^{2}}+a \right )}}\right ) \sqrt {-\frac {i \left (b \cos \left (e x +d \right )-c \cos \left (e x +d \right )-\sqrt {a^{2}-b^{2}+c^{2}}\, \sin \left (e x +d \right )-a \sin \left (e x +d \right )-b +c \right )}{\left (i \sin \left (e x +d \right )-\cos \left (e x +d \right )+1\right ) \left (i b -i c +\sqrt {a^{2}-b^{2}+c^{2}}+a \right )}}\, \sqrt {-\frac {i \left (\sqrt {a^{2}-b^{2}+c^{2}}\, \sin \left (e x +d \right )+b \cos \left (e x +d \right )-c \cos \left (e x +d \right )-a \sin \left (e x +d \right )-b +c \right )}{\left (i \sin \left (e x +d \right )-\cos \left (e x +d \right )+1\right ) \left (i b -i c -\sqrt {a^{2}-b^{2}+c^{2}}+a \right )}}\, \sqrt {-\frac {\left (i \cos \left (e x +d \right )+i-\sin \left (e x +d \right )\right )^{2} \left (i b -i c -\sqrt {a^{2}-b^{2}+c^{2}}-a \right )}{\left (1+\cos \left (e x +d \right )\right ) \left (i b -i c +\sqrt {a^{2}-b^{2}+c^{2}}+a \right )}}\, \left (1+\cos \left (e x +d \right )\right )^{2} \left (\cos \left (e x +d \right )-1\right )^{2} \left (i b \cos \left (e x +d \right )-i c \cos \left (e x +d \right )-i \sqrt {a^{2}-b^{2}+c^{2}}\, \sin \left (e x +d \right )-i \sin \left (e x +d \right ) a +\cos \left (e x +d \right ) \sqrt {a^{2}-b^{2}+c^{2}}+a \cos \left (e x +d \right )+b \sin \left (e x +d \right )-c \sin \left (e x +d \right )\right ) \sqrt {2}\, \sqrt {a +c \cot \left (e x +d \right )+b \csc \left (e x +d \right )}}{e \sin \left (e x +d \right )^{\frac {7}{2}} \left (b +c \cos \left (e x +d \right )+a \sin \left (e x +d \right )\right ) \left (i b -i c -\sqrt {a^{2}-b^{2}+c^{2}}-a \right )}\)	\(716\)

[In]

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

[Out]

-2*I/e*EllipticF(1/2*2^(1/2)*(-(I*cos(e*x+d)+I-sin(e*x+d))^2/(1+cos(e*x+d))*(I*b-I*c-(a^2-b^2+c^2)^(1/2)-a)/(I
*b-I*c+(a^2-b^2+c^2)^(1/2)+a))^(1/2),((I*b-I*c+(a^2-b^2+c^2)^(1/2)+a)*(I*b-I*c+(a^2-b^2+c^2)^(1/2)-a)/(I*b-I*c
-(a^2-b^2+c^2)^(1/2)-a)/(I*b-I*c-(a^2-b^2+c^2)^(1/2)+a))^(1/2))*(-I*(b*cos(e*x+d)-c*cos(e*x+d)-(a^2-b^2+c^2)^(
1/2)*sin(e*x+d)-a*sin(e*x+d)-b+c)/(I*sin(e*x+d)-cos(e*x+d)+1)/(I*b-I*c+(a^2-b^2+c^2)^(1/2)+a))^(1/2)*(-I*((a^2
-b^2+c^2)^(1/2)*sin(e*x+d)+b*cos(e*x+d)-c*cos(e*x+d)-a*sin(e*x+d)-b+c)/(I*sin(e*x+d)-cos(e*x+d)+1)/(I*b-I*c-(a
^2-b^2+c^2)^(1/2)+a))^(1/2)*(-(I*cos(e*x+d)+I-sin(e*x+d))^2/(1+cos(e*x+d))*(I*b-I*c-(a^2-b^2+c^2)^(1/2)-a)/(I*
b-I*c+(a^2-b^2+c^2)^(1/2)+a))^(1/2)*(1+cos(e*x+d))^2*(cos(e*x+d)-1)^2*(I*b*cos(e*x+d)-I*c*cos(e*x+d)-I*(a^2-b^
2+c^2)^(1/2)*sin(e*x+d)-I*sin(e*x+d)*a+cos(e*x+d)*(a^2-b^2+c^2)^(1/2)+a*cos(e*x+d)+b*sin(e*x+d)-c*sin(e*x+d))*
2^(1/2)*(a+c*cot(e*x+d)+b*csc(e*x+d))^(1/2)/sin(e*x+d)^(7/2)/(b+c*cos(e*x+d)+a*sin(e*x+d))/(I*b-I*c-(a^2-b^2+c
^2)^(1/2)-a)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.10 (sec) , antiderivative size = 506, normalized size of antiderivative = 4.29 \[ \int \frac {1}{\sqrt {a+c \cot (d+e x)+b \csc (d+e x)} \sqrt {\sin (d+e x)}} \, dx=\frac {\sqrt {2} {\left (a + i \, c\right )} \sqrt {i \, a + c} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (3 \, a^{4} - 4 \, a^{2} b^{2} + 4 \, b^{2} c^{2} + 6 i \, a c^{3} - 3 \, c^{4} + 2 i \, {\left (3 \, a^{3} - 4 \, a b^{2}\right )} c\right )}}{3 \, {\left (a^{4} + 2 \, a^{2} c^{2} + c^{4}\right )}}, -\frac {8 \, {\left (-9 i \, a^{5} b + 8 i \, a^{3} b^{3} + 27 i \, a b c^{4} - 9 \, b c^{5} + 2 \, {\left (9 \, a^{2} b + 4 \, b^{3}\right )} c^{3} + 6 i \, {\left (3 \, a^{3} b - 4 \, a b^{3}\right )} c^{2} + 3 \, {\left (9 \, a^{4} b - 8 \, a^{2} b^{3}\right )} c\right )}}{27 \, {\left (a^{6} + 3 \, a^{4} c^{2} + 3 \, a^{2} c^{4} + c^{6}\right )}}, \frac {-2 i \, a b + 2 \, b c + 3 \, {\left (a^{2} + c^{2}\right )} \cos \left (e x + d\right ) - 3 \, {\left (i \, a^{2} + i \, c^{2}\right )} \sin \left (e x + d\right )}{3 \, {\left (a^{2} + c^{2}\right )}}\right ) + \sqrt {2} {\left (a - i \, c\right )} \sqrt {-i \, a + c} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (3 \, a^{4} - 4 \, a^{2} b^{2} + 4 \, b^{2} c^{2} - 6 i \, a c^{3} - 3 \, c^{4} - 2 i \, {\left (3 \, a^{3} - 4 \, a b^{2}\right )} c\right )}}{3 \, {\left (a^{4} + 2 \, a^{2} c^{2} + c^{4}\right )}}, -\frac {8 \, {\left (9 i \, a^{5} b - 8 i \, a^{3} b^{3} - 27 i \, a b c^{4} - 9 \, b c^{5} + 2 \, {\left (9 \, a^{2} b + 4 \, b^{3}\right )} c^{3} - 6 i \, {\left (3 \, a^{3} b - 4 \, a b^{3}\right )} c^{2} + 3 \, {\left (9 \, a^{4} b - 8 \, a^{2} b^{3}\right )} c\right )}}{27 \, {\left (a^{6} + 3 \, a^{4} c^{2} + 3 \, a^{2} c^{4} + c^{6}\right )}}, \frac {2 i \, a b + 2 \, b c + 3 \, {\left (a^{2} + c^{2}\right )} \cos \left (e x + d\right ) - 3 \, {\left (-i \, a^{2} - i \, c^{2}\right )} \sin \left (e x + d\right )}{3 \, {\left (a^{2} + c^{2}\right )}}\right )}{{\left (a^{2} + c^{2}\right )} e} \]

[In]

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

[Out]

(sqrt(2)*(a + I*c)*sqrt(I*a + c)*weierstrassPInverse(4/3*(3*a^4 - 4*a^2*b^2 + 4*b^2*c^2 + 6*I*a*c^3 - 3*c^4 +
2*I*(3*a^3 - 4*a*b^2)*c)/(a^4 + 2*a^2*c^2 + c^4), -8/27*(-9*I*a^5*b + 8*I*a^3*b^3 + 27*I*a*b*c^4 - 9*b*c^5 + 2
*(9*a^2*b + 4*b^3)*c^3 + 6*I*(3*a^3*b - 4*a*b^3)*c^2 + 3*(9*a^4*b - 8*a^2*b^3)*c)/(a^6 + 3*a^4*c^2 + 3*a^2*c^4
+ c^6), 1/3*(-2*I*a*b + 2*b*c + 3*(a^2 + c^2)*cos(e*x + d) - 3*(I*a^2 + I*c^2)*sin(e*x + d))/(a^2 + c^2)) + s
qrt(2)*(a - I*c)*sqrt(-I*a + c)*weierstrassPInverse(4/3*(3*a^4 - 4*a^2*b^2 + 4*b^2*c^2 - 6*I*a*c^3 - 3*c^4 - 2
*I*(3*a^3 - 4*a*b^2)*c)/(a^4 + 2*a^2*c^2 + c^4), -8/27*(9*I*a^5*b - 8*I*a^3*b^3 - 27*I*a*b*c^4 - 9*b*c^5 + 2*(
9*a^2*b + 4*b^3)*c^3 - 6*I*(3*a^3*b - 4*a*b^3)*c^2 + 3*(9*a^4*b - 8*a^2*b^3)*c)/(a^6 + 3*a^4*c^2 + 3*a^2*c^4 +
c^6), 1/3*(2*I*a*b + 2*b*c + 3*(a^2 + c^2)*cos(e*x + d) - 3*(-I*a^2 - I*c^2)*sin(e*x + d))/(a^2 + c^2)))/((a^
2 + c^2)*e)

Sympy [F]

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

[In]

integrate(1/(a+c*cot(e*x+d)+b*csc(e*x+d))**(1/2)/sin(e*x+d)**(1/2),x)

[Out]

Integral(1/(sqrt(a + b*csc(d + e*x) + c*cot(d + e*x))*sqrt(sin(d + e*x))), x)

Maxima [F]

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

[In]

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

[Out]

integrate(1/(sqrt(c*cot(e*x + d) + b*csc(e*x + d) + a)*sqrt(sin(e*x + d))), x)

Giac [F]

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

[In]

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

[Out]

integrate(1/(sqrt(c*cot(e*x + d) + b*csc(e*x + d) + a)*sqrt(sin(e*x + d))), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+c \cot (d+e x)+b \csc (d+e x)} \sqrt {\sin (d+e x)}} \, dx=\int \frac {1}{\sqrt {\sin \left (d+e\,x\right )}\,\sqrt {a+c\,\mathrm {cot}\left (d+e\,x\right )+\frac {b}{\sin \left (d+e\,x\right )}}} \,d x \]

[In]

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

[Out]

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