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

Optimal. Leaf size=118 \[ \frac {2 \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}} F\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {\sin (d+e x)} \sqrt {a+b \csc (d+e x)+c \cot (d+e x)}} \]

[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)

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Rubi [A]  time = 0.15, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3164, 3127, 2661} \[ \frac {2 \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}} F\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {\sin (d+e x)} \sqrt {a+b \csc (d+e x)+c \cot (d+e x)}} \]

Antiderivative was successfully verified.

[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 2661

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

Rule 3127

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]*Cos[d + e*x - ArcTan[b, c]])/(a + Sqrt[b^2 + c^2])], 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 3164

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*} \int \frac {1}{\sqrt {a+c \cot (d+e x)+b \csc (d+e x)} \sqrt {\sin (d+e x)}} \, dx &=\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 F\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*}

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Mathematica [C]  time = 2.84, size = 519, normalized size = 4.40 \[ \frac {4 \left (i \sqrt {a^2-b^2+c^2}-i a-b+c\right ) (\cos (d+e x)+i \sin (d+e x)) \sqrt {-\frac {i \left (\sqrt {a^2-b^2+c^2}+a+(b-c) \tan \left (\frac {1}{2} (d+e x)\right )\right )}{\left (\sqrt {a^2-b^2+c^2}+a-i b+i c\right ) \left (\tan \left (\frac {1}{2} (d+e x)\right )-i\right )}} \sqrt {-\frac {i \left (\sqrt {a^2-b^2+c^2}-a+(c-b) \tan \left (\frac {1}{2} (d+e x)\right )\right )}{\left (\sqrt {a^2-b^2+c^2}-a+i b-i c\right ) \left (\tan \left (\frac {1}{2} (d+e x)\right )-i\right )}} \sqrt {\frac {\left (\sqrt {a^2-b^2+c^2}-a-i b+i c\right ) (-\cos (d+e x)+i \sin (d+e x))}{\sqrt {a^2-b^2+c^2}-a+i b-i c}} F\left (\sin ^{-1}\left (\sqrt {\frac {\left (-a-i b+i c+\sqrt {a^2-b^2+c^2}\right ) (i \sin (d+e x)-\cos (d+e x))}{-a+i b-i c+\sqrt {a^2-b^2+c^2}}}\right )|\frac {i b+\sqrt {a^2-b^2+c^2}}{i b-\sqrt {a^2-b^2+c^2}}\right )}{e \left (-\sqrt {a^2-b^2+c^2}+a+i b-i c\right ) \sqrt {\sin (d+e x)} \sqrt {a+b \csc (d+e x)+c \cot (d+e x)}} \]

Warning: Unable to verify antiderivative.

[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]])

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fricas [F]  time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\sqrt {c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a} \sqrt {\sin \left (e x + d\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a} \sqrt {\sin \left (e x + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 1.16, size = 705, normalized size = 5.97 \[ \frac {4 i \sqrt {\frac {b +c \cos \left (e x +d \right )+a \sin \left (e x +d \right )}{\sin \left (e x +d \right )}}\, \sqrt {\frac {\left (i \sin \left (e x +d \right )+\cos \left (e x +d \right )\right ) \left (i b -i c -\sqrt {a^{2}-b^{2}+c^{2}}-a \right )}{i b -i c +\sqrt {a^{2}-b^{2}+c^{2}}+a}}\, \sqrt {-\frac {i \left (\cos \left (e x +d \right ) \sqrt {a^{2}-b^{2}+c^{2}}-b \sin \left (e x +d \right )+c \sin \left (e x +d \right )-a \cos \left (e x +d \right )+\sqrt {a^{2}-b^{2}+c^{2}}-a \right )}{\left (i \cos \left (e x +d \right )+\sin \left (e x +d \right )+i\right ) \left (i b -i c -\sqrt {a^{2}-b^{2}+c^{2}}+a \right )}}\, \sqrt {\frac {i \left (b \sin \left (e x +d \right )-c \sin \left (e x +d \right )+\cos \left (e x +d \right ) \sqrt {a^{2}-b^{2}+c^{2}}+a \cos \left (e x +d \right )+\sqrt {a^{2}-b^{2}+c^{2}}+a \right )}{\left (i \cos \left (e x +d \right )+\sin \left (e x +d \right )+i\right ) \left (i b -i c +\sqrt {a^{2}-b^{2}+c^{2}}+a \right )}}\, \left (\cos \left (e x +d \right )+1\right )^{2} \EllipticF \left (\sqrt {\frac {\left (i \sin \left (e x +d \right )+\cos \left (e x +d \right )\right ) \left (i b -i c -\sqrt {a^{2}-b^{2}+c^{2}}-a \right )}{i b -i c +\sqrt {a^{2}-b^{2}+c^{2}}+a}}, \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 ) \left (\cos \left (e x +d \right )-1\right )^{2} \left (i \sqrt {a^{2}-b^{2}+c^{2}}\, \sin \left (e x +d \right )+i \sin \left (e x +d \right ) a -i \cos \left (e x +d \right ) b +i \cos \left (e x +d \right ) c -b \sin \left (e x +d \right )+c \sin \left (e x +d \right )-\cos \left (e x +d \right ) \sqrt {a^{2}-b^{2}+c^{2}}-a \cos \left (e x +d \right )\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 )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*I/e*((b+c*cos(e*x+d)+a*sin(e*x+d))/sin(e*x+d))^(1/2)*((I*sin(e*x+d)+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*(cos(e*x+d)*(a^2-b^2+c^2)^(1/2)-b*sin(e*x+d)+c*sin(e*x+d)-a*cos
(e*x+d)+(a^2-b^2+c^2)^(1/2)-a)/(I*cos(e*x+d)+sin(e*x+d)+I)/(I*b-I*c-(a^2-b^2+c^2)^(1/2)+a))^(1/2)*(I*(b*sin(e*
x+d)-c*sin(e*x+d)+cos(e*x+d)*(a^2-b^2+c^2)^(1/2)+a*cos(e*x+d)+(a^2-b^2+c^2)^(1/2)+a)/(I*cos(e*x+d)+sin(e*x+d)+
I)/(I*b-I*c+(a^2-b^2+c^2)^(1/2)+a))^(1/2)*(cos(e*x+d)+1)^2*EllipticF(((I*sin(e*x+d)+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))*(cos(e*x+d)-1)^2*(I*(a^2-b
^2+c^2)^(1/2)*sin(e*x+d)+I*sin(e*x+d)*a-I*cos(e*x+d)*b+I*cos(e*x+d)*c-b*sin(e*x+d)+c*sin(e*x+d)-cos(e*x+d)*(a^
2-b^2+c^2)^(1/2)-a*cos(e*x+d))/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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a} \sqrt {\sin \left (e x + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {\sin \left (d+e\,x\right )}\,\sqrt {a+c\,\mathrm {cot}\left (d+e\,x\right )+\frac {b}{\sin \left (d+e\,x\right )}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + b \csc {\left (d + e x \right )} + c \cot {\left (d + e x \right )}} \sqrt {\sin {\left (d + e x \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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