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 \[ \frac{2 \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}} \text{EllipticF}\left (\frac{1}{2} \left (-\tan ^{-1}(c,a)+d+e x\right ),\frac{2 \sqrt{a^2+c^2}}{\sqrt{a^2+c^2}+b}\right )}{e \sqrt{\sin (d+e x)} \sqrt{a+b \csc (d+e x)+c \cot (d+e x)}} \]

[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]])

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Rubi [A]  time = 0.148861, antiderivative size = 118, normalized size of antiderivative = 1., 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 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]

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 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]

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*}

Mathematica [C]  time = 2.88962, size = 519, normalized size = 4.4 \[ \frac{4 \left (i \sqrt{a^2-b^2+c^2}-i a-b+c\right ) (\cos (d+e x)+i \sin (d+e x)) \sqrt{-\frac{i \left (\sqrt{a^2-b^2+c^2}+a+(b-c) \tan \left (\frac{1}{2} (d+e x)\right )\right )}{\left (\sqrt{a^2-b^2+c^2}+a-i b+i c\right ) \left (\tan \left (\frac{1}{2} (d+e x)\right )-i\right )}} \sqrt{-\frac{i \left (\sqrt{a^2-b^2+c^2}-a+(c-b) \tan \left (\frac{1}{2} (d+e x)\right )\right )}{\left (\sqrt{a^2-b^2+c^2}-a+i b-i c\right ) \left (\tan \left (\frac{1}{2} (d+e x)\right )-i\right )}} \sqrt{\frac{\left (\sqrt{a^2-b^2+c^2}-a-i b+i c\right ) (-\cos (d+e x)+i \sin (d+e x))}{\sqrt{a^2-b^2+c^2}-a+i b-i c}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{\left (\sqrt{a^2-b^2+c^2}-a-i b+i c\right ) (-\cos (d+e x)+i \sin (d+e x))}{\sqrt{a^2-b^2+c^2}-a+i b-i c}}\right ),\frac{\sqrt{a^2-b^2+c^2}+i b}{-\sqrt{a^2-b^2+c^2}+i b}\right )}{e \left (-\sqrt{a^2-b^2+c^2}+a+i b-i c\right ) \sqrt{\sin (d+e x)} \sqrt{a+b \csc (d+e x)+c \cot (d+e x)}} \]

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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Maple [C]  time = 0.403, size = 705, normalized size = 6. \begin{align*}{\frac{4\,i \left ( \cos \left ( ex+d \right ) +1 \right ) ^{2} \left ( \cos \left ( ex+d \right ) -1 \right ) ^{2}}{e \left ( b+c\cos \left ( ex+d \right ) +a\sin \left ( ex+d \right ) \right ) }{\it EllipticF} \left ( \sqrt{{(i\sin \left ( ex+d \right ) +\cos \left ( ex+d \right ) ) \left ( ib-ic-\sqrt{{a}^{2}-{b}^{2}+{c}^{2}}-a \right ) \left ( ib-ic+\sqrt{{a}^{2}-{b}^{2}+{c}^{2}}+a \right ) ^{-1}}},\sqrt{{ \left ( ib-ic+\sqrt{{a}^{2}-{b}^{2}+{c}^{2}}+a \right ) \left ( ib-ic+\sqrt{{a}^{2}-{b}^{2}+{c}^{2}}-a \right ) \left ( ib-ic-\sqrt{{a}^{2}-{b}^{2}+{c}^{2}}-a \right ) ^{-1} \left ( ib-ic-\sqrt{{a}^{2}-{b}^{2}+{c}^{2}}+a \right ) ^{-1}}} \right ) \sqrt{{\frac{b+c\cos \left ( ex+d \right ) +a\sin \left ( ex+d \right ) }{\sin \left ( ex+d \right ) }}}\sqrt{{(i\sin \left ( ex+d \right ) +\cos \left ( ex+d \right ) ) \left ( ib-ic-\sqrt{{a}^{2}-{b}^{2}+{c}^{2}}-a \right ) \left ( ib-ic+\sqrt{{a}^{2}-{b}^{2}+{c}^{2}}+a \right ) ^{-1}}}\sqrt{{\frac{-i}{i\cos \left ( ex+d \right ) +i+\sin \left ( ex+d \right ) } \left ( \cos \left ( ex+d \right ) \sqrt{{a}^{2}-{b}^{2}+{c}^{2}}-a\cos \left ( ex+d \right ) -b\sin \left ( ex+d \right ) +c\sin \left ( ex+d \right ) +\sqrt{{a}^{2}-{b}^{2}+{c}^{2}}-a \right ) \left ( ib-ic-\sqrt{{a}^{2}-{b}^{2}+{c}^{2}}+a \right ) ^{-1}}}\sqrt{{\frac{i}{i\cos \left ( ex+d \right ) +i+\sin \left ( ex+d \right ) } \left ( b\sin \left ( ex+d \right ) -c\sin \left ( ex+d \right ) +\cos \left ( ex+d \right ) \sqrt{{a}^{2}-{b}^{2}+{c}^{2}}+a\cos \left ( ex+d \right ) +\sqrt{{a}^{2}-{b}^{2}+{c}^{2}}+a \right ) \left ( ib-ic+\sqrt{{a}^{2}-{b}^{2}+{c}^{2}}+a \right ) ^{-1}}} \left ( i\sqrt{{a}^{2}-{b}^{2}+{c}^{2}}\sin \left ( ex+d \right ) -i\cos \left ( ex+d \right ) b+i\cos \left ( ex+d \right ) c+i\sin \left ( ex+d \right ) a-\cos \left ( ex+d \right ) \sqrt{{a}^{2}-{b}^{2}+{c}^{2}}-a\cos \left ( ex+d \right ) -b\sin \left ( ex+d \right ) +c\sin \left ( ex+d \right ) \right ) \left ( ib-ic-\sqrt{{a}^{2}-{b}^{2}+{c}^{2}}-a \right ) ^{-1} \left ( \sin \left ( ex+d \right ) \right ) ^{-{\frac{7}{2}}}} \end{align*}

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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)