3.206 \(\int \csc ^2(e+f x) \sqrt{a+b \sin (e+f x)} \, dx\)

Optimal. Leaf size=213 \[ -\frac{\cot (e+f x) \sqrt{a+b \sin (e+f x)}}{f}+\frac{a \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{f \sqrt{a+b \sin (e+f x)}}-\frac{\sqrt{a+b \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}+\frac{b \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{f \sqrt{a+b \sin (e+f x)}} \]

[Out]

-((Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/f) - (EllipticE[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[e
+ f*x]])/(f*Sqrt[(a + b*Sin[e + f*x])/(a + b)]) + (a*EllipticF[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*
Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]]) + (b*EllipticPi[2, (e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqr
t[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]])

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Rubi [A]  time = 0.480974, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {2796, 3060, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac{\cot (e+f x) \sqrt{a+b \sin (e+f x)}}{f}+\frac{a \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{f \sqrt{a+b \sin (e+f x)}}-\frac{\sqrt{a+b \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}+\frac{b \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{f \sqrt{a+b \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]],x]

[Out]

-((Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]])/f) - (EllipticE[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[e
+ f*x]])/(f*Sqrt[(a + b*Sin[e + f*x])/(a + b)]) + (a*EllipticF[(e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*
Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]]) + (b*EllipticPi[2, (e - Pi/2 + f*x)/2, (2*b)/(a + b)]*Sqr
t[(a + b*Sin[e + f*x])/(a + b)])/(f*Sqrt[a + b*Sin[e + f*x]])

Rule 2796

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -S
imp[(b*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n)/(f*(m + 1)*(a^2 - b^2)), x] + Dist[1/
((m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[a*c*(m + 1) + b*d*n
+ (a*d*(m + 1) - b*c*(m + 2))*Sin[e + f*x] - b*d*(m + n + 2)*Sin[e + f*x]^2, 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] && LtQ[m, -1] && LtQ[0, n, 1] && Inte
gersQ[2*m, 2*n]

Rule 3060

Int[((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[
(e_.) + (f_.)*(x_)])), x_Symbol] :> Dist[C/(b*d), Int[Sqrt[a + b*Sin[e + f*x]], x], x] - Dist[1/(b*d), Int[Sim
p[a*c*C - A*b*d + (b*c*C + a*C*d)*Sin[e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /;
FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 2655

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

Rule 2653

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

Rule 3002

Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[
(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[B/d, Int[(a + b*Sin[e + f*x])^m, x], x] - Dist[(B*c - A*d)/d, Int[(a +
b*Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 2663

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

Rule 2807

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

Rule 2805

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2*EllipticPi[(2*b)/(a + b), (1*(e - Pi/2 + f*x))/2, (2*d)/(c + d)])/(f*(a + b)*Sqrt[c + d]), 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] && GtQ[c + d, 0]

Rubi steps

\begin{align*} \int \csc ^2(e+f x) \sqrt{a+b \sin (e+f x)} \, dx &=-\frac{\cot (e+f x) \sqrt{a+b \sin (e+f x)}}{f}+\int \frac{\csc (e+f x) \left (\frac{b}{2}-\frac{1}{2} b \sin ^2(e+f x)\right )}{\sqrt{a+b \sin (e+f x)}} \, dx\\ &=-\frac{\cot (e+f x) \sqrt{a+b \sin (e+f x)}}{f}-\frac{1}{2} \int \sqrt{a+b \sin (e+f x)} \, dx-\frac{\int \frac{\csc (e+f x) \left (-\frac{b^2}{2}-\frac{1}{2} a b \sin (e+f x)\right )}{\sqrt{a+b \sin (e+f x)}} \, dx}{b}\\ &=-\frac{\cot (e+f x) \sqrt{a+b \sin (e+f x)}}{f}+\frac{1}{2} a \int \frac{1}{\sqrt{a+b \sin (e+f x)}} \, dx+\frac{1}{2} b \int \frac{\csc (e+f x)}{\sqrt{a+b \sin (e+f x)}} \, dx-\frac{\sqrt{a+b \sin (e+f x)} \int \sqrt{\frac{a}{a+b}+\frac{b \sin (e+f x)}{a+b}} \, dx}{2 \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}\\ &=-\frac{\cot (e+f x) \sqrt{a+b \sin (e+f x)}}{f}-\frac{E\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 b}{a+b}\right ) \sqrt{a+b \sin (e+f x)}}{f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}+\frac{\left (a \sqrt{\frac{a+b \sin (e+f x)}{a+b}}\right ) \int \frac{1}{\sqrt{\frac{a}{a+b}+\frac{b \sin (e+f x)}{a+b}}} \, dx}{2 \sqrt{a+b \sin (e+f x)}}+\frac{\left (b \sqrt{\frac{a+b \sin (e+f x)}{a+b}}\right ) \int \frac{\csc (e+f x)}{\sqrt{\frac{a}{a+b}+\frac{b \sin (e+f x)}{a+b}}} \, dx}{2 \sqrt{a+b \sin (e+f x)}}\\ &=-\frac{\cot (e+f x) \sqrt{a+b \sin (e+f x)}}{f}-\frac{E\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 b}{a+b}\right ) \sqrt{a+b \sin (e+f x)}}{f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}+\frac{a F\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 b}{a+b}\right ) \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}{f \sqrt{a+b \sin (e+f x)}}+\frac{b \Pi \left (2;\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 b}{a+b}\right ) \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}{f \sqrt{a+b \sin (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 8.8882, size = 312, normalized size = 1.46 \[ \frac{-4 \cot (e+f x) \sqrt{a+b \sin (e+f x)}-\frac{2 b \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left (2;\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (e+f x)}}+\frac{2 i \sec (e+f x) \sqrt{-\frac{b (\sin (e+f x)-1)}{a+b}} \sqrt{-\frac{b (\sin (e+f x)+1)}{a-b}} \left (b \left (b \Pi \left (\frac{a+b}{a};i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (e+f x)}\right )|\frac{a+b}{a-b}\right )-2 a F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (e+f x)}\right )|\frac{a+b}{a-b}\right )\right )-2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (e+f x)}\right )|\frac{a+b}{a-b}\right )\right )}{a b \sqrt{-\frac{1}{a+b}}}}{4 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[e + f*x]^2*Sqrt[a + b*Sin[e + f*x]],x]

[Out]

(((2*I)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[e + f*x]]], (a + b)/(a - b)] + b*
(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[e + f*x]]], (a + b)/(a - b)] + b*EllipticPi[(a +
b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[e + f*x]]], (a + b)/(a - b)]))*Sec[e + f*x]*Sqrt[-((b*(-1 +
Sin[e + f*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[e + f*x]))/(a - b))])/(a*b*Sqrt[-(a + b)^(-1)]) - 4*Cot[e + f*x]*
Sqrt[a + b*Sin[e + f*x]] - (2*b*EllipticPi[2, (-2*e + Pi - 2*f*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[e + f*x])/
(a + b)])/Sqrt[a + b*Sin[e + f*x]])/(4*f)

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Maple [A]  time = 1.047, size = 456, normalized size = 2.1 \begin{align*} -{\frac{1}{ab\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) f} \left ( a{b}^{2}\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+\sqrt{{\frac{b\sin \left ( fx+e \right ) }{a-b}}+{\frac{a}{a-b}}}\sqrt{-{\frac{b\sin \left ( fx+e \right ) }{a+b}}+{\frac{b}{a+b}}}\sqrt{-{\frac{b\sin \left ( fx+e \right ) }{a-b}}-{\frac{b}{a-b}}} \left ({\it EllipticF} \left ( \sqrt{{\frac{b\sin \left ( fx+e \right ) }{a-b}}+{\frac{a}{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ){a}^{2}b-{\it EllipticF} \left ( \sqrt{{\frac{b\sin \left ( fx+e \right ) }{a-b}}+{\frac{a}{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ) a{b}^{2}+{\it EllipticPi} \left ( \sqrt{{\frac{b\sin \left ( fx+e \right ) }{a-b}}+{\frac{a}{a-b}}},{\frac{a-b}{a}},\sqrt{{\frac{a-b}{a+b}}} \right ) a{b}^{2}-{\it EllipticPi} \left ( \sqrt{{\frac{b\sin \left ( fx+e \right ) }{a-b}}+{\frac{a}{a-b}}},{\frac{a-b}{a}},\sqrt{{\frac{a-b}{a+b}}} \right ){b}^{3}-{\it EllipticE} \left ( \sqrt{{\frac{b\sin \left ( fx+e \right ) }{a-b}}+{\frac{a}{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ){a}^{3}+{\it EllipticE} \left ( \sqrt{{\frac{b\sin \left ( fx+e \right ) }{a-b}}+{\frac{a}{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ) a{b}^{2} \right ) \sin \left ( fx+e \right ) +{a}^{2}b \left ( \cos \left ( fx+e \right ) \right ) ^{2} \right ){\frac{1}{\sqrt{a+b\sin \left ( fx+e \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*b^2*sin(f*x+e)*cos(f*x+e)^2+(b/(a-b)*sin(f*x+e)+1/(a-b)*a)^(1/2)*(-b/(a+b)*sin(f*x+e)+b/(a+b))^(1/2)*(-b/(
a-b)*sin(f*x+e)-b/(a-b))^(1/2)*(EllipticF((b/(a-b)*sin(f*x+e)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1/2))*a^2*b-Elli
pticF((b/(a-b)*sin(f*x+e)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1/2))*a*b^2+EllipticPi((b/(a-b)*sin(f*x+e)+1/(a-b)*a
)^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a*b^2-EllipticPi((b/(a-b)*sin(f*x+e)+1/(a-b)*a)^(1/2),(a-b)/a,((a-b)/(a+b
))^(1/2))*b^3-EllipticE((b/(a-b)*sin(f*x+e)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1/2))*a^3+EllipticE((b/(a-b)*sin(f
*x+e)+1/(a-b)*a)^(1/2),((a-b)/(a+b))^(1/2))*a*b^2)*sin(f*x+e)+a^2*b*cos(f*x+e)^2)/a/b/sin(f*x+e)/cos(f*x+e)/(a
+b*sin(f*x+e))^(1/2)/f

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sin \left (f x + e\right ) + a} \csc \left (f x + e\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*sin(f*x + e) + a)*csc(f*x + e)^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{b \sin \left (f x + e\right ) + a} \csc \left (f x + e\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*sin(f*x + e) + a)*csc(f*x + e)^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \sin{\left (e + f x \right )}} \csc ^{2}{\left (e + f x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)**2*(a+b*sin(f*x+e))**(1/2),x)

[Out]

Integral(sqrt(a + b*sin(e + f*x))*csc(e + f*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sin \left (f x + e\right ) + a} \csc \left (f x + e\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*sin(f*x + e) + a)*csc(f*x + e)^2, x)