3.29 \(\int \frac{\csc (e+f x) \sqrt{a+b \sin (e+f x)}}{c+c \sin (e+f x)} \, dx\)

Optimal. Leaf size=238 \[ \frac{\cos (e+f x) \sqrt{a+b \sin (e+f x)}}{f (c \sin (e+f x)+c)}-\frac{(a-b) \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{c f \sqrt{a+b \sin (e+f x)}}+\frac{\sqrt{a+b \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{c f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}+\frac{2 a \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{c f \sqrt{a+b \sin (e+f x)}} \]

[Out]

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

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Rubi [A]  time = 0.503375, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2935, 2807, 2805, 2768, 2752, 2663, 2661, 2655, 2653} \[ \frac{\cos (e+f x) \sqrt{a+b \sin (e+f x)}}{f (c \sin (e+f x)+c)}-\frac{(a-b) \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{c f \sqrt{a+b \sin (e+f x)}}+\frac{\sqrt{a+b \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{c f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}+\frac{2 a \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{c f \sqrt{a+b \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2935

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

Rule 2768

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b
^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(a*f*(b*c - a*d)*(a + b*Sin[e + f*x])), x] + Dist[d/(a*(b*c - a*
d)), Int[(c + d*Sin[e + f*x])^n*(a*n - b*(n + 1)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, 0] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 2752

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

Rubi steps

\begin{align*} \int \frac{\csc (e+f x) \sqrt{a+b \sin (e+f x)}}{c+c \sin (e+f x)} \, dx &=(-a+b) \int \frac{1}{\sqrt{a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx+\frac{a \int \frac{\csc (e+f x)}{\sqrt{a+b \sin (e+f x)}} \, dx}{c}\\ &=\frac{\cos (e+f x) \sqrt{a+b \sin (e+f x)}}{f (c+c \sin (e+f x))}-\frac{b \int \frac{-\frac{c}{2}-\frac{1}{2} c \sin (e+f x)}{\sqrt{a+b \sin (e+f x)}} \, dx}{c^2}+\frac{\left (a \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}{c \sqrt{a+b \sin (e+f x)}}\\ &=\frac{2 a \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}}}{c f \sqrt{a+b \sin (e+f x)}}+\frac{\cos (e+f x) \sqrt{a+b \sin (e+f x)}}{f (c+c \sin (e+f x))}+\frac{\int \sqrt{a+b \sin (e+f x)} \, dx}{2 c}-\frac{(a-b) \int \frac{1}{\sqrt{a+b \sin (e+f x)}} \, dx}{2 c}\\ &=\frac{2 a \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}}}{c f \sqrt{a+b \sin (e+f x)}}+\frac{\cos (e+f x) \sqrt{a+b \sin (e+f x)}}{f (c+c \sin (e+f x))}+\frac{\sqrt{a+b \sin (e+f x)} \int \sqrt{\frac{a}{a+b}+\frac{b \sin (e+f x)}{a+b}} \, dx}{2 c \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}-\frac{\left ((a-b) \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 c \sqrt{a+b \sin (e+f x)}}\\ &=\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)}}{c f \sqrt{\frac{a+b \sin (e+f x)}{a+b}}}-\frac{(a-b) 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}}}{c f \sqrt{a+b \sin (e+f x)}}+\frac{2 a \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}}}{c f \sqrt{a+b \sin (e+f x)}}+\frac{\cos (e+f x) \sqrt{a+b \sin (e+f x)}}{f (c+c \sin (e+f x))}\\ \end{align*}

Mathematica [C]  time = 6.54579, size = 611, normalized size = 2.57 \[ -\frac{2 \sin \left (\frac{1}{2} (e+f x)\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \sqrt{a+b \sin (e+f x)}}{f (c \sin (e+f x)+c)}+\frac{\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2 \left (\frac{2 i b \cos (e+f x) \cos (2 (e+f x)) \sqrt{\frac{b-b \sin (e+f x)}{a+b}} \sqrt{-\frac{b \sin (e+f x)+b}{a-b}} \left (2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (e+f x)}\right )|\frac{a+b}{a-b}\right )+b \left (2 a F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (e+f x)}\right )|\frac{a+b}{a-b}\right )-b \Pi \left (\frac{a+b}{a};i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (e+f x)}\right )|\frac{a+b}{a-b}\right )\right )\right )}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\sin ^2(e+f x)} \left (-2 a^2+4 a (a+b \sin (e+f x))-2 (a+b \sin (e+f x))^2+b^2\right ) \sqrt{-\frac{a^2-2 a (a+b \sin (e+f x))+(a+b \sin (e+f x))^2-b^2}{b^2}}}+\frac{2 \sin (2 (e+f x)) \cot (e+f x) \sqrt{a+b \sin (e+f x)}}{1-\sin ^2(e+f x)}-\frac{4 b \sqrt{\frac{a+b \sin (e+f x)}{a+b}} F\left (\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (e+f x)}}+\frac{2 (-4 a-b) \sqrt{\frac{a+b \sin (e+f x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (e+f x)}}\right )}{4 f (c \sin (e+f x)+c)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[a + b*Sin[e + f*x]])/(f*(c + c*Sin[e + f*x]))
+ ((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*((-4*b*EllipticF[(-e + Pi/2 - f*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Si
n[e + f*x])/(a + b)])/Sqrt[a + b*Sin[e + f*x]] + (2*(-4*a - b)*EllipticPi[2, (-e + Pi/2 - f*x)/2, (2*b)/(a + b
)]*Sqrt[(a + b*Sin[e + f*x])/(a + b)])/Sqrt[a + b*Sin[e + f*x]] + ((2*I)*b*Cos[e + f*x]*Cos[2*(e + f*x)]*(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)]))*Sqrt[(b - b*Sin[e + f*x])/(a + b)]*Sqrt[-((
b + b*Sin[e + f*x])/(a - b))])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Sin[e + f*x]^2]*(-2*a^2 + b^2 + 4*a*(a + b*Sin[
e + f*x]) - 2*(a + b*Sin[e + f*x])^2)*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Sin[e + f*x]) + (a + b*Sin[e + f*x])^2)/b
^2)]) + (2*Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*Sin[2*(e + f*x)])/(1 - Sin[e + f*x]^2)))/(4*f*(c + c*Sin[e +
f*x]))

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Maple [A]  time = 5.068, size = 593, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-(-b*sin(f*x+e)-a)*cos(f*x+e)^2)^(1/2)/c*((-a+b)*(-(-b*sin(f*x+e)^2-a*sin(f*x+e)+b*sin(f*x+e)+a)/(a-b)/((-b*s
in(f*x+e)-a)*(-1+sin(f*x+e))*(1+sin(f*x+e)))^(1/2)-2*b/(2*a-2*b)*(a/b-1)*((a+b*sin(f*x+e))/(a-b))^(1/2)*(b*(1-
sin(f*x+e))/(a+b))^(1/2)*((-sin(f*x+e)-1)*b/(a-b))^(1/2)/(-(-b*sin(f*x+e)-a)*cos(f*x+e)^2)^(1/2)*EllipticF(((a
+b*sin(f*x+e))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))-b/(a-b)*(a/b-1)*((a+b*sin(f*x+e))/(a-b))^(1/2)*(b*(1-sin(f*x+
e))/(a+b))^(1/2)*((-sin(f*x+e)-1)*b/(a-b))^(1/2)/(-(-b*sin(f*x+e)-a)*cos(f*x+e)^2)^(1/2)*((-a/b-1)*EllipticE((
(a+b*sin(f*x+e))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))+EllipticF(((a+b*sin(f*x+e))/(a-b))^(1/2),((a-b)/(a+b))^(1/2
))))-2*(a/b-1)*((a+b*sin(f*x+e))/(a-b))^(1/2)*(b*(1-sin(f*x+e))/(a+b))^(1/2)*((-sin(f*x+e)-1)*b/(a-b))^(1/2)/(
-(-b*sin(f*x+e)-a)*cos(f*x+e)^2)^(1/2)*b*EllipticPi(((a+b*sin(f*x+e))/(a-b))^(1/2),-(-a/b+1)/a*b,((a-b)/(a+b))
^(1/2)))/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 \frac{\sqrt{b \sin \left (f x + e\right ) + a}}{{\left (c \sin \left (f x + e\right ) + c\right )} \sin \left (f x + e\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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