3.3.6 \(\int \csc ^2(e+f x) \sqrt {a+b \sin (e+f x)} \, dx\) [206]

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

[Out]

-cot(f*x+e)*(a+b*sin(f*x+e))^(1/2)/f+(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticE(c
os(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(b/(a+b))^(1/2))*(a+b*sin(f*x+e))^(1/2)/f/((a+b*sin(f*x+e))/(a+b))^(1/2)-a*(s
in(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticF(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(b/(a+
b))^(1/2))*((a+b*sin(f*x+e))/(a+b))^(1/2)/f/(a+b*sin(f*x+e))^(1/2)-b*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1
/2*e+1/4*Pi+1/2*f*x)*EllipticPi(cos(1/2*e+1/4*Pi+1/2*f*x),2,2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin(f*x+e))/(a+b))^
(1/2)/f/(a+b*sin(f*x+e))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.33, antiderivative size = 213, normalized size of antiderivative = 1.00, 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 = {2875, 3139, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \begin {gather*} -\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)}} \end {gather*}

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 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/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 2734

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/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

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 2742

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/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2875

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-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] && In
tegersQ[2*m, 2*n]

Rule 2884

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(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 2886

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/
(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] && N
eQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]

Rule 3081

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 3139

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]

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] Result contains complex when optimal does not.
time = 16.07, size = 312, normalized size = 1.46 \begin {gather*} \frac {\frac {2 i \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 ) \sec (e+f x) \sqrt {-\frac {b (-1+\sin (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sin (e+f x))}{a-b}}}{a b \sqrt {-\frac {1}{a+b}}}-4 \cot (e+f x) \sqrt {a+b \sin (e+f x)}-\frac {2 b \Pi \left (2;\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{\sqrt {a+b \sin (e+f x)}}}{4 f} \end {gather*}

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)

________________________________________________________________________________________

Maple [A]
time = 3.12, size = 456, normalized size = 2.14

method	result	size
default	\(-\frac {a \,b^{2} \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+\sqrt {\frac {b \sin \left (f x +e \right )}{a -b}+\frac {a}{a -b}}\, \sqrt {-\frac {b \sin \left (f x +e \right )}{a -b}-\frac {b}{a -b}}\, \sqrt {-\frac {b \sin \left (f x +e \right )}{a +b}+\frac {b}{a +b}}\, \left (\EllipticF \left (\sqrt {\frac {b \sin \left (f x +e \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{2} b -\EllipticF \left (\sqrt {\frac {b \sin \left (f x +e \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{2}+\EllipticPi \left (\sqrt {\frac {b \sin \left (f x +e \right )}{a -b}+\frac {a}{a -b}}, \frac {a -b}{a}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{2}-\EllipticPi \left (\sqrt {\frac {b \sin \left (f x +e \right )}{a -b}+\frac {a}{a -b}}, \frac {a -b}{a}, \sqrt {\frac {a -b}{a +b}}\right ) b^{3}-\EllipticE \left (\sqrt {\frac {b \sin \left (f x +e \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3}+\EllipticE \left (\sqrt {\frac {b \sin \left (f x +e \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{2}\right ) \sin \left (f x +e \right )+a^{2} b \left (\cos ^{2}\left (f x +e \right )\right )}{a b \sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {a +b \sin \left (f x +e \right )}\, f}\)	\(456\)

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,method=_RETURNVERBOSE)

[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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a + b \sin {\left (e + f x \right )}} \csc ^{2}{\left (e + f x \right )}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {a+b\,\sin \left (e+f\,x\right )}}{{\sin \left (e+f\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________