3.302 \(\int \frac{\sqrt{e \csc (c+d x)}}{(a+a \sec (c+d x))^2} \, dx\)

Optimal. Leaf size=201 \[ \frac{20 \sqrt{\sin (c+d x)} \text{EllipticF}\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right ),2\right ) \sqrt{e \csc (c+d x)}}{21 a^2 d}+\frac{4 \csc ^3(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}-\frac{4 \csc (c+d x) \sqrt{e \csc (c+d x)}}{3 a^2 d}-\frac{2 \cot ^3(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}-\frac{2 \cot (c+d x) \csc ^2(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}+\frac{16 \cot (c+d x) \sqrt{e \csc (c+d x)}}{21 a^2 d} \]

[Out]

(16*Cot[c + d*x]*Sqrt[e*Csc[c + d*x]])/(21*a^2*d) - (2*Cot[c + d*x]^3*Sqrt[e*Csc[c + d*x]])/(7*a^2*d) - (4*Csc
[c + d*x]*Sqrt[e*Csc[c + d*x]])/(3*a^2*d) - (2*Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[e*Csc[c + d*x]])/(7*a^2*d) + (
4*Csc[c + d*x]^3*Sqrt[e*Csc[c + d*x]])/(7*a^2*d) + (20*Sqrt[e*Csc[c + d*x]]*EllipticF[(c - Pi/2 + d*x)/2, 2]*S
qrt[Sin[c + d*x]])/(21*a^2*d)

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Rubi [A]  time = 0.448139, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {3878, 3872, 2875, 2873, 2567, 2636, 2641, 2564, 14} \[ \frac{4 \csc ^3(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}-\frac{4 \csc (c+d x) \sqrt{e \csc (c+d x)}}{3 a^2 d}-\frac{2 \cot ^3(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}-\frac{2 \cot (c+d x) \csc ^2(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}+\frac{16 \cot (c+d x) \sqrt{e \csc (c+d x)}}{21 a^2 d}+\frac{20 \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \csc (c+d x)}}{21 a^2 d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[e*Csc[c + d*x]]/(a + a*Sec[c + d*x])^2,x]

[Out]

(16*Cot[c + d*x]*Sqrt[e*Csc[c + d*x]])/(21*a^2*d) - (2*Cot[c + d*x]^3*Sqrt[e*Csc[c + d*x]])/(7*a^2*d) - (4*Csc
[c + d*x]*Sqrt[e*Csc[c + d*x]])/(3*a^2*d) - (2*Cot[c + d*x]*Csc[c + d*x]^2*Sqrt[e*Csc[c + d*x]])/(7*a^2*d) + (
4*Csc[c + d*x]^3*Sqrt[e*Csc[c + d*x]])/(7*a^2*d) + (20*Sqrt[e*Csc[c + d*x]]*EllipticF[(c - Pi/2 + d*x)/2, 2]*S
qrt[Sin[c + d*x]])/(21*a^2*d)

Rule 3878

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*((g_.)*sec[(e_.) + (f_.)*(x_)])^(p_), x_Symbol] :> Dist[g^Int
Part[p]*(g*Sec[e + f*x])^FracPart[p]*Cos[e + f*x]^FracPart[p], Int[(a + b*Csc[e + f*x])^m/Cos[e + f*x]^p, x],
x] /; FreeQ[{a, b, e, f, g, m, p}, x] &&  !IntegerQ[p]

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rule 2875

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Dist[(a/g)^(2*m), Int[((g*Cos[e + f*x])^(2*m + p)*(d*Sin[e + f*x])^n)/(a - b*Sin[e +
f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]

Rule 2873

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2567

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*(a*Cos[e +
f*x])^(m - 1)*(b*Sin[e + f*x])^(n + 1))/(b*f*(n + 1)), x] + Dist[(a^2*(m - 1))/(b^2*(n + 1)), Int[(a*Cos[e +
f*x])^(m - 2)*(b*Sin[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && (Intege
rsQ[2*m, 2*n] || EqQ[m + n, 0])

Rule 2636

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1))/(b*d*(n +
1)), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*n]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rule 2564

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{\sqrt{e \csc (c+d x)}}{(a+a \sec (c+d x))^2} \, dx &=\left (\sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{1}{(a+a \sec (c+d x))^2 \sqrt{\sin (c+d x)}} \, dx\\ &=\left (\sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{\cos ^2(c+d x)}{(-a-a \cos (c+d x))^2 \sqrt{\sin (c+d x)}} \, dx\\ &=\frac{\left (\sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{\cos ^2(c+d x) (-a+a \cos (c+d x))^2}{\sin ^{\frac{9}{2}}(c+d x)} \, dx}{a^4}\\ &=\frac{\left (\sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \left (\frac{a^2 \cos ^2(c+d x)}{\sin ^{\frac{9}{2}}(c+d x)}-\frac{2 a^2 \cos ^3(c+d x)}{\sin ^{\frac{9}{2}}(c+d x)}+\frac{a^2 \cos ^4(c+d x)}{\sin ^{\frac{9}{2}}(c+d x)}\right ) \, dx}{a^4}\\ &=\frac{\left (\sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{\cos ^2(c+d x)}{\sin ^{\frac{9}{2}}(c+d x)} \, dx}{a^2}+\frac{\left (\sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{\cos ^4(c+d x)}{\sin ^{\frac{9}{2}}(c+d x)} \, dx}{a^2}-\frac{\left (2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{\cos ^3(c+d x)}{\sin ^{\frac{9}{2}}(c+d x)} \, dx}{a^2}\\ &=-\frac{2 \cot ^3(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}-\frac{2 \cot (c+d x) \csc ^2(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}-\frac{\left (2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sin ^{\frac{5}{2}}(c+d x)} \, dx}{7 a^2}-\frac{\left (6 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{\cos ^2(c+d x)}{\sin ^{\frac{5}{2}}(c+d x)} \, dx}{7 a^2}-\frac{\left (2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1-x^2}{x^{9/2}} \, dx,x,\sin (c+d x)\right )}{a^2 d}\\ &=\frac{16 \cot (c+d x) \sqrt{e \csc (c+d x)}}{21 a^2 d}-\frac{2 \cot ^3(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}-\frac{2 \cot (c+d x) \csc ^2(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}-\frac{\left (2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{21 a^2}+\frac{\left (4 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{7 a^2}-\frac{\left (2 \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^{9/2}}-\frac{1}{x^{5/2}}\right ) \, dx,x,\sin (c+d x)\right )}{a^2 d}\\ &=\frac{16 \cot (c+d x) \sqrt{e \csc (c+d x)}}{21 a^2 d}-\frac{2 \cot ^3(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}-\frac{4 \csc (c+d x) \sqrt{e \csc (c+d x)}}{3 a^2 d}-\frac{2 \cot (c+d x) \csc ^2(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}+\frac{4 \csc ^3(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}+\frac{20 \sqrt{e \csc (c+d x)} F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{21 a^2 d}\\ \end{align*}

Mathematica [A]  time = 0.690762, size = 82, normalized size = 0.41 \[ -\frac{4 \csc ^3(c+d x) \sqrt{e \csc (c+d x)} \left (5 \sin ^{\frac{7}{2}}(c+d x) \text{EllipticF}\left (\frac{1}{4} (-2 c-2 d x+\pi ),2\right )+2 \sin ^4\left (\frac{1}{2} (c+d x)\right ) (11 \cos (c+d x)+8)\right )}{21 a^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[e*Csc[c + d*x]]/(a + a*Sec[c + d*x])^2,x]

[Out]

(-4*Csc[c + d*x]^3*Sqrt[e*Csc[c + d*x]]*(2*(8 + 11*Cos[c + d*x])*Sin[(c + d*x)/2]^4 + 5*EllipticF[(-2*c + Pi -
2*d*x)/4, 2]*Sin[c + d*x]^(7/2)))/(21*a^2*d)

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Maple [C]  time = 0.24, size = 474, normalized size = 2.4 \begin{align*} -{\frac{\sqrt{2} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) ^{3}}{21\,d{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}\sqrt{{\frac{e}{\sin \left ( dx+c \right ) }}} \left ( 10\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) \sqrt{{\frac{-i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }}}\sqrt{{\frac{i\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) -i}{\sin \left ( dx+c \right ) }}}\sqrt{-{\frac{i\cos \left ( dx+c \right ) -\sin \left ( dx+c \right ) -i}{\sin \left ( dx+c \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) -i}{\sin \left ( dx+c \right ) }}},{\frac{\sqrt{2}}{2}} \right ) +20\,i\cos \left ( dx+c \right ) \sqrt{{\frac{i\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) -i}{\sin \left ( dx+c \right ) }}}\sqrt{{\frac{-i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }}}\sqrt{-{\frac{i\cos \left ( dx+c \right ) -\sin \left ( dx+c \right ) -i}{\sin \left ( dx+c \right ) }}}\sin \left ( dx+c \right ){\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) -i}{\sin \left ( dx+c \right ) }}},{\frac{\sqrt{2}}{2}} \right ) +10\,i\sqrt{{\frac{i\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) -i}{\sin \left ( dx+c \right ) }}}\sqrt{{\frac{-i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }}}\sqrt{-{\frac{i\cos \left ( dx+c \right ) -\sin \left ( dx+c \right ) -i}{\sin \left ( dx+c \right ) }}}\sin \left ( dx+c \right ){\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) -i}{\sin \left ( dx+c \right ) }}},{\frac{\sqrt{2}}{2}} \right ) +11\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sqrt{2}-3\,\cos \left ( dx+c \right ) \sqrt{2}-8\,\sqrt{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21/a^2/d*2^(1/2)*(e/sin(d*x+c))^(1/2)*(cos(d*x+c)+1)^2*(-1+cos(d*x+c))^3*(10*I*cos(d*x+c)^2*sin(d*x+c)*(-I*
(-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2)*(-(I*cos(d*x+c)-sin(d*x+c)-I)
/sin(d*x+c))^(1/2)*EllipticF(((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2),1/2*2^(1/2))+20*I*cos(d*x+c)*((I*c
os(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2)*(-I*(-1+cos(d*x+c))/sin(d*x+c))^(1/2)*(-(I*cos(d*x+c)-sin(d*x+c)-I)/
sin(d*x+c))^(1/2)*sin(d*x+c)*EllipticF(((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2),1/2*2^(1/2))+10*I*((I*co
s(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2)*(-I*(-1+cos(d*x+c))/sin(d*x+c))^(1/2)*(-(I*cos(d*x+c)-sin(d*x+c)-I)/s
in(d*x+c))^(1/2)*sin(d*x+c)*EllipticF(((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2),1/2*2^(1/2))+11*cos(d*x+c
)^2*2^(1/2)-3*cos(d*x+c)*2^(1/2)-8*2^(1/2))/sin(d*x+c)^7

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*csc(d*x+c))^(1/2)/(a+a*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*csc(d*x+c))^(1/2)/(a+a*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

integral(sqrt(e*csc(d*x + c))/(a^2*sec(d*x + c)^2 + 2*a^2*sec(d*x + c) + a^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sqrt{e \csc{\left (c + d x \right )}}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec{\left (c + d x \right )} + 1}\, dx}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*csc(d*x+c))**(1/2)/(a+a*sec(d*x+c))**2,x)

[Out]

Integral(sqrt(e*csc(c + d*x))/(sec(c + d*x)**2 + 2*sec(c + d*x) + 1), x)/a**2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*csc(d*x+c))^(1/2)/(a+a*sec(d*x+c))^2,x, algorithm="giac")

[Out]

integrate(sqrt(e*csc(d*x + c))/(a*sec(d*x + c) + a)^2, x)