3.296 \(\int \frac{1}{\sqrt{e \csc (c+d x)} (a+a \sec (c+d x))} \, dx\)

Optimal. Leaf size=99 \[ -\frac{2 \csc (c+d x)}{a d \sqrt{e \csc (c+d x)}}+\frac{2 \cot (c+d x)}{a d \sqrt{e \csc (c+d x)}}+\frac{4 E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{a d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}} \]

[Out]

(2*Cot[c + d*x])/(a*d*Sqrt[e*Csc[c + d*x]]) - (2*Csc[c + d*x])/(a*d*Sqrt[e*Csc[c + d*x]]) + (4*EllipticE[(c -
Pi/2 + d*x)/2, 2])/(a*d*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])

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Rubi [A]  time = 0.210942, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {3878, 3872, 2839, 2564, 30, 2567, 2639} \[ -\frac{2 \csc (c+d x)}{a d \sqrt{e \csc (c+d x)}}+\frac{2 \cot (c+d x)}{a d \sqrt{e \csc (c+d x)}}+\frac{4 E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{a d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[e*Csc[c + d*x]]*(a + a*Sec[c + d*x])),x]

[Out]

(2*Cot[c + d*x])/(a*d*Sqrt[e*Csc[c + d*x]]) - (2*Csc[c + d*x])/(a*d*Sqrt[e*Csc[c + d*x]]) + (4*EllipticE[(c -
Pi/2 + d*x)/2, 2])/(a*d*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])

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 2839

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^2, 0]

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 2639

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

Rubi steps

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

Mathematica [C]  time = 0.602061, size = 95, normalized size = 0.96 \[ \frac{6 (\cot (c+d x)-\csc (c+d x)+2 i)-4 \sqrt{1-e^{2 i (c+d x)}} (\cot (c+d x)+i) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},e^{2 i (c+d x)}\right )}{3 a d \sqrt{e \csc (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[e*Csc[c + d*x]]*(a + a*Sec[c + d*x])),x]

[Out]

(6*(2*I + Cot[c + d*x] - Csc[c + d*x]) - 4*Sqrt[1 - E^((2*I)*(c + d*x))]*(I + Cot[c + d*x])*Hypergeometric2F1[
1/2, 3/4, 7/4, E^((2*I)*(c + d*x))])/(3*a*d*Sqrt[e*Csc[c + d*x]])

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Maple [C]  time = 0.214, size = 524, normalized size = 5.3 \begin{align*} -{\frac{\sqrt{2}}{ad\sin \left ( dx+c \right ) } \left ( 4\,\sqrt{{\frac{-i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }}}\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\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) +i}{\sin \left ( dx+c \right ) }}}{\it EllipticE} \left ( \sqrt{{\frac{i\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) -i}{\sin \left ( dx+c \right ) }}},1/2\,\sqrt{2} \right ) -2\,\sqrt{{\frac{-i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }}}\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\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 ) }}},1/2\,\sqrt{2} \right ) +4\,\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 ) }}}{\it EllipticE} \left ( \sqrt{{\frac{i\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) -i}{\sin \left ( dx+c \right ) }}},1/2\,\sqrt{2} \right ) -2\,\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 ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) -i}{\sin \left ( dx+c \right ) }}},1/2\,\sqrt{2} \right ) +\cos \left ( dx+c \right ) \sqrt{2}-\sqrt{2} \right ){\frac{1}{\sqrt{{\frac{e}{\sin \left ( dx+c \right ) }}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a/d*2^(1/2)*(4*(-I*(-1+cos(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)*((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)*EllipticE(((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2),1
/2*2^(1/2))-2*(-I*(-1+cos(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)*((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))+4*((I*cos(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)*EllipticE(((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2),1/2*2^(1/2))-2*((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)*EllipticF(((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2),1/2*2^(1/2))+cos(d*x+c)*2^(1/2)-2^(
1/2))/(e/sin(d*x+c))^(1/2)/sin(d*x+c)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 e \csc \left (d x + c\right ) \sec \left (d x + c\right ) + a e \csc \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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