3.238 \(\int \frac{\csc ^2(a+b x)}{\sqrt{d \cos (a+b x)}} \, dx\)

Optimal. Leaf size=64 \[ \frac{\sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b \sqrt{d \cos (a+b x)}}-\frac{\csc (a+b x) \sqrt{d \cos (a+b x)}}{b d} \]

[Out]

-((Sqrt[d*Cos[a + b*x]]*Csc[a + b*x])/(b*d)) + (Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2])/(b*Sqrt[d*Cos[a
+ b*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.0578789, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2570, 2642, 2641} \[ \frac{\sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b \sqrt{d \cos (a+b x)}}-\frac{\csc (a+b x) \sqrt{d \cos (a+b x)}}{b d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^2/Sqrt[d*Cos[a + b*x]],x]

[Out]

-((Sqrt[d*Cos[a + b*x]]*Csc[a + b*x])/(b*d)) + (Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2])/(b*Sqrt[d*Cos[a
+ b*x]])

Rule 2570

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

Rule 2642

Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

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]

Rubi steps

\begin{align*} \int \frac{\csc ^2(a+b x)}{\sqrt{d \cos (a+b x)}} \, dx &=-\frac{\sqrt{d \cos (a+b x)} \csc (a+b x)}{b d}+\frac{1}{2} \int \frac{1}{\sqrt{d \cos (a+b x)}} \, dx\\ &=-\frac{\sqrt{d \cos (a+b x)} \csc (a+b x)}{b d}+\frac{\sqrt{\cos (a+b x)} \int \frac{1}{\sqrt{\cos (a+b x)}} \, dx}{2 \sqrt{d \cos (a+b x)}}\\ &=-\frac{\sqrt{d \cos (a+b x)} \csc (a+b x)}{b d}+\frac{\sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b \sqrt{d \cos (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.0836877, size = 47, normalized size = 0.73 \[ \frac{\sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )-\cot (a+b x)}{b \sqrt{d \cos (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^2/Sqrt[d*Cos[a + b*x]],x]

[Out]

(-Cot[a + b*x] + Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2])/(b*Sqrt[d*Cos[a + b*x]])

________________________________________________________________________________________

Maple [B]  time = 0.27, size = 188, normalized size = 2.9 \begin{align*}{\frac{d}{2\,b}\sqrt{d \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}}\sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \left ( 2\, \left ( 2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) ^{3/2}\sqrt{ \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}{\it EllipticF} \left ( \cos \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) \cos \left ( 1/2\,bx+a/2 \right ) -4\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+4\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) \left ( \cos \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-1} \left ( -2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}d+ \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{d \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^2/(d*cos(b*x+a))^(1/2),x)

[Out]

1/2*(d*(2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)/cos(1/2*b*x+1/2*a)/(-2*sin(1/2*b*x+1/2*a)^4*d+si
n(1/2*b*x+1/2*a)^2*d)^(3/2)*d*sin(1/2*b*x+1/2*a)*(2*(2*sin(1/2*b*x+1/2*a)^2-1)^(3/2)*(sin(1/2*b*x+1/2*a)^2)^(1
/2)*EllipticF(cos(1/2*b*x+1/2*a),2^(1/2))*cos(1/2*b*x+1/2*a)-4*sin(1/2*b*x+1/2*a)^4+4*sin(1/2*b*x+1/2*a)^2-1)/
(d*(2*cos(1/2*b*x+1/2*a)^2-1))^(1/2)/b

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^2/(d*cos(b*x+a))^(1/2),x, algorithm="maxima")

[Out]

integrate(csc(b*x + a)^2/sqrt(d*cos(b*x + a)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^2/(d*cos(b*x+a))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*cos(b*x + a))*csc(b*x + a)^2/(d*cos(b*x + a)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)**2/(d*cos(b*x+a))**(1/2),x)

[Out]

Integral(csc(a + b*x)**2/sqrt(d*cos(a + b*x)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^2/(d*cos(b*x+a))^(1/2),x, algorithm="giac")

[Out]

integrate(csc(b*x + a)^2/sqrt(d*cos(b*x + a)), x)