3.236 \(\int (d \cos (a+b x))^{3/2} \csc ^2(a+b x) \, dx\)

Optimal. Leaf size=66 \[ -\frac{d^2 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b \sqrt{d \cos (a+b x)}}-\frac{d \csc (a+b x) \sqrt{d \cos (a+b x)}}{b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0637282, antiderivative size = 66, 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 = {2567, 2642, 2641} \[ -\frac{d^2 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b \sqrt{d \cos (a+b x)}}-\frac{d \csc (a+b x) \sqrt{d \cos (a+b x)}}{b} \]

Antiderivative was successfully verified.

[In]

Int[(d*Cos[a + b*x])^(3/2)*Csc[a + b*x]^2,x]

[Out]

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

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 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 (d \cos (a+b x))^{3/2} \csc ^2(a+b x) \, dx &=-\frac{d \sqrt{d \cos (a+b x)} \csc (a+b x)}{b}-\frac{1}{2} d^2 \int \frac{1}{\sqrt{d \cos (a+b x)}} \, dx\\ &=-\frac{d \sqrt{d \cos (a+b x)} \csc (a+b x)}{b}-\frac{\left (d^2 \sqrt{\cos (a+b x)}\right ) \int \frac{1}{\sqrt{\cos (a+b x)}} \, dx}{2 \sqrt{d \cos (a+b x)}}\\ &=-\frac{d \sqrt{d \cos (a+b x)} \csc (a+b x)}{b}-\frac{d^2 \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 = 0.0975214, size = 56, normalized size = 0.85 \[ -\frac{(d \cos (a+b x))^{3/2} \left (F\left (\left .\frac{1}{2} (a+b x)\right |2\right )+\sqrt{\cos (a+b x)} \csc (a+b x)\right )}{b \cos ^{\frac{3}{2}}(a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Cos[a + b*x])^(3/2)*Csc[a + b*x]^2,x]

[Out]

-(((d*Cos[a + b*x])^(3/2)*(Sqrt[Cos[a + b*x]]*Csc[a + b*x] + EllipticF[(a + b*x)/2, 2]))/(b*Cos[a + b*x]^(3/2)
))

________________________________________________________________________________________

Maple [B]  time = 0.266, size = 190, normalized size = 2.9 \begin{align*} -{\frac{{d}^{3}}{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 ( -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}}} \left ( \cos \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-1}{\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((d*cos(b*x+a))^(3/2)*csc(b*x+a)^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)*d^3/(-2*sin(1/2*b*x+1/2*a)^4*d+sin(1/2*b*x+1/2*
a)^2*d)^(3/2)/cos(1/2*b*x+1/2*a)*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 \left (d \cos \left (b x + a\right )\right )^{\frac{3}{2}} \csc \left (b x + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^(3/2)*csc(b*x+a)^2,x, algorithm="maxima")

[Out]

integrate((d*cos(b*x + a))^(3/2)*csc(b*x + a)^2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^(3/2)*csc(b*x+a)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [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((d*cos(b*x+a))**(3/2)*csc(b*x+a)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cos \left (b x + a\right )\right )^{\frac{3}{2}} \csc \left (b x + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^(3/2)*csc(b*x+a)^2,x, algorithm="giac")

[Out]

integrate((d*cos(b*x + a))^(3/2)*csc(b*x + a)^2, x)