3.235 \(\int (d \cos (a+b x))^{5/2} \csc ^2(a+b x) \, dx\)

Optimal. Leaf size=66 \[ -\frac{3 d^2 E\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{d \cos (a+b x)}}{b \sqrt{\cos (a+b x)}}-\frac{d \csc (a+b x) (d \cos (a+b x))^{3/2}}{b} \]

[Out]

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

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Rubi [A]  time = 0.0627215, 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, 2640, 2639} \[ -\frac{3 d^2 E\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{d \cos (a+b x)}}{b \sqrt{\cos (a+b x)}}-\frac{d \csc (a+b x) (d \cos (a+b x))^{3/2}}{b} \]

Antiderivative was successfully verified.

[In]

Int[(d*Cos[a + b*x])^(5/2)*Csc[a + b*x]^2,x]

[Out]

-((d*(d*Cos[a + b*x])^(3/2)*Csc[a + b*x])/b) - (3*d^2*Sqrt[d*Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2])/(b*Sqrt[
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 2640

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

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 (d \cos (a+b x))^{5/2} \csc ^2(a+b x) \, dx &=-\frac{d (d \cos (a+b x))^{3/2} \csc (a+b x)}{b}-\frac{1}{2} \left (3 d^2\right ) \int \sqrt{d \cos (a+b x)} \, dx\\ &=-\frac{d (d \cos (a+b x))^{3/2} \csc (a+b x)}{b}-\frac{\left (3 d^2 \sqrt{d \cos (a+b x)}\right ) \int \sqrt{\cos (a+b x)} \, dx}{2 \sqrt{\cos (a+b x)}}\\ &=-\frac{d (d \cos (a+b x))^{3/2} \csc (a+b x)}{b}-\frac{3 d^2 \sqrt{d \cos (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b \sqrt{\cos (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.14493, size = 58, normalized size = 0.88 \[ -\frac{(d \cos (a+b x))^{5/2} \left (3 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )+\cos ^{\frac{3}{2}}(a+b x) \csc (a+b x)\right )}{b \cos ^{\frac{5}{2}}(a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Cos[a + b*x])^(5/2)*Csc[a + b*x]^2,x]

[Out]

-(((d*Cos[a + b*x])^(5/2)*(Cos[a + b*x]^(3/2)*Csc[a + b*x] + 3*EllipticE[(a + b*x)/2, 2]))/(b*Cos[a + b*x]^(5/
2)))

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Maple [B]  time = 0.396, size = 203, normalized size = 3.1 \begin{align*}{\frac{{d}^{4}}{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 ( 6\, \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 EllipticE} \left ( \cos \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) \cos \left ( 1/2\,bx+a/2 \right ) +8\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{6}-12\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+6\, \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))^(5/2)*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^4/(-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)*(6*(2*sin(1/2*b*x+1/2*a)^2-1)^(3/2)*(sin(1/2*b*x+1/2*a)^2)^
(1/2)*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2))*cos(1/2*b*x+1/2*a)+8*sin(1/2*b*x+1/2*a)^6-12*sin(1/2*b*x+1/2*a)^4+
6*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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cos \left (b x + a\right )\right )^{\frac{5}{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))^(5/2)*csc(b*x+a)^2,x, algorithm="maxima")

[Out]

integrate((d*cos(b*x + a))^(5/2)*csc(b*x + a)^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{d \cos \left (b x + a\right )} d^{2} \cos \left (b x + a\right )^{2} \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))^(5/2)*csc(b*x+a)^2,x, algorithm="fricas")

[Out]

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

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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))**(5/2)*csc(b*x+a)**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cos \left (b x + a\right )\right )^{\frac{5}{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))^(5/2)*csc(b*x+a)^2,x, algorithm="giac")

[Out]

integrate((d*cos(b*x + a))^(5/2)*csc(b*x + a)^2, x)