3.522 \(\int \csc ^2(e+f x) (d \csc (e+f x))^{3/2} \, dx\)

Optimal. Leaf size=103 \[ -\frac{6 d^2 E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right )}{5 f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}-\frac{2 \cos (e+f x) (d \csc (e+f x))^{5/2}}{5 d f}-\frac{6 d \cos (e+f x) \sqrt{d \csc (e+f x)}}{5 f} \]

[Out]

(-6*d*Cos[e + f*x]*Sqrt[d*Csc[e + f*x]])/(5*f) - (2*Cos[e + f*x]*(d*Csc[e + f*x])^(5/2))/(5*d*f) - (6*d^2*Elli
pticE[(e - Pi/2 + f*x)/2, 2])/(5*f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])

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Rubi [A]  time = 0.0580863, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {16, 3768, 3771, 2639} \[ -\frac{6 d^2 E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right )}{5 f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}-\frac{2 \cos (e+f x) (d \csc (e+f x))^{5/2}}{5 d f}-\frac{6 d \cos (e+f x) \sqrt{d \csc (e+f x)}}{5 f} \]

Antiderivative was successfully verified.

[In]

Int[Csc[e + f*x]^2*(d*Csc[e + f*x])^(3/2),x]

[Out]

(-6*d*Cos[e + f*x]*Sqrt[d*Csc[e + f*x]])/(5*f) - (2*Cos[e + f*x]*(d*Csc[e + f*x])^(5/2))/(5*d*f) - (6*d^2*Elli
pticE[(e - Pi/2 + f*x)/2, 2])/(5*f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 3768

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

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

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 \csc ^2(e+f x) (d \csc (e+f x))^{3/2} \, dx &=\frac{\int (d \csc (e+f x))^{7/2} \, dx}{d^2}\\ &=-\frac{2 \cos (e+f x) (d \csc (e+f x))^{5/2}}{5 d f}+\frac{3}{5} \int (d \csc (e+f x))^{3/2} \, dx\\ &=-\frac{6 d \cos (e+f x) \sqrt{d \csc (e+f x)}}{5 f}-\frac{2 \cos (e+f x) (d \csc (e+f x))^{5/2}}{5 d f}-\frac{1}{5} \left (3 d^2\right ) \int \frac{1}{\sqrt{d \csc (e+f x)}} \, dx\\ &=-\frac{6 d \cos (e+f x) \sqrt{d \csc (e+f x)}}{5 f}-\frac{2 \cos (e+f x) (d \csc (e+f x))^{5/2}}{5 d f}-\frac{\left (3 d^2\right ) \int \sqrt{\sin (e+f x)} \, dx}{5 \sqrt{d \csc (e+f x)} \sqrt{\sin (e+f x)}}\\ &=-\frac{6 d \cos (e+f x) \sqrt{d \csc (e+f x)}}{5 f}-\frac{2 \cos (e+f x) (d \csc (e+f x))^{5/2}}{5 d f}-\frac{6 d^2 E\left (\left .\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )\right |2\right )}{5 f \sqrt{d \csc (e+f x)} \sqrt{\sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 0.243958, size = 68, normalized size = 0.66 \[ \frac{(d \csc (e+f x))^{5/2} \left (-7 \cos (e+f x)+3 \cos (3 (e+f x))+12 \sin ^{\frac{5}{2}}(e+f x) E\left (\left .\frac{1}{4} (-2 e-2 f x+\pi )\right |2\right )\right )}{10 d f} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[e + f*x]^2*(d*Csc[e + f*x])^(3/2),x]

[Out]

((d*Csc[e + f*x])^(5/2)*(-7*Cos[e + f*x] + 3*Cos[3*(e + f*x)] + 12*EllipticE[(-2*e + Pi - 2*f*x)/4, 2]*Sin[e +
f*x]^(5/2)))/(10*d*f)

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Maple [C]  time = 0.135, size = 1054, normalized size = 10.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(f*x+e)^2*(d*csc(f*x+e))^(3/2),x)

[Out]

-1/5/f*2^(1/2)*(d/sin(f*x+e))^(3/2)*(6*cos(f*x+e)^3*(-I*(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*(-(I*cos(f*x+e)-sin(
f*x+e)-I)/sin(f*x+e))^(1/2)*EllipticE(((I*cos(f*x+e)+sin(f*x+e)-I)/sin(f*x+e))^(1/2),1/2*2^(1/2))*((I*cos(f*x+
e)+sin(f*x+e)-I)/sin(f*x+e))^(1/2)-3*cos(f*x+e)^3*(-I*(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*(-(I*cos(f*x+e)-sin(f*
x+e)-I)/sin(f*x+e))^(1/2)*((I*cos(f*x+e)+sin(f*x+e)-I)/sin(f*x+e))^(1/2)*EllipticF(((I*cos(f*x+e)+sin(f*x+e)-I
)/sin(f*x+e))^(1/2),1/2*2^(1/2))+6*(-(I*cos(f*x+e)-sin(f*x+e)-I)/sin(f*x+e))^(1/2)*EllipticE(((I*cos(f*x+e)+si
n(f*x+e)-I)/sin(f*x+e))^(1/2),1/2*2^(1/2))*cos(f*x+e)^2*(-I*(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*((I*cos(f*x+e)+s
in(f*x+e)-I)/sin(f*x+e))^(1/2)-3*(-(I*cos(f*x+e)-sin(f*x+e)-I)/sin(f*x+e))^(1/2)*EllipticF(((I*cos(f*x+e)+sin(
f*x+e)-I)/sin(f*x+e))^(1/2),1/2*2^(1/2))*cos(f*x+e)^2*(-I*(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*((I*cos(f*x+e)+sin
(f*x+e)-I)/sin(f*x+e))^(1/2)-6*cos(f*x+e)*(-I*(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*((I*cos(f*x+e)+sin(f*x+e)-I)/s
in(f*x+e))^(1/2)*(-(I*cos(f*x+e)-sin(f*x+e)-I)/sin(f*x+e))^(1/2)*EllipticE(((I*cos(f*x+e)+sin(f*x+e)-I)/sin(f*
x+e))^(1/2),1/2*2^(1/2))+3*cos(f*x+e)*(-I*(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*((I*cos(f*x+e)+sin(f*x+e)-I)/sin(f
*x+e))^(1/2)*(-(I*cos(f*x+e)-sin(f*x+e)-I)/sin(f*x+e))^(1/2)*EllipticF(((I*cos(f*x+e)+sin(f*x+e)-I)/sin(f*x+e)
)^(1/2),1/2*2^(1/2))-6*(-I*(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*((I*cos(f*x+e)+sin(f*x+e)-I)/sin(f*x+e))^(1/2)*(-
(I*cos(f*x+e)-sin(f*x+e)-I)/sin(f*x+e))^(1/2)*EllipticE(((I*cos(f*x+e)+sin(f*x+e)-I)/sin(f*x+e))^(1/2),1/2*2^(
1/2))+3*(-I*(-1+cos(f*x+e))/sin(f*x+e))^(1/2)*((I*cos(f*x+e)+sin(f*x+e)-I)/sin(f*x+e))^(1/2)*(-(I*cos(f*x+e)-s
in(f*x+e)-I)/sin(f*x+e))^(1/2)*EllipticF(((I*cos(f*x+e)+sin(f*x+e)-I)/sin(f*x+e))^(1/2),1/2*2^(1/2))-3*2^(1/2)
*cos(f*x+e)^2+2^(1/2)*cos(f*x+e)+3*2^(1/2))/sin(f*x+e)

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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(csc(f*x+e)^2*(d*csc(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{d \csc \left (f x + e\right )} d \csc \left (f x + e\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^2*(d*csc(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*csc(f*x + e))*d*csc(f*x + e)^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \csc{\left (e + f x \right )}\right )^{\frac{3}{2}} \csc ^{2}{\left (e + f x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)**2*(d*csc(f*x+e))**(3/2),x)

[Out]

Integral((d*csc(e + f*x))**(3/2)*csc(e + f*x)**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^2*(d*csc(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate((d*csc(f*x + e))^(3/2)*csc(f*x + e)^2, x)