3.536 \(\int \frac {\csc ^4(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=77 \[ \frac {2 \sqrt {\sin (e+f x)} F\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {d \csc (e+f x)}}{3 d^2 f}-\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d^3 f} \]

[Out]

-2/3*cos(f*x+e)*(d*csc(f*x+e))^(3/2)/d^3/f-2/3*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*E
llipticF(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2))*(d*csc(f*x+e))^(1/2)*sin(f*x+e)^(1/2)/d^2/f

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Rubi [A]  time = 0.04, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {16, 3768, 3771, 2641} \[ \frac {2 \sqrt {\sin (e+f x)} F\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {d \csc (e+f x)}}{3 d^2 f}-\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d^3 f} \]

Antiderivative was successfully verified.

[In]

Int[Csc[e + f*x]^4/(d*Csc[e + f*x])^(3/2),x]

[Out]

(-2*Cos[e + f*x]*(d*Csc[e + f*x])^(3/2))/(3*d^3*f) + (2*Sqrt[d*Csc[e + f*x]]*EllipticF[(e - Pi/2 + f*x)/2, 2]*
Sqrt[Sin[e + f*x]])/(3*d^2*f)

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 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]

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]

Rubi steps

\begin {align*} \int \frac {\csc ^4(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx &=\frac {\int (d \csc (e+f x))^{5/2} \, dx}{d^4}\\ &=-\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d^3 f}+\frac {\int \sqrt {d \csc (e+f x)} \, dx}{3 d^2}\\ &=-\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d^3 f}+\frac {\left (\sqrt {d \csc (e+f x)} \sqrt {\sin (e+f x)}\right ) \int \frac {1}{\sqrt {\sin (e+f x)}} \, dx}{3 d^2}\\ &=-\frac {2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d^3 f}+\frac {2 \sqrt {d \csc (e+f x)} F\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right ) \sqrt {\sin (e+f x)}}{3 d^2 f}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 60, normalized size = 0.78 \[ -\frac {2 \csc ^3(e+f x) \left (\cos (e+f x)+\sin ^{\frac {3}{2}}(e+f x) F\left (\left .\frac {1}{4} (-2 e-2 f x+\pi )\right |2\right )\right )}{3 f (d \csc (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[e + f*x]^4/(d*Csc[e + f*x])^(3/2),x]

[Out]

(-2*Csc[e + f*x]^3*(Cos[e + f*x] + EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sin[e + f*x]^(3/2)))/(3*f*(d*Csc[e + f*
x])^(3/2))

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fricas [F]  time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \csc \left (f x + e\right )} \csc \left (f x + e\right )^{2}}{d^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^4/(d*csc(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*csc(f*x + e))*csc(f*x + e)^2/d^2, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (f x + e\right )^{4}}{\left (d \csc \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^4/(d*csc(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate(csc(f*x + e)^4/(d*csc(f*x + e))^(3/2), x)

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maple [C]  time = 0.17, size = 319, normalized size = 4.14 \[ \frac {\left (\cos \left (f x +e \right )+1\right )^{2} \left (-1+\cos \left (f x +e \right )\right )^{2} \left (i \EllipticF \left (\sqrt {\frac {i \cos \left (f x +e \right )+\sin \left (f x +e \right )-i}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {i \cos \left (f x +e \right )+\sin \left (f x +e \right )-i}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {i \cos \left (f x +e \right )-\sin \left (f x +e \right )-i}{\sin \left (f x +e \right )}}\, \sin \left (f x +e \right ) \cos \left (f x +e \right )+i \sin \left (f x +e \right ) \sqrt {-\frac {i \cos \left (f x +e \right )-\sin \left (f x +e \right )-i}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (f x +e \right )+\sin \left (f x +e \right )-i}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {i \cos \left (f x +e \right )+\sin \left (f x +e \right )-i}{\sin \left (f x +e \right )}}-\cos \left (f x +e \right ) \sqrt {2}\right ) \sqrt {2}}{3 f \sin \left (f x +e \right )^{7} \left (\frac {d}{\sin \left (f x +e \right )}\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(f*x+e)^4/(d*csc(f*x+e))^(3/2),x)

[Out]

1/3/f*(cos(f*x+e)+1)^2*(-1+cos(f*x+e))^2*(I*EllipticF(((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)*(-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)*sin(f*x+e)*cos(f*x+e)+I*sin(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))-cos(f*x+e)*2^(1/2))/sin(f*x+e)^7/(d/sin(f*x+e))^(3/2)*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc \left (f x + e\right )^{4}}{\left (d \csc \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^4/(d*csc(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate(csc(f*x + e)^4/(d*csc(f*x + e))^(3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\sin \left (e+f\,x\right )}^4\,{\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(e + f*x)^4*(d/sin(e + f*x))^(3/2)),x)

[Out]

int(1/(sin(e + f*x)^4*(d/sin(e + f*x))^(3/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{4}{\left (e + f x \right )}}{\left (d \csc {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)**4/(d*csc(f*x+e))**(3/2),x)

[Out]

Integral(csc(e + f*x)**4/(d*csc(e + f*x))**(3/2), x)

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