3.24 \(\int \frac{1}{(c \csc (a+b x))^{7/2}} \, dx\)

Optimal. Leaf size=105 \[ \frac{10 \sqrt{\sin (a+b x)} \text{EllipticF}\left (\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right ),2\right ) \sqrt{c \csc (a+b x)}}{21 b c^4}-\frac{10 \cos (a+b x)}{21 b c^3 \sqrt{c \csc (a+b x)}}-\frac{2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}} \]

[Out]

(-2*Cos[a + b*x])/(7*b*c*(c*Csc[a + b*x])^(5/2)) - (10*Cos[a + b*x])/(21*b*c^3*Sqrt[c*Csc[a + b*x]]) + (10*Sqr
t[c*Csc[a + b*x]]*EllipticF[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(21*b*c^4)

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Rubi [A]  time = 0.0491781, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3769, 3771, 2641} \[ -\frac{10 \cos (a+b x)}{21 b c^3 \sqrt{c \csc (a+b x)}}+\frac{10 \sqrt{\sin (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{c \csc (a+b x)}}{21 b c^4}-\frac{2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(c*Csc[a + b*x])^(-7/2),x]

[Out]

(-2*Cos[a + b*x])/(7*b*c*(c*Csc[a + b*x])^(5/2)) - (10*Cos[a + b*x])/(21*b*c^3*Sqrt[c*Csc[a + b*x]]) + (10*Sqr
t[c*Csc[a + b*x]]*EllipticF[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(21*b*c^4)

Rule 3769

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Csc[c + d*x])^(n + 1))/(b*d*n), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[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 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{1}{(c \csc (a+b x))^{7/2}} \, dx &=-\frac{2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}}+\frac{5 \int \frac{1}{(c \csc (a+b x))^{3/2}} \, dx}{7 c^2}\\ &=-\frac{2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}}-\frac{10 \cos (a+b x)}{21 b c^3 \sqrt{c \csc (a+b x)}}+\frac{5 \int \sqrt{c \csc (a+b x)} \, dx}{21 c^4}\\ &=-\frac{2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}}-\frac{10 \cos (a+b x)}{21 b c^3 \sqrt{c \csc (a+b x)}}+\frac{\left (5 \sqrt{c \csc (a+b x)} \sqrt{\sin (a+b x)}\right ) \int \frac{1}{\sqrt{\sin (a+b x)}} \, dx}{21 c^4}\\ &=-\frac{2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}}-\frac{10 \cos (a+b x)}{21 b c^3 \sqrt{c \csc (a+b x)}}+\frac{10 \sqrt{c \csc (a+b x)} F\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right ) \sqrt{\sin (a+b x)}}{21 b c^4}\\ \end{align*}

Mathematica [A]  time = 0.143812, size = 70, normalized size = 0.67 \[ -\frac{\sqrt{c \csc (a+b x)} \left (40 \sqrt{\sin (a+b x)} \text{EllipticF}\left (\frac{1}{4} (-2 a-2 b x+\pi ),2\right )+26 \sin (2 (a+b x))-3 \sin (4 (a+b x))\right )}{84 b c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Csc[a + b*x])^(-7/2),x]

[Out]

-(Sqrt[c*Csc[a + b*x]]*(40*EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]] + 26*Sin[2*(a + b*x)] - 3*Si
n[4*(a + b*x)]))/(84*b*c^4)

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Maple [C]  time = 0.275, size = 216, normalized size = 2.1 \begin{align*} -{\frac{\sqrt{2}}{21\,b \left ( -1+\cos \left ( bx+a \right ) \right ) \left ( \sin \left ( bx+a \right ) \right ) ^{3}} \left ( 5\,i\sqrt{{\frac{-i \left ( -1+\cos \left ( bx+a \right ) \right ) }{\sin \left ( bx+a \right ) }}}\sin \left ( bx+a \right ) \sqrt{-{\frac{i\cos \left ( bx+a \right ) -\sin \left ( bx+a \right ) -i}{\sin \left ( bx+a \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{i\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) -i}{\sin \left ( bx+a \right ) }}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{{\frac{i\cos \left ( bx+a \right ) +\sin \left ( bx+a \right ) -i}{\sin \left ( bx+a \right ) }}}-3\,\sqrt{2} \left ( \cos \left ( bx+a \right ) \right ) ^{4}+3\,\sqrt{2} \left ( \cos \left ( bx+a \right ) \right ) ^{3}+8\,\sqrt{2} \left ( \cos \left ( bx+a \right ) \right ) ^{2}-8\,\sqrt{2}\cos \left ( bx+a \right ) \right ) \left ({\frac{c}{\sin \left ( bx+a \right ) }} \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*csc(b*x+a))^(7/2),x)

[Out]

-1/21/b*2^(1/2)*(5*I*(-I*(-1+cos(b*x+a))/sin(b*x+a))^(1/2)*sin(b*x+a)*(-(I*cos(b*x+a)-sin(b*x+a)-I)/sin(b*x+a)
)^(1/2)*EllipticF(((I*cos(b*x+a)+sin(b*x+a)-I)/sin(b*x+a))^(1/2),1/2*2^(1/2))*((I*cos(b*x+a)+sin(b*x+a)-I)/sin
(b*x+a))^(1/2)-3*2^(1/2)*cos(b*x+a)^4+3*2^(1/2)*cos(b*x+a)^3+8*2^(1/2)*cos(b*x+a)^2-8*2^(1/2)*cos(b*x+a))/(-1+
cos(b*x+a))/(c/sin(b*x+a))^(7/2)/sin(b*x+a)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c \csc \left (b x + a\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*csc(b*x+a))^(7/2),x, algorithm="maxima")

[Out]

integrate((c*csc(b*x + a))^(-7/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c \csc \left (b x + a\right )}}{c^{4} \csc \left (b x + a\right )^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*csc(b*x+a))^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*csc(b*x + a))/(c^4*csc(b*x + a)^4), 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(1/(c*csc(b*x+a))**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c \csc \left (b x + a\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*csc(b*x+a))^(7/2),x, algorithm="giac")

[Out]

integrate((c*csc(b*x + a))^(-7/2), x)