3.31 \(\int \frac{1}{(c \sin (a+b x))^{5/2}} \, dx\)

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

[Out]

(-2*Cos[a + b*x])/(3*b*c*(c*Sin[a + b*x])^(3/2)) + (2*EllipticF[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(3*
b*c^2*Sqrt[c*Sin[a + b*x]])

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Rubi [A]  time = 0.0325813, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2636, 2642, 2641} \[ \frac{2 \sqrt{\sin (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{3 b c^2 \sqrt{c \sin (a+b x)}}-\frac{2 \cos (a+b x)}{3 b c (c \sin (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x])^(-5/2),x]

[Out]

(-2*Cos[a + b*x])/(3*b*c*(c*Sin[a + b*x])^(3/2)) + (2*EllipticF[(a - Pi/2 + b*x)/2, 2]*Sqrt[Sin[a + b*x]])/(3*
b*c^2*Sqrt[c*Sin[a + b*x]])

Rule 2636

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

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

Mathematica [A]  time = 0.0752905, size = 55, normalized size = 0.71 \[ -\frac{2 \left (\cos (a+b x)+\sin ^{\frac{3}{2}}(a+b x) F\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{3 b c (c \sin (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Sin[a + b*x])^(-5/2),x]

[Out]

(-2*(Cos[a + b*x] + EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sin[a + b*x]^(3/2)))/(3*b*c*(c*Sin[a + b*x])^(3/2))

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Maple [A]  time = 0.04, size = 105, normalized size = 1.4 \begin{align*} -{\frac{1}{3\,{c}^{2} \left ( \sin \left ( bx+a \right ) \right ) ^{2}\cos \left ( bx+a \right ) b} \left ( \sqrt{-\sin \left ( bx+a \right ) +1}\sqrt{2\,\sin \left ( bx+a \right ) +2} \left ( \sin \left ( bx+a \right ) \right ) ^{{\frac{5}{2}}}{\it EllipticF} \left ( \sqrt{-\sin \left ( bx+a \right ) +1},{\frac{\sqrt{2}}{2}} \right ) -2\, \left ( \sin \left ( bx+a \right ) \right ) ^{3}+2\,\sin \left ( bx+a \right ) \right ){\frac{1}{\sqrt{c\sin \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*sin(b*x+a))^(5/2),x)

[Out]

-1/3/c^2*((-sin(b*x+a)+1)^(1/2)*(2*sin(b*x+a)+2)^(1/2)*sin(b*x+a)^(5/2)*EllipticF((-sin(b*x+a)+1)^(1/2),1/2*2^
(1/2))-2*sin(b*x+a)^3+2*sin(b*x+a))/sin(b*x+a)^2/cos(b*x+a)/(c*sin(b*x+a))^(1/2)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c \sin \left (b x + a\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*sin(b*x+a))^(5/2),x, algorithm="maxima")

[Out]

integrate((c*sin(b*x + a))^(-5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{c \sin \left (b x + a\right )}}{{\left (c^{3} \cos \left (b x + a\right )^{2} - c^{3}\right )} \sin \left (b x + a\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*sin(b*x+a))^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(c*sin(b*x + a))/((c^3*cos(b*x + a)^2 - c^3)*sin(b*x + a)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c \sin{\left (a + b x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*sin(b*x+a))**(5/2),x)

[Out]

Integral((c*sin(a + b*x))**(-5/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c \sin \left (b x + a\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*sin(b*x+a))^(5/2),x, algorithm="giac")

[Out]

integrate((c*sin(b*x + a))^(-5/2), x)