3.18 \(\int (c \csc (a+b x))^{5/2} \, dx\)

Optimal. Leaf size=75 \[ \frac {2 c^2 \sqrt {\sin (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {c \csc (a+b x)}}{3 b}-\frac {2 c \cos (a+b x) (c \csc (a+b x))^{3/2}}{3 b} \]

[Out]

-2/3*c*cos(b*x+a)*(c*csc(b*x+a))^(3/2)/b-2/3*c^2*(sin(1/2*a+1/4*Pi+1/2*b*x)^2)^(1/2)/sin(1/2*a+1/4*Pi+1/2*b*x)
*EllipticF(cos(1/2*a+1/4*Pi+1/2*b*x),2^(1/2))*(c*csc(b*x+a))^(1/2)*sin(b*x+a)^(1/2)/b

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Rubi [A]  time = 0.03, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3768, 3771, 2641} \[ \frac {2 c^2 \sqrt {\sin (a+b x)} F\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {c \csc (a+b x)}}{3 b}-\frac {2 c \cos (a+b x) (c \csc (a+b x))^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(c*Csc[a + b*x])^(5/2),x]

[Out]

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

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

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Mathematica [A]  time = 0.16, size = 55, normalized size = 0.73 \[ -\frac {(c \csc (a+b x))^{5/2} \left (\sin (2 (a+b x))+2 \sin ^{\frac {5}{2}}(a+b x) F\left (\left .\frac {1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Csc[a + b*x])^(5/2),x]

[Out]

-1/3*((c*Csc[a + b*x])^(5/2)*(2*EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sin[a + b*x]^(5/2) + Sin[2*(a + b*x)]))/b

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fricas [F]  time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c \csc \left (b x + a\right )} c^{2} \csc \left (b x + a\right )^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \csc \left (b x + a\right )\right )^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*csc(b*x + a))^(5/2), x)

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maple [C]  time = 1.04, size = 313, normalized size = 4.17 \[ \frac {\left (-1+\cos \left (b x +a \right )\right )^{2} \left (i \cos \left (b x +a \right ) \sqrt {\frac {-i \cos \left (b x +a \right )+\sin \left (b x +a \right )+i}{\sin \left (b x +a \right )}}\, \sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {-\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}}\, \sin \left (b x +a \right ) \EllipticF \left (\sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )+i \sqrt {\frac {-i \cos \left (b x +a \right )+\sin \left (b x +a \right )+i}{\sin \left (b x +a \right )}}\, \sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {-\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}}\, \sin \left (b x +a \right ) \EllipticF \left (\sqrt {\frac {i \cos \left (b x +a \right )-i+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )-\cos \left (b x +a \right ) \sqrt {2}\right ) \left (\cos \left (b x +a \right )+1\right )^{2} \left (\frac {c}{\sin \left (b x +a \right )}\right )^{\frac {5}{2}} \sqrt {2}}{3 b \sin \left (b x +a \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \csc \left (b x + a\right )\right )^{\frac {5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*csc(b*x + a))^(5/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {c}{\sin \left (a+b\,x\right )}\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c/sin(a + b*x))^(5/2),x)

[Out]

int((c/sin(a + b*x))^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \csc {\left (a + b x \right )}\right )^{\frac {5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c*csc(a + b*x))**(5/2), x)

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