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

Optimal. Leaf size=103 \[ -\frac{6 c^3 \cos (a+b x) \sqrt{c \csc (a+b x)}}{5 b}-\frac{6 c^4 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{5 b \sqrt{\sin (a+b x)} \sqrt{c \csc (a+b x)}}-\frac{2 c \cos (a+b x) (c \csc (a+b x))^{5/2}}{5 b} \]

[Out]

(-6*c^3*Cos[a + b*x]*Sqrt[c*Csc[a + b*x]])/(5*b) - (2*c*Cos[a + b*x]*(c*Csc[a + b*x])^(5/2))/(5*b) - (6*c^4*El
lipticE[(a - Pi/2 + b*x)/2, 2])/(5*b*Sqrt[c*Csc[a + b*x]]*Sqrt[Sin[a + b*x]])

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Rubi [A]  time = 0.0528121, antiderivative size = 103, 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 = {3768, 3771, 2639} \[ -\frac{6 c^3 \cos (a+b x) \sqrt{c \csc (a+b x)}}{5 b}-\frac{6 c^4 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{5 b \sqrt{\sin (a+b x)} \sqrt{c \csc (a+b x)}}-\frac{2 c \cos (a+b x) (c \csc (a+b x))^{5/2}}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[(c*Csc[a + b*x])^(7/2),x]

[Out]

(-6*c^3*Cos[a + b*x]*Sqrt[c*Csc[a + b*x]])/(5*b) - (2*c*Cos[a + b*x]*(c*Csc[a + b*x])^(5/2))/(5*b) - (6*c^4*El
lipticE[(a - Pi/2 + b*x)/2, 2])/(5*b*Sqrt[c*Csc[a + b*x]]*Sqrt[Sin[a + b*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]

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 (c \csc (a+b x))^{7/2} \, dx &=-\frac{2 c \cos (a+b x) (c \csc (a+b x))^{5/2}}{5 b}+\frac{1}{5} \left (3 c^2\right ) \int (c \csc (a+b x))^{3/2} \, dx\\ &=-\frac{6 c^3 \cos (a+b x) \sqrt{c \csc (a+b x)}}{5 b}-\frac{2 c \cos (a+b x) (c \csc (a+b x))^{5/2}}{5 b}-\frac{1}{5} \left (3 c^4\right ) \int \frac{1}{\sqrt{c \csc (a+b x)}} \, dx\\ &=-\frac{6 c^3 \cos (a+b x) \sqrt{c \csc (a+b x)}}{5 b}-\frac{2 c \cos (a+b x) (c \csc (a+b x))^{5/2}}{5 b}-\frac{\left (3 c^4\right ) \int \sqrt{\sin (a+b x)} \, dx}{5 \sqrt{c \csc (a+b x)} \sqrt{\sin (a+b x)}}\\ &=-\frac{6 c^3 \cos (a+b x) \sqrt{c \csc (a+b x)}}{5 b}-\frac{2 c \cos (a+b x) (c \csc (a+b x))^{5/2}}{5 b}-\frac{6 c^4 E\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right )}{5 b \sqrt{c \csc (a+b x)} \sqrt{\sin (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.196421, size = 67, normalized size = 0.65 \[ \frac{(c \csc (a+b x))^{7/2} \left (-10 \sin (2 (a+b x))+3 \sin (4 (a+b x))+24 \sin ^{\frac{7}{2}}(a+b x) E\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{20 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Csc[a + b*x])^(7/2),x]

[Out]

((c*Csc[a + b*x])^(7/2)*(24*EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sin[a + b*x]^(7/2) - 10*Sin[2*(a + b*x)] + 3*S
in[4*(a + b*x)]))/(20*b)

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Maple [C]  time = 0.335, size = 1029, normalized size = 10. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5/b*2^(1/2)*(3*cos(b*x+a)^3*(-I*(-1+cos(b*x+a))/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)*((I*cos(b*x+a)+sin(b*x+a)-I)/sin(b
*x+a))^(1/2)-6*cos(b*x+a)^3*(-I*(-1+cos(b*x+a))/sin(b*x+a))^(1/2)*((-I*cos(b*x+a)+sin(b*x+a)+I)/sin(b*x+a))^(1
/2)*((I*cos(b*x+a)+sin(b*x+a)-I)/sin(b*x+a))^(1/2)*EllipticE(((I*cos(b*x+a)+sin(b*x+a)-I)/sin(b*x+a))^(1/2),1/
2*2^(1/2))+3*EllipticF(((I*cos(b*x+a)+sin(b*x+a)-I)/sin(b*x+a))^(1/2),1/2*2^(1/2))*(-I*(-1+cos(b*x+a))/sin(b*x
+a))^(1/2)*cos(b*x+a)^2*((-I*cos(b*x+a)+sin(b*x+a)+I)/sin(b*x+a))^(1/2)*((I*cos(b*x+a)+sin(b*x+a)-I)/sin(b*x+a
))^(1/2)-6*(-I*(-1+cos(b*x+a))/sin(b*x+a))^(1/2)*cos(b*x+a)^2*EllipticE(((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)*((I*cos(b*x+a)+sin(b*x+a)-I)/sin(b*x+a))
^(1/2)-3*EllipticF(((I*cos(b*x+a)+sin(b*x+a)-I)/sin(b*x+a))^(1/2),1/2*2^(1/2))*(-I*(-1+cos(b*x+a))/sin(b*x+a))
^(1/2)*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)+sin(b*x+a)-I)/sin(b*x+a))^(1/
2)+6*(-I*(-1+cos(b*x+a))/sin(b*x+a))^(1/2)*cos(b*x+a)*EllipticE(((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)*((I*cos(b*x+a)+sin(b*x+a)-I)/sin(b*x+a))^(1/2)-3
*EllipticF(((I*cos(b*x+a)+sin(b*x+a)-I)/sin(b*x+a))^(1/2),1/2*2^(1/2))*(-I*(-1+cos(b*x+a))/sin(b*x+a))^(1/2)*(
(-I*cos(b*x+a)+sin(b*x+a)+I)/sin(b*x+a))^(1/2)*((I*cos(b*x+a)+sin(b*x+a)-I)/sin(b*x+a))^(1/2)+6*(-I*(-1+cos(b*
x+a))/sin(b*x+a))^(1/2)*EllipticE(((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)*((I*cos(b*x+a)+sin(b*x+a)-I)/sin(b*x+a))^(1/2)+3*2^(1/2)*cos(b*x+a)^2-2^(1/2)*
cos(b*x+a)-3*2^(1/2))*sin(b*x+a)*(c/sin(b*x+a))^(7/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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((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 (\sqrt{c \csc \left (b x + a\right )} c^{3} \csc \left (b x + a\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*csc(b*x + a))*c^3*csc(b*x + a)^3, 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((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 \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((c*csc(b*x+a))^(7/2),x, algorithm="giac")

[Out]

integrate((c*csc(b*x + a))^(7/2), x)