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

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

[Out]

-2/5*c*cos(b*x+a)*(c*csc(b*x+a))^(5/2)/b-6/5*c^3*cos(b*x+a)*(c*csc(b*x+a))^(1/2)/b+6/5*c^4*(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)*EllipticE(cos(1/2*a+1/4*Pi+1/2*b*x),2^(1/2))/b/(c*csc(b*x+a))^(1/2)
/sin(b*x+a)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 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]

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))^{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*}

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Mathematica [A]  time = 0.18, 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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fricas [F]  time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c \csc \left (b x + a\right )} c^{3} \csc \left (b x + a\right )^{3}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \csc \left (b x + a\right )\right )^{\frac {7}{2}}\,{d x} \]

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)

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maple [C]  time = 1.23, size = 1028, normalized size = 9.98 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5/b*(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)*E
llipticE(((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))*((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^
(1/2)-3*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)-I+sin(b*x+a))/sin(b*x+a))^(1/2)*EllipticF(((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/
2))+6*cos(b*x+a)^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)*Ellip
ticE(((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))*((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2
)-3*cos(b*x+a)^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)*Ellipti
cF(((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))*((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2)-
6*cos(b*x+a)*(-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)*EllipticE((
(I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))*((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2)+3*co
s(b*x+a)*(-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)*EllipticF(((I*c
os(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))*((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2)-6*(-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)*EllipticE(((I*cos(b*x+a)-I+sin
(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))*((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2)+3*(-I*(-1+cos(b*x+a))/s
in(b*x+a))^(1/2)*((-I*cos(b*x+a)+sin(b*x+a)+I)/sin(b*x+a))^(1/2)*EllipticF(((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*
x+a))^(1/2),1/2*2^(1/2))*((I*cos(b*x+a)-I+sin(b*x+a))/sin(b*x+a))^(1/2)-3*cos(b*x+a)^2*2^(1/2)+cos(b*x+a)*2^(1
/2)+3*2^(1/2))*sin(b*x+a)*(c/sin(b*x+a))^(7/2)*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \csc \left (b x + a\right )\right )^{\frac {7}{2}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {c}{\sin \left (a+b\,x\right )}\right )}^{7/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c/sin(a + b*x))^(7/2),x)

[Out]

int((c/sin(a + b*x))^(7/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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