3.122 \(\int \frac{\csc ^3(a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx\)

Optimal. Leaf size=107 \[ \frac{32 \sin (a+b x)}{45 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{8 \cos (a+b x)}{15 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{64 \cos (a+b x)}{45 b \sqrt{\sin (2 a+2 b x)}}-\frac{\csc ^3(a+b x)}{9 b \sin ^{\frac{3}{2}}(2 a+2 b x)} \]

[Out]

(-8*Cos[a + b*x])/(15*b*Sin[2*a + 2*b*x]^(5/2)) - Csc[a + b*x]^3/(9*b*Sin[2*a + 2*b*x]^(3/2)) + (32*Sin[a + b*
x])/(45*b*Sin[2*a + 2*b*x]^(3/2)) - (64*Cos[a + b*x])/(45*b*Sqrt[Sin[2*a + 2*b*x]])

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Rubi [A]  time = 0.115507, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {4300, 4308, 4303, 4304, 4291} \[ \frac{32 \sin (a+b x)}{45 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{8 \cos (a+b x)}{15 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{64 \cos (a+b x)}{45 b \sqrt{\sin (2 a+2 b x)}}-\frac{\csc ^3(a+b x)}{9 b \sin ^{\frac{3}{2}}(2 a+2 b x)} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^3/Sin[2*a + 2*b*x]^(5/2),x]

[Out]

(-8*Cos[a + b*x])/(15*b*Sin[2*a + 2*b*x]^(5/2)) - Csc[a + b*x]^3/(9*b*Sin[2*a + 2*b*x]^(3/2)) + (32*Sin[a + b*
x])/(45*b*Sin[2*a + 2*b*x]^(3/2)) - (64*Cos[a + b*x])/(45*b*Sqrt[Sin[2*a + 2*b*x]])

Rule 4300

Int[((e_.)*sin[(a_.) + (b_.)*(x_)])^(m_)*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> Simp[((e*Sin[a + b
*x])^m*(g*Sin[c + d*x])^(p + 1))/(2*b*g*(m + p + 1)), x] + Dist[(m + 2*p + 2)/(e^2*(m + p + 1)), Int[(e*Sin[a
+ b*x])^(m + 2)*(g*Sin[c + d*x])^p, x], x] /; FreeQ[{a, b, c, d, e, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b,
2] &&  !IntegerQ[p] && LtQ[m, -1] && NeQ[m + 2*p + 2, 0] && NeQ[m + p + 1, 0] && IntegersQ[2*m, 2*p]

Rule 4308

Int[((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_)/sin[(a_.) + (b_.)*(x_)], x_Symbol] :> Dist[2*g, Int[Cos[a + b*x]*(g*S
in[c + d*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, g, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] &&  !IntegerQ
[p] && IntegerQ[2*p]

Rule 4303

Int[cos[(a_.) + (b_.)*(x_)]*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> Simp[(Cos[a + b*x]*(g*Sin[c + d
*x])^(p + 1))/(2*b*g*(p + 1)), x] + Dist[(2*p + 3)/(2*g*(p + 1)), Int[Sin[a + b*x]*(g*Sin[c + d*x])^(p + 1), x
], x] /; FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] &&  !IntegerQ[p] && LtQ[p, -1] && Integ
erQ[2*p]

Rule 4304

Int[sin[(a_.) + (b_.)*(x_)]*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> -Simp[(Sin[a + b*x]*(g*Sin[c +
d*x])^(p + 1))/(2*b*g*(p + 1)), x] + Dist[(2*p + 3)/(2*g*(p + 1)), Int[Cos[a + b*x]*(g*Sin[c + d*x])^(p + 1),
x], x] /; FreeQ[{a, b, c, d, g}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] &&  !IntegerQ[p] && LtQ[p, -1] && Inte
gerQ[2*p]

Rule 4291

Int[(cos[(a_.) + (b_.)*(x_)]*(e_.))^(m_.)*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> -Simp[((e*Cos[a +
b*x])^m*(g*Sin[c + d*x])^(p + 1))/(b*g*m), x] /; FreeQ[{a, b, c, d, e, g, m, p}, x] && EqQ[b*c - a*d, 0] && E
qQ[d/b, 2] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0]

Rubi steps

\begin{align*} \int \frac{\csc ^3(a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx &=-\frac{\csc ^3(a+b x)}{9 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{4}{3} \int \frac{\csc (a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx\\ &=-\frac{\csc ^3(a+b x)}{9 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{8}{3} \int \frac{\cos (a+b x)}{\sin ^{\frac{7}{2}}(2 a+2 b x)} \, dx\\ &=-\frac{8 \cos (a+b x)}{15 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{\csc ^3(a+b x)}{9 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{32}{15} \int \frac{\sin (a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx\\ &=-\frac{8 \cos (a+b x)}{15 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{\csc ^3(a+b x)}{9 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{32 \sin (a+b x)}{45 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{64}{45} \int \frac{\cos (a+b x)}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx\\ &=-\frac{8 \cos (a+b x)}{15 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{\csc ^3(a+b x)}{9 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{32 \sin (a+b x)}{45 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{64 \cos (a+b x)}{45 b \sqrt{\sin (2 a+2 b x)}}\\ \end{align*}

Mathematica [A]  time = 0.091329, size = 62, normalized size = 0.58 \[ -\frac{\sqrt{\sin (2 (a+b x))} \left (5 \csc ^5(a+b x)+17 \csc ^3(a+b x)+113 \csc (a+b x)-15 \tan (a+b x) \sec (a+b x)\right )}{180 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^3/Sin[2*a + 2*b*x]^(5/2),x]

[Out]

-(Sqrt[Sin[2*(a + b*x)]]*(113*Csc[a + b*x] + 17*Csc[a + b*x]^3 + 5*Csc[a + b*x]^5 - 15*Sec[a + b*x]*Tan[a + b*
x]))/(180*b)

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Maple [C]  time = 18.636, size = 560, normalized size = 5.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^3/sin(2*b*x+2*a)^(5/2),x)

[Out]

-1/2880*(-tan(1/2*b*x+1/2*a)/(tan(1/2*b*x+1/2*a)^2-1))^(1/2)/tan(1/2*b*x+1/2*a)^5*(5*(tan(1/2*b*x+1/2*a)*(tan(
1/2*b*x+1/2*a)^2-1))^(1/2)*tan(1/2*b*x+1/2*a)^10+192*(tan(1/2*b*x+1/2*a)+1)^(1/2)*(-2*tan(1/2*b*x+1/2*a)+2)^(1
/2)*(-tan(1/2*b*x+1/2*a))^(1/2)*EllipticE((tan(1/2*b*x+1/2*a)+1)^(1/2),1/2*2^(1/2))*(tan(1/2*b*x+1/2*a)*(tan(1
/2*b*x+1/2*a)^2-1))^(1/2)*tan(1/2*b*x+1/2*a)^4-96*(tan(1/2*b*x+1/2*a)+1)^(1/2)*(-2*tan(1/2*b*x+1/2*a)+2)^(1/2)
*(-tan(1/2*b*x+1/2*a))^(1/2)*EllipticF((tan(1/2*b*x+1/2*a)+1)^(1/2),1/2*2^(1/2))*(tan(1/2*b*x+1/2*a)*(tan(1/2*
b*x+1/2*a)^2-1))^(1/2)*tan(1/2*b*x+1/2*a)^4-7*tan(1/2*b*x+1/2*a)^8*(tan(1/2*b*x+1/2*a)*(tan(1/2*b*x+1/2*a)^2-1
))^(1/2)+96*(tan(1/2*b*x+1/2*a)^3-tan(1/2*b*x+1/2*a))^(1/2)*tan(1/2*b*x+1/2*a)^6+2*(tan(1/2*b*x+1/2*a)*(tan(1/
2*b*x+1/2*a)^2-1))^(1/2)*tan(1/2*b*x+1/2*a)^6-96*(tan(1/2*b*x+1/2*a)^3-tan(1/2*b*x+1/2*a))^(1/2)*tan(1/2*b*x+1
/2*a)^4+2*(tan(1/2*b*x+1/2*a)*(tan(1/2*b*x+1/2*a)^2-1))^(1/2)*tan(1/2*b*x+1/2*a)^4-7*(tan(1/2*b*x+1/2*a)*(tan(
1/2*b*x+1/2*a)^2-1))^(1/2)*tan(1/2*b*x+1/2*a)^2+5*(tan(1/2*b*x+1/2*a)*(tan(1/2*b*x+1/2*a)^2-1))^(1/2))/(tan(1/
2*b*x+1/2*a)^3-tan(1/2*b*x+1/2*a))^(1/2)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (b x + a\right )^{3}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^3/sin(2*b*x+2*a)^(5/2),x, algorithm="maxima")

[Out]

integrate(csc(b*x + a)^3/sin(2*b*x + 2*a)^(5/2), x)

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Fricas [A]  time = 0.527352, size = 356, normalized size = 3.33 \begin{align*} -\frac{\sqrt{2}{\left (128 \, \cos \left (b x + a\right )^{6} - 288 \, \cos \left (b x + a\right )^{4} + 180 \, \cos \left (b x + a\right )^{2} - 15\right )} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 128 \,{\left (\cos \left (b x + a\right )^{6} - 2 \, \cos \left (b x + a\right )^{4} + \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )}{180 \,{\left (b \cos \left (b x + a\right )^{6} - 2 \, b \cos \left (b x + a\right )^{4} + b \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^3/sin(2*b*x+2*a)^(5/2),x, algorithm="fricas")

[Out]

-1/180*(sqrt(2)*(128*cos(b*x + a)^6 - 288*cos(b*x + a)^4 + 180*cos(b*x + a)^2 - 15)*sqrt(cos(b*x + a)*sin(b*x
+ a)) + 128*(cos(b*x + a)^6 - 2*cos(b*x + a)^4 + cos(b*x + a)^2)*sin(b*x + a))/((b*cos(b*x + a)^6 - 2*b*cos(b*
x + a)^4 + b*cos(b*x + a)^2)*sin(b*x + a))

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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(csc(b*x+a)**3/sin(2*b*x+2*a)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (b x + a\right )^{3}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^3/sin(2*b*x+2*a)^(5/2),x, algorithm="giac")

[Out]

integrate(csc(b*x + a)^3/sin(2*b*x + 2*a)^(5/2), x)