3.105 \(\int \csc ^2(a+b x) \sin ^{\frac{9}{2}}(2 a+2 b x) \, dx\)

Optimal. Leaf size=106 \[ \frac{6 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{5 b}-\frac{2 \sin ^{\frac{7}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{7 b}-\frac{2 \sin ^{\frac{3}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{5 b}+\frac{\sin ^{\frac{11}{2}}(2 a+2 b x) \csc ^2(a+b x)}{7 b} \]

[Out]

(6*EllipticE[a - Pi/4 + b*x, 2])/(5*b) - (2*Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x]^(3/2))/(5*b) - (2*Cos[2*a + 2*b*
x]*Sin[2*a + 2*b*x]^(7/2))/(7*b) + (Csc[a + b*x]^2*Sin[2*a + 2*b*x]^(11/2))/(7*b)

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Rubi [A]  time = 0.0584837, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {4300, 2635, 2639} \[ \frac{6 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{5 b}-\frac{2 \sin ^{\frac{7}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{7 b}-\frac{2 \sin ^{\frac{3}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{5 b}+\frac{\sin ^{\frac{11}{2}}(2 a+2 b x) \csc ^2(a+b x)}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^2*Sin[2*a + 2*b*x]^(9/2),x]

[Out]

(6*EllipticE[a - Pi/4 + b*x, 2])/(5*b) - (2*Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x]^(3/2))/(5*b) - (2*Cos[2*a + 2*b*
x]*Sin[2*a + 2*b*x]^(7/2))/(7*b) + (Csc[a + b*x]^2*Sin[2*a + 2*b*x]^(11/2))/(7*b)

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 2635

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

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 \csc ^2(a+b x) \sin ^{\frac{9}{2}}(2 a+2 b x) \, dx &=\frac{\csc ^2(a+b x) \sin ^{\frac{11}{2}}(2 a+2 b x)}{7 b}+\frac{18}{7} \int \sin ^{\frac{9}{2}}(2 a+2 b x) \, dx\\ &=-\frac{2 \cos (2 a+2 b x) \sin ^{\frac{7}{2}}(2 a+2 b x)}{7 b}+\frac{\csc ^2(a+b x) \sin ^{\frac{11}{2}}(2 a+2 b x)}{7 b}+2 \int \sin ^{\frac{5}{2}}(2 a+2 b x) \, dx\\ &=-\frac{2 \cos (2 a+2 b x) \sin ^{\frac{3}{2}}(2 a+2 b x)}{5 b}-\frac{2 \cos (2 a+2 b x) \sin ^{\frac{7}{2}}(2 a+2 b x)}{7 b}+\frac{\csc ^2(a+b x) \sin ^{\frac{11}{2}}(2 a+2 b x)}{7 b}+\frac{6}{5} \int \sqrt{\sin (2 a+2 b x)} \, dx\\ &=\frac{6 E\left (\left .a-\frac{\pi }{4}+b x\right |2\right )}{5 b}-\frac{2 \cos (2 a+2 b x) \sin ^{\frac{3}{2}}(2 a+2 b x)}{5 b}-\frac{2 \cos (2 a+2 b x) \sin ^{\frac{7}{2}}(2 a+2 b x)}{7 b}+\frac{\csc ^2(a+b x) \sin ^{\frac{11}{2}}(2 a+2 b x)}{7 b}\\ \end{align*}

Mathematica [A]  time = 0.2952, size = 66, normalized size = 0.62 \[ \frac{\sqrt{\sin (2 (a+b x))} (15 \sin (2 (a+b x))-14 \sin (4 (a+b x))-5 \sin (6 (a+b x)))+84 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{70 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^2*Sin[2*a + 2*b*x]^(9/2),x]

[Out]

(84*EllipticE[a - Pi/4 + b*x, 2] + Sqrt[Sin[2*(a + b*x)]]*(15*Sin[2*(a + b*x)] - 14*Sin[4*(a + b*x)] - 5*Sin[6
*(a + b*x)]))/(70*b)

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Maple [A]  time = 7.493, size = 204, normalized size = 1.9 \begin{align*} 8\,{\frac{\sqrt{2}}{b} \left ({\frac{\sqrt{2} \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{7/2}}{56}}-{\frac{\sqrt{2} \left ( 6\,\sqrt{\sin \left ( 2\,bx+2\,a \right ) +1}\sqrt{-2\,\sin \left ( 2\,bx+2\,a \right ) +2}\sqrt{-\sin \left ( 2\,bx+2\,a \right ) }{\it EllipticE} \left ( \sqrt{\sin \left ( 2\,bx+2\,a \right ) +1},1/2\,\sqrt{2} \right ) -3\,\sqrt{\sin \left ( 2\,bx+2\,a \right ) +1}\sqrt{-2\,\sin \left ( 2\,bx+2\,a \right ) +2}\sqrt{-\sin \left ( 2\,bx+2\,a \right ) }{\it EllipticF} \left ( \sqrt{\sin \left ( 2\,bx+2\,a \right ) +1},1/2\,\sqrt{2} \right ) -2\, \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{4}+2\, \left ( \sin \left ( 2\,bx+2\,a \right ) \right ) ^{2} \right ) }{80\,\cos \left ( 2\,bx+2\,a \right ) \sqrt{\sin \left ( 2\,bx+2\,a \right ) }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^2*sin(2*b*x+2*a)^(9/2),x)

[Out]

8*2^(1/2)*(1/56*2^(1/2)*sin(2*b*x+2*a)^(7/2)-1/80*2^(1/2)*(6*(sin(2*b*x+2*a)+1)^(1/2)*(-2*sin(2*b*x+2*a)+2)^(1
/2)*(-sin(2*b*x+2*a))^(1/2)*EllipticE((sin(2*b*x+2*a)+1)^(1/2),1/2*2^(1/2))-3*(sin(2*b*x+2*a)+1)^(1/2)*(-2*sin
(2*b*x+2*a)+2)^(1/2)*(-sin(2*b*x+2*a))^(1/2)*EllipticF((sin(2*b*x+2*a)+1)^(1/2),1/2*2^(1/2))-2*sin(2*b*x+2*a)^
4+2*sin(2*b*x+2*a)^2)/cos(2*b*x+2*a)/sin(2*b*x+2*a)^(1/2))/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc \left (b x + a\right )^{2} \sin \left (2 \, b x + 2 \, a\right )^{\frac{9}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^2*sin(2*b*x+2*a)^(9/2),x, algorithm="maxima")

[Out]

integrate(csc(b*x + a)^2*sin(2*b*x + 2*a)^(9/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (\cos \left (2 \, b x + 2 \, a\right )^{4} - 2 \, \cos \left (2 \, b x + 2 \, a\right )^{2} + 1\right )} \csc \left (b x + a\right )^{2} \sqrt{\sin \left (2 \, b x + 2 \, a\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^2*sin(2*b*x+2*a)^(9/2),x, algorithm="fricas")

[Out]

integral((cos(2*b*x + 2*a)^4 - 2*cos(2*b*x + 2*a)^2 + 1)*csc(b*x + a)^2*sqrt(sin(2*b*x + 2*a)), 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(csc(b*x+a)**2*sin(2*b*x+2*a)**(9/2),x)

[Out]

Timed out

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Giac [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)^2*sin(2*b*x+2*a)^(9/2),x, algorithm="giac")

[Out]

Timed out