3.116 \(\int \csc ^3(a+b x) \sin ^{\frac{7}{2}}(2 a+2 b x) \, dx\)

Optimal. Leaf size=164 \[ \frac{4 \sin (a+b x) \sin ^{\frac{5}{2}}(2 a+2 b x)}{3 b}+\frac{5 \sin (a+b x) \sqrt{\sin (2 a+2 b x)}}{2 b}-\frac{5 \sin ^{\frac{3}{2}}(2 a+2 b x) \cos (a+b x)}{3 b}-\frac{5 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{4 b}+\frac{\sin ^{\frac{9}{2}}(2 a+2 b x) \csc ^3(a+b x)}{3 b}-\frac{5 \log \left (\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right )}{4 b} \]

[Out]

(-5*ArcSin[Cos[a + b*x] - Sin[a + b*x]])/(4*b) - (5*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]])
/(4*b) + (5*Sin[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(2*b) - (5*Cos[a + b*x]*Sin[2*a + 2*b*x]^(3/2))/(3*b) + (4*Si
n[a + b*x]*Sin[2*a + 2*b*x]^(5/2))/(3*b) + (Csc[a + b*x]^3*Sin[2*a + 2*b*x]^(9/2))/(3*b)

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Rubi [A]  time = 0.156032, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {4300, 4308, 4301, 4302, 4306} \[ \frac{4 \sin (a+b x) \sin ^{\frac{5}{2}}(2 a+2 b x)}{3 b}+\frac{5 \sin (a+b x) \sqrt{\sin (2 a+2 b x)}}{2 b}-\frac{5 \sin ^{\frac{3}{2}}(2 a+2 b x) \cos (a+b x)}{3 b}-\frac{5 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{4 b}+\frac{\sin ^{\frac{9}{2}}(2 a+2 b x) \csc ^3(a+b x)}{3 b}-\frac{5 \log \left (\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right )}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^(7/2),x]

[Out]

(-5*ArcSin[Cos[a + b*x] - Sin[a + b*x]])/(4*b) - (5*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*a + 2*b*x]]])
/(4*b) + (5*Sin[a + b*x]*Sqrt[Sin[2*a + 2*b*x]])/(2*b) - (5*Cos[a + b*x]*Sin[2*a + 2*b*x]^(3/2))/(3*b) + (4*Si
n[a + b*x]*Sin[2*a + 2*b*x]^(5/2))/(3*b) + (Csc[a + b*x]^3*Sin[2*a + 2*b*x]^(9/2))/(3*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 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 4301

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

Rule 4302

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

Rule 4306

Int[sin[(a_.) + (b_.)*(x_)]/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> -Simp[ArcSin[Cos[a + b*x] - Sin[a + b*
x]]/d, x] - Simp[Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[c + d*x]]]/d, x] /; FreeQ[{a, b, c, d}, x] && EqQ[
b*c - a*d, 0] && EqQ[d/b, 2]

Rubi steps

\begin{align*} \int \csc ^3(a+b x) \sin ^{\frac{7}{2}}(2 a+2 b x) \, dx &=\frac{\csc ^3(a+b x) \sin ^{\frac{9}{2}}(2 a+2 b x)}{3 b}+4 \int \csc (a+b x) \sin ^{\frac{7}{2}}(2 a+2 b x) \, dx\\ &=\frac{\csc ^3(a+b x) \sin ^{\frac{9}{2}}(2 a+2 b x)}{3 b}+8 \int \cos (a+b x) \sin ^{\frac{5}{2}}(2 a+2 b x) \, dx\\ &=\frac{4 \sin (a+b x) \sin ^{\frac{5}{2}}(2 a+2 b x)}{3 b}+\frac{\csc ^3(a+b x) \sin ^{\frac{9}{2}}(2 a+2 b x)}{3 b}+\frac{20}{3} \int \sin (a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x) \, dx\\ &=-\frac{5 \cos (a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x)}{3 b}+\frac{4 \sin (a+b x) \sin ^{\frac{5}{2}}(2 a+2 b x)}{3 b}+\frac{\csc ^3(a+b x) \sin ^{\frac{9}{2}}(2 a+2 b x)}{3 b}+5 \int \cos (a+b x) \sqrt{\sin (2 a+2 b x)} \, dx\\ &=\frac{5 \sin (a+b x) \sqrt{\sin (2 a+2 b x)}}{2 b}-\frac{5 \cos (a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x)}{3 b}+\frac{4 \sin (a+b x) \sin ^{\frac{5}{2}}(2 a+2 b x)}{3 b}+\frac{\csc ^3(a+b x) \sin ^{\frac{9}{2}}(2 a+2 b x)}{3 b}+\frac{5}{2} \int \frac{\sin (a+b x)}{\sqrt{\sin (2 a+2 b x)}} \, dx\\ &=-\frac{5 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{4 b}-\frac{5 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}\right )}{4 b}+\frac{5 \sin (a+b x) \sqrt{\sin (2 a+2 b x)}}{2 b}-\frac{5 \cos (a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x)}{3 b}+\frac{4 \sin (a+b x) \sin ^{\frac{5}{2}}(2 a+2 b x)}{3 b}+\frac{\csc ^3(a+b x) \sin ^{\frac{9}{2}}(2 a+2 b x)}{3 b}\\ \end{align*}

Mathematica [A]  time = 0.219607, size = 84, normalized size = 0.51 \[ \frac{2 \sqrt{\sin (2 (a+b x))} (6 \sin (a+b x)+\sin (3 (a+b x)))-5 \left (\sin ^{-1}(\cos (a+b x)-\sin (a+b x))+\log \left (\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right )\right )}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^(7/2),x]

[Out]

(-5*(ArcSin[Cos[a + b*x] - Sin[a + b*x]] + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]) + 2*Sqrt
[Sin[2*(a + b*x)]]*(6*Sin[a + b*x] + Sin[3*(a + b*x)]))/(4*b)

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Maple [C]  time = 19.125, size = 973, normalized size = 5.9 \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)^(7/2),x)

[Out]

32/5*(-tan(1/2*b*x+1/2*a)/(tan(1/2*b*x+1/2*a)^2-1))^(1/2)*(2*(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)-1)*(tan(1/2*b*x+1/2*a)+1))^(1/2)*tan(1/2*b*x+1/2*a)^4-(tan(1/2*b*x+1/2*a)+1)^(1/2)*(-2*t
an(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))*(ta
n(1/2*b*x+1/2*a)*(tan(1/2*b*x+1/2*a)-1)*(tan(1/2*b*x+1/2*a)+1))^(1/2)*tan(1/2*b*x+1/2*a)^4-4*(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)-1)*(tan(1/2*b*x+1/2*a)+1))^(1/2)*tan(1/2*b*x+1/2*a)^2+2*
(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)-1)*(tan(1/2*b*x+1/2*a)+1))^(1/2)*tan(1/
2*b*x+1/2*a)^2+2*(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)+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)-1)*(tan(1/2*b*x+1/2*a)+1))^(1/2)-(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)-1)*(tan(1/2*b*x+1/2*a)+1))^(1/2)-4*(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)-1)*(tan(1/2*b*x+1/2*a)+1))^(1/2
)*tan(1/2*b*x+1/2*a)^4+2*(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)^2+2*(tan(1/2*b*x+1
/2*a)*(tan(1/2*b*x+1/2*a)-1)*(tan(1/2*b*x+1/2*a)+1))^(1/2)*tan(1/2*b*x+1/2*a)^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)^3-tan(1/2*b*x+1/2*a))^(1/2)/(tan(1/2*b*x+1/2*a)*(tan(1/2*b*x+1/2*a)
-1)*(tan(1/2*b*x+1/2*a)+1))^(1/2)/(tan(1/2*b*x+1/2*a)-1)/(tan(1/2*b*x+1/2*a)+1)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc \left (b x + a\right )^{3} \sin \left (2 \, b x + 2 \, a\right )^{\frac{7}{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)^(7/2),x, algorithm="maxima")

[Out]

integrate(csc(b*x + a)^3*sin(2*b*x + 2*a)^(7/2), x)

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Fricas [A]  time = 0.560269, size = 772, normalized size = 4.71 \begin{align*} \frac{8 \, \sqrt{2}{\left (4 \, \cos \left (b x + a\right )^{2} + 5\right )} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )} \sin \left (b x + a\right ) + 10 \, \arctan \left (-\frac{\sqrt{2} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )}{\left (\cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} + \cos \left (b x + a\right ) \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 1}\right ) - 10 \, \arctan \left (-\frac{2 \, \sqrt{2} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )} - \cos \left (b x + a\right ) - \sin \left (b x + a\right )}{\cos \left (b x + a\right ) - \sin \left (b x + a\right )}\right ) + 5 \, \log \left (-32 \, \cos \left (b x + a\right )^{4} + 4 \, \sqrt{2}{\left (4 \, \cos \left (b x + a\right )^{3} -{\left (4 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right ) - 5 \, \cos \left (b x + a\right )\right )} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 32 \, \cos \left (b x + a\right )^{2} + 16 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 1\right )}{16 \, b} \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)^(7/2),x, algorithm="fricas")

[Out]

1/16*(8*sqrt(2)*(4*cos(b*x + a)^2 + 5)*sqrt(cos(b*x + a)*sin(b*x + a))*sin(b*x + a) + 10*arctan(-(sqrt(2)*sqrt
(cos(b*x + a)*sin(b*x + a))*(cos(b*x + a) - sin(b*x + a)) + cos(b*x + a)*sin(b*x + a))/(cos(b*x + a)^2 + 2*cos
(b*x + a)*sin(b*x + a) - 1)) - 10*arctan(-(2*sqrt(2)*sqrt(cos(b*x + a)*sin(b*x + a)) - cos(b*x + a) - sin(b*x
+ a))/(cos(b*x + a) - sin(b*x + a))) + 5*log(-32*cos(b*x + a)^4 + 4*sqrt(2)*(4*cos(b*x + a)^3 - (4*cos(b*x + a
)^2 + 1)*sin(b*x + a) - 5*cos(b*x + a))*sqrt(cos(b*x + a)*sin(b*x + a)) + 32*cos(b*x + a)^2 + 16*cos(b*x + a)*
sin(b*x + a) + 1))/b

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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)**(7/2),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)^3*sin(2*b*x+2*a)^(7/2),x, algorithm="giac")

[Out]

Timed out