3.21 \(\int \csc ^4(2 a+2 b x) \sin ^2(a+b x) \, dx\)

Optimal. Leaf size=42 \[ \frac{\tan ^3(a+b x)}{48 b}+\frac{\tan (a+b x)}{8 b}-\frac{\cot (a+b x)}{16 b} \]

[Out]

-Cot[a + b*x]/(16*b) + Tan[a + b*x]/(8*b) + Tan[a + b*x]^3/(48*b)

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Rubi [A]  time = 0.062722, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4288, 2620, 270} \[ \frac{\tan ^3(a+b x)}{48 b}+\frac{\tan (a+b x)}{8 b}-\frac{\cot (a+b x)}{16 b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[2*a + 2*b*x]^4*Sin[a + b*x]^2,x]

[Out]

-Cot[a + b*x]/(16*b) + Tan[a + b*x]/(8*b) + Tan[a + b*x]^3/(48*b)

Rule 4288

Int[((f_.)*sin[(a_.) + (b_.)*(x_)])^(n_.)*sin[(c_.) + (d_.)*(x_)]^(p_.), x_Symbol] :> Dist[2^p/f^p, Int[Cos[a
+ b*x]^p*(f*Sin[a + b*x])^(n + p), x], x] /; FreeQ[{a, b, c, d, f, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2]
&& IntegerQ[p]

Rule 2620

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \csc ^4(2 a+2 b x) \sin ^2(a+b x) \, dx &=\frac{1}{16} \int \csc ^2(a+b x) \sec ^4(a+b x) \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{x^2} \, dx,x,\tan (a+b x)\right )}{16 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (2+\frac{1}{x^2}+x^2\right ) \, dx,x,\tan (a+b x)\right )}{16 b}\\ &=-\frac{\cot (a+b x)}{16 b}+\frac{\tan (a+b x)}{8 b}+\frac{\tan ^3(a+b x)}{48 b}\\ \end{align*}

Mathematica [A]  time = 0.0538419, size = 48, normalized size = 1.14 \[ \frac{5 \tan (a+b x)}{48 b}-\frac{\cot (a+b x)}{16 b}+\frac{\tan (a+b x) \sec ^2(a+b x)}{48 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[2*a + 2*b*x]^4*Sin[a + b*x]^2,x]

[Out]

-Cot[a + b*x]/(16*b) + (5*Tan[a + b*x])/(48*b) + (Sec[a + b*x]^2*Tan[a + b*x])/(48*b)

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Maple [A]  time = 0.057, size = 51, normalized size = 1.2 \begin{align*}{\frac{1}{16\,b} \left ({\frac{1}{3\,\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{3}}}+{\frac{4}{3\,\sin \left ( bx+a \right ) \cos \left ( bx+a \right ) }}-{\frac{8\,\cot \left ( bx+a \right ) }{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(2*b*x+2*a)^4*sin(b*x+a)^2,x)

[Out]

1/16/b*(1/3/sin(b*x+a)/cos(b*x+a)^3+4/3/sin(b*x+a)/cos(b*x+a)-8/3*cot(b*x+a))

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Maxima [B]  time = 1.07467, size = 416, normalized size = 9.9 \begin{align*} -\frac{{\left (2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \sin \left (8 \, b x + 8 \, a\right ) + 2 \,{\left (2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \sin \left (6 \, b x + 6 \, a\right ) - 2 \, \cos \left (8 \, b x + 8 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) - 4 \, \cos \left (6 \, b x + 6 \, a\right ) \sin \left (2 \, b x + 2 \, a\right )}{3 \,{\left (b \cos \left (8 \, b x + 8 \, a\right )^{2} + 4 \, b \cos \left (6 \, b x + 6 \, a\right )^{2} + 4 \, b \cos \left (2 \, b x + 2 \, a\right )^{2} + b \sin \left (8 \, b x + 8 \, a\right )^{2} + 4 \, b \sin \left (6 \, b x + 6 \, a\right )^{2} - 8 \, b \sin \left (6 \, b x + 6 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) + 4 \, b \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \,{\left (2 \, b \cos \left (6 \, b x + 6 \, a\right ) - 2 \, b \cos \left (2 \, b x + 2 \, a\right ) - b\right )} \cos \left (8 \, b x + 8 \, a\right ) - 4 \,{\left (2 \, b \cos \left (2 \, b x + 2 \, a\right ) + b\right )} \cos \left (6 \, b x + 6 \, a\right ) + 4 \, b \cos \left (2 \, b x + 2 \, a\right ) + 4 \,{\left (b \sin \left (6 \, b x + 6 \, a\right ) - b \sin \left (2 \, b x + 2 \, a\right )\right )} \sin \left (8 \, b x + 8 \, a\right ) + b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(2*b*x+2*a)^4*sin(b*x+a)^2,x, algorithm="maxima")

[Out]

-1/3*((2*cos(2*b*x + 2*a) + 1)*sin(8*b*x + 8*a) + 2*(2*cos(2*b*x + 2*a) + 1)*sin(6*b*x + 6*a) - 2*cos(8*b*x +
8*a)*sin(2*b*x + 2*a) - 4*cos(6*b*x + 6*a)*sin(2*b*x + 2*a))/(b*cos(8*b*x + 8*a)^2 + 4*b*cos(6*b*x + 6*a)^2 +
4*b*cos(2*b*x + 2*a)^2 + b*sin(8*b*x + 8*a)^2 + 4*b*sin(6*b*x + 6*a)^2 - 8*b*sin(6*b*x + 6*a)*sin(2*b*x + 2*a)
+ 4*b*sin(2*b*x + 2*a)^2 + 2*(2*b*cos(6*b*x + 6*a) - 2*b*cos(2*b*x + 2*a) - b)*cos(8*b*x + 8*a) - 4*(2*b*cos(
2*b*x + 2*a) + b)*cos(6*b*x + 6*a) + 4*b*cos(2*b*x + 2*a) + 4*(b*sin(6*b*x + 6*a) - b*sin(2*b*x + 2*a))*sin(8*
b*x + 8*a) + b)

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Fricas [A]  time = 0.469919, size = 109, normalized size = 2.6 \begin{align*} -\frac{8 \, \cos \left (b x + a\right )^{4} - 4 \, \cos \left (b x + a\right )^{2} - 1}{48 \, b \cos \left (b x + a\right )^{3} \sin \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(2*b*x+2*a)^4*sin(b*x+a)^2,x, algorithm="fricas")

[Out]

-1/48*(8*cos(b*x + a)^4 - 4*cos(b*x + a)^2 - 1)/(b*cos(b*x + a)^3*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(2*b*x+2*a)**4*sin(b*x+a)**2,x)

[Out]

Timed out

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Giac [B]  time = 1.69862, size = 1396, normalized size = 33.24 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(2*b*x+2*a)^4*sin(b*x+a)^2,x, algorithm="giac")

[Out]

-1/384*(24*(tan(1/2*a)^12 + 6*tan(1/2*a)^10 + 15*tan(1/2*a)^8 + 20*tan(1/2*a)^6 + 15*tan(1/2*a)^4 + 6*tan(1/2*
a)^2 + 1)/((tan(b*x + 4*a)*tan(1/2*a)^6 - 15*tan(b*x + 4*a)*tan(1/2*a)^4 + 6*tan(1/2*a)^5 + 15*tan(b*x + 4*a)*
tan(1/2*a)^2 - 20*tan(1/2*a)^3 - tan(b*x + 4*a) + 6*tan(1/2*a))*(tan(1/2*a)^6 - 15*tan(1/2*a)^4 + 15*tan(1/2*a
)^2 - 1)) + (108*tan(b*x + 4*a)^2*tan(1/2*a)^34 - 18*tan(b*x + 4*a)*tan(1/2*a)^35 + tan(1/2*a)^36 + 4464*tan(b
*x + 4*a)^2*tan(1/2*a)^32 - 1182*tan(b*x + 4*a)*tan(1/2*a)^33 + 126*tan(1/2*a)^34 - 28608*tan(b*x + 4*a)^2*tan
(1/2*a)^30 + 26208*tan(b*x + 4*a)*tan(1/2*a)^31 - 3159*tan(1/2*a)^32 + 14544*tan(b*x + 4*a)^2*tan(1/2*a)^28 -
81216*tan(b*x + 4*a)*tan(1/2*a)^29 + 29232*tan(1/2*a)^30 + 197136*tan(b*x + 4*a)^2*tan(1/2*a)^26 - 130344*tan(
b*x + 4*a)*tan(1/2*a)^27 - 26460*tan(1/2*a)^28 - 230160*tan(b*x + 4*a)^2*tan(1/2*a)^24 + 657288*tan(b*x + 4*a)
*tan(1/2*a)^25 - 228600*tan(1/2*a)^26 - 753984*tan(b*x + 4*a)^2*tan(1/2*a)^22 + 419328*tan(b*x + 4*a)*tan(1/2*
a)^23 + 237588*tan(1/2*a)^24 + 604368*tan(b*x + 4*a)^2*tan(1/2*a)^20 - 2188128*tan(b*x + 4*a)*tan(1/2*a)^21 +
944208*tan(1/2*a)^22 + 1957128*tan(b*x + 4*a)^2*tan(1/2*a)^18 - 1928412*tan(b*x + 4*a)*tan(1/2*a)^19 - 142434*
tan(1/2*a)^20 + 604368*tan(b*x + 4*a)^2*tan(1/2*a)^16 + 1928412*tan(b*x + 4*a)*tan(1/2*a)^17 - 1358860*tan(1/2
*a)^18 - 753984*tan(b*x + 4*a)^2*tan(1/2*a)^14 + 2188128*tan(b*x + 4*a)*tan(1/2*a)^15 - 142434*tan(1/2*a)^16 -
230160*tan(b*x + 4*a)^2*tan(1/2*a)^12 - 419328*tan(b*x + 4*a)*tan(1/2*a)^13 + 944208*tan(1/2*a)^14 + 197136*t
an(b*x + 4*a)^2*tan(1/2*a)^10 - 657288*tan(b*x + 4*a)*tan(1/2*a)^11 + 237588*tan(1/2*a)^12 + 14544*tan(b*x + 4
*a)^2*tan(1/2*a)^8 + 130344*tan(b*x + 4*a)*tan(1/2*a)^9 - 228600*tan(1/2*a)^10 - 28608*tan(b*x + 4*a)^2*tan(1/
2*a)^6 + 81216*tan(b*x + 4*a)*tan(1/2*a)^7 - 26460*tan(1/2*a)^8 + 4464*tan(b*x + 4*a)^2*tan(1/2*a)^4 - 26208*t
an(b*x + 4*a)*tan(1/2*a)^5 + 29232*tan(1/2*a)^6 + 108*tan(b*x + 4*a)^2*tan(1/2*a)^2 + 1182*tan(b*x + 4*a)*tan(
1/2*a)^3 - 3159*tan(1/2*a)^4 + 18*tan(b*x + 4*a)*tan(1/2*a) + 126*tan(1/2*a)^2 + 1)/((27*tan(1/2*a)^15 - 270*t
an(1/2*a)^13 + 981*tan(1/2*a)^11 - 1540*tan(1/2*a)^9 + 981*tan(1/2*a)^7 - 270*tan(1/2*a)^5 + 27*tan(1/2*a)^3)*
(6*tan(b*x + 4*a)*tan(1/2*a)^5 - tan(1/2*a)^6 - 20*tan(b*x + 4*a)*tan(1/2*a)^3 + 15*tan(1/2*a)^4 + 6*tan(b*x +
4*a)*tan(1/2*a) - 15*tan(1/2*a)^2 + 1)^3))/b