3.150 \(\int \cos ^2(a+b x) \csc ^5(2 a+2 b x) \, dx\)
Optimal. Leaf size=60 \[ \frac{\tan ^2(a+b x)}{64 b}-\frac{\cot ^4(a+b x)}{128 b}-\frac{3 \cot ^2(a+b x)}{64 b}+\frac{3 \log (\tan (a+b x))}{32 b} \]
[Out]
(-3*Cot[a + b*x]^2)/(64*b) - Cot[a + b*x]^4/(128*b) + (3*Log[Tan[a + b*x]])/(32*b) + Tan[a + b*x]^2/(64*b)
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Rubi [A] time = 0.067544, antiderivative size = 60, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used =
{4287, 2620, 266, 43} \[ \frac{\tan ^2(a+b x)}{64 b}-\frac{\cot ^4(a+b x)}{128 b}-\frac{3 \cot ^2(a+b x)}{64 b}+\frac{3 \log (\tan (a+b x))}{32 b} \]
Antiderivative was successfully verified.
[In]
Int[Cos[a + b*x]^2*Csc[2*a + 2*b*x]^5,x]
[Out]
(-3*Cot[a + b*x]^2)/(64*b) - Cot[a + b*x]^4/(128*b) + (3*Log[Tan[a + b*x]])/(32*b) + Tan[a + b*x]^2/(64*b)
Rule 4287
Int[(cos[(a_.) + (b_.)*(x_)]*(e_.))^(m_.)*sin[(c_.) + (d_.)*(x_)]^(p_.), x_Symbol] :> Dist[2^p/e^p, Int[(e*Cos
[a + b*x])^(m + p)*Sin[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, e, m}, 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 266
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Rule 43
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Rubi steps
\begin{align*} \int \cos ^2(a+b x) \csc ^5(2 a+2 b x) \, dx &=\frac{1}{32} \int \csc ^5(a+b x) \sec ^3(a+b x) \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{x^5} \, dx,x,\tan (a+b x)\right )}{32 b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(1+x)^3}{x^3} \, dx,x,\tan ^2(a+b x)\right )}{64 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{1}{x^3}+\frac{3}{x^2}+\frac{3}{x}\right ) \, dx,x,\tan ^2(a+b x)\right )}{64 b}\\ &=-\frac{3 \cot ^2(a+b x)}{64 b}-\frac{\cot ^4(a+b x)}{128 b}+\frac{3 \log (\tan (a+b x))}{32 b}+\frac{\tan ^2(a+b x)}{64 b}\\ \end{align*}
Mathematica [A] time = 0.367386, size = 54, normalized size = 0.9 \[ -\frac{\csc ^4(a+b x)+4 \csc ^2(a+b x)-2 \sec ^2(a+b x)-12 \log (\sin (a+b x))+12 \log (\cos (a+b x))}{128 b} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[a + b*x]^2*Csc[2*a + 2*b*x]^5,x]
[Out]
-(4*Csc[a + b*x]^2 + Csc[a + b*x]^4 + 12*Log[Cos[a + b*x]] - 12*Log[Sin[a + b*x]] - 2*Sec[a + b*x]^2)/(128*b)
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Maple [A] time = 0.027, size = 69, normalized size = 1.2 \begin{align*} -{\frac{1}{128\,b \left ( \sin \left ( bx+a \right ) \right ) ^{4} \left ( \cos \left ( bx+a \right ) \right ) ^{2}}}+{\frac{3}{128\,b \left ( \sin \left ( bx+a \right ) \right ) ^{2} \left ( \cos \left ( bx+a \right ) \right ) ^{2}}}-{\frac{3}{64\,b \left ( \sin \left ( bx+a \right ) \right ) ^{2}}}+{\frac{3\,\ln \left ( \tan \left ( bx+a \right ) \right ) }{32\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(b*x+a)^2/sin(2*b*x+2*a)^5,x)
[Out]
-1/128/b/sin(b*x+a)^4/cos(b*x+a)^2+3/128/b/sin(b*x+a)^2/cos(b*x+a)^2-3/64/b/sin(b*x+a)^2+3/32*ln(tan(b*x+a))/b
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Maxima [B] time = 1.50663, size = 4304, normalized size = 71.73 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(b*x+a)^2/sin(2*b*x+2*a)^5,x, algorithm="maxima")
[Out]
1/64*(4*(3*cos(10*b*x + 10*a) - 6*cos(8*b*x + 8*a) - 2*cos(6*b*x + 6*a) - 6*cos(4*b*x + 4*a) + 3*cos(2*b*x + 2
*a))*cos(12*b*x + 12*a) + 4*(9*cos(8*b*x + 8*a) + 16*cos(6*b*x + 6*a) + 9*cos(4*b*x + 4*a) - 12*cos(2*b*x + 2*
a) + 3)*cos(10*b*x + 10*a) - 24*cos(10*b*x + 10*a)^2 - 4*(22*cos(6*b*x + 6*a) - 12*cos(4*b*x + 4*a) - 9*cos(2*
b*x + 2*a) + 6)*cos(8*b*x + 8*a) + 24*cos(8*b*x + 8*a)^2 - 8*(11*cos(4*b*x + 4*a) - 8*cos(2*b*x + 2*a) + 1)*co
s(6*b*x + 6*a) - 32*cos(6*b*x + 6*a)^2 + 12*(3*cos(2*b*x + 2*a) - 2)*cos(4*b*x + 4*a) + 24*cos(4*b*x + 4*a)^2
- 24*cos(2*b*x + 2*a)^2 + 3*(2*(2*cos(10*b*x + 10*a) + cos(8*b*x + 8*a) - 4*cos(6*b*x + 6*a) + cos(4*b*x + 4*a
) + 2*cos(2*b*x + 2*a) - 1)*cos(12*b*x + 12*a) - cos(12*b*x + 12*a)^2 - 4*(cos(8*b*x + 8*a) - 4*cos(6*b*x + 6*
a) + cos(4*b*x + 4*a) + 2*cos(2*b*x + 2*a) - 1)*cos(10*b*x + 10*a) - 4*cos(10*b*x + 10*a)^2 + 2*(4*cos(6*b*x +
6*a) - cos(4*b*x + 4*a) - 2*cos(2*b*x + 2*a) + 1)*cos(8*b*x + 8*a) - cos(8*b*x + 8*a)^2 + 8*(cos(4*b*x + 4*a)
+ 2*cos(2*b*x + 2*a) - 1)*cos(6*b*x + 6*a) - 16*cos(6*b*x + 6*a)^2 - 2*(2*cos(2*b*x + 2*a) - 1)*cos(4*b*x + 4
*a) - cos(4*b*x + 4*a)^2 - 4*cos(2*b*x + 2*a)^2 + 2*(2*sin(10*b*x + 10*a) + sin(8*b*x + 8*a) - 4*sin(6*b*x + 6
*a) + sin(4*b*x + 4*a) + 2*sin(2*b*x + 2*a))*sin(12*b*x + 12*a) - sin(12*b*x + 12*a)^2 - 4*(sin(8*b*x + 8*a) -
4*sin(6*b*x + 6*a) + sin(4*b*x + 4*a) + 2*sin(2*b*x + 2*a))*sin(10*b*x + 10*a) - 4*sin(10*b*x + 10*a)^2 + 2*(
4*sin(6*b*x + 6*a) - sin(4*b*x + 4*a) - 2*sin(2*b*x + 2*a))*sin(8*b*x + 8*a) - sin(8*b*x + 8*a)^2 + 8*(sin(4*b
*x + 4*a) + 2*sin(2*b*x + 2*a))*sin(6*b*x + 6*a) - 16*sin(6*b*x + 6*a)^2 - sin(4*b*x + 4*a)^2 - 4*sin(4*b*x +
4*a)*sin(2*b*x + 2*a) - 4*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x + 2*a) - 1)*log(cos(2*b*x)^2 + 2*cos(2*b*x)*cos(2*a
) + cos(2*a)^2 + sin(2*b*x)^2 - 2*sin(2*b*x)*sin(2*a) + sin(2*a)^2) - 3*(2*(2*cos(10*b*x + 10*a) + cos(8*b*x +
8*a) - 4*cos(6*b*x + 6*a) + cos(4*b*x + 4*a) + 2*cos(2*b*x + 2*a) - 1)*cos(12*b*x + 12*a) - cos(12*b*x + 12*a
)^2 - 4*(cos(8*b*x + 8*a) - 4*cos(6*b*x + 6*a) + cos(4*b*x + 4*a) + 2*cos(2*b*x + 2*a) - 1)*cos(10*b*x + 10*a)
- 4*cos(10*b*x + 10*a)^2 + 2*(4*cos(6*b*x + 6*a) - cos(4*b*x + 4*a) - 2*cos(2*b*x + 2*a) + 1)*cos(8*b*x + 8*a
) - cos(8*b*x + 8*a)^2 + 8*(cos(4*b*x + 4*a) + 2*cos(2*b*x + 2*a) - 1)*cos(6*b*x + 6*a) - 16*cos(6*b*x + 6*a)^
2 - 2*(2*cos(2*b*x + 2*a) - 1)*cos(4*b*x + 4*a) - cos(4*b*x + 4*a)^2 - 4*cos(2*b*x + 2*a)^2 + 2*(2*sin(10*b*x
+ 10*a) + sin(8*b*x + 8*a) - 4*sin(6*b*x + 6*a) + sin(4*b*x + 4*a) + 2*sin(2*b*x + 2*a))*sin(12*b*x + 12*a) -
sin(12*b*x + 12*a)^2 - 4*(sin(8*b*x + 8*a) - 4*sin(6*b*x + 6*a) + sin(4*b*x + 4*a) + 2*sin(2*b*x + 2*a))*sin(1
0*b*x + 10*a) - 4*sin(10*b*x + 10*a)^2 + 2*(4*sin(6*b*x + 6*a) - sin(4*b*x + 4*a) - 2*sin(2*b*x + 2*a))*sin(8*
b*x + 8*a) - sin(8*b*x + 8*a)^2 + 8*(sin(4*b*x + 4*a) + 2*sin(2*b*x + 2*a))*sin(6*b*x + 6*a) - 16*sin(6*b*x +
6*a)^2 - sin(4*b*x + 4*a)^2 - 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) - 4*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x + 2*a)
- 1)*log(cos(b*x)^2 + 2*cos(b*x)*cos(a) + cos(a)^2 + sin(b*x)^2 - 2*sin(b*x)*sin(a) + sin(a)^2) - 3*(2*(2*cos(
10*b*x + 10*a) + cos(8*b*x + 8*a) - 4*cos(6*b*x + 6*a) + cos(4*b*x + 4*a) + 2*cos(2*b*x + 2*a) - 1)*cos(12*b*x
+ 12*a) - cos(12*b*x + 12*a)^2 - 4*(cos(8*b*x + 8*a) - 4*cos(6*b*x + 6*a) + cos(4*b*x + 4*a) + 2*cos(2*b*x +
2*a) - 1)*cos(10*b*x + 10*a) - 4*cos(10*b*x + 10*a)^2 + 2*(4*cos(6*b*x + 6*a) - cos(4*b*x + 4*a) - 2*cos(2*b*x
+ 2*a) + 1)*cos(8*b*x + 8*a) - cos(8*b*x + 8*a)^2 + 8*(cos(4*b*x + 4*a) + 2*cos(2*b*x + 2*a) - 1)*cos(6*b*x +
6*a) - 16*cos(6*b*x + 6*a)^2 - 2*(2*cos(2*b*x + 2*a) - 1)*cos(4*b*x + 4*a) - cos(4*b*x + 4*a)^2 - 4*cos(2*b*x
+ 2*a)^2 + 2*(2*sin(10*b*x + 10*a) + sin(8*b*x + 8*a) - 4*sin(6*b*x + 6*a) + sin(4*b*x + 4*a) + 2*sin(2*b*x +
2*a))*sin(12*b*x + 12*a) - sin(12*b*x + 12*a)^2 - 4*(sin(8*b*x + 8*a) - 4*sin(6*b*x + 6*a) + sin(4*b*x + 4*a)
+ 2*sin(2*b*x + 2*a))*sin(10*b*x + 10*a) - 4*sin(10*b*x + 10*a)^2 + 2*(4*sin(6*b*x + 6*a) - sin(4*b*x + 4*a)
- 2*sin(2*b*x + 2*a))*sin(8*b*x + 8*a) - sin(8*b*x + 8*a)^2 + 8*(sin(4*b*x + 4*a) + 2*sin(2*b*x + 2*a))*sin(6*
b*x + 6*a) - 16*sin(6*b*x + 6*a)^2 - sin(4*b*x + 4*a)^2 - 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) - 4*sin(2*b*x +
2*a)^2 + 4*cos(2*b*x + 2*a) - 1)*log(cos(b*x)^2 - 2*cos(b*x)*cos(a) + cos(a)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(a
) + sin(a)^2) + 4*(3*sin(10*b*x + 10*a) - 6*sin(8*b*x + 8*a) - 2*sin(6*b*x + 6*a) - 6*sin(4*b*x + 4*a) + 3*sin
(2*b*x + 2*a))*sin(12*b*x + 12*a) + 4*(9*sin(8*b*x + 8*a) + 16*sin(6*b*x + 6*a) + 9*sin(4*b*x + 4*a) - 12*sin(
2*b*x + 2*a))*sin(10*b*x + 10*a) - 24*sin(10*b*x + 10*a)^2 - 4*(22*sin(6*b*x + 6*a) - 12*sin(4*b*x + 4*a) - 9*
sin(2*b*x + 2*a))*sin(8*b*x + 8*a) + 24*sin(8*b*x + 8*a)^2 - 8*(11*sin(4*b*x + 4*a) - 8*sin(2*b*x + 2*a))*sin(
6*b*x + 6*a) - 32*sin(6*b*x + 6*a)^2 + 24*sin(4*b*x + 4*a)^2 + 36*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) - 24*sin(2
*b*x + 2*a)^2 + 12*cos(2*b*x + 2*a))/(b*cos(12*b*x + 12*a)^2 + 4*b*cos(10*b*x + 10*a)^2 + b*cos(8*b*x + 8*a)^2
+ 16*b*cos(6*b*x + 6*a)^2 + b*cos(4*b*x + 4*a)^2 + 4*b*cos(2*b*x + 2*a)^2 + b*sin(12*b*x + 12*a)^2 + 4*b*sin(
10*b*x + 10*a)^2 + b*sin(8*b*x + 8*a)^2 + 16*b*sin(6*b*x + 6*a)^2 + b*sin(4*b*x + 4*a)^2 + 4*b*sin(4*b*x + 4*a
)*sin(2*b*x + 2*a) + 4*b*sin(2*b*x + 2*a)^2 - 2*(2*b*cos(10*b*x + 10*a) + b*cos(8*b*x + 8*a) - 4*b*cos(6*b*x +
6*a) + b*cos(4*b*x + 4*a) + 2*b*cos(2*b*x + 2*a) - b)*cos(12*b*x + 12*a) + 4*(b*cos(8*b*x + 8*a) - 4*b*cos(6*
b*x + 6*a) + b*cos(4*b*x + 4*a) + 2*b*cos(2*b*x + 2*a) - b)*cos(10*b*x + 10*a) - 2*(4*b*cos(6*b*x + 6*a) - b*c
os(4*b*x + 4*a) - 2*b*cos(2*b*x + 2*a) + b)*cos(8*b*x + 8*a) - 8*(b*cos(4*b*x + 4*a) + 2*b*cos(2*b*x + 2*a) -
b)*cos(6*b*x + 6*a) + 2*(2*b*cos(2*b*x + 2*a) - b)*cos(4*b*x + 4*a) - 4*b*cos(2*b*x + 2*a) - 2*(2*b*sin(10*b*x
+ 10*a) + b*sin(8*b*x + 8*a) - 4*b*sin(6*b*x + 6*a) + b*sin(4*b*x + 4*a) + 2*b*sin(2*b*x + 2*a))*sin(12*b*x +
12*a) + 4*(b*sin(8*b*x + 8*a) - 4*b*sin(6*b*x + 6*a) + b*sin(4*b*x + 4*a) + 2*b*sin(2*b*x + 2*a))*sin(10*b*x
+ 10*a) - 2*(4*b*sin(6*b*x + 6*a) - b*sin(4*b*x + 4*a) - 2*b*sin(2*b*x + 2*a))*sin(8*b*x + 8*a) - 8*(b*sin(4*b
*x + 4*a) + 2*b*sin(2*b*x + 2*a))*sin(6*b*x + 6*a) + b)
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Fricas [B] time = 0.516827, size = 369, normalized size = 6.15 \begin{align*} \frac{6 \, \cos \left (b x + a\right )^{4} - 9 \, \cos \left (b x + a\right )^{2} - 6 \,{\left (\cos \left (b x + a\right )^{6} - 2 \, \cos \left (b x + a\right )^{4} + \cos \left (b x + a\right )^{2}\right )} \log \left (\cos \left (b x + a\right )^{2}\right ) + 6 \,{\left (\cos \left (b x + a\right )^{6} - 2 \, \cos \left (b x + a\right )^{4} + \cos \left (b x + a\right )^{2}\right )} \log \left (-\frac{1}{4} \, \cos \left (b x + a\right )^{2} + \frac{1}{4}\right ) + 2}{128 \,{\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(b*x+a)^2/sin(2*b*x+2*a)^5,x, algorithm="fricas")
[Out]
1/128*(6*cos(b*x + a)^4 - 9*cos(b*x + a)^2 - 6*(cos(b*x + a)^6 - 2*cos(b*x + a)^4 + cos(b*x + a)^2)*log(cos(b*
x + a)^2) + 6*(cos(b*x + a)^6 - 2*cos(b*x + a)^4 + cos(b*x + a)^2)*log(-1/4*cos(b*x + a)^2 + 1/4) + 2)/(b*cos(
b*x + a)^6 - 2*b*cos(b*x + a)^4 + b*cos(b*x + a)^2)
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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(cos(b*x+a)**2/sin(2*b*x+2*a)**5,x)
[Out]
Timed out
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Giac [B] time = 1.29568, size = 313, normalized size = 5.22 \begin{align*} \frac{\frac{20 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac{{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac{\frac{18 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac{111 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac{36 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac{72 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} - 1}{{\left (\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + \frac{{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}}\right )}^{2}} + 96 \, \log \left (\frac{{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) - 192 \, \log \left ({\left | -\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1 \right |}\right )}{2048 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(b*x+a)^2/sin(2*b*x+2*a)^5,x, algorithm="giac")
[Out]
1/2048*(20*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) - (cos(b*x + a) - 1)^2/(cos(b*x + a) + 1)^2 + (18*(cos(b*x +
a) - 1)/(cos(b*x + a) + 1) + 111*(cos(b*x + a) - 1)^2/(cos(b*x + a) + 1)^2 + 36*(cos(b*x + a) - 1)^3/(cos(b*x
+ a) + 1)^3 + 72*(cos(b*x + a) - 1)^4/(cos(b*x + a) + 1)^4 - 1)/((cos(b*x + a) - 1)/(cos(b*x + a) + 1) + (cos(
b*x + a) - 1)^2/(cos(b*x + a) + 1)^2)^2 + 96*log(abs(-cos(b*x + a) + 1)/abs(cos(b*x + a) + 1)) - 192*log(abs(-
(cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 1)))/b