\(\int \cos (a+b x) \csc ^5(2 a+2 b x) \, dx\) [140]
Optimal result
Integrand size = 18, antiderivative size = 89 \[
\int \cos (a+b x) \csc ^5(2 a+2 b x) \, dx=-\frac {35 \text {arctanh}(\cos (a+b x))}{256 b}+\frac {35 \sec (a+b x)}{256 b}+\frac {35 \sec ^3(a+b x)}{768 b}-\frac {7 \csc ^2(a+b x) \sec ^3(a+b x)}{256 b}-\frac {\csc ^4(a+b x) \sec ^3(a+b x)}{128 b}
\]
[Out]
-35/256*arctanh(cos(b*x+a))/b+35/256*sec(b*x+a)/b+35/768*sec(b*x+a)^3/b-7/256*csc(b*x+a)^2*sec(b*x+a)^3/b-1/12
8*csc(b*x+a)^4*sec(b*x+a)^3/b
Rubi [A] (verified)
Time = 0.09 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of
steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {4372, 2702, 294, 308, 213}
\[
\int \cos (a+b x) \csc ^5(2 a+2 b x) \, dx=-\frac {35 \text {arctanh}(\cos (a+b x))}{256 b}+\frac {35 \sec ^3(a+b x)}{768 b}+\frac {35 \sec (a+b x)}{256 b}-\frac {\csc ^4(a+b x) \sec ^3(a+b x)}{128 b}-\frac {7 \csc ^2(a+b x) \sec ^3(a+b x)}{256 b}
\]
[In]
Int[Cos[a + b*x]*Csc[2*a + 2*b*x]^5,x]
[Out]
(-35*ArcTanh[Cos[a + b*x]])/(256*b) + (35*Sec[a + b*x])/(256*b) + (35*Sec[a + b*x]^3)/(768*b) - (7*Csc[a + b*x
]^2*Sec[a + b*x]^3)/(256*b) - (Csc[a + b*x]^4*Sec[a + b*x]^3)/(128*b)
Rule 213
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Rule 294
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]
Rule 308
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]
Rule 2702
Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] && !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])
Rule 4372
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]
Rubi steps \begin{align*}
\text {integral}& = \frac {1}{32} \int \csc ^5(a+b x) \sec ^4(a+b x) \, dx \\ & = \frac {\text {Subst}\left (\int \frac {x^8}{\left (-1+x^2\right )^3} \, dx,x,\sec (a+b x)\right )}{32 b} \\ & = -\frac {\csc ^4(a+b x) \sec ^3(a+b x)}{128 b}+\frac {7 \text {Subst}\left (\int \frac {x^6}{\left (-1+x^2\right )^2} \, dx,x,\sec (a+b x)\right )}{128 b} \\ & = -\frac {7 \csc ^2(a+b x) \sec ^3(a+b x)}{256 b}-\frac {\csc ^4(a+b x) \sec ^3(a+b x)}{128 b}+\frac {35 \text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{256 b} \\ & = -\frac {7 \csc ^2(a+b x) \sec ^3(a+b x)}{256 b}-\frac {\csc ^4(a+b x) \sec ^3(a+b x)}{128 b}+\frac {35 \text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (a+b x)\right )}{256 b} \\ & = \frac {35 \sec (a+b x)}{256 b}+\frac {35 \sec ^3(a+b x)}{768 b}-\frac {7 \csc ^2(a+b x) \sec ^3(a+b x)}{256 b}-\frac {\csc ^4(a+b x) \sec ^3(a+b x)}{128 b}+\frac {35 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{256 b} \\ & = -\frac {35 \text {arctanh}(\cos (a+b x))}{256 b}+\frac {35 \sec (a+b x)}{256 b}+\frac {35 \sec ^3(a+b x)}{768 b}-\frac {7 \csc ^2(a+b x) \sec ^3(a+b x)}{256 b}-\frac {\csc ^4(a+b x) \sec ^3(a+b x)}{128 b} \\
\end{align*}
Mathematica [B] (verified)
Leaf count is larger than twice the leaf count of optimal. \(268\) vs. \(2(89)=178\).
Time = 0.73 (sec) , antiderivative size = 268, normalized size of antiderivative = 3.01
\[
\int \cos (a+b x) \csc ^5(2 a+2 b x) \, dx=-\frac {\csc ^{10}(a+b x) \left (-204+658 \cos (2 (a+b x))-228 \cos (3 (a+b x))+140 \cos (4 (a+b x))-76 \cos (5 (a+b x))-210 \cos (6 (a+b x))+76 \cos (7 (a+b x))-315 \cos (3 (a+b x)) \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )-105 \cos (5 (a+b x)) \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )+105 \cos (7 (a+b x)) \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )+3 \cos (a+b x) \left (76+105 \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )-105 \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )\right )+315 \cos (3 (a+b x)) \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )+105 \cos (5 (a+b x)) \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )-105 \cos (7 (a+b x)) \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )\right )}{768 b \left (\csc ^2\left (\frac {1}{2} (a+b x)\right )-\sec ^2\left (\frac {1}{2} (a+b x)\right )\right )^3}
\]
[In]
Integrate[Cos[a + b*x]*Csc[2*a + 2*b*x]^5,x]
[Out]
-1/768*(Csc[a + b*x]^10*(-204 + 658*Cos[2*(a + b*x)] - 228*Cos[3*(a + b*x)] + 140*Cos[4*(a + b*x)] - 76*Cos[5*
(a + b*x)] - 210*Cos[6*(a + b*x)] + 76*Cos[7*(a + b*x)] - 315*Cos[3*(a + b*x)]*Log[Cos[(a + b*x)/2]] - 105*Cos
[5*(a + b*x)]*Log[Cos[(a + b*x)/2]] + 105*Cos[7*(a + b*x)]*Log[Cos[(a + b*x)/2]] + 3*Cos[a + b*x]*(76 + 105*Lo
g[Cos[(a + b*x)/2]] - 105*Log[Sin[(a + b*x)/2]]) + 315*Cos[3*(a + b*x)]*Log[Sin[(a + b*x)/2]] + 105*Cos[5*(a +
b*x)]*Log[Sin[(a + b*x)/2]] - 105*Cos[7*(a + b*x)]*Log[Sin[(a + b*x)/2]]))/(b*(Csc[(a + b*x)/2]^2 - Sec[(a +
b*x)/2]^2)^3)
Maple [A] (verified)
Time = 5.29 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00
| | |
| method | result | size |
| | |
| default |
\(\frac {-\frac {1}{4 \sin \left (x b +a \right )^{4} \cos \left (x b +a \right )^{3}}+\frac {7}{12 \sin \left (x b +a \right )^{2} \cos \left (x b +a \right )^{3}}-\frac {35}{24 \sin \left (x b +a \right )^{2} \cos \left (x b +a \right )}+\frac {35}{8 \cos \left (x b +a \right )}+\frac {35 \ln \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}{8}}{32 b}\) |
\(89\) |
| risch |
\(\frac {105 \,{\mathrm e}^{13 i \left (x b +a \right )}-70 \,{\mathrm e}^{11 i \left (x b +a \right )}-329 \,{\mathrm e}^{9 i \left (x b +a \right )}+204 \,{\mathrm e}^{7 i \left (x b +a \right )}-329 \,{\mathrm e}^{5 i \left (x b +a \right )}-70 \,{\mathrm e}^{3 i \left (x b +a \right )}+105 \,{\mathrm e}^{i \left (x b +a \right )}}{384 b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{4} \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )^{3}}+\frac {35 \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{256 b}-\frac {35 \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{256 b}\) |
\(145\) |
| | |
|
|
|
|
[In]
int(cos(b*x+a)/sin(2*b*x+2*a)^5,x,method=_RETURNVERBOSE)
[Out]
1/32/b*(-1/4/sin(b*x+a)^4/cos(b*x+a)^3+7/12/sin(b*x+a)^2/cos(b*x+a)^3-35/24/sin(b*x+a)^2/cos(b*x+a)+35/8/cos(b
*x+a)+35/8*ln(csc(b*x+a)-cot(b*x+a)))
Fricas [A] (verification not implemented)
none
Time = 0.25 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.66
\[
\int \cos (a+b x) \csc ^5(2 a+2 b x) \, dx=\frac {210 \, \cos \left (b x + a\right )^{6} - 350 \, \cos \left (b x + a\right )^{4} + 112 \, \cos \left (b x + a\right )^{2} - 105 \, {\left (\cos \left (b x + a\right )^{7} - 2 \, \cos \left (b x + a\right )^{5} + \cos \left (b x + a\right )^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 105 \, {\left (\cos \left (b x + a\right )^{7} - 2 \, \cos \left (b x + a\right )^{5} + \cos \left (b x + a\right )^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 16}{1536 \, {\left (b \cos \left (b x + a\right )^{7} - 2 \, b \cos \left (b x + a\right )^{5} + b \cos \left (b x + a\right )^{3}\right )}}
\]
[In]
integrate(cos(b*x+a)/sin(2*b*x+2*a)^5,x, algorithm="fricas")
[Out]
1/1536*(210*cos(b*x + a)^6 - 350*cos(b*x + a)^4 + 112*cos(b*x + a)^2 - 105*(cos(b*x + a)^7 - 2*cos(b*x + a)^5
+ cos(b*x + a)^3)*log(1/2*cos(b*x + a) + 1/2) + 105*(cos(b*x + a)^7 - 2*cos(b*x + a)^5 + cos(b*x + a)^3)*log(-
1/2*cos(b*x + a) + 1/2) + 16)/(b*cos(b*x + a)^7 - 2*b*cos(b*x + a)^5 + b*cos(b*x + a)^3)
Sympy [F(-1)]
Timed out. \[
\int \cos (a+b x) \csc ^5(2 a+2 b x) \, dx=\text {Timed out}
\]
[In]
integrate(cos(b*x+a)/sin(2*b*x+2*a)**5,x)
[Out]
Timed out
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 3846 vs. \(2 (79) = 158\).
Time = 0.33 (sec) , antiderivative size = 3846, normalized size of antiderivative = 43.21
\[
\int \cos (a+b x) \csc ^5(2 a+2 b x) \, dx=\text {Too large to display}
\]
[In]
integrate(cos(b*x+a)/sin(2*b*x+2*a)^5,x, algorithm="maxima")
[Out]
1/1536*(4*(105*cos(13*b*x + 13*a) - 70*cos(11*b*x + 11*a) - 329*cos(9*b*x + 9*a) + 204*cos(7*b*x + 7*a) - 329*
cos(5*b*x + 5*a) - 70*cos(3*b*x + 3*a) + 105*cos(b*x + a))*cos(14*b*x + 14*a) - 420*(cos(12*b*x + 12*a) + 3*co
s(10*b*x + 10*a) - 3*cos(8*b*x + 8*a) - 3*cos(6*b*x + 6*a) + 3*cos(4*b*x + 4*a) + cos(2*b*x + 2*a) - 1)*cos(13
*b*x + 13*a) + 4*(70*cos(11*b*x + 11*a) + 329*cos(9*b*x + 9*a) - 204*cos(7*b*x + 7*a) + 329*cos(5*b*x + 5*a) +
70*cos(3*b*x + 3*a) - 105*cos(b*x + a))*cos(12*b*x + 12*a) + 280*(3*cos(10*b*x + 10*a) - 3*cos(8*b*x + 8*a) -
3*cos(6*b*x + 6*a) + 3*cos(4*b*x + 4*a) + cos(2*b*x + 2*a) - 1)*cos(11*b*x + 11*a) + 12*(329*cos(9*b*x + 9*a)
- 204*cos(7*b*x + 7*a) + 329*cos(5*b*x + 5*a) + 70*cos(3*b*x + 3*a) - 105*cos(b*x + a))*cos(10*b*x + 10*a) -
1316*(3*cos(8*b*x + 8*a) + 3*cos(6*b*x + 6*a) - 3*cos(4*b*x + 4*a) - cos(2*b*x + 2*a) + 1)*cos(9*b*x + 9*a) +
12*(204*cos(7*b*x + 7*a) - 329*cos(5*b*x + 5*a) - 70*cos(3*b*x + 3*a) + 105*cos(b*x + a))*cos(8*b*x + 8*a) + 8
16*(3*cos(6*b*x + 6*a) - 3*cos(4*b*x + 4*a) - cos(2*b*x + 2*a) + 1)*cos(7*b*x + 7*a) - 84*(47*cos(5*b*x + 5*a)
+ 10*cos(3*b*x + 3*a) - 15*cos(b*x + a))*cos(6*b*x + 6*a) + 1316*(3*cos(4*b*x + 4*a) + cos(2*b*x + 2*a) - 1)*
cos(5*b*x + 5*a) + 420*(2*cos(3*b*x + 3*a) - 3*cos(b*x + a))*cos(4*b*x + 4*a) + 280*(cos(2*b*x + 2*a) - 1)*cos
(3*b*x + 3*a) - 420*cos(2*b*x + 2*a)*cos(b*x + a) + 105*(2*(cos(12*b*x + 12*a) + 3*cos(10*b*x + 10*a) - 3*cos(
8*b*x + 8*a) - 3*cos(6*b*x + 6*a) + 3*cos(4*b*x + 4*a) + cos(2*b*x + 2*a) - 1)*cos(14*b*x + 14*a) - cos(14*b*x
+ 14*a)^2 - 2*(3*cos(10*b*x + 10*a) - 3*cos(8*b*x + 8*a) - 3*cos(6*b*x + 6*a) + 3*cos(4*b*x + 4*a) + cos(2*b*
x + 2*a) - 1)*cos(12*b*x + 12*a) - cos(12*b*x + 12*a)^2 + 6*(3*cos(8*b*x + 8*a) + 3*cos(6*b*x + 6*a) - 3*cos(4
*b*x + 4*a) - cos(2*b*x + 2*a) + 1)*cos(10*b*x + 10*a) - 9*cos(10*b*x + 10*a)^2 - 6*(3*cos(6*b*x + 6*a) - 3*co
s(4*b*x + 4*a) - cos(2*b*x + 2*a) + 1)*cos(8*b*x + 8*a) - 9*cos(8*b*x + 8*a)^2 + 6*(3*cos(4*b*x + 4*a) + cos(2
*b*x + 2*a) - 1)*cos(6*b*x + 6*a) - 9*cos(6*b*x + 6*a)^2 - 6*(cos(2*b*x + 2*a) - 1)*cos(4*b*x + 4*a) - 9*cos(4
*b*x + 4*a)^2 - cos(2*b*x + 2*a)^2 + 2*(sin(12*b*x + 12*a) + 3*sin(10*b*x + 10*a) - 3*sin(8*b*x + 8*a) - 3*sin
(6*b*x + 6*a) + 3*sin(4*b*x + 4*a) + sin(2*b*x + 2*a))*sin(14*b*x + 14*a) - sin(14*b*x + 14*a)^2 - 2*(3*sin(10
*b*x + 10*a) - 3*sin(8*b*x + 8*a) - 3*sin(6*b*x + 6*a) + 3*sin(4*b*x + 4*a) + sin(2*b*x + 2*a))*sin(12*b*x + 1
2*a) - sin(12*b*x + 12*a)^2 + 6*(3*sin(8*b*x + 8*a) + 3*sin(6*b*x + 6*a) - 3*sin(4*b*x + 4*a) - sin(2*b*x + 2*
a))*sin(10*b*x + 10*a) - 9*sin(10*b*x + 10*a)^2 - 6*(3*sin(6*b*x + 6*a) - 3*sin(4*b*x + 4*a) - sin(2*b*x + 2*a
))*sin(8*b*x + 8*a) - 9*sin(8*b*x + 8*a)^2 + 6*(3*sin(4*b*x + 4*a) + sin(2*b*x + 2*a))*sin(6*b*x + 6*a) - 9*si
n(6*b*x + 6*a)^2 - 9*sin(4*b*x + 4*a)^2 - 6*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) - sin(2*b*x + 2*a)^2 + 2*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) - 10
5*(2*(cos(12*b*x + 12*a) + 3*cos(10*b*x + 10*a) - 3*cos(8*b*x + 8*a) - 3*cos(6*b*x + 6*a) + 3*cos(4*b*x + 4*a)
+ cos(2*b*x + 2*a) - 1)*cos(14*b*x + 14*a) - cos(14*b*x + 14*a)^2 - 2*(3*cos(10*b*x + 10*a) - 3*cos(8*b*x + 8
*a) - 3*cos(6*b*x + 6*a) + 3*cos(4*b*x + 4*a) + cos(2*b*x + 2*a) - 1)*cos(12*b*x + 12*a) - cos(12*b*x + 12*a)^
2 + 6*(3*cos(8*b*x + 8*a) + 3*cos(6*b*x + 6*a) - 3*cos(4*b*x + 4*a) - cos(2*b*x + 2*a) + 1)*cos(10*b*x + 10*a)
- 9*cos(10*b*x + 10*a)^2 - 6*(3*cos(6*b*x + 6*a) - 3*cos(4*b*x + 4*a) - cos(2*b*x + 2*a) + 1)*cos(8*b*x + 8*a
) - 9*cos(8*b*x + 8*a)^2 + 6*(3*cos(4*b*x + 4*a) + cos(2*b*x + 2*a) - 1)*cos(6*b*x + 6*a) - 9*cos(6*b*x + 6*a)
^2 - 6*(cos(2*b*x + 2*a) - 1)*cos(4*b*x + 4*a) - 9*cos(4*b*x + 4*a)^2 - cos(2*b*x + 2*a)^2 + 2*(sin(12*b*x + 1
2*a) + 3*sin(10*b*x + 10*a) - 3*sin(8*b*x + 8*a) - 3*sin(6*b*x + 6*a) + 3*sin(4*b*x + 4*a) + sin(2*b*x + 2*a))
*sin(14*b*x + 14*a) - sin(14*b*x + 14*a)^2 - 2*(3*sin(10*b*x + 10*a) - 3*sin(8*b*x + 8*a) - 3*sin(6*b*x + 6*a)
+ 3*sin(4*b*x + 4*a) + sin(2*b*x + 2*a))*sin(12*b*x + 12*a) - sin(12*b*x + 12*a)^2 + 6*(3*sin(8*b*x + 8*a) +
3*sin(6*b*x + 6*a) - 3*sin(4*b*x + 4*a) - sin(2*b*x + 2*a))*sin(10*b*x + 10*a) - 9*sin(10*b*x + 10*a)^2 - 6*(3
*sin(6*b*x + 6*a) - 3*sin(4*b*x + 4*a) - sin(2*b*x + 2*a))*sin(8*b*x + 8*a) - 9*sin(8*b*x + 8*a)^2 + 6*(3*sin(
4*b*x + 4*a) + sin(2*b*x + 2*a))*sin(6*b*x + 6*a) - 9*sin(6*b*x + 6*a)^2 - 9*sin(4*b*x + 4*a)^2 - 6*sin(4*b*x
+ 4*a)*sin(2*b*x + 2*a) - sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) - 1)*log(cos(b*x)^2 - 2*cos(b*x)*cos(a) + co
s(a)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(a) + sin(a)^2) + 4*(105*sin(13*b*x + 13*a) - 70*sin(11*b*x + 11*a) - 329*
sin(9*b*x + 9*a) + 204*sin(7*b*x + 7*a) - 329*sin(5*b*x + 5*a) - 70*sin(3*b*x + 3*a) + 105*sin(b*x + a))*sin(1
4*b*x + 14*a) - 420*(sin(12*b*x + 12*a) + 3*sin(10*b*x + 10*a) - 3*sin(8*b*x + 8*a) - 3*sin(6*b*x + 6*a) + 3*s
in(4*b*x + 4*a) + sin(2*b*x + 2*a))*sin(13*b*x + 13*a) + 4*(70*sin(11*b*x + 11*a) + 329*sin(9*b*x + 9*a) - 204
*sin(7*b*x + 7*a) + 329*sin(5*b*x + 5*a) + 70*sin(3*b*x + 3*a) - 105*sin(b*x + a))*sin(12*b*x + 12*a) + 280*(3
*sin(10*b*x + 10*a) - 3*sin(8*b*x + 8*a) - 3*sin(6*b*x + 6*a) + 3*sin(4*b*x + 4*a) + sin(2*b*x + 2*a))*sin(11*
b*x + 11*a) + 12*(329*sin(9*b*x + 9*a) - 204*sin(7*b*x + 7*a) + 329*sin(5*b*x + 5*a) + 70*sin(3*b*x + 3*a) - 1
05*sin(b*x + a))*sin(10*b*x + 10*a) - 1316*(3*sin(8*b*x + 8*a) + 3*sin(6*b*x + 6*a) - 3*sin(4*b*x + 4*a) - sin
(2*b*x + 2*a))*sin(9*b*x + 9*a) + 12*(204*sin(7*b*x + 7*a) - 329*sin(5*b*x + 5*a) - 70*sin(3*b*x + 3*a) + 105*
sin(b*x + a))*sin(8*b*x + 8*a) + 816*(3*sin(6*b*x + 6*a) - 3*sin(4*b*x + 4*a) - sin(2*b*x + 2*a))*sin(7*b*x +
7*a) - 84*(47*sin(5*b*x + 5*a) + 10*sin(3*b*x + 3*a) - 15*sin(b*x + a))*sin(6*b*x + 6*a) + 1316*(3*sin(4*b*x +
4*a) + sin(2*b*x + 2*a))*sin(5*b*x + 5*a) + 420*(2*sin(3*b*x + 3*a) - 3*sin(b*x + a))*sin(4*b*x + 4*a) + 280*
sin(3*b*x + 3*a)*sin(2*b*x + 2*a) - 420*sin(2*b*x + 2*a)*sin(b*x + a) + 420*cos(b*x + a))/(b*cos(14*b*x + 14*a
)^2 + b*cos(12*b*x + 12*a)^2 + 9*b*cos(10*b*x + 10*a)^2 + 9*b*cos(8*b*x + 8*a)^2 + 9*b*cos(6*b*x + 6*a)^2 + 9*
b*cos(4*b*x + 4*a)^2 + b*cos(2*b*x + 2*a)^2 + b*sin(14*b*x + 14*a)^2 + b*sin(12*b*x + 12*a)^2 + 9*b*sin(10*b*x
+ 10*a)^2 + 9*b*sin(8*b*x + 8*a)^2 + 9*b*sin(6*b*x + 6*a)^2 + 9*b*sin(4*b*x + 4*a)^2 + 6*b*sin(4*b*x + 4*a)*s
in(2*b*x + 2*a) + b*sin(2*b*x + 2*a)^2 - 2*(b*cos(12*b*x + 12*a) + 3*b*cos(10*b*x + 10*a) - 3*b*cos(8*b*x + 8*
a) - 3*b*cos(6*b*x + 6*a) + 3*b*cos(4*b*x + 4*a) + b*cos(2*b*x + 2*a) - b)*cos(14*b*x + 14*a) + 2*(3*b*cos(10*
b*x + 10*a) - 3*b*cos(8*b*x + 8*a) - 3*b*cos(6*b*x + 6*a) + 3*b*cos(4*b*x + 4*a) + b*cos(2*b*x + 2*a) - b)*cos
(12*b*x + 12*a) - 6*(3*b*cos(8*b*x + 8*a) + 3*b*cos(6*b*x + 6*a) - 3*b*cos(4*b*x + 4*a) - b*cos(2*b*x + 2*a) +
b)*cos(10*b*x + 10*a) + 6*(3*b*cos(6*b*x + 6*a) - 3*b*cos(4*b*x + 4*a) - b*cos(2*b*x + 2*a) + b)*cos(8*b*x +
8*a) - 6*(3*b*cos(4*b*x + 4*a) + b*cos(2*b*x + 2*a) - b)*cos(6*b*x + 6*a) + 6*(b*cos(2*b*x + 2*a) - b)*cos(4*b
*x + 4*a) - 2*b*cos(2*b*x + 2*a) - 2*(b*sin(12*b*x + 12*a) + 3*b*sin(10*b*x + 10*a) - 3*b*sin(8*b*x + 8*a) - 3
*b*sin(6*b*x + 6*a) + 3*b*sin(4*b*x + 4*a) + b*sin(2*b*x + 2*a))*sin(14*b*x + 14*a) + 2*(3*b*sin(10*b*x + 10*a
) - 3*b*sin(8*b*x + 8*a) - 3*b*sin(6*b*x + 6*a) + 3*b*sin(4*b*x + 4*a) + b*sin(2*b*x + 2*a))*sin(12*b*x + 12*a
) - 6*(3*b*sin(8*b*x + 8*a) + 3*b*sin(6*b*x + 6*a) - 3*b*sin(4*b*x + 4*a) - b*sin(2*b*x + 2*a))*sin(10*b*x + 1
0*a) + 6*(3*b*sin(6*b*x + 6*a) - 3*b*sin(4*b*x + 4*a) - b*sin(2*b*x + 2*a))*sin(8*b*x + 8*a) - 6*(3*b*sin(4*b*
x + 4*a) + b*sin(2*b*x + 2*a))*sin(6*b*x + 6*a) + b)
Giac [A] (verification not implemented)
none
Time = 0.29 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.96
\[
\int \cos (a+b x) \csc ^5(2 a+2 b x) \, dx=\frac {\frac {6 \, {\left (11 \, \cos \left (b x + a\right )^{3} - 13 \, \cos \left (b x + a\right )\right )}}{{\left (\cos \left (b x + a\right )^{2} - 1\right )}^{2}} + \frac {16 \, {\left (9 \, \cos \left (b x + a\right )^{2} + 1\right )}}{\cos \left (b x + a\right )^{3}} - 105 \, \log \left (\cos \left (b x + a\right ) + 1\right ) + 105 \, \log \left (-\cos \left (b x + a\right ) + 1\right )}{1536 \, b}
\]
[In]
integrate(cos(b*x+a)/sin(2*b*x+2*a)^5,x, algorithm="giac")
[Out]
1/1536*(6*(11*cos(b*x + a)^3 - 13*cos(b*x + a))/(cos(b*x + a)^2 - 1)^2 + 16*(9*cos(b*x + a)^2 + 1)/cos(b*x + a
)^3 - 105*log(cos(b*x + a) + 1) + 105*log(-cos(b*x + a) + 1))/b
Mupad [B] (verification not implemented)
Time = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.88
\[
\int \cos (a+b x) \csc ^5(2 a+2 b x) \, dx=\frac {\frac {35\,{\cos \left (a+b\,x\right )}^6}{256}-\frac {175\,{\cos \left (a+b\,x\right )}^4}{768}+\frac {7\,{\cos \left (a+b\,x\right )}^2}{96}+\frac {1}{96}}{b\,\left ({\cos \left (a+b\,x\right )}^7-2\,{\cos \left (a+b\,x\right )}^5+{\cos \left (a+b\,x\right )}^3\right )}-\frac {35\,\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{256\,b}
\]
[In]
int(cos(a + b*x)/sin(2*a + 2*b*x)^5,x)
[Out]
((7*cos(a + b*x)^2)/96 - (175*cos(a + b*x)^4)/768 + (35*cos(a + b*x)^6)/256 + 1/96)/(b*(cos(a + b*x)^3 - 2*cos
(a + b*x)^5 + cos(a + b*x)^7)) - (35*atanh(cos(a + b*x)))/(256*b)