\(\int \csc ^2(a+b x) \sin (2 a+2 b x) \, dx\) [52]
Optimal result
Integrand size = 18, antiderivative size = 12 \[
\int \csc ^2(a+b x) \sin (2 a+2 b x) \, dx=\frac {2 \log (\sin (a+b x))}{b}
\]
[Out]
2*ln(sin(b*x+a))/b
Rubi [A] (verified)
Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number
of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4373, 3556}
\[
\int \csc ^2(a+b x) \sin (2 a+2 b x) \, dx=\frac {2 \log (\sin (a+b x))}{b}
\]
[In]
Int[Csc[a + b*x]^2*Sin[2*a + 2*b*x],x]
[Out]
(2*Log[Sin[a + b*x]])/b
Rule 3556
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]
Rule 4373
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]
Rubi steps \begin{align*}
\text {integral}& = 2 \int \cot (a+b x) \, dx \\ & = \frac {2 \log (\sin (a+b x))}{b} \\
\end{align*}
Mathematica [B] (verified)
Leaf count is larger than twice the leaf count of optimal. \(25\) vs. \(2(12)=24\).
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 2.08
\[
\int \csc ^2(a+b x) \sin (2 a+2 b x) \, dx=2 \left (\frac {\log (\cos (a+b x))}{b}+\frac {\log (\tan (a+b x))}{b}\right )
\]
[In]
Integrate[Csc[a + b*x]^2*Sin[2*a + 2*b*x],x]
[Out]
2*(Log[Cos[a + b*x]]/b + Log[Tan[a + b*x]]/b)
Maple [A] (verified)
Time = 0.59 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08
| | |
| method | result | size |
| | |
| default |
\(\frac {2 \ln \left (\sin \left (x b +a \right )\right )}{b}\) |
\(13\) |
| risch |
\(-2 i x -\frac {4 i a}{b}+\frac {2 \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}{b}\) |
\(30\) |
| | |
|
|
|
|
[In]
int(csc(b*x+a)^2*sin(2*b*x+2*a),x,method=_RETURNVERBOSE)
[Out]
2*ln(sin(b*x+a))/b
Fricas [A] (verification not implemented)
none
Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17
\[
\int \csc ^2(a+b x) \sin (2 a+2 b x) \, dx=\frac {2 \, \log \left (\frac {1}{2} \, \sin \left (b x + a\right )\right )}{b}
\]
[In]
integrate(csc(b*x+a)^2*sin(2*b*x+2*a),x, algorithm="fricas")
[Out]
2*log(1/2*sin(b*x + a))/b
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 6307 vs. \(2 (10) = 20\).
Time = 133.80 (sec) , antiderivative size = 18894, normalized size of antiderivative = 1574.50
\[
\int \csc ^2(a+b x) \sin (2 a+2 b x) \, dx=\text {Too large to display}
\]
[In]
integrate(csc(b*x+a)**2*sin(2*b*x+2*a),x)
[Out]
4*Piecewise((zoo*x, Eq(a, 0) & Eq(b, 0)), (log(sin(b*x))/b, Eq(a, 0)), (0, Eq(b, 0)), (-4*b*x*tan(a/2)**5*tan(
b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*
b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/
2) - b*tan(b*x/2)) - 4*b*x*tan(a/2)**4*tan(b*x/2)**2/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 -
b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*ta
n(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + 4*b*x*tan(a/2)**4/(b*tan(a/2)**6*tan(b*x/2)
+ b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*
b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + 8*b*x*tan(a
/2)**3*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(
b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2
- b*tan(a/2) - b*tan(b*x/2)) + 4*b*x*tan(a/2)**2*tan(b*x/2)**2/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(
b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan
(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - 4*b*x*tan(a/2)**2/(b*tan(a/2)**6
*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x
/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) -
4*b*x*tan(a/2)*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)
**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b
*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + log(tan(a/2) + tan(b*x/2))*tan(a/2)**6*tan(b*x/2)/(b*tan(a/2)**6*tan(b
*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2
- 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + log(ta
n(a/2) + tan(b*x/2))*tan(a/2)**5*tan(b*x/2)**2/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan
(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/
2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - log(tan(a/2) + tan(b*x/2))*tan(a/2)**5/(b*tan(a/2
)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan
(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)
) - 7*log(tan(a/2) + tan(b*x/2))*tan(a/2)**4*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**
2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2
*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - 6*log(tan(a/2) + tan(b*x/2))*tan(a/2)**3
*tan(b*x/2)**2/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x
/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 -
b*tan(a/2) - b*tan(b*x/2)) + 6*log(tan(a/2) + tan(b*x/2))*tan(a/2)**3/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**
5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 -
b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + 7*log(tan(a/2) + tan(b*x/2
))*tan(a/2)**2*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)
**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b
*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + log(tan(a/2) + tan(b*x/2))*tan(a/2)*tan(b*x/2)**2/(b*tan(a/2)**6*tan(b
*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2
- 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - log(ta
n(a/2) + tan(b*x/2))*tan(a/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(
a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*t
an(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - log(tan(a/2) + tan(b*x/2))*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) +
b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*t
an(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + log(tan(b*x/2)
- 1/tan(a/2))*tan(a/2)**6*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5
+ b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*ta
n(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)**5*tan(b*x/2)**2/(b*
tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)
**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan
(b*x/2)) - log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)**5/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 -
b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan
(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - 7*log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)**4*t
an(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) +
2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan
(a/2) - b*tan(b*x/2)) - 6*log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)**3*tan(b*x/2)**2/(b*tan(a/2)**6*tan(b*x/2) + b
*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*ta
n(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + 6*log(tan(b*x/2
) - 1/tan(a/2))*tan(a/2)**3/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/
2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan
(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + 7*log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)**2*tan(b*x/2)/(b*tan(a/2)**6
*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x
/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) +
log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)*tan(b*x/2)**2/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 -
b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan
(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)/(b*tan
(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3
*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*
x/2)) - log(tan(b*x/2) - 1/tan(a/2))*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*ta
n(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x
/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - log(tan(b*x/2)**2 + 1)*tan(a/2)**6*tan(b*x/2)/(b
*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2
)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*ta
n(b*x/2)) - log(tan(b*x/2)**2 + 1)*tan(a/2)**5*tan(b*x/2)**2/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x
/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/
2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + log(tan(b*x/2)**2 + 1)*tan(a/2)**5/
(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a
/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*
tan(b*x/2)) + 7*log(tan(b*x/2)**2 + 1)*tan(a/2)**4*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*
x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a
/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + 6*log(tan(b*x/2)**2 + 1)*tan(a/2)*
*3*tan(b*x/2)**2/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b
*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2
- b*tan(a/2) - b*tan(b*x/2)) - 6*log(tan(b*x/2)**2 + 1)*tan(a/2)**3/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*
tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b
*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - 7*log(tan(b*x/2)**2 + 1)*tan
(a/2)**2*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*ta
n(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)*
*2 - b*tan(a/2) - b*tan(b*x/2)) - log(tan(b*x/2)**2 + 1)*tan(a/2)*tan(b*x/2)**2/(b*tan(a/2)**6*tan(b*x/2) + b*
tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan
(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + log(tan(b*x/2)**
2 + 1)*tan(a/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*
x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 -
b*tan(a/2) - b*tan(b*x/2)) + log(tan(b*x/2)**2 + 1)*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(
b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan
(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - tan(a/2)**6*tan(b*x/2)/(b*tan(a/
2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*ta
n(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2
)) - tan(a/2)**4*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/
2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan
(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + tan(a/2)**2*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan
(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*ta
n(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + tan(b*x/2)/(b*tan(a/2)**6*tan(b
*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2
- 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)), True))*
cos(a)**2 - 2*Piecewise((zoo*x, Eq(a, 0) & Eq(b, 0)), (log(sin(b*x))/b, Eq(a, 0)), (0, Eq(b, 0)), (-4*b*x*tan(
a/2)**5*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan
(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**
2 - b*tan(a/2) - b*tan(b*x/2)) - 4*b*x*tan(a/2)**4*tan(b*x/2)**2/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan
(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*ta
n(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + 4*b*x*tan(a/2)**4/(b*tan(a/2)**
6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*
x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) +
8*b*x*tan(a/2)**3*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(
a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*t
an(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + 4*b*x*tan(a/2)**2*tan(b*x/2)**2/(b*tan(a/2)**6*tan(b*x/2) + b*tan(
a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2
)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - 4*b*x*tan(a/2)**2/(b
*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2
)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*ta
n(b*x/2)) - 4*b*x*tan(a/2)*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5
+ b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*ta
n(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + log(tan(a/2) + tan(b*x/2))*tan(a/2)**6*tan(b*x/2)/(b*tan(a
/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*t
an(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/
2)) + log(tan(a/2) + tan(b*x/2))*tan(a/2)**5*tan(b*x/2)**2/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2
)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)
**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - log(tan(a/2) + tan(b*x/2))*tan(a/2)**
5/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan
(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) -
b*tan(b*x/2)) - 7*log(tan(a/2) + tan(b*x/2))*tan(a/2)**4*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*
tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b
*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - 6*log(tan(a/2) + tan(b*x/2))
*tan(a/2)**3*tan(b*x/2)**2/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2
)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(
b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + 6*log(tan(a/2) + tan(b*x/2))*tan(a/2)**3/(b*tan(a/2)**6*tan(b*x/2) +
b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*t
an(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + 7*log(tan(a/2)
+ tan(b*x/2))*tan(a/2)**2*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5
+ b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*ta
n(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + log(tan(a/2) + tan(b*x/2))*tan(a/2)*tan(b*x/2)**2/(b*tan(a
/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*t
an(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/
2)) - log(tan(a/2) + tan(b*x/2))*tan(a/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)
**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) +
b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - log(tan(a/2) + tan(b*x/2))*tan(b*x/2)/(b*tan(a/2)**6*t
an(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2
)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + lo
g(tan(b*x/2) - 1/tan(a/2))*tan(a/2)**6*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*
tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b
*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)**5*tan(b
*x/2)**2/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) +
2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(
a/2) - b*tan(b*x/2)) - log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)**5/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(
b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan
(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - 7*log(tan(b*x/2) - 1/tan(a/2))*t
an(a/2)**4*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*
tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2
)**2 - b*tan(a/2) - b*tan(b*x/2)) - 6*log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)**3*tan(b*x/2)**2/(b*tan(a/2)**6*ta
n(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)
**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + 6*l
og(tan(b*x/2) - 1/tan(a/2))*tan(a/2)**3/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**
5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*
tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + 7*log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)**2*tan(b*x/2)/(b
*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2
)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*ta
n(b*x/2)) + log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)*tan(b*x/2)**2/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(
b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan
(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - log(tan(b*x/2) - 1/tan(a/2))*tan
(a/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b
*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2
) - b*tan(b*x/2)) - log(tan(b*x/2) - 1/tan(a/2))*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/
2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2
)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - log(tan(b*x/2)**2 + 1)*tan(a/2)**6*t
an(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) +
2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan
(a/2) - b*tan(b*x/2)) - log(tan(b*x/2)**2 + 1)*tan(a/2)**5*tan(b*x/2)**2/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2
)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**
3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + log(tan(b*x/2)**2 + 1)*
tan(a/2)**5/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2)
+ 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*t
an(a/2) - b*tan(b*x/2)) + 7*log(tan(b*x/2)**2 + 1)*tan(a/2)**4*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/
2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)*
*3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + 6*log(tan(b*x/2)**2 +
1)*tan(a/2)**3*tan(b*x/2)**2/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a
/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*ta
n(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - 6*log(tan(b*x/2)**2 + 1)*tan(a/2)**3/(b*tan(a/2)**6*tan(b*x/2) + b*
tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan
(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - 7*log(tan(b*x/2)
**2 + 1)*tan(a/2)**2*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*ta
n(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)
*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - log(tan(b*x/2)**2 + 1)*tan(a/2)*tan(b*x/2)**2/(b*tan(a/2)**6*tan
(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)*
*2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + log(
tan(b*x/2)**2 + 1)*tan(a/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/
2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan
(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + log(tan(b*x/2)**2 + 1)*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(
a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2
)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - tan(a/2)**6*tan(b*x/
2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*ta
n(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) -
b*tan(b*x/2)) - tan(a/2)**4*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**
5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*
tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + tan(a/2)**2*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan
(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/
2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + tan(b*x/2)/(b*tan(a
/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*t
an(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/
2)), True)) + 4*Piecewise((zoo*x, Eq(a, 0) & Eq(b, 0)), (-x - cos(b*x)/(b*sin(b*x)), Eq(a, 0)), (x/sin(a)**2,
Eq(b, 0)), (-2*b*x*tan(a/2)**7*tan(b*x/2)/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*ta
n(a/2)**6 + 2*b*tan(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan
(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)) - 2*b*x*tan(a/2)**6*tan(b
*x/2)**2/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan(a/2)**5*tan(b
*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x
/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)) + 2*b*x*tan(a/2)**6/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(
a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan
(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x
/2)) + 14*b*x*tan(a/2)**5*tan(b*x/2)/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2
)**6 + 2*b*tan(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/
2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)) + 12*b*x*tan(a/2)**4*tan(b*x/2
)**2/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan(a/2)**5*tan(b*x/2
) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)*
*2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)) - 12*b*x*tan(a/2)**4/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2
)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/
2)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)
) - 14*b*x*tan(a/2)**3*tan(b*x/2)/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**
6 + 2*b*tan(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/2)
+ 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)) - 2*b*x*tan(a/2)**2*tan(b*x/2)**2
/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan(a/2)**5*tan(b*x/2) +
4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 -
2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)) + 2*b*x*tan(a/2)**2/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)**6*
tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2)**4
- 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)) + 2
*b*x*tan(a/2)*tan(b*x/2)/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*t
an(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan
(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)) - 8*log(tan(a/2) + tan(b*x/2))*tan(a/2)**6
*tan(b*x/2)/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan(a/2)**5*ta
n(b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(
b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)) - 8*log(tan(a/2) + tan(b*x/2))*tan(a/2)**5*tan(b*x/2)**
2/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan(a/2)**5*tan(b*x/2) +
4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2
- 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)) + 8*log(tan(a/2) + tan(b*x/2))*tan(a/2)**5/(2*b*tan(a/2)**7*tan(b
*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x
/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*
tan(a/2)*tan(b*x/2)) + 16*log(tan(a/2) + tan(b*x/2))*tan(a/2)**4*tan(b*x/2)/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*
tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b
*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan
(b*x/2)) + 8*log(tan(a/2) + tan(b*x/2))*tan(a/2)**3*tan(b*x/2)**2/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)**
6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2)*
*4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)) -
8*log(tan(a/2) + tan(b*x/2))*tan(a/2)**3/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*ta
n(a/2)**6 + 2*b*tan(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan
(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)) - 8*log(tan(a/2) + tan(b*
x/2))*tan(a/2)**2*tan(b*x/2)/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2
*b*tan(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b
*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)) - 8*log(tan(b*x/2) - 1/tan(a/2))*tan(a
/2)**6*tan(b*x/2)/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan(a/2)
**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**
2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)) - 8*log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)**5*tan(
b*x/2)**2/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan(a/2)**5*tan(
b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*
x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)) + 8*log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)**5/(2*b*tan(a/2
)**7*tan(b*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)*
*4*tan(b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)
**2 - 2*b*tan(a/2)*tan(b*x/2)) + 16*log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)**4*tan(b*x/2)/(2*b*tan(a/2)**7*tan(b
*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x
/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*
tan(a/2)*tan(b*x/2)) + 8*log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)**3*tan(b*x/2)**2/(2*b*tan(a/2)**7*tan(b*x/2) +
2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 -
4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)
*tan(b*x/2)) - 8*log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)**3/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)**6*tan(b*
x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b
*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)) - 8*log(t
an(b*x/2) - 1/tan(a/2))*tan(a/2)**2*tan(b*x/2)/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2
*b*tan(a/2)**6 + 2*b*tan(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**
3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)) + 8*log(tan(b*x/2)**
2 + 1)*tan(a/2)**6*tan(b*x/2)/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 +
2*b*tan(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*
b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)) + 8*log(tan(b*x/2)**2 + 1)*tan(a/2)**
5*tan(b*x/2)**2/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan(a/2)**
5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*
tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)) - 8*log(tan(b*x/2)**2 + 1)*tan(a/2)**5/(2*b*tan(a/2
)**7*tan(b*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)*
*4*tan(b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)
**2 - 2*b*tan(a/2)*tan(b*x/2)) - 16*log(tan(b*x/2)**2 + 1)*tan(a/2)**4*tan(b*x/2)/(2*b*tan(a/2)**7*tan(b*x/2)
+ 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2
- 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/
2)*tan(b*x/2)) - 8*log(tan(b*x/2)**2 + 1)*tan(a/2)**3*tan(b*x/2)**2/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)
**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2
)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2))
+ 8*log(tan(b*x/2)**2 + 1)*tan(a/2)**3/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(
a/2)**6 + 2*b*tan(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b
*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)) + 8*log(tan(b*x/2)**2 + 1)*
tan(a/2)**2*tan(b*x/2)/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan
(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a
/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)) - tan(a/2)**8*tan(b*x/2)/(2*b*tan(a/2)**7*ta
n(b*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(
b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2
*b*tan(a/2)*tan(b*x/2)) + 2*tan(a/2)**4*tan(b*x/2)/(2*b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2
- 2*b*tan(a/2)**6 + 2*b*tan(a/2)**5*tan(b*x/2) + 4*b*tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/
2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)) - tan(b*x/2)/(2*
b*tan(a/2)**7*tan(b*x/2) + 2*b*tan(a/2)**6*tan(b*x/2)**2 - 2*b*tan(a/2)**6 + 2*b*tan(a/2)**5*tan(b*x/2) + 4*b*
tan(a/2)**4*tan(b*x/2)**2 - 4*b*tan(a/2)**4 - 2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b
*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)), True))*sin(a)*cos(a) - 2*Piecewise((zoo*x, Eq(a, 0) & Eq(b, 0)), (tan
(b*x/2)/(2*b) - 1/(2*b*tan(b*x/2)), Eq(a, 0)), (x/sin(a)**2, Eq(b, 0)), (-tan(a/2)**4*tan(b*x/2)/(2*b*tan(a/2)
**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/2)) - 2*tan(a/2)**2*ta
n(b*x/2)/(2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*tan(b*x/
2)) - tan(b*x/2)/(2*b*tan(a/2)**3*tan(b*x/2) + 2*b*tan(a/2)**2*tan(b*x/2)**2 - 2*b*tan(a/2)**2 - 2*b*tan(a/2)*
tan(b*x/2)), True))*sin(a)*cos(a)
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (12) = 24\).
Time = 0.21 (sec) , antiderivative size = 81, normalized size of antiderivative = 6.75
\[
\int \csc ^2(a+b x) \sin (2 a+2 b x) \, dx=\frac {\log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right ) + \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right )}{b}
\]
[In]
integrate(csc(b*x+a)^2*sin(2*b*x+2*a),x, algorithm="maxima")
[Out]
(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) + 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))/b
Giac [A] (verification not implemented)
none
Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08
\[
\int \csc ^2(a+b x) \sin (2 a+2 b x) \, dx=\frac {2 \, \log \left ({\left | \sin \left (b x + a\right ) \right |}\right )}{b}
\]
[In]
integrate(csc(b*x+a)^2*sin(2*b*x+2*a),x, algorithm="giac")
[Out]
2*log(abs(sin(b*x + a)))/b
Mupad [B] (verification not implemented)
Time = 19.67 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08
\[
\int \csc ^2(a+b x) \sin (2 a+2 b x) \, dx=\frac {\ln \left ({\sin \left (a+b\,x\right )}^2\right )}{b}
\]
[In]
int(sin(2*a + 2*b*x)/sin(a + b*x)^2,x)
[Out]
log(sin(a + b*x)^2)/b