3.142 \(\int \cot (c-b x) \cot (a+b x) \, dx\)
Optimal. Leaf size=34 \[ -\frac {\cot (a+c) \log (\sin (c-b x))}{b}+\frac {\cot (a+c) \log (\sin (a+b x))}{b}+x \]
[Out]
x-cot(a+c)*ln(-sin(b*x-c))/b+cot(a+c)*ln(sin(b*x+a))/b
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Rubi [A] time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used =
{4613, 4611, 3475} \[ -\frac {\cot (a+c) \log (\sin (c-b x))}{b}+\frac {\cot (a+c) \log (\sin (a+b x))}{b}+x \]
Antiderivative was successfully verified.
[In]
Int[Cot[c - b*x]*Cot[a + b*x],x]
[Out]
x - (Cot[a + c]*Log[Sin[c - b*x]])/b + (Cot[a + c]*Log[Sin[a + b*x]])/b
Rule 3475
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]
Rule 4611
Int[Csc[(a_.) + (b_.)*(x_)]*Csc[(c_) + (d_.)*(x_)], x_Symbol] :> Dist[Csc[(b*c - a*d)/b], Int[Cot[a + b*x], x]
, x] - Dist[Csc[(b*c - a*d)/d], Int[Cot[c + d*x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ
[b*c - a*d, 0]
Rule 4613
Int[Cot[(a_.) + (b_.)*(x_)]*Cot[(c_) + (d_.)*(x_)], x_Symbol] :> -Simp[(b*x)/d, x] + Dist[Cos[(b*c - a*d)/d],
Int[Csc[a + b*x]*Csc[c + d*x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]
Rubi steps
\begin {align*} \int \cot (c-b x) \cot (a+b x) \, dx &=x+\cos (a+c) \int \csc (c-b x) \csc (a+b x) \, dx\\ &=x+\cot (a+c) \int \cot (c-b x) \, dx+\cot (a+c) \int \cot (a+b x) \, dx\\ &=x-\frac {\cot (a+c) \log (\sin (c-b x))}{b}+\frac {\cot (a+c) \log (\sin (a+b x))}{b}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 30, normalized size = 0.88 \[ \frac {\cot (a+c) (\log (-\sin (a+b x))-\log (\sin (c-b x)))}{b}+x \]
Antiderivative was successfully verified.
[In]
Integrate[Cot[c - b*x]*Cot[a + b*x],x]
[Out]
x + (Cot[a + c]*(-Log[Sin[c - b*x]] + Log[-Sin[a + b*x]]))/b
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fricas [B] time = 2.94, size = 118, normalized size = 3.47 \[ \frac {2 \, b x \sin \left (2 \, a + 2 \, c\right ) - {\left (\cos \left (2 \, a + 2 \, c\right ) + 1\right )} \log \left (-\frac {\cos \left (2 \, b x + 2 \, a\right ) \cos \left (2 \, a + 2 \, c\right ) + \sin \left (2 \, b x + 2 \, a\right ) \sin \left (2 \, a + 2 \, c\right ) - 1}{\cos \left (2 \, a + 2 \, c\right ) + 1}\right ) + {\left (\cos \left (2 \, a + 2 \, c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right )}{2 \, b \sin \left (2 \, a + 2 \, c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(-cot(b*x-c)*cot(b*x+a),x, algorithm="fricas")
[Out]
1/2*(2*b*x*sin(2*a + 2*c) - (cos(2*a + 2*c) + 1)*log(-(cos(2*b*x + 2*a)*cos(2*a + 2*c) + sin(2*b*x + 2*a)*sin(
2*a + 2*c) - 1)/(cos(2*a + 2*c) + 1)) + (cos(2*a + 2*c) + 1)*log(-1/2*cos(2*b*x + 2*a) + 1/2))/(b*sin(2*a + 2*
c))
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(-cot(b*x-c)*cot(b*x+a),x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (2*pi/x/2)>(-2*pi/
x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)-2/b*(-1/2*b*x+(ta
n(a/2)^2*tan(c/2)^4-2*tan(a/2)^2*tan(c/2)^2+tan(a/2)^2-4*tan(a/2)*tan(c/2)^3+4*tan(a/2)*tan(c/2)-tan(c/2)^4+2*
tan(c/2)^2-1)/(-4*tan(a/2)^2*tan(c/2)^3+4*tan(a/2)^2*tan(c/2)-4*tan(a/2)*tan(c/2)^4+8*tan(a/2)*tan(c/2)^2-4*ta
n(a/2)+4*tan(c/2)^3-4*tan(c/2))*ln(abs(tan(b*x)*tan(c/2)^2-tan(b*x)+2*tan(c/2)))+(tan(a/2)^4*tan(c/2)^2-tan(a/
2)^4-4*tan(a/2)^3*tan(c/2)-2*tan(a/2)^2*tan(c/2)^2+2*tan(a/2)^2+4*tan(a/2)*tan(c/2)+tan(c/2)^2-1)/(4*tan(a/2)^
4*tan(c/2)+4*tan(a/2)^3*tan(c/2)^2-4*tan(a/2)^3-8*tan(a/2)^2*tan(c/2)-4*tan(a/2)*tan(c/2)^2+4*tan(a/2)+4*tan(c
/2))*ln(abs(tan(b*x)*tan(a/2)^2-tan(b*x)-2*tan(a/2))))
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maple [C] time = 0.18, size = 149, normalized size = 4.38 \[ x -\frac {i \ln \left (-{\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{2 i \left (b x +a \right )}\right ) {\mathrm e}^{2 i \left (a +c \right )}}{b \left ({\mathrm e}^{2 i \left (a +c \right )}-1\right )}-\frac {i \ln \left (-{\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{2 i \left (b x +a \right )}\right )}{b \left ({\mathrm e}^{2 i \left (a +c \right )}-1\right )}+\frac {i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) {\mathrm e}^{2 i \left (a +c \right )}}{b \left ({\mathrm e}^{2 i \left (a +c \right )}-1\right )}+\frac {i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}{b \left ({\mathrm e}^{2 i \left (a +c \right )}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-cot(b*x-c)*cot(b*x+a),x)
[Out]
x-I/b/(exp(2*I*(a+c))-1)*ln(-exp(2*I*(a+c))+exp(2*I*(b*x+a)))*exp(2*I*(a+c))-I/b/(exp(2*I*(a+c))-1)*ln(-exp(2*
I*(a+c))+exp(2*I*(b*x+a)))+I/b/(exp(2*I*(a+c))-1)*ln(exp(2*I*(b*x+a))-1)*exp(2*I*(a+c))+I/b/(exp(2*I*(a+c))-1)
*ln(exp(2*I*(b*x+a))-1)
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maxima [B] time = 0.44, size = 432, normalized size = 12.71 \[ \frac {{\left (b \cos \left (2 \, a + 2 \, c\right )^{2} + b \sin \left (2 \, a + 2 \, c\right )^{2} - 2 \, b \cos \left (2 \, a + 2 \, c\right ) + b\right )} x - {\left (\cos \left (2 \, a + 2 \, c\right )^{2} + \sin \left (2 \, a + 2 \, c\right )^{2} - 1\right )} \arctan \left (\sin \left (b x\right ) + \sin \relax (a), \cos \left (b x\right ) - \cos \relax (a)\right ) - {\left (\cos \left (2 \, a + 2 \, c\right )^{2} + \sin \left (2 \, a + 2 \, c\right )^{2} - 1\right )} \arctan \left (\sin \left (b x\right ) - \sin \relax (a), \cos \left (b x\right ) + \cos \relax (a)\right ) + {\left (\cos \left (2 \, a + 2 \, c\right )^{2} + \sin \left (2 \, a + 2 \, c\right )^{2} - 1\right )} \arctan \left (\sin \left (b x\right ) + \sin \relax (c), \cos \left (b x\right ) + \cos \relax (c)\right ) + {\left (\cos \left (2 \, a + 2 \, c\right )^{2} + \sin \left (2 \, a + 2 \, c\right )^{2} - 1\right )} \arctan \left (\sin \left (b x\right ) - \sin \relax (c), \cos \left (b x\right ) - \cos \relax (c)\right ) + \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \relax (a) + \cos \relax (a)^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \relax (a) + \sin \relax (a)^{2}\right ) \sin \left (2 \, a + 2 \, c\right ) + \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \relax (a) + \cos \relax (a)^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \relax (a) + \sin \relax (a)^{2}\right ) \sin \left (2 \, a + 2 \, c\right ) - \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \relax (c) + \cos \relax (c)^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \relax (c) + \sin \relax (c)^{2}\right ) \sin \left (2 \, a + 2 \, c\right ) - \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \relax (c) + \cos \relax (c)^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \relax (c) + \sin \relax (c)^{2}\right ) \sin \left (2 \, a + 2 \, c\right )}{b \cos \left (2 \, a + 2 \, c\right )^{2} + b \sin \left (2 \, a + 2 \, c\right )^{2} - 2 \, b \cos \left (2 \, a + 2 \, c\right ) + b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(-cot(b*x-c)*cot(b*x+a),x, algorithm="maxima")
[Out]
((b*cos(2*a + 2*c)^2 + b*sin(2*a + 2*c)^2 - 2*b*cos(2*a + 2*c) + b)*x - (cos(2*a + 2*c)^2 + sin(2*a + 2*c)^2 -
1)*arctan2(sin(b*x) + sin(a), cos(b*x) - cos(a)) - (cos(2*a + 2*c)^2 + sin(2*a + 2*c)^2 - 1)*arctan2(sin(b*x)
- sin(a), cos(b*x) + cos(a)) + (cos(2*a + 2*c)^2 + sin(2*a + 2*c)^2 - 1)*arctan2(sin(b*x) + sin(c), cos(b*x)
+ cos(c)) + (cos(2*a + 2*c)^2 + sin(2*a + 2*c)^2 - 1)*arctan2(sin(b*x) - sin(c), cos(b*x) - cos(c)) + 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)*sin(2*a + 2*c) + 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)*sin(2*a + 2*c) - log(cos(b*x)
^2 + 2*cos(b*x)*cos(c) + cos(c)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(c) + sin(c)^2)*sin(2*a + 2*c) - log(cos(b*x)^2
- 2*cos(b*x)*cos(c) + cos(c)^2 + sin(b*x)^2 - 2*sin(b*x)*sin(c) + sin(c)^2)*sin(2*a + 2*c))/(b*cos(2*a + 2*c)
^2 + b*sin(2*a + 2*c)^2 - 2*b*cos(2*a + 2*c) + b)
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mupad [B] time = 5.05, size = 200, normalized size = 5.88 \[ \frac {\frac {x}{2}+x\,\left ({\sin \left (a+c\right )}^2-\frac {1}{2}\right )}{{\sin \left (a+c\right )}^2}+\frac {\frac {\sin \left (2\,a+2\,c\right )\,\ln \left ({\sin \left (2\,a+2\,c\right )}^2\,2{}\mathrm {i}+{\sin \left (a+b\,x\right )}^2\,2{}\mathrm {i}-{\sin \left (3\,a+2\,c+b\,x\right )}^2\,2{}\mathrm {i}+\sin \left (4\,a+4\,c\right )-\sin \left (6\,a+4\,c+2\,b\,x\right )+\sin \left (2\,a+2\,b\,x\right )\right )}{2}-\frac {\sin \left (2\,a+2\,c\right )\,\ln \left (-{\sin \left (2\,a+c+b\,x\right )}^2\,2{}\mathrm {i}+{\sin \left (2\,a+2\,c\right )}^2\,2{}\mathrm {i}+{\sin \left (c-b\,x\right )}^2\,2{}\mathrm {i}+\sin \left (4\,a+4\,c\right )-\sin \left (4\,a+2\,c+2\,b\,x\right )-\sin \left (2\,c-2\,b\,x\right )\right )}{2}}{b\,{\sin \left (a+c\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(a + b*x)*cot(c - b*x),x)
[Out]
(x/2 + x*(sin(a + c)^2 - 1/2))/sin(a + c)^2 + ((sin(2*a + 2*c)*log(sin(4*a + 4*c) - sin(6*a + 4*c + 2*b*x) + s
in(2*a + 2*b*x) + sin(2*a + 2*c)^2*2i + sin(a + b*x)^2*2i - sin(3*a + 2*c + b*x)^2*2i))/2 - (sin(2*a + 2*c)*lo
g(sin(4*a + 4*c) - sin(4*a + 2*c + 2*b*x) - sin(2*c - 2*b*x) - sin(2*a + c + b*x)^2*2i + sin(2*a + 2*c)^2*2i +
sin(c - b*x)^2*2i))/2)/(b*sin(a + c)^2)
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sympy [B] time = 24.89, size = 7499, normalized size = 220.56 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(-cot(b*x-c)*cot(b*x+a),x)
[Out]
-Piecewise((x/(zoo*cot(c) + zoo + cot(c)/tan(c) + zoo/tan(c)), Eq(b, 0) & Eq(a, -atan(tan(c)) - pi*floor((c -
pi/2)/pi) - pi*floor(c/pi - 1/2))), (b*x*tan(c)**5/(-b*tan(c)**5 + b*tan(c)**4*tan(b*x) - 2*b*tan(c)**3 + 2*b*
tan(c)**2*tan(b*x) - b*tan(c) + b*tan(b*x)) - b*x*tan(c)**4*tan(b*x)/(-b*tan(c)**5 + b*tan(c)**4*tan(b*x) - 2*
b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) - b*tan(c) + b*tan(b*x)) - b*x*tan(c)**3/(-b*tan(c)**5 + b*tan(c)**4*tan(
b*x) - 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) - b*tan(c) + b*tan(b*x)) + b*x*tan(c)**2*tan(b*x)/(-b*tan(c)**5
+ b*tan(c)**4*tan(b*x) - 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) - b*tan(c) + b*tan(b*x)) - 2*log(-tan(c) + tan
(b*x))*tan(c)**4/(-b*tan(c)**5 + b*tan(c)**4*tan(b*x) - 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) - b*tan(c) + b*
tan(b*x)) + 2*log(-tan(c) + tan(b*x))*tan(c)**3*tan(b*x)/(-b*tan(c)**5 + b*tan(c)**4*tan(b*x) - 2*b*tan(c)**3
+ 2*b*tan(c)**2*tan(b*x) - b*tan(c) + b*tan(b*x)) + log(tan(b*x)**2 + 1)*tan(c)**4/(-b*tan(c)**5 + b*tan(c)**4
*tan(b*x) - 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) - b*tan(c) + b*tan(b*x)) - log(tan(b*x)**2 + 1)*tan(c)**3*t
an(b*x)/(-b*tan(c)**5 + b*tan(c)**4*tan(b*x) - 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) - b*tan(c) + b*tan(b*x))
- tan(c)**6/(-b*tan(c)**5 + b*tan(c)**4*tan(b*x) - 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) - b*tan(c) + b*tan(
b*x)) - tan(c)**4/(-b*tan(c)**5 + b*tan(c)**4*tan(b*x) - 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) - b*tan(c) + b
*tan(b*x)), Eq(a, -atan(tan(c)) - pi*floor((c - pi/2)/pi) - pi*floor(c/pi - 1/2))), (x/(-cot(a)*cot(c) + zoo*c
ot(a) + zoo*cot(c) + zoo), Eq(b, 0)), (-2*b*x*tan(a)**3*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2
*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)
) - 2*b*x*tan(a)**2*tan(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)*
*2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - 2*b*x*tan(a)**2*tan(c)/(2*b*tan(
a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*
tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - 2*b*x*tan(a)*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*t
an(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) +
2*log(tan(a) + tan(b*x))*tan(a)**3*tan(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**
3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) + 2*log(tan(a) + ta
n(b*x))*tan(a)**3*tan(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*ta
n(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - 2*log(-tan(c) + tan(b*x))*tan(a)**3*t
an(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a
)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - 2*log(-tan(c) + tan(b*x))*tan(a)*tan(c)**3/(2*b*tan(a
)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*t
an(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - log(tan(b*x)**2 + 1)*tan(a)**3*tan(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan
(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 +
2*b*tan(c)) + log(tan(b*x)**2 + 1)*tan(a)*tan(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*t
an(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)), True)) + Pi
ecewise((zoo*x/(zoo*cot(c) + zoo + cot(c)/tan(c) + zoo/tan(c)), Eq(b, 0) & Eq(a, -atan(tan(c)) - pi*floor((c -
pi/2)/pi) - pi*floor(c/pi - 1/2))), (-b*x*tan(c)**5/(-b*tan(c)**5 + b*tan(c)**4*tan(b*x) - 2*b*tan(c)**3 + 2*
b*tan(c)**2*tan(b*x) - b*tan(c) + b*tan(b*x)) + b*x*tan(c)**4*tan(b*x)/(-b*tan(c)**5 + b*tan(c)**4*tan(b*x) -
2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) - b*tan(c) + b*tan(b*x)) + b*x*tan(c)**3/(-b*tan(c)**5 + b*tan(c)**4*ta
n(b*x) - 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) - b*tan(c) + b*tan(b*x)) - b*x*tan(c)**2*tan(b*x)/(-b*tan(c)**
5 + b*tan(c)**4*tan(b*x) - 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) - b*tan(c) + b*tan(b*x)) + 2*log(-tan(c) + t
an(b*x))*tan(c)**4/(-b*tan(c)**5 + b*tan(c)**4*tan(b*x) - 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) - b*tan(c) +
b*tan(b*x)) - 2*log(-tan(c) + tan(b*x))*tan(c)**3*tan(b*x)/(-b*tan(c)**5 + b*tan(c)**4*tan(b*x) - 2*b*tan(c)**
3 + 2*b*tan(c)**2*tan(b*x) - b*tan(c) + b*tan(b*x)) - log(tan(b*x)**2 + 1)*tan(c)**4/(-b*tan(c)**5 + b*tan(c)*
*4*tan(b*x) - 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) - b*tan(c) + b*tan(b*x)) + log(tan(b*x)**2 + 1)*tan(c)**3
*tan(b*x)/(-b*tan(c)**5 + b*tan(c)**4*tan(b*x) - 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) - b*tan(c) + b*tan(b*x
)) - tan(c)**4/(-b*tan(c)**5 + b*tan(c)**4*tan(b*x) - 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) - b*tan(c) + b*ta
n(b*x)) - tan(c)**2/(-b*tan(c)**5 + b*tan(c)**4*tan(b*x) - 2*b*tan(c)**3 + 2*b*tan(c)**2*tan(b*x) - b*tan(c) +
b*tan(b*x)), Eq(a, -atan(tan(c)) - pi*floor((c - pi/2)/pi) - pi*floor(c/pi - 1/2))), (zoo*x/(-cot(a)*cot(c) +
zoo*cot(a) + zoo*cot(c) + zoo), Eq(b, 0)), (2*b*x*tan(a)**3*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**
3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*t
an(c)) + 2*b*x*tan(a)**2*tan(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*ta
n(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) + 2*b*x*tan(a)**2*tan(c)/(2*b
*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 +
2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) + 2*b*x*tan(a)*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 +
2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c
)) + 2*log(tan(a) + tan(b*x))*tan(a)*tan(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)
**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) + 2*log(tan(a) +
tan(b*x))*tan(a)*tan(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan
(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - 2*log(-tan(c) + tan(b*x))*tan(a)**3*ta
n(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*ta
n(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - 2*log(-tan(c) + tan(b*x))*tan(a)*tan(c)/(2*b*tan(a)**3*ta
n(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) +
2*b*tan(c)**3 + 2*b*tan(c)) + log(tan(b*x)**2 + 1)*tan(a)**3*tan(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3
+ 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan
(c)) - log(tan(b*x)**2 + 1)*tan(a)*tan(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**
3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)), True))*cot(a)*cot(
c) + Piecewise((zoo*x/(zoo*cot(c) + zoo + cot(c)/tan(c) + zoo/tan(c)), Eq(b, 0) & Eq(a, -atan(tan(c)) - pi*flo
or((c - pi/2)/pi) - pi*floor(c/pi - 1/2))), (4*b*x*tan(c)**4/(-2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) - 4*b*ta
n(c)**3 + 4*b*tan(c)**2*tan(b*x) - 2*b*tan(c) + 2*b*tan(b*x)) - 4*b*x*tan(c)**3*tan(b*x)/(-2*b*tan(c)**5 + 2*b
*tan(c)**4*tan(b*x) - 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) - 2*b*tan(c) + 2*b*tan(b*x)) + 2*log(-tan(c) + ta
n(b*x))*tan(c)**5/(-2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) - 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) - 2*b*tan(
c) + 2*b*tan(b*x)) - 2*log(-tan(c) + tan(b*x))*tan(c)**4*tan(b*x)/(-2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) - 4
*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) - 2*b*tan(c) + 2*b*tan(b*x)) - 2*log(-tan(c) + tan(b*x))*tan(c)**3/(-2*b
*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) - 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) - 2*b*tan(c) + 2*b*tan(b*x)) + 2*
log(-tan(c) + tan(b*x))*tan(c)**2*tan(b*x)/(-2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) - 4*b*tan(c)**3 + 4*b*tan(
c)**2*tan(b*x) - 2*b*tan(c) + 2*b*tan(b*x)) - log(tan(b*x)**2 + 1)*tan(c)**5/(-2*b*tan(c)**5 + 2*b*tan(c)**4*t
an(b*x) - 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) - 2*b*tan(c) + 2*b*tan(b*x)) + log(tan(b*x)**2 + 1)*tan(c)**4
*tan(b*x)/(-2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) - 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) - 2*b*tan(c) + 2*b
*tan(b*x)) + log(tan(b*x)**2 + 1)*tan(c)**3/(-2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) - 4*b*tan(c)**3 + 4*b*tan
(c)**2*tan(b*x) - 2*b*tan(c) + 2*b*tan(b*x)) - log(tan(b*x)**2 + 1)*tan(c)**2*tan(b*x)/(-2*b*tan(c)**5 + 2*b*t
an(c)**4*tan(b*x) - 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) - 2*b*tan(c) + 2*b*tan(b*x)) - 2*tan(c)**5/(-2*b*ta
n(c)**5 + 2*b*tan(c)**4*tan(b*x) - 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) - 2*b*tan(c) + 2*b*tan(b*x)) - 2*tan
(c)**3/(-2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) - 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) - 2*b*tan(c) + 2*b*ta
n(b*x)), Eq(a, -atan(tan(c)) - pi*floor((c - pi/2)/pi) - pi*floor(c/pi - 1/2))), (zoo*x/(-cot(a)*cot(c) + zoo*
cot(a) + zoo*cot(c) + zoo), Eq(b, 0)), (-2*b*x*tan(a)**3*tan(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b
*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c))
+ 2*b*x*tan(a)*tan(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*ta
n(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - 2*log(tan(a) + tan(b*x))*tan(a)**2*ta
n(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)
*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - 2*log(tan(a) + tan(b*x))*tan(a)**2*tan(c)/(2*b*tan(a)*
*3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan
(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - 2*log(-tan(c) + tan(b*x))*tan(a)**3*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2
*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)
**3 + 2*b*tan(c)) - 2*log(-tan(c) + tan(b*x))*tan(a)*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*
tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) +
log(tan(b*x)**2 + 1)*tan(a)**3*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 +
2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) + log(tan(b*x)**2 + 1)
*tan(a)**2*tan(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c)
+ 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) + log(tan(b*x)**2 + 1)*tan(a)**2*tan(c)/(2*
b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2
+ 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) + log(tan(b*x)**2 + 1)*tan(a)*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 +
2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c
)**3 + 2*b*tan(c)), True))*cot(a) - Piecewise((zoo*x/(zoo*cot(c) + zoo + cot(c)/tan(c) + zoo/tan(c)), Eq(b, 0)
& Eq(a, -atan(tan(c)) - pi*floor((c - pi/2)/pi) - pi*floor(c/pi - 1/2))), (4*b*x*tan(c)**4/(-2*b*tan(c)**5 +
2*b*tan(c)**4*tan(b*x) - 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) - 2*b*tan(c) + 2*b*tan(b*x)) - 4*b*x*tan(c)**3
*tan(b*x)/(-2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) - 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) - 2*b*tan(c) + 2*b
*tan(b*x)) + 2*log(-tan(c) + tan(b*x))*tan(c)**5/(-2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) - 4*b*tan(c)**3 + 4*
b*tan(c)**2*tan(b*x) - 2*b*tan(c) + 2*b*tan(b*x)) - 2*log(-tan(c) + tan(b*x))*tan(c)**4*tan(b*x)/(-2*b*tan(c)*
*5 + 2*b*tan(c)**4*tan(b*x) - 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) - 2*b*tan(c) + 2*b*tan(b*x)) - 2*log(-tan
(c) + tan(b*x))*tan(c)**3/(-2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) - 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) -
2*b*tan(c) + 2*b*tan(b*x)) + 2*log(-tan(c) + tan(b*x))*tan(c)**2*tan(b*x)/(-2*b*tan(c)**5 + 2*b*tan(c)**4*tan(
b*x) - 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) - 2*b*tan(c) + 2*b*tan(b*x)) - log(tan(b*x)**2 + 1)*tan(c)**5/(-
2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) - 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) - 2*b*tan(c) + 2*b*tan(b*x)) +
log(tan(b*x)**2 + 1)*tan(c)**4*tan(b*x)/(-2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) - 4*b*tan(c)**3 + 4*b*tan(c)
**2*tan(b*x) - 2*b*tan(c) + 2*b*tan(b*x)) + log(tan(b*x)**2 + 1)*tan(c)**3/(-2*b*tan(c)**5 + 2*b*tan(c)**4*tan
(b*x) - 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) - 2*b*tan(c) + 2*b*tan(b*x)) - log(tan(b*x)**2 + 1)*tan(c)**2*t
an(b*x)/(-2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) - 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) - 2*b*tan(c) + 2*b*t
an(b*x)) - 2*tan(c)**5/(-2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) - 4*b*tan(c)**3 + 4*b*tan(c)**2*tan(b*x) - 2*b
*tan(c) + 2*b*tan(b*x)) - 2*tan(c)**3/(-2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) - 4*b*tan(c)**3 + 4*b*tan(c)**2
*tan(b*x) - 2*b*tan(c) + 2*b*tan(b*x)), Eq(a, -atan(tan(c)) - pi*floor((c - pi/2)/pi) - pi*floor(c/pi - 1/2)))
, (zoo*x/(-cot(a)*cot(c) + zoo*cot(a) + zoo*cot(c) + zoo), Eq(b, 0)), (-2*b*x*tan(a)**3*tan(c)/(2*b*tan(a)**3*
tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a)
+ 2*b*tan(c)**3 + 2*b*tan(c)) + 2*b*x*tan(a)*tan(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)*
*2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - 2*log(
tan(a) + tan(b*x))*tan(a)**2*tan(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*
b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - 2*log(tan(a) + tan(b*x)
)*tan(a)**2*tan(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) +
2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - 2*log(-tan(c) + tan(b*x))*tan(a)**3*tan(c)*
*2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(
c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - 2*log(-tan(c) + tan(b*x))*tan(a)*tan(c)**2/(2*b*tan(a)**3*t
an(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a)
+ 2*b*tan(c)**3 + 2*b*tan(c)) + log(tan(b*x)**2 + 1)*tan(a)**3*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)
**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b
*tan(c)) + log(tan(b*x)**2 + 1)*tan(a)**2*tan(c)**3/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*t
an(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) + log(tan(b*
x)**2 + 1)*tan(a)**2*tan(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2
*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) + log(tan(b*x)**2 + 1)*tan(a)*tan(c)
**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan
(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)), True))*cot(c)
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