∫ cot(a + bx) cot(c + bx) dx

3.141 \(\int \cot (a+b x) \cot (c+b x) \, dx\)

Optimal. Leaf size=39 \[ -\frac {\cot (a-c) \log (\sin (a+b x))}{b}+\frac {\cot (a-c) \log (\sin (b x+c))}{b}-x \]

[Out]

-x-cot(a-c)*ln(sin(b*x+a))/b+cot(a-c)*ln(sin(b*x+c))/b

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Rubi [A] time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4613, 4611, 3475} \[ -\frac {\cot (a-c) \log (\sin (a+b x))}{b}+\frac {\cot (a-c) \log (\sin (b x+c))}{b}-x \]

Antiderivative was successfully veriﬁed.

[In]

Int[Cot[a + b*x]*Cot[c + b*x],x]

[Out]

-x - (Cot[a - c]*Log[Sin[a + b*x]])/b + (Cot[a - c]*Log[Sin[c + 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 (a+b x) \cot (c+b x) \, dx &=-x+\cos (a-c) \int \csc (a+b x) \csc (c+b x) \, dx\\ &=-x-\cot (a-c) \int \cot (a+b x) \, dx+\cot (a-c) \int \cot (c+b x) \, dx\\ &=-x-\frac {\cot (a-c) \log (\sin (a+b x))}{b}+\frac {\cot (a-c) \log (\sin (c+b x))}{b}\\ \end {align*}

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Mathematica [A] time = 0.51, size = 31, normalized size = 0.79 \[ \frac {\cot (a-c) (\log (\sin (b x+c))-\log (\sin (a+b x)))}{b}-x \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[Cot[a + b*x]*Cot[c + b*x],x]

[Out]

-x + (Cot[a - c]*(-Log[Sin[a + b*x]] + Log[Sin[c + b*x]]))/b

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fricas [B] time = 1.53, size = 118, normalized size = 3.03 \[ -\frac {2 \, b x \sin \left (-2 \, a + 2 \, c\right ) - {\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \log \left (-\frac {\cos \left (2 \, b x + 2 \, c\right ) \cos \left (-2 \, a + 2 \, c\right ) + \sin \left (2 \, b x + 2 \, c\right ) \sin \left (-2 \, a + 2 \, c\right ) - 1}{\cos \left (-2 \, a + 2 \, c\right ) + 1}\right ) + {\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, c\right ) + \frac {1}{2}\right )}{2 \, b \sin \left (-2 \, a + 2 \, c\right )} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)*cot(b*x+c),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*c)*cos(-2*a + 2*c) + sin(2*b*x + 2*c)*

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*c) + 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+a)*cot(b*x+c),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+(tan

(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*t

an(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*tan(

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 = 0.19, size = 177, normalized size = 4.54 \[ -x -\frac {i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) {\mathrm e}^{2 i a}}{b \left ({\mathrm e}^{2 i a}-{\mathrm e}^{2 i c}\right )}-\frac {i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right ) {\mathrm e}^{2 i c}}{b \left ({\mathrm e}^{2 i a}-{\mathrm e}^{2 i c}\right )}+\frac {i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-{\mathrm e}^{2 i \left (a -c \right )}\right ) {\mathrm e}^{2 i a}}{b \left ({\mathrm e}^{2 i a}-{\mathrm e}^{2 i c}\right )}+\frac {i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-{\mathrm e}^{2 i \left (a -c \right )}\right ) {\mathrm e}^{2 i c}}{b \left ({\mathrm e}^{2 i a}-{\mathrm e}^{2 i c}\right )} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cot(b*x+a)*cot(b*x+c),x)

[Out]

-x-I/b/(exp(2*I*a)-exp(2*I*c))*ln(exp(2*I*(b*x+a))-1)*exp(2*I*a)-I/b/(exp(2*I*a)-exp(2*I*c))*ln(exp(2*I*(b*x+a

))-1)*exp(2*I*c)+I/b/(exp(2*I*a)-exp(2*I*c))*ln(exp(2*I*(b*x+a))-exp(2*I*(a-c)))*exp(2*I*a)+I/b/(exp(2*I*a)-ex

p(2*I*c))*ln(exp(2*I*(b*x+a))-exp(2*I*(a-c)))*exp(2*I*c)

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maxima [B] time = 0.93, size = 549, normalized size = 14.08 \[ -\frac {{\left (2 \, b \cos \left (2 \, a\right ) \cos \left (2 \, c\right ) - b \cos \left (2 \, c\right )^{2} + 2 \, b \sin \left (2 \, a\right ) \sin \left (2 \, c\right ) - b \sin \left (2 \, c\right )^{2} - {\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )} b\right )} x + {\left (\cos \left (2 \, a\right )^{2} - \cos \left (2 \, c\right )^{2} + \sin \left (2 \, a\right )^{2} - \sin \left (2 \, c\right )^{2}\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\right )^{2} - \cos \left (2 \, c\right )^{2} + \sin \left (2 \, a\right )^{2} - \sin \left (2 \, c\right )^{2}\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\right )^{2} - \cos \left (2 \, c\right )^{2} + \sin \left (2 \, a\right )^{2} - \sin \left (2 \, c\right )^{2}\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\right )^{2} - \cos \left (2 \, c\right )^{2} + \sin \left (2 \, a\right )^{2} - \sin \left (2 \, c\right )^{2}\right )} \arctan \left (\sin \left (b x\right ) - \sin \relax (c), \cos \left (b x\right ) + \cos \relax (c)\right ) - {\left (\cos \left (2 \, c\right ) \sin \left (2 \, a\right ) - \cos \left (2 \, a\right ) \sin \left (2 \, c\right )\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 ) - {\left (\cos \left (2 \, c\right ) \sin \left (2 \, a\right ) - \cos \left (2 \, a\right ) \sin \left (2 \, c\right )\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 ) + {\left (\cos \left (2 \, c\right ) \sin \left (2 \, a\right ) - \cos \left (2 \, a\right ) \sin \left (2 \, c\right )\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 ) + {\left (\cos \left (2 \, c\right ) \sin \left (2 \, a\right ) - \cos \left (2 \, a\right ) \sin \left (2 \, c\right )\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 )}{2 \, b \cos \left (2 \, a\right ) \cos \left (2 \, c\right ) - b \cos \left (2 \, c\right )^{2} + 2 \, b \sin \left (2 \, a\right ) \sin \left (2 \, c\right ) - b \sin \left (2 \, c\right )^{2} - {\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )} b} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)*cot(b*x+c),x, algorithm="maxima")

[Out]

-((2*b*cos(2*a)*cos(2*c) - b*cos(2*c)^2 + 2*b*sin(2*a)*sin(2*c) - b*sin(2*c)^2 - (cos(2*a)^2 + sin(2*a)^2)*b)*

x + (cos(2*a)^2 - cos(2*c)^2 + sin(2*a)^2 - sin(2*c)^2)*arctan2(sin(b*x) + sin(a), cos(b*x) - cos(a)) + (cos(2

*a)^2 - cos(2*c)^2 + sin(2*a)^2 - sin(2*c)^2)*arctan2(sin(b*x) - sin(a), cos(b*x) + cos(a)) - (cos(2*a)^2 - co

s(2*c)^2 + sin(2*a)^2 - sin(2*c)^2)*arctan2(sin(b*x) + sin(c), cos(b*x) - cos(c)) - (cos(2*a)^2 - cos(2*c)^2 +

sin(2*a)^2 - sin(2*c)^2)*arctan2(sin(b*x) - sin(c), cos(b*x) + cos(c)) - (cos(2*c)*sin(2*a) - cos(2*a)*sin(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) - (cos(2*c)*sin

(2*a) - cos(2*a)*sin(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) + si

n(a)^2) + (cos(2*c)*sin(2*a) - cos(2*a)*sin(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) + (cos(2*c)*sin(2*a) - cos(2*a)*sin(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))/(2*b*cos(2*a)*cos(2*c) - b*cos(2*c)^2 + 2*b*sin(2*a)*si

n(2*c) - b*sin(2*c)^2 - (cos(2*a)^2 + sin(2*a)^2)*b)

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mupad [B] time = 4.82, size = 207, normalized size = 5.31 \[ -\frac {\frac {x}{2}+x\,\left ({\sin \left (a-c\right )}^2-\frac {1}{2}\right )}{{\sin \left (a-c\right )}^2}-\frac {\frac {\sin \left (2\,a-2\,c\right )\,\ln \left ({\sin \left (2\,a-2\,c\right )}^2\,2{}\mathrm {i}+{\sin \left (a+b\,x\right )}^2\,2{}\mathrm {i}-{\sin \left (3\,a-2\,c+b\,x\right )}^2\,2{}\mathrm {i}+\sin \left (4\,a-4\,c\right )-\sin \left (6\,a-4\,c+2\,b\,x\right )+\sin \left (2\,a+2\,b\,x\right )\right )}{2}-\frac {\sin \left (2\,a-2\,c\right )\,\ln \left ({\sin \left (2\,a-2\,c\right )}^2\,2{}\mathrm {i}+{\sin \left (c+b\,x\right )}^2\,2{}\mathrm {i}-{\sin \left (2\,a-c+b\,x\right )}^2\,2{}\mathrm {i}+\sin \left (4\,a-4\,c\right )-\sin \left (4\,a-2\,c+2\,b\,x\right )+\sin \left (2\,c+2\,b\,x\right )\right )}{2}}{b\,{\sin \left (a-c\right )}^2} \]

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) +

sin(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)*

log(sin(4*a - 4*c) - sin(4*a - 2*c + 2*b*x) + sin(2*c + 2*b*x) + sin(2*a - 2*c)^2*2i + sin(c + b*x)^2*2i - sin

(2*a - c + b*x)^2*2i))/2)/(b*sin(a - c)^2)

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sympy [B] time = 24.28, size = 7485, normalized size = 191.92 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(b*x+a)*cot(b*x+c),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*ta

n(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*t

an(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*t

an(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*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)**

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)), E

q(a, atan(tan(c)) + pi*floor((c - pi/2)/pi) + pi*floor(c/pi - 1/2))), (x/(cot(a)*cot(c) + zoo*cot(a) + zoo*cot

n(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*ta

n(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)**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) + tan(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*log(tan(c) + tan(b*x))*tan(a)**3*tan(c)**3/(2*b*ta

n(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*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*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)), True)) + Piecewise((zoo*x/(z

oo*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*fl

oor(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*t

an(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*ta

n(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*t

an(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)) - 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*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*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*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)*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)**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*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*ta

n(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)) - lo

g(tan(b*x)**2 + 1)*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)) + 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*t

an(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)*t

an(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*t

an(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*floo

r(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*ta

n(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*tan(b*x)/(2*b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) + 4*b*tan(c)**3 + 4*b*t

an(c)**2*tan(b*x) + 2*b*tan(c) + 2*b*tan(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*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)) - 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*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*log(tan(c) + tan(b*x))*ta

n(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*t

an(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*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)), True))*cot(c)

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