3.146 \(\int \csc (c-b x) \csc (a+b x) \, dx\)

Optimal. Leaf size=33 \[ \frac {\csc (a+c) \log (\sin (a+b x))}{b}-\frac {\csc (a+c) \log (\sin (c-b x))}{b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4611, 3475} \[ \frac {\csc (a+c) \log (\sin (a+b x))}{b}-\frac {\csc (a+c) \log (\sin (c-b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c - b*x]*Csc[a + b*x],x]

[Out]

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

Rubi steps

\begin {align*} \int \csc (c-b x) \csc (a+b x) \, dx &=\csc (a+c) \int \cot (c-b x) \, dx+\csc (a+c) \int \cot (a+b x) \, dx\\ &=-\frac {\csc (a+c) \log (\sin (c-b x))}{b}+\frac {\csc (a+c) \log (\sin (a+b x))}{b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.22, size = 29, normalized size = 0.88 \[ -\frac {\csc (a+c) (\log (\sin (c-b x))-\log (-\sin (a+b x)))}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c - b*x]*Csc[a + b*x],x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.78, size = 96, normalized size = 2.91 \[ \frac {\log \left (-\frac {1}{4} \, \cos \left (b x + a\right )^{2} + \frac {1}{4}\right ) - \log \left (-\frac {2 \, \cos \left (b x + a\right ) \cos \left (a + c\right ) \sin \left (b x + a\right ) \sin \left (a + c\right ) + {\left (2 \, \cos \left (a + c\right )^{2} - 1\right )} \cos \left (b x + a\right )^{2} - \cos \left (a + c\right )^{2}}{\cos \left (a + c\right )^{2} + 2 \, \cos \left (a + c\right ) + 1}\right )}{2 \, b \sin \left (a + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-csc(b*x-c)*csc(b*x+a),x, algorithm="fricas")

[Out]

1/2*(log(-1/4*cos(b*x + a)^2 + 1/4) - log(-(2*cos(b*x + a)*cos(a + c)*sin(b*x + a)*sin(a + c) + (2*cos(a + c)^
2 - 1)*cos(b*x + a)^2 - cos(a + c)^2)/(cos(a + c)^2 + 2*cos(a + c) + 1)))/(b*sin(a + c))

________________________________________________________________________________________

giac [B]  time = 0.23, size = 397, normalized size = 12.03 \[ \frac {\frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{4} \tan \left (\frac {1}{2} \, c\right )^{4} - 4 \, \tan \left (\frac {1}{2} \, a\right )^{3} \tan \left (\frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, a\right )^{4} - 4 \, \tan \left (\frac {1}{2} \, a\right )^{3} \tan \left (\frac {1}{2} \, c\right ) - 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, c\right )^{4} - 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + 1\right )} \log \left ({\left | \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right )^{2} - 4 \, \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + 2 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (b x + a\right ) - 2 \, \tan \left (\frac {1}{2} \, a\right ) - 2 \, \tan \left (\frac {1}{2} \, c\right ) \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{4} \tan \left (\frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, a\right )^{3} \tan \left (\frac {1}{2} \, c\right )^{4} - \tan \left (\frac {1}{2} \, a\right )^{4} \tan \left (\frac {1}{2} \, c\right ) - 6 \, \tan \left (\frac {1}{2} \, a\right )^{3} \tan \left (\frac {1}{2} \, c\right )^{2} - 6 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{4} + \tan \left (\frac {1}{2} \, a\right )^{3} + 6 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 6 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )} - \frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} \log \left ({\left | \tan \left (b x + a\right ) \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-csc(b*x-c)*csc(b*x+a),x, algorithm="giac")

[Out]

1/2*((tan(1/2*a)^4*tan(1/2*c)^4 - 4*tan(1/2*a)^3*tan(1/2*c)^3 - tan(1/2*a)^4 - 4*tan(1/2*a)^3*tan(1/2*c) - 4*t
an(1/2*a)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*a)*tan(1/2*c) + 1)*log(abs(tan(b*x + a)*tan(1/2*a)^2*tan(1/2
*c)^2 - tan(b*x + a)*tan(1/2*a)^2 - 4*tan(b*x + a)*tan(1/2*a)*tan(1/2*c) + 2*tan(1/2*a)^2*tan(1/2*c) - tan(b*x
+ a)*tan(1/2*c)^2 + 2*tan(1/2*a)*tan(1/2*c)^2 + tan(b*x + a) - 2*tan(1/2*a) - 2*tan(1/2*c)))/(tan(1/2*a)^4*ta
n(1/2*c)^3 + tan(1/2*a)^3*tan(1/2*c)^4 - tan(1/2*a)^4*tan(1/2*c) - 6*tan(1/2*a)^3*tan(1/2*c)^2 - 6*tan(1/2*a)^
2*tan(1/2*c)^3 - tan(1/2*a)*tan(1/2*c)^4 + tan(1/2*a)^3 + 6*tan(1/2*a)^2*tan(1/2*c) + 6*tan(1/2*a)*tan(1/2*c)^
2 + tan(1/2*c)^3 - tan(1/2*a) - tan(1/2*c)) - (tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*a)^2 + tan(1/2*c)^2 + 1)*lo
g(abs(tan(b*x + a)))/(tan(1/2*a)^2*tan(1/2*c) + tan(1/2*a)*tan(1/2*c)^2 - tan(1/2*a) - tan(1/2*c)))/b

________________________________________________________________________________________

maple [B]  time = 0.48, size = 81, normalized size = 2.45 \[ -\frac {\ln \left (\tan \left (b x +a \right ) \cos \relax (a ) \cos \relax (c )-\tan \left (b x +a \right ) \sin \relax (a ) \sin \relax (c )-\cos \relax (a ) \sin \relax (c )-\sin \relax (a ) \cos \relax (c )\right )}{b \left (\sin \relax (a ) \cos \relax (c )+\cos \relax (a ) \sin \relax (c )\right )}+\frac {\ln \left (\tan \left (b x +a \right )\right )}{b \left (\sin \relax (a ) \cos \relax (c )+\cos \relax (a ) \sin \relax (c )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-csc(b*x-c)*csc(b*x+a),x)

[Out]

-1/b/(sin(a)*cos(c)+cos(a)*sin(c))*ln(tan(b*x+a)*cos(a)*cos(c)-tan(b*x+a)*sin(a)*sin(c)-cos(a)*sin(c)-sin(a)*c
os(c))+1/b/(sin(a)*cos(c)+cos(a)*sin(c))*ln(tan(b*x+a))

________________________________________________________________________________________

maxima [B]  time = 0.50, size = 536, normalized size = 16.24 \[ -\frac {2 \, {\left (\cos \left (2 \, a + 2 \, c\right ) \cos \left (a + c\right ) + \sin \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) - \cos \left (a + c\right )\right )} \arctan \left (\sin \left (b x\right ) + \sin \relax (a), \cos \left (b x\right ) - \cos \relax (a)\right ) + 2 \, {\left (\cos \left (2 \, a + 2 \, c\right ) \cos \left (a + c\right ) + \sin \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) - \cos \left (a + c\right )\right )} \arctan \left (\sin \left (b x\right ) - \sin \relax (a), \cos \left (b x\right ) + \cos \relax (a)\right ) - 2 \, {\left (\cos \left (2 \, a + 2 \, c\right ) \cos \left (a + c\right ) + \sin \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) - \cos \left (a + c\right )\right )} \arctan \left (\sin \left (b x\right ) + \sin \relax (c), \cos \left (b x\right ) + \cos \relax (c)\right ) - 2 \, {\left (\cos \left (2 \, a + 2 \, c\right ) \cos \left (a + c\right ) + \sin \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) - \cos \left (a + c\right )\right )} \arctan \left (\sin \left (b x\right ) - \sin \relax (c), \cos \left (b x\right ) - \cos \relax (c)\right ) - {\left (\cos \left (a + c\right ) \sin \left (2 \, a + 2 \, c\right ) - \cos \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) + \sin \left (a + 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 (a + c\right ) \sin \left (2 \, a + 2 \, c\right ) - \cos \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) + \sin \left (a + 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 (a + c\right ) \sin \left (2 \, a + 2 \, c\right ) - \cos \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) + \sin \left (a + 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 (a + c\right ) \sin \left (2 \, a + 2 \, c\right ) - \cos \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) + \sin \left (a + 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 )}{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(-csc(b*x-c)*csc(b*x+a),x, algorithm="maxima")

[Out]

-(2*(cos(2*a + 2*c)*cos(a + c) + sin(2*a + 2*c)*sin(a + c) - cos(a + c))*arctan2(sin(b*x) + sin(a), cos(b*x) -
cos(a)) + 2*(cos(2*a + 2*c)*cos(a + c) + sin(2*a + 2*c)*sin(a + c) - cos(a + c))*arctan2(sin(b*x) - sin(a), c
os(b*x) + cos(a)) - 2*(cos(2*a + 2*c)*cos(a + c) + sin(2*a + 2*c)*sin(a + c) - cos(a + c))*arctan2(sin(b*x) +
sin(c), cos(b*x) + cos(c)) - 2*(cos(2*a + 2*c)*cos(a + c) + sin(2*a + 2*c)*sin(a + c) - cos(a + c))*arctan2(si
n(b*x) - sin(c), cos(b*x) - cos(c)) - (cos(a + c)*sin(2*a + 2*c) - cos(2*a + 2*c)*sin(a + c) + sin(a + 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(a + c)*sin(2*a
+ 2*c) - cos(2*a + 2*c)*sin(a + c) + sin(a + 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(a + c)*sin(2*a + 2*c) - cos(2*a + 2*c)*sin(a + c) + sin(a + 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(a + c)*sin(2*a + 2*c)
- cos(2*a + 2*c)*sin(a + c) + sin(a + 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))/(b*cos(2*a + 2*c)^2 + b*sin(2*a + 2*c)^2 - 2*b*cos(2*a + 2*c) + b)

________________________________________________________________________________________

mupad [B]  time = 7.87, size = 249, normalized size = 7.55 \[ \frac {2\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}}\,\left (\ln \left (\frac {2\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}}\,\left (-4\,b\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}+2\,b\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}+2\,b\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}\right )}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}-1\right )}+{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}\,4{}\mathrm {i}\right )-\ln \left (\frac {2\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}}\,\left (-4\,b\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}+2\,b\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}+2\,b\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}\right )}{b-b\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}}+{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}\,4{}\mathrm {i}\right )\right )}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(a + b*x)*sin(c - b*x)),x)

[Out]

(2*(-exp(a*2i + c*2i))^(1/2)*(log((2*(-exp(a*2i)*exp(c*2i))^(1/2)*(2*b*exp(a*2i)*exp(b*x*2i) - 4*b*exp(a*2i)*e
xp(c*2i) + 2*b*exp(a*4i)*exp(c*2i)*exp(b*x*2i)))/(b*(exp(a*2i)*exp(c*2i) - 1)) + exp(a*1i)*exp(a*2i)*exp(c*1i)
*exp(b*x*2i)*4i) - log((2*(-exp(a*2i)*exp(c*2i))^(1/2)*(2*b*exp(a*2i)*exp(b*x*2i) - 4*b*exp(a*2i)*exp(c*2i) +
2*b*exp(a*4i)*exp(c*2i)*exp(b*x*2i)))/(b - b*exp(a*2i)*exp(c*2i)) + exp(a*1i)*exp(a*2i)*exp(c*1i)*exp(b*x*2i)*
4i)))/(b*(exp(a*2i + c*2i) - 1))

________________________________________________________________________________________

sympy [A]  time = 129.08, size = 1838, normalized size = 55.70 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-csc(b*x-c)*csc(b*x+a),x)

[Out]

Piecewise((-tan(c/2)**4*tan(b*x/2)/(-2*b*tan(c/2)**3*tan(b*x/2) + 2*b*tan(c/2)**2*tan(b*x/2)**2 - 2*b*tan(c/2)
**2 + 2*b*tan(c/2)*tan(b*x/2)) - 2*tan(c/2)**2*tan(b*x/2)/(-2*b*tan(c/2)**3*tan(b*x/2) + 2*b*tan(c/2)**2*tan(b
*x/2)**2 - 2*b*tan(c/2)**2 + 2*b*tan(c/2)*tan(b*x/2)) - tan(b*x/2)/(-2*b*tan(c/2)**3*tan(b*x/2) + 2*b*tan(c/2)
**2*tan(b*x/2)**2 - 2*b*tan(c/2)**2 + 2*b*tan(c/2)*tan(b*x/2)), Eq(a, 2*atan(1/tan(c/2)))), (tan(c/2)**4*tan(b
*x/2)/(-2*b*tan(c/2)**3*tan(b*x/2) + 2*b*tan(c/2)**2*tan(b*x/2)**2 - 2*b*tan(c/2)**2 + 2*b*tan(c/2)*tan(b*x/2)
) + 2*tan(c/2)**2*tan(b*x/2)/(-2*b*tan(c/2)**3*tan(b*x/2) + 2*b*tan(c/2)**2*tan(b*x/2)**2 - 2*b*tan(c/2)**2 +
2*b*tan(c/2)*tan(b*x/2)) + tan(b*x/2)/(-2*b*tan(c/2)**3*tan(b*x/2) + 2*b*tan(c/2)**2*tan(b*x/2)**2 - 2*b*tan(c
/2)**2 + 2*b*tan(c/2)*tan(b*x/2)), Eq(a, -2*atan(tan(c/2)) - 2*pi*floor((c/2 - pi/2)/pi) - 2*pi*floor(c/(2*pi)
- 1/2))), (x/(sin(a)*sin(c)), Eq(b, 0)), (log(tan(a/2) + tan(b*x/2))*tan(a/2)/(2*b) + log(tan(a/2) + tan(b*x/
2))/(2*b*tan(a/2)) + log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)/(2*b) + log(tan(b*x/2) - 1/tan(a/2))/(2*b*tan(a/2))
- log(tan(b*x/2))*tan(a/2)/(2*b) - log(tan(b*x/2))/(2*b*tan(a/2)), Eq(c, 0)), (-log(-tan(c/2) + tan(b*x/2))*t
an(c/2)/(2*b) - log(-tan(c/2) + tan(b*x/2))/(2*b*tan(c/2)) - log(tan(b*x/2) + 1/tan(c/2))*tan(c/2)/(2*b) - log
(tan(b*x/2) + 1/tan(c/2))/(2*b*tan(c/2)) + log(tan(b*x/2))*tan(c/2)/(2*b) + log(tan(b*x/2))/(2*b*tan(c/2)), Eq
(a, 0)), (-log(tan(a/2) + tan(b*x/2))*tan(a/2)**2*tan(c/2)**2/(2*b*tan(a/2)**2*tan(c/2) + 2*b*tan(a/2)*tan(c/2
)**2 - 2*b*tan(a/2) - 2*b*tan(c/2)) - log(tan(a/2) + tan(b*x/2))*tan(a/2)**2/(2*b*tan(a/2)**2*tan(c/2) + 2*b*t
an(a/2)*tan(c/2)**2 - 2*b*tan(a/2) - 2*b*tan(c/2)) - log(tan(a/2) + tan(b*x/2))*tan(c/2)**2/(2*b*tan(a/2)**2*t
an(c/2) + 2*b*tan(a/2)*tan(c/2)**2 - 2*b*tan(a/2) - 2*b*tan(c/2)) - log(tan(a/2) + tan(b*x/2))/(2*b*tan(a/2)**
2*tan(c/2) + 2*b*tan(a/2)*tan(c/2)**2 - 2*b*tan(a/2) - 2*b*tan(c/2)) + log(-tan(c/2) + tan(b*x/2))*tan(a/2)**2
*tan(c/2)**2/(2*b*tan(a/2)**2*tan(c/2) + 2*b*tan(a/2)*tan(c/2)**2 - 2*b*tan(a/2) - 2*b*tan(c/2)) + log(-tan(c/
2) + tan(b*x/2))*tan(a/2)**2/(2*b*tan(a/2)**2*tan(c/2) + 2*b*tan(a/2)*tan(c/2)**2 - 2*b*tan(a/2) - 2*b*tan(c/2
)) + log(-tan(c/2) + tan(b*x/2))*tan(c/2)**2/(2*b*tan(a/2)**2*tan(c/2) + 2*b*tan(a/2)*tan(c/2)**2 - 2*b*tan(a/
2) - 2*b*tan(c/2)) + log(-tan(c/2) + tan(b*x/2))/(2*b*tan(a/2)**2*tan(c/2) + 2*b*tan(a/2)*tan(c/2)**2 - 2*b*ta
n(a/2) - 2*b*tan(c/2)) - log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)**2*tan(c/2)**2/(2*b*tan(a/2)**2*tan(c/2) + 2*b*
tan(a/2)*tan(c/2)**2 - 2*b*tan(a/2) - 2*b*tan(c/2)) - log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)**2/(2*b*tan(a/2)**
2*tan(c/2) + 2*b*tan(a/2)*tan(c/2)**2 - 2*b*tan(a/2) - 2*b*tan(c/2)) - log(tan(b*x/2) - 1/tan(a/2))*tan(c/2)**
2/(2*b*tan(a/2)**2*tan(c/2) + 2*b*tan(a/2)*tan(c/2)**2 - 2*b*tan(a/2) - 2*b*tan(c/2)) - log(tan(b*x/2) - 1/tan
(a/2))/(2*b*tan(a/2)**2*tan(c/2) + 2*b*tan(a/2)*tan(c/2)**2 - 2*b*tan(a/2) - 2*b*tan(c/2)) + log(tan(b*x/2) +
1/tan(c/2))*tan(a/2)**2*tan(c/2)**2/(2*b*tan(a/2)**2*tan(c/2) + 2*b*tan(a/2)*tan(c/2)**2 - 2*b*tan(a/2) - 2*b*
tan(c/2)) + log(tan(b*x/2) + 1/tan(c/2))*tan(a/2)**2/(2*b*tan(a/2)**2*tan(c/2) + 2*b*tan(a/2)*tan(c/2)**2 - 2*
b*tan(a/2) - 2*b*tan(c/2)) + log(tan(b*x/2) + 1/tan(c/2))*tan(c/2)**2/(2*b*tan(a/2)**2*tan(c/2) + 2*b*tan(a/2)
*tan(c/2)**2 - 2*b*tan(a/2) - 2*b*tan(c/2)) + log(tan(b*x/2) + 1/tan(c/2))/(2*b*tan(a/2)**2*tan(c/2) + 2*b*tan
(a/2)*tan(c/2)**2 - 2*b*tan(a/2) - 2*b*tan(c/2)), True))

________________________________________________________________________________________