3.196 \(\int \csc ^2(c+b x) \sin (a+b x) \, dx\)

Optimal. Leaf size=36 \[ -\frac {\cos (a-c) \tanh ^{-1}(\cos (b x+c))}{b}-\frac {\sin (a-c) \csc (b x+c)}{b} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4582, 2606, 8, 3770} \[ -\frac {\cos (a-c) \tanh ^{-1}(\cos (b x+c))}{b}-\frac {\sin (a-c) \csc (b x+c)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + b*x]^2*Sin[a + b*x],x]

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4582

Int[Csc[w_]^(n_.)*Sin[v_], x_Symbol] :> Dist[Sin[v - w], Int[Cot[w]*Csc[w]^(n - 1), x], x] + Dist[Cos[v - w],
Int[Csc[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]

Rubi steps

\begin {align*} \int \csc ^2(c+b x) \sin (a+b x) \, dx &=\cos (a-c) \int \csc (c+b x) \, dx+\sin (a-c) \int \cot (c+b x) \csc (c+b x) \, dx\\ &=-\frac {\tanh ^{-1}(\cos (c+b x)) \cos (a-c)}{b}-\frac {\sin (a-c) \operatorname {Subst}(\int 1 \, dx,x,\csc (c+b x))}{b}\\ &=-\frac {\tanh ^{-1}(\cos (c+b x)) \cos (a-c)}{b}-\frac {\csc (c+b x) \sin (a-c)}{b}\\ \end {align*}

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Mathematica [C]  time = 0.10, size = 90, normalized size = 2.50 \[ -\frac {\sin (a-c) \csc (b x+c)}{b}-\frac {2 i \cos (a-c) \tan ^{-1}\left (\frac {(\cos (c)-i \sin (c)) \left (\cos (c) \cos \left (\frac {b x}{2}\right )-\sin (c) \sin \left (\frac {b x}{2}\right )\right )}{\sin (c) \cos \left (\frac {b x}{2}\right )+i \cos (c) \cos \left (\frac {b x}{2}\right )}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + b*x]^2*Sin[a + b*x],x]

[Out]

((-2*I)*ArcTan[((Cos[c] - I*Sin[c])*(Cos[c]*Cos[(b*x)/2] - Sin[c]*Sin[(b*x)/2]))/(I*Cos[c]*Cos[(b*x)/2] + Cos[
(b*x)/2]*Sin[c])]*Cos[a - c])/b - (Csc[c + b*x]*Sin[a - c])/b

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fricas [A]  time = 0.51, size = 71, normalized size = 1.97 \[ -\frac {\cos \left (-a + c\right ) \log \left (\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) - \cos \left (-a + c\right ) \log \left (-\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) - 2 \, \sin \left (-a + c\right )}{2 \, b \sin \left (b x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+c)^2*sin(b*x+a),x, algorithm="fricas")

[Out]

-1/2*(cos(-a + c)*log(1/2*cos(b*x + c) + 1/2)*sin(b*x + c) - cos(-a + c)*log(-1/2*cos(b*x + c) + 1/2)*sin(b*x
+ c) - 2*sin(-a + c))/(b*sin(b*x + 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(csc(b*x+c)^2*sin(b*x+a),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)2/b*((-tan((b*x+c)/2)*tan(a/2)^2*tan(c/2)+tan((b*x+c)/2)*tan(a/2)*tan(c/2)^2-tan((b*x+c)/2)*tan(a/2)+tan((
b*x+c)/2)*tan(c/2))/(2*tan(a/2)^2*tan(c/2)^2+2*tan(a/2)^2+2*tan(c/2)^2+2)+(-tan((b*x+c)/2)*tan(a/2)^2*tan(c/2)
^2+tan((b*x+c)/2)*tan(a/2)^2-4*tan((b*x+c)/2)*tan(a/2)*tan(c/2)+tan((b*x+c)/2)*tan(c/2)^2-tan((b*x+c)/2)-tan(a
/2)^2*tan(c/2)+tan(a/2)*tan(c/2)^2-tan(a/2)+tan(c/2))/(2*tan(a/2)^2*tan(c/2)^2+2*tan(a/2)^2+2*tan(c/2)^2+2)/ta
n((b*x+c)/2)+(tan(a/2)^2*tan(c/2)^2-tan(a/2)^2+4*tan(a/2)*tan(c/2)-tan(c/2)^2+1)/(2*tan(a/2)^2*tan(c/2)^2+2*ta
n(a/2)^2+2*tan(c/2)^2+2)*ln(abs(tan((b*x+c)/2))))

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maple [B]  time = 2.23, size = 890, normalized size = 24.72 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+c)^2*sin(b*x+a),x)

[Out]

-8/b/(-4*cos(a)^2*cos(c)^2-4*cos(a)^2*sin(c)^2-4*cos(c)^2*sin(a)^2-4*sin(a)^2*sin(c)^2)/(-cos(a)*sin(c)*tan(1/
2*b*x+1/2*a)^2+cos(c)*sin(a)*tan(1/2*b*x+1/2*a)^2+2*tan(1/2*b*x+1/2*a)*cos(a)*cos(c)+2*tan(1/2*b*x+1/2*a)*sin(
a)*sin(c)+cos(a)*sin(c)-sin(a)*cos(c))*tan(1/2*b*x+1/2*a)*cos(a)*cos(c)-8/b/(-4*cos(a)^2*cos(c)^2-4*cos(a)^2*s
in(c)^2-4*cos(c)^2*sin(a)^2-4*sin(a)^2*sin(c)^2)/(-cos(a)*sin(c)*tan(1/2*b*x+1/2*a)^2+cos(c)*sin(a)*tan(1/2*b*
x+1/2*a)^2+2*tan(1/2*b*x+1/2*a)*cos(a)*cos(c)+2*tan(1/2*b*x+1/2*a)*sin(a)*sin(c)+cos(a)*sin(c)-sin(a)*cos(c))*
tan(1/2*b*x+1/2*a)*sin(a)*sin(c)-8/b/(-4*cos(a)^2*cos(c)^2-4*cos(a)^2*sin(c)^2-4*cos(c)^2*sin(a)^2-4*sin(a)^2*
sin(c)^2)/(-cos(a)*sin(c)*tan(1/2*b*x+1/2*a)^2+cos(c)*sin(a)*tan(1/2*b*x+1/2*a)^2+2*tan(1/2*b*x+1/2*a)*cos(a)*
cos(c)+2*tan(1/2*b*x+1/2*a)*sin(a)*sin(c)+cos(a)*sin(c)-sin(a)*cos(c))*cos(a)*sin(c)+8/b/(-4*cos(a)^2*cos(c)^2
-4*cos(a)^2*sin(c)^2-4*cos(c)^2*sin(a)^2-4*sin(a)^2*sin(c)^2)/(-cos(a)*sin(c)*tan(1/2*b*x+1/2*a)^2+cos(c)*sin(
a)*tan(1/2*b*x+1/2*a)^2+2*tan(1/2*b*x+1/2*a)*cos(a)*cos(c)+2*tan(1/2*b*x+1/2*a)*sin(a)*sin(c)+cos(a)*sin(c)-si
n(a)*cos(c))*sin(a)*cos(c)-8/b/(-4*cos(a)^2*cos(c)^2-4*cos(a)^2*sin(c)^2-4*cos(c)^2*sin(a)^2-4*sin(a)^2*sin(c)
^2)/(-cos(a)^2*cos(c)^2-cos(a)^2*sin(c)^2-cos(c)^2*sin(a)^2-sin(a)^2*sin(c)^2)^(1/2)*arctan(1/2*(2*(sin(a)*cos
(c)-cos(a)*sin(c))*tan(1/2*b*x+1/2*a)+2*cos(a)*cos(c)+2*sin(a)*sin(c))/(-cos(a)^2*cos(c)^2-cos(a)^2*sin(c)^2-c
os(c)^2*sin(a)^2-sin(a)^2*sin(c)^2)^(1/2))*cos(a)*cos(c)-8/b/(-4*cos(a)^2*cos(c)^2-4*cos(a)^2*sin(c)^2-4*cos(c
)^2*sin(a)^2-4*sin(a)^2*sin(c)^2)/(-cos(a)^2*cos(c)^2-cos(a)^2*sin(c)^2-cos(c)^2*sin(a)^2-sin(a)^2*sin(c)^2)^(
1/2)*arctan(1/2*(2*(sin(a)*cos(c)-cos(a)*sin(c))*tan(1/2*b*x+1/2*a)+2*cos(a)*cos(c)+2*sin(a)*sin(c))/(-cos(a)^
2*cos(c)^2-cos(a)^2*sin(c)^2-cos(c)^2*sin(a)^2-sin(a)^2*sin(c)^2)^(1/2))*sin(a)*sin(c)

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maxima [B]  time = 0.36, size = 454, normalized size = 12.61 \[ -\frac {2 \, {\left (\cos \left (b x + 2 \, a\right ) - \cos \left (b x + 2 \, c\right )\right )} \cos \left (2 \, b x + a + 2 \, c\right ) - 2 \, \cos \left (b x + 2 \, a\right ) \cos \relax (a) + 2 \, \cos \left (b x + 2 \, c\right ) \cos \relax (a) + {\left (\cos \left (2 \, b x + a + 2 \, c\right )^{2} \cos \left (-a + c\right ) - 2 \, \cos \left (2 \, b x + a + 2 \, c\right ) \cos \relax (a) \cos \left (-a + c\right ) + \cos \left (-a + c\right ) \sin \left (2 \, b x + a + 2 \, c\right )^{2} - 2 \, \cos \left (-a + c\right ) \sin \left (2 \, b x + a + 2 \, c\right ) \sin \relax (a) + {\left (\cos \relax (a)^{2} + \sin \relax (a)^{2}\right )} \cos \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 (2 \, b x + a + 2 \, c\right )^{2} \cos \left (-a + c\right ) - 2 \, \cos \left (2 \, b x + a + 2 \, c\right ) \cos \relax (a) \cos \left (-a + c\right ) + \cos \left (-a + c\right ) \sin \left (2 \, b x + a + 2 \, c\right )^{2} - 2 \, \cos \left (-a + c\right ) \sin \left (2 \, b x + a + 2 \, c\right ) \sin \relax (a) + {\left (\cos \relax (a)^{2} + \sin \relax (a)^{2}\right )} \cos \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 ) + 2 \, {\left (\sin \left (b x + 2 \, a\right ) - \sin \left (b x + 2 \, c\right )\right )} \sin \left (2 \, b x + a + 2 \, c\right ) - 2 \, \sin \left (b x + 2 \, a\right ) \sin \relax (a) + 2 \, \sin \left (b x + 2 \, c\right ) \sin \relax (a)}{2 \, {\left (b \cos \left (2 \, b x + a + 2 \, c\right )^{2} - 2 \, b \cos \left (2 \, b x + a + 2 \, c\right ) \cos \relax (a) + b \sin \left (2 \, b x + a + 2 \, c\right )^{2} - 2 \, b \sin \left (2 \, b x + a + 2 \, c\right ) \sin \relax (a) + {\left (\cos \relax (a)^{2} + \sin \relax (a)^{2}\right )} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+c)^2*sin(b*x+a),x, algorithm="maxima")

[Out]

-1/2*(2*(cos(b*x + 2*a) - cos(b*x + 2*c))*cos(2*b*x + a + 2*c) - 2*cos(b*x + 2*a)*cos(a) + 2*cos(b*x + 2*c)*co
s(a) + (cos(2*b*x + a + 2*c)^2*cos(-a + c) - 2*cos(2*b*x + a + 2*c)*cos(a)*cos(-a + c) + cos(-a + c)*sin(2*b*x
+ a + 2*c)^2 - 2*cos(-a + c)*sin(2*b*x + a + 2*c)*sin(a) + (cos(a)^2 + sin(a)^2)*cos(-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(2*b*x + a + 2*c)^2*cos(-a +
c) - 2*cos(2*b*x + a + 2*c)*cos(a)*cos(-a + c) + cos(-a + c)*sin(2*b*x + a + 2*c)^2 - 2*cos(-a + c)*sin(2*b*x
+ a + 2*c)*sin(a) + (cos(a)^2 + sin(a)^2)*cos(-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) + 2*(sin(b*x + 2*a) - sin(b*x + 2*c))*sin(2*b*x + a + 2*c) - 2*sin(b*x +
2*a)*sin(a) + 2*sin(b*x + 2*c)*sin(a))/(b*cos(2*b*x + a + 2*c)^2 - 2*b*cos(2*b*x + a + 2*c)*cos(a) + b*sin(2*b
*x + a + 2*c)^2 - 2*b*sin(2*b*x + a + 2*c)*sin(a) + (cos(a)^2 + sin(a)^2)*b)

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mupad [B]  time = 5.21, size = 252, normalized size = 7.00 \[ -\frac {\ln \left (-{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+1\right )\,1{}\mathrm {i}}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )}{2\,b\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}}+\frac {\ln \left (-{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )+\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+1\right )\,1{}\mathrm {i}}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )}{2\,b\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)/sin(c + b*x)^2,x)

[Out]

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

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sympy [B]  time = 101.78, size = 3266, normalized size = 90.72 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+c)**2*sin(b*x+a),x)

[Out]

Piecewise((0, Eq(b, 0) & (Eq(b, 0) | Eq(c, 0))), (log(tan(b*x/2))/b, Eq(c, 0)), (-log(tan(c/2) + tan(b*x/2))*t
an(c/2)**4*tan(b*x/2)/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan
(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)) - log(tan(c/2) + tan(b*x/2))*tan(c/2)**3*tan(b*x/2)**2/(b*tan(c/2)**4*
tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2
)) + log(tan(c/2) + tan(b*x/2))*tan(c/2)**3/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/
2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)) + 2*log(tan(c/2) + tan(b*x/2))*tan(c/2)**2*tan(b
*x/2)/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*t
an(c/2) - b*tan(b*x/2)) + log(tan(c/2) + tan(b*x/2))*tan(c/2)*tan(b*x/2)**2/(b*tan(c/2)**4*tan(b*x/2) + b*tan(
c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)) - log(tan(c/2) +
tan(b*x/2))*tan(c/2)/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan
(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)) - log(tan(c/2) + tan(b*x/2))*tan(b*x/2)/(b*tan(c/2)**4*tan(b*x/2) + b*
tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)) + log(tan(b*
x/2) - 1/tan(c/2))*tan(c/2)**4*tan(b*x/2)/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)
**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)) + log(tan(b*x/2) - 1/tan(c/2))*tan(c/2)**3*tan(b*x
/2)**2/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*
tan(c/2) - b*tan(b*x/2)) - log(tan(b*x/2) - 1/tan(c/2))*tan(c/2)**3/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*
tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)) - 2*log(tan(b*x/2) - 1/t
an(c/2))*tan(c/2)**2*tan(b*x/2)/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*ta
n(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)) - log(tan(b*x/2) - 1/tan(c/2))*tan(c/2)*tan(b*x/2)**2/(b*tan
(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*
tan(b*x/2)) + log(tan(b*x/2) - 1/tan(c/2))*tan(c/2)/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 -
b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)) + log(tan(b*x/2) - 1/tan(c/2))*tan(b*x/2
)/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c
/2) - b*tan(b*x/2)) + tan(c/2)**4*tan(b*x/2)/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c
/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)) - 2*tan(c/2)**3/(b*tan(c/2)**4*tan(b*x/2) + b*t
an(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)) - 2*tan(c/2)/
(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan(c/2)*tan(b*x/2)**2 - b*tan(c/2
) - b*tan(b*x/2)) - tan(b*x/2)/(b*tan(c/2)**4*tan(b*x/2) + b*tan(c/2)**3*tan(b*x/2)**2 - b*tan(c/2)**3 + b*tan
(c/2)*tan(b*x/2)**2 - b*tan(c/2) - b*tan(b*x/2)), True))*cos(a) + Piecewise((zoo*x, Eq(b, 0) & Eq(c, 0)), (-1/
(b*sin(b*x)), Eq(c, 0)), (x/sin(c)**2, Eq(b, 0)), (4*log(tan(c/2) + tan(b*x/2))*tan(c/2)**4*tan(b*x/2)/(2*b*ta
n(c/2)**5*tan(b*x/2) + 2*b*tan(c/2)**4*tan(b*x/2)**2 - 2*b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 - 2*b*t
an(c/2)**2 - 2*b*tan(c/2)*tan(b*x/2)) + 4*log(tan(c/2) + tan(b*x/2))*tan(c/2)**3*tan(b*x/2)**2/(2*b*tan(c/2)**
5*tan(b*x/2) + 2*b*tan(c/2)**4*tan(b*x/2)**2 - 2*b*tan(c/2)**4 + 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)) - 4*log(tan(c/2) + tan(b*x/2))*tan(c/2)**3/(2*b*tan(c/2)**5*tan(b*x/2) + 2*b*tan
(c/2)**4*tan(b*x/2)**2 - 2*b*tan(c/2)**4 + 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)) - 4*log(tan(c/2) + tan(b*x/2))*tan(c/2)**2*tan(b*x/2)/(2*b*tan(c/2)**5*tan(b*x/2) + 2*b*tan(c/2)**4*ta
n(b*x/2)**2 - 2*b*tan(c/2)**4 + 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)) - 4
*log(tan(b*x/2) - 1/tan(c/2))*tan(c/2)**4*tan(b*x/2)/(2*b*tan(c/2)**5*tan(b*x/2) + 2*b*tan(c/2)**4*tan(b*x/2)*
*2 - 2*b*tan(c/2)**4 + 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)) - 4*log(tan(
b*x/2) - 1/tan(c/2))*tan(c/2)**3*tan(b*x/2)**2/(2*b*tan(c/2)**5*tan(b*x/2) + 2*b*tan(c/2)**4*tan(b*x/2)**2 - 2
*b*tan(c/2)**4 + 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)) + 4*log(tan(b*x/2)
- 1/tan(c/2))*tan(c/2)**3/(2*b*tan(c/2)**5*tan(b*x/2) + 2*b*tan(c/2)**4*tan(b*x/2)**2 - 2*b*tan(c/2)**4 + 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)) + 4*log(tan(b*x/2) - 1/tan(c/2))*tan(c
/2)**2*tan(b*x/2)/(2*b*tan(c/2)**5*tan(b*x/2) + 2*b*tan(c/2)**4*tan(b*x/2)**2 - 2*b*tan(c/2)**4 + 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(c/2)**6*tan(b*x/2)/(2*b*tan(c/2)**5*tan(b
*x/2) + 2*b*tan(c/2)**4*tan(b*x/2)**2 - 2*b*tan(c/2)**4 + 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)**5/(2*b*tan(c/2)**5*tan(b*x/2) + 2*b*tan(c/2)**4*tan(b*x/2)**2 - 2*b*tan(c
/2)**4 + 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(c/2)**4*tan(b*x/2)/(
2*b*tan(c/2)**5*tan(b*x/2) + 2*b*tan(c/2)**4*tan(b*x/2)**2 - 2*b*tan(c/2)**4 + 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(c/2)**2*tan(b*x/2)/(2*b*tan(c/2)**5*tan(b*x/2) + 2*b*tan(c/2
)**4*tan(b*x/2)**2 - 2*b*tan(c/2)**4 + 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*b*tan(c/2)**5*tan(b*x/2) + 2*b*tan(c/2)**4*tan(b*x/2)**2 - 2*b*tan(c/2)**4 + 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)**5*tan(b*x/2) + 2*b*t
an(c/2)**4*tan(b*x/2)**2 - 2*b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 - 2*b*tan(c/2)**2 - 2*b*tan(c/2)*ta
n(b*x/2)), True))*sin(a)

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