\(\int \cot ^2(c+b x) \sin (a+b x) \, dx\) [234]

Optimal result

Integrand size = 15, antiderivative size = 46 \[ \int \cot ^2(c+b x) \sin (a+b x) \, dx=-\frac {\text {arctanh}(\cos (c+b x)) \cos (a-c)}{b}+\frac {\cos (a+b x)}{b}-\frac {\csc (c+b x) \sin (a-c)}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4674, 4673, 2718, 3855, 2686, 8} \[ \int \cot ^2(c+b x) \sin (a+b x) \, dx=-\frac {\cos (a-c) \text {arctanh}(\cos (b x+c))}{b}-\frac {\sin (a-c) \csc (b x+c)}{b}+\frac {\cos (a+b x)}{b} \]

[In]

Int[Cot[c + b*x]^2*Sin[a + b*x],x]

[Out]

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

Rule 8

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

Rule 2686

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 2718

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

Rule 3855

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

Rule 4673

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

Rule 4674

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

Rubi steps \begin{align*} \text {integral}& = \sin (a-c) \int \cot (c+b x) \csc (c+b x) \, dx+\int \cos (a+b x) \cot (c+b x) \, dx \\ & = \cos (a-c) \int \csc (c+b x) \, dx-\frac {\sin (a-c) \text {Subst}(\int 1 \, dx,x,\csc (c+b x))}{b}-\int \sin (a+b x) \, dx \\ & = -\frac {\text {arctanh}(\cos (c+b x)) \cos (a-c)}{b}+\frac {\cos (a+b x)}{b}-\frac {\csc (c+b x) \sin (a-c)}{b} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.15 (sec) , antiderivative size = 111, normalized size of antiderivative = 2.41 \[ \int \cot ^2(c+b x) \sin (a+b x) \, dx=-\frac {2 i \arctan \left (\frac {(\cos (c)-i \sin (c)) \left (\cos (c) \cos \left (\frac {b x}{2}\right )-\sin (c) \sin \left (\frac {b x}{2}\right )\right )}{i \cos (c) \cos \left (\frac {b x}{2}\right )+\cos \left (\frac {b x}{2}\right ) \sin (c)}\right ) \cos (a-c)}{b}+\frac {\cos (a) \cos (b x)}{b}-\frac {\csc (c+b x) \sin (a-c)}{b}-\frac {\sin (a) \sin (b x)}{b} \]

[In]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 143, normalized size of antiderivative = 3.11

method	result	size
risch	\(\frac {{\mathrm e}^{i \left (x b +a \right )}}{2 b}+\frac {{\mathrm e}^{-i \left (x b +a \right )}}{2 b}+\frac {{\mathrm e}^{i \left (x b +3 a \right )}-{\mathrm e}^{i \left (x b +a +2 c \right )}}{b \left (-{\mathrm e}^{2 i \left (x b +a +c \right )}+{\mathrm e}^{2 i a}\right )}+\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{b}-\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{b}\)	\(143\)

[In]

int(cot(b*x+c)^2*sin(b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/2*exp(I*(b*x+a))/b+1/2/b*exp(-I*(b*x+a))+1/b/(-exp(2*I*(b*x+a+c))+exp(2*I*a))*(exp(I*(b*x+3*a))-exp(I*(b*x+a
+2*c)))+ln(exp(I*(b*x+a))-exp(I*(a-c)))/b*cos(a-c)-ln(exp(I*(b*x+a))+exp(I*(a-c)))/b*cos(a-c)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (46) = 92\).

Time = 0.25 (sec) , antiderivative size = 316, normalized size of antiderivative = 6.87 \[ \int \cot ^2(c+b x) \sin (a+b x) \, dx=\frac {4 \, {\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + \frac {\sqrt {2} {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) + {\left (\cos \left (-2 \, a + 2 \, c\right )^{2} + 2 \, \cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \sin \left (b x + a\right )\right )} \log \left (-\frac {2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \frac {2 \, \sqrt {2} {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right )\right )}}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} - \cos \left (-2 \, a + 2 \, c\right ) + 3}{2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \cos \left (-2 \, a + 2 \, c\right ) - 1}\right )}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} + 4 \, {\left (\cos \left (b x + a\right )^{2} + 1\right )} \sin \left (-2 \, a + 2 \, c\right )}{4 \, {\left (b \cos \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) + {\left (b \cos \left (-2 \, a + 2 \, c\right ) + b\right )} \sin \left (b x + a\right )\right )}} \]

[In]

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

[Out]

1/4*(4*(cos(-2*a + 2*c) + 1)*cos(b*x + a)*sin(b*x + a) + sqrt(2)*((cos(-2*a + 2*c) + 1)*cos(b*x + a)*sin(-2*a
+ 2*c) + (cos(-2*a + 2*c)^2 + 2*cos(-2*a + 2*c) + 1)*sin(b*x + a))*log(-(2*cos(b*x + a)^2*cos(-2*a + 2*c) - 2*
cos(b*x + a)*sin(b*x + a)*sin(-2*a + 2*c) - 2*sqrt(2)*((cos(-2*a + 2*c) + 1)*cos(b*x + a) - sin(b*x + a)*sin(-
2*a + 2*c))/sqrt(cos(-2*a + 2*c) + 1) - cos(-2*a + 2*c) + 3)/(2*cos(b*x + a)^2*cos(-2*a + 2*c) - 2*cos(b*x + a
)*sin(b*x + a)*sin(-2*a + 2*c) - cos(-2*a + 2*c) - 1))/sqrt(cos(-2*a + 2*c) + 1) + 4*(cos(b*x + a)^2 + 1)*sin(
-2*a + 2*c))/(b*cos(b*x + a)*sin(-2*a + 2*c) + (b*cos(-2*a + 2*c) + b)*sin(b*x + a))

Sympy [F]

\[ \int \cot ^2(c+b x) \sin (a+b x) \, dx=\int \sin {\left (a + b x \right )} \cot ^{2}{\left (b x + c \right )}\, dx \]

[In]

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

[Out]

Integral(sin(a + b*x)*cot(b*x + c)**2, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 612 vs. \(2 (46) = 92\).

Time = 0.31 (sec) , antiderivative size = 612, normalized size of antiderivative = 13.30 \[ \int \cot ^2(c+b x) \sin (a+b x) \, dx=\frac {{\left (\cos \left (3 \, b x + a + 2 \, c\right ) - \cos \left (b x + a\right )\right )} \cos \left (4 \, b x + 2 \, a + 2 \, c\right ) - {\left (3 \, \cos \left (2 \, b x + 2 \, a\right ) - 3 \, \cos \left (2 \, b x + 2 \, c\right ) + 1\right )} \cos \left (3 \, b x + a + 2 \, c\right ) + 3 \, \cos \left (2 \, b x + 2 \, a\right ) \cos \left (b x + a\right ) - 3 \, \cos \left (2 \, b x + 2 \, c\right ) \cos \left (b x + a\right ) - {\left (\cos \left (3 \, b x + a + 2 \, c\right )^{2} \cos \left (-a + c\right ) - 2 \, \cos \left (3 \, b x + a + 2 \, c\right ) \cos \left (b x + a\right ) \cos \left (-a + c\right ) + \cos \left (b x + a\right )^{2} \cos \left (-a + c\right ) + \cos \left (-a + c\right ) \sin \left (3 \, b x + a + 2 \, c\right )^{2} - 2 \, \cos \left (-a + c\right ) \sin \left (3 \, b x + a + 2 \, c\right ) \sin \left (b x + a\right ) + \cos \left (-a + c\right ) \sin \left (b x + a\right )^{2}\right )} \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) + {\left (\cos \left (3 \, b x + a + 2 \, c\right )^{2} \cos \left (-a + c\right ) - 2 \, \cos \left (3 \, b x + a + 2 \, c\right ) \cos \left (b x + a\right ) \cos \left (-a + c\right ) + \cos \left (b x + a\right )^{2} \cos \left (-a + c\right ) + \cos \left (-a + c\right ) \sin \left (3 \, b x + a + 2 \, c\right )^{2} - 2 \, \cos \left (-a + c\right ) \sin \left (3 \, b x + a + 2 \, c\right ) \sin \left (b x + a\right ) + \cos \left (-a + c\right ) \sin \left (b x + a\right )^{2}\right )} \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) + {\left (\sin \left (3 \, b x + a + 2 \, c\right ) - \sin \left (b x + a\right )\right )} \sin \left (4 \, b x + 2 \, a + 2 \, c\right ) - 3 \, {\left (\sin \left (2 \, b x + 2 \, a\right ) - \sin \left (2 \, b x + 2 \, c\right )\right )} \sin \left (3 \, b x + a + 2 \, c\right ) + 3 \, \sin \left (2 \, b x + 2 \, a\right ) \sin \left (b x + a\right ) - 3 \, \sin \left (2 \, b x + 2 \, c\right ) \sin \left (b x + a\right ) + \cos \left (b x + a\right )}{2 \, {\left (b \cos \left (3 \, b x + a + 2 \, c\right )^{2} - 2 \, b \cos \left (3 \, b x + a + 2 \, c\right ) \cos \left (b x + a\right ) + b \cos \left (b x + a\right )^{2} + b \sin \left (3 \, b x + a + 2 \, c\right )^{2} - 2 \, b \sin \left (3 \, b x + a + 2 \, c\right ) \sin \left (b x + a\right ) + b \sin \left (b x + a\right )^{2}\right )}} \]

[In]

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

[Out]

1/2*((cos(3*b*x + a + 2*c) - cos(b*x + a))*cos(4*b*x + 2*a + 2*c) - (3*cos(2*b*x + 2*a) - 3*cos(2*b*x + 2*c) +
1)*cos(3*b*x + a + 2*c) + 3*cos(2*b*x + 2*a)*cos(b*x + a) - 3*cos(2*b*x + 2*c)*cos(b*x + a) - (cos(3*b*x + a
+ 2*c)^2*cos(-a + c) - 2*cos(3*b*x + a + 2*c)*cos(b*x + a)*cos(-a + c) + cos(b*x + a)^2*cos(-a + c) + cos(-a +
c)*sin(3*b*x + a + 2*c)^2 - 2*cos(-a + c)*sin(3*b*x + a + 2*c)*sin(b*x + a) + cos(-a + c)*sin(b*x + a)^2)*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(3*b*x + a + 2*c
)^2*cos(-a + c) - 2*cos(3*b*x + a + 2*c)*cos(b*x + a)*cos(-a + c) + cos(b*x + a)^2*cos(-a + c) + cos(-a + c)*s
in(3*b*x + a + 2*c)^2 - 2*cos(-a + c)*sin(3*b*x + a + 2*c)*sin(b*x + a) + cos(-a + c)*sin(b*x + a)^2)*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(3*b*x + a + 2*c) - s
in(b*x + a))*sin(4*b*x + 2*a + 2*c) - 3*(sin(2*b*x + 2*a) - sin(2*b*x + 2*c))*sin(3*b*x + a + 2*c) + 3*sin(2*b
*x + 2*a)*sin(b*x + a) - 3*sin(2*b*x + 2*c)*sin(b*x + a) + cos(b*x + a))/(b*cos(3*b*x + a + 2*c)^2 - 2*b*cos(3
*b*x + a + 2*c)*cos(b*x + a) + b*cos(b*x + a)^2 + b*sin(3*b*x + a + 2*c)^2 - 2*b*sin(3*b*x + a + 2*c)*sin(b*x
+ a) + b*sin(b*x + a)^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 577 vs. \(2 (46) = 92\).

Time = 0.31 (sec) , antiderivative size = 577, normalized size of antiderivative = 12.54 \[ \int \cot ^2(c+b x) \sin (a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 25.77 (sec) , antiderivative size = 290, normalized size of antiderivative = 6.30 \[ \int \cot ^2(c+b x) \sin (a+b x) \, dx=\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}-b\,x\,1{}\mathrm {i}}}{2\,b}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}}{2\,b}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )\,1{}\mathrm {i}}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}\,1{}\mathrm {i}-{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}\right )}-\frac {\ln \left (-{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+1\right )\,1{}\mathrm {i}}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )}{2\,b\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}}+\frac {\ln \left (-{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )+\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+1\right )\,1{}\mathrm {i}}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )}{2\,b\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}} \]

[In]

int(cot(c + b*x)^2*sin(a + b*x),x)

[Out]

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