\(\int \tan (a+b x) \tan (c+b x) \, dx\) [139]

Optimal result

Integrand size = 13, antiderivative size = 39 \[ \int \tan (a+b x) \tan (c+b x) \, dx=-x-\frac {\cot (a-c) \log (\cos (a+b x))}{b}+\frac {\cot (a-c) \log (\cos (c+b x))}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4708, 4706, 3556} \[ \int \tan (a+b x) \tan (c+b x) \, dx=-\frac {\cot (a-c) \log (\cos (a+b x))}{b}+\frac {\cot (a-c) \log (\cos (b x+c))}{b}-x \]

[In]

Int[Tan[a + b*x]*Tan[c + b*x],x]

[Out]

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

Rule 3556

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

Rule 4706

Int[Sec[(a_.) + (b_.)*(x_)]*Sec[(c_) + (d_.)*(x_)], x_Symbol] :> Dist[-Csc[(b*c - a*d)/d], Int[Tan[a + b*x], x
], x] + Dist[Csc[(b*c - a*d)/b], Int[Tan[c + d*x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && Ne
Q[b*c - a*d, 0]

Rule 4708

Int[Tan[(a_.) + (b_.)*(x_)]*Tan[(c_) + (d_.)*(x_)], x_Symbol] :> Simp[(-b)*(x/d), x] + Dist[(b/d)*Cos[(b*c - a
*d)/d], Int[Sec[a + b*x]*Sec[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*} \text {integral}& = -x+\cos (a-c) \int \sec (a+b x) \sec (c+b x) \, dx \\ & = -x+\cot (a-c) \int \tan (a+b x) \, dx-\cot (a-c) \int \tan (c+b x) \, dx \\ & = -x-\frac {\cot (a-c) \log (\cos (a+b x))}{b}+\frac {\cot (a-c) \log (\cos (c+b x))}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.79 \[ \int \tan (a+b x) \tan (c+b x) \, dx=-x+\frac {\cot (a-c) (-\log (\cos (a+b x))+\log (\cos (c+b x)))}{b} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.36 (sec) , antiderivative size = 173, normalized size of antiderivative = 4.44

method	result	size
risch	\(-x +\frac {i \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+{\mathrm e}^{2 i \left (a -c \right )}\right ) {\mathrm e}^{2 i a}}{b \left ({\mathrm e}^{2 i a}-{\mathrm e}^{2 i c}\right )}+\frac {i \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+{\mathrm e}^{2 i \left (a -c \right )}\right ) {\mathrm e}^{2 i c}}{b \left ({\mathrm e}^{2 i a}-{\mathrm e}^{2 i c}\right )}-\frac {i \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) {\mathrm e}^{2 i a}}{b \left ({\mathrm e}^{2 i a}-{\mathrm e}^{2 i c}\right )}-\frac {i \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) {\mathrm e}^{2 i c}}{b \left ({\mathrm e}^{2 i a}-{\mathrm e}^{2 i c}\right )}\)	\(173\)

[In]

int(tan(b*x+a)*tan(b*x+c),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (39) = 78\).

Time = 0.25 (sec) , antiderivative size = 145, normalized size of antiderivative = 3.72 \[ \int \tan (a+b x) \tan (c+b x) \, dx=-\frac {2 \, b x \sin \left (-2 \, a + 2 \, c\right ) - {\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \log \left (-\frac {{\left (\cos \left (-2 \, a + 2 \, c\right ) - 1\right )} \tan \left (b x + c\right )^{2} - 2 \, \sin \left (-2 \, a + 2 \, c\right ) \tan \left (b x + c\right ) - \cos \left (-2 \, a + 2 \, c\right ) - 1}{{\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \tan \left (b x + c\right )^{2} + \cos \left (-2 \, a + 2 \, c\right ) + 1}\right ) + {\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \log \left (\frac {1}{\tan \left (b x + c\right )^{2} + 1}\right )}{2 \, b \sin \left (-2 \, a + 2 \, c\right )} \]

[In]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2020 vs. \(2 (31) = 62\).

Time = 2.88 (sec) , antiderivative size = 7672, normalized size of antiderivative = 196.72 \[ \int \tan (a+b x) \tan (c+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((0, Eq(a, 0) & Eq(b, 0) & Eq(c, 0)), (-2*b*x*tan(c)/(2*b*tan(c)**2 + 2*b) - 2*log(tan(b*x) - 1/tan(c
))/(2*b*tan(c)**2 + 2*b) + log(tan(b*x)**2 + 1)/(2*b*tan(c)**2 + 2*b), Eq(a, 0)), (-2*b*x*tan(a)/(2*b*tan(a)**
2 + 2*b) - 2*log(tan(b*x) - 1/tan(a))/(2*b*tan(a)**2 + 2*b) + log(tan(b*x)**2 + 1)/(2*b*tan(a)**2 + 2*b), Eq(c
, 0)), (-4*b*x*tan(c)**2*tan(b*x)/(2*b*tan(c)**5*tan(b*x) - 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) - 4*b*tan(c
)**2 + 2*b*tan(c)*tan(b*x) - 2*b) + 4*b*x*tan(c)/(2*b*tan(c)**5*tan(b*x) - 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b
*x) - 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) - 2*b) + 2*log(tan(b*x) - 1/tan(c))*tan(c)**3*tan(b*x)/(2*b*tan(c)**
5*tan(b*x) - 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) - 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) - 2*b) - 2*log(tan(b
*x) - 1/tan(c))*tan(c)**2/(2*b*tan(c)**5*tan(b*x) - 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) - 4*b*tan(c)**2 + 2
*b*tan(c)*tan(b*x) - 2*b) - 2*log(tan(b*x) - 1/tan(c))*tan(c)*tan(b*x)/(2*b*tan(c)**5*tan(b*x) - 2*b*tan(c)**4
+ 4*b*tan(c)**3*tan(b*x) - 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) - 2*b) + 2*log(tan(b*x) - 1/tan(c))/(2*b*tan(c
)**5*tan(b*x) - 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) - 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) - 2*b) - log(tan(
b*x)**2 + 1)*tan(c)**3*tan(b*x)/(2*b*tan(c)**5*tan(b*x) - 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) - 4*b*tan(c)*
*2 + 2*b*tan(c)*tan(b*x) - 2*b) + log(tan(b*x)**2 + 1)*tan(c)**2/(2*b*tan(c)**5*tan(b*x) - 2*b*tan(c)**4 + 4*b
*tan(c)**3*tan(b*x) - 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) - 2*b) + log(tan(b*x)**2 + 1)*tan(c)*tan(b*x)/(2*b*t
an(c)**5*tan(b*x) - 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) - 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) - 2*b) - log(
tan(b*x)**2 + 1)/(2*b*tan(c)**5*tan(b*x) - 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) - 4*b*tan(c)**2 + 2*b*tan(c)
*tan(b*x) - 2*b) - 2*tan(c)**2/(2*b*tan(c)**5*tan(b*x) - 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) - 4*b*tan(c)**
2 + 2*b*tan(c)*tan(b*x) - 2*b) - 2/(2*b*tan(c)**5*tan(b*x) - 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) - 4*b*tan(
c)**2 + 2*b*tan(c)*tan(b*x) - 2*b), Eq(a, atan(tan(c)) + pi*floor((c - pi/2)/pi))), (0, Eq(b, 0)), (-2*b*x*tan
(a)**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(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*log(tan(b*x) - 1/tan(a))*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
(b*x) - 1/tan(a))*tan(a)/(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(b*x) - 1/tan(c))*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(b*x) - 1/tan(c))*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)**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)) + log(tan(b*x)**2 + 1)*
tan(a)/(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(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)), True))*tan(a) + Piecewise((0, Eq(a, 0) & Eq(b, 0) & Eq(c, 0)), (-2*b*x*tan(c)/(2*b*tan(c)*
*2 + 2*b) - 2*log(tan(b*x) - 1/tan(c))/(2*b*tan(c)**2 + 2*b) + log(tan(b*x)**2 + 1)/(2*b*tan(c)**2 + 2*b), Eq(
a, 0)), (-2*b*x*tan(a)/(2*b*tan(a)**2 + 2*b) - 2*log(tan(b*x) - 1/tan(a))/(2*b*tan(a)**2 + 2*b) + log(tan(b*x)
**2 + 1)/(2*b*tan(a)**2 + 2*b), Eq(c, 0)), (-4*b*x*tan(c)**2*tan(b*x)/(2*b*tan(c)**5*tan(b*x) - 2*b*tan(c)**4
+ 4*b*tan(c)**3*tan(b*x) - 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) - 2*b) + 4*b*x*tan(c)/(2*b*tan(c)**5*tan(b*x) -
2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) - 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) - 2*b) + 2*log(tan(b*x) - 1/tan(
c))*tan(c)**3*tan(b*x)/(2*b*tan(c)**5*tan(b*x) - 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) - 4*b*tan(c)**2 + 2*b*
tan(c)*tan(b*x) - 2*b) - 2*log(tan(b*x) - 1/tan(c))*tan(c)**2/(2*b*tan(c)**5*tan(b*x) - 2*b*tan(c)**4 + 4*b*ta
n(c)**3*tan(b*x) - 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) - 2*b) - 2*log(tan(b*x) - 1/tan(c))*tan(c)*tan(b*x)/(2*
b*tan(c)**5*tan(b*x) - 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) - 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) - 2*b) + 2
*log(tan(b*x) - 1/tan(c))/(2*b*tan(c)**5*tan(b*x) - 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) - 4*b*tan(c)**2 + 2
*b*tan(c)*tan(b*x) - 2*b) - log(tan(b*x)**2 + 1)*tan(c)**3*tan(b*x)/(2*b*tan(c)**5*tan(b*x) - 2*b*tan(c)**4 +
4*b*tan(c)**3*tan(b*x) - 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) - 2*b) + log(tan(b*x)**2 + 1)*tan(c)**2/(2*b*tan(
c)**5*tan(b*x) - 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) - 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) - 2*b) + log(tan
(b*x)**2 + 1)*tan(c)*tan(b*x)/(2*b*tan(c)**5*tan(b*x) - 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) - 4*b*tan(c)**2
+ 2*b*tan(c)*tan(b*x) - 2*b) - log(tan(b*x)**2 + 1)/(2*b*tan(c)**5*tan(b*x) - 2*b*tan(c)**4 + 4*b*tan(c)**3*t
an(b*x) - 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) - 2*b) - 2*tan(c)**2/(2*b*tan(c)**5*tan(b*x) - 2*b*tan(c)**4 + 4
*b*tan(c)**3*tan(b*x) - 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) - 2*b) - 2/(2*b*tan(c)**5*tan(b*x) - 2*b*tan(c)**4
+ 4*b*tan(c)**3*tan(b*x) - 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) - 2*b), Eq(a, atan(tan(c)) + pi*floor((c - pi/
2)/pi))), (0, Eq(b, 0)), (-2*b*x*tan(a)**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(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(b*x) - 1/tan(a))*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(b*x) - 1/tan(a))*tan(a)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 - 2*b*ta
n(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(b*x) - 1/tan(c))*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(b*x) - 1/
tan(c))*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)**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*ta
n(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*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(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)), True))*tan(c) + Piecewise((0, Eq(a, 0) & Eq(b, 0) & Eq
(c, 0)), (-2*b*x*tan(c)/(2*b*tan(c)**3 + 2*b*tan(c)) - 2*log(tan(b*x) - 1/tan(c))/(2*b*tan(c)**3 + 2*b*tan(c))
- log(tan(b*x)**2 + 1)*tan(c)**2/(2*b*tan(c)**3 + 2*b*tan(c)), Eq(a, 0)), (-2*b*x*tan(a)/(2*b*tan(a)**3 + 2*b
*tan(a)) - 2*log(tan(b*x) - 1/tan(a))/(2*b*tan(a)**3 + 2*b*tan(a)) - log(tan(b*x)**2 + 1)*tan(a)**2/(2*b*tan(a
)**3 + 2*b*tan(a)), Eq(c, 0)), (b*x*tan(c)**4*tan(b*x)/(b*tan(c)**6*tan(b*x) - b*tan(c)**5 + 2*b*tan(c)**4*tan
(b*x) - 2*b*tan(c)**3 + b*tan(c)**2*tan(b*x) - b*tan(c)) - b*x*tan(c)**3/(b*tan(c)**6*tan(b*x) - b*tan(c)**5 +
2*b*tan(c)**4*tan(b*x) - 2*b*tan(c)**3 + b*tan(c)**2*tan(b*x) - b*tan(c)) - b*x*tan(c)**2*tan(b*x)/(b*tan(c)*
*6*tan(b*x) - b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) - 2*b*tan(c)**3 + b*tan(c)**2*tan(b*x) - b*tan(c)) + b*x*ta
n(c)/(b*tan(c)**6*tan(b*x) - b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) - 2*b*tan(c)**3 + b*tan(c)**2*tan(b*x) - b*t
an(c)) + 2*log(tan(b*x) - 1/tan(c))*tan(c)**3*tan(b*x)/(b*tan(c)**6*tan(b*x) - b*tan(c)**5 + 2*b*tan(c)**4*tan
(b*x) - 2*b*tan(c)**3 + b*tan(c)**2*tan(b*x) - b*tan(c)) - 2*log(tan(b*x) - 1/tan(c))*tan(c)**2/(b*tan(c)**6*t
an(b*x) - b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) - 2*b*tan(c)**3 + b*tan(c)**2*tan(b*x) - b*tan(c)) - log(tan(b*
x)**2 + 1)*tan(c)**3*tan(b*x)/(b*tan(c)**6*tan(b*x) - b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) - 2*b*tan(c)**3 + b
*tan(c)**2*tan(b*x) - b*tan(c)) + log(tan(b*x)**2 + 1)*tan(c)**2/(b*tan(c)**6*tan(b*x) - b*tan(c)**5 + 2*b*tan
(c)**4*tan(b*x) - 2*b*tan(c)**3 + b*tan(c)**2*tan(b*x) - b*tan(c)) - tan(c)**2/(b*tan(c)**6*tan(b*x) - b*tan(c
)**5 + 2*b*tan(c)**4*tan(b*x) - 2*b*tan(c)**3 + b*tan(c)**2*tan(b*x) - b*tan(c)) - 1/(b*tan(c)**6*tan(b*x) - b
*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) - 2*b*tan(c)**3 + b*tan(c)**2*tan(b*x) - b*tan(c)), Eq(a, atan(tan(c)) + p
i*floor((c - pi/2)/pi))), (0, Eq(b, 0)), (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*t
an(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*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*t
an(c)**3 - 2*b*tan(c)) + 2*b*x*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(b*x) - 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*t
an(a)*tan(c)**2 + 2*b*tan(a) - 2*b*tan(c)**3 - 2*b*tan(c)) - 2*log(tan(b*x) - 1/tan(a))/(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(b*x) - 1/tan(c))*tan(a)**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(b*x) - 1/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)**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(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)) + Piecewise((x, Eq(a, 0) & Eq(b, 0) & Eq(c, 0)), (2*b*x/(2*b*tan(c)**2 + 2*b) - 2*log(tan(
b*x) - 1/tan(c))*tan(c)/(2*b*tan(c)**2 + 2*b) + log(tan(b*x)**2 + 1)*tan(c)/(2*b*tan(c)**2 + 2*b), Eq(a, 0)),
(2*b*x/(2*b*tan(a)**2 + 2*b) - 2*log(tan(b*x) - 1/tan(a))*tan(a)/(2*b*tan(a)**2 + 2*b) + log(tan(b*x)**2 + 1)*
tan(a)/(2*b*tan(a)**2 + 2*b), Eq(c, 0)), (-b*x*tan(c)**3*tan(b*x)/(b*tan(c)**5*tan(b*x) - b*tan(c)**4 + 2*b*ta
n(c)**3*tan(b*x) - 2*b*tan(c)**2 + b*tan(c)*tan(b*x) - b) + b*x*tan(c)**2/(b*tan(c)**5*tan(b*x) - b*tan(c)**4
+ 2*b*tan(c)**3*tan(b*x) - 2*b*tan(c)**2 + b*tan(c)*tan(b*x) - b) + b*x*tan(c)*tan(b*x)/(b*tan(c)**5*tan(b*x)
- b*tan(c)**4 + 2*b*tan(c)**3*tan(b*x) - 2*b*tan(c)**2 + b*tan(c)*tan(b*x) - b) - b*x/(b*tan(c)**5*tan(b*x) -
b*tan(c)**4 + 2*b*tan(c)**3*tan(b*x) - 2*b*tan(c)**2 + b*tan(c)*tan(b*x) - b) - 2*log(tan(b*x) - 1/tan(c))*tan
(c)**2*tan(b*x)/(b*tan(c)**5*tan(b*x) - b*tan(c)**4 + 2*b*tan(c)**3*tan(b*x) - 2*b*tan(c)**2 + b*tan(c)*tan(b*
x) - b) + 2*log(tan(b*x) - 1/tan(c))*tan(c)/(b*tan(c)**5*tan(b*x) - b*tan(c)**4 + 2*b*tan(c)**3*tan(b*x) - 2*b
*tan(c)**2 + b*tan(c)*tan(b*x) - b) + log(tan(b*x)**2 + 1)*tan(c)**2*tan(b*x)/(b*tan(c)**5*tan(b*x) - b*tan(c)
**4 + 2*b*tan(c)**3*tan(b*x) - 2*b*tan(c)**2 + b*tan(c)*tan(b*x) - b) - log(tan(b*x)**2 + 1)*tan(c)/(b*tan(c)*
*5*tan(b*x) - b*tan(c)**4 + 2*b*tan(c)**3*tan(b*x) - 2*b*tan(c)**2 + b*tan(c)*tan(b*x) - b) - tan(c)**3/(b*tan
(c)**5*tan(b*x) - b*tan(c)**4 + 2*b*tan(c)**3*tan(b*x) - 2*b*tan(c)**2 + b*tan(c)*tan(b*x) - b) - tan(c)/(b*ta
n(c)**5*tan(b*x) - b*tan(c)**4 + 2*b*tan(c)**3*tan(b*x) - 2*b*tan(c)**2 + b*tan(c)*tan(b*x) - b), Eq(a, atan(t
an(c)) + pi*floor((c - pi/2)/pi))), (x, Eq(b, 0)), (-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*b*x*tan(a)/(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(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(b*x
) - 1/tan(a))*tan(a)**2*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(b*x) - 1/tan(a))*t
an(a)**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(b*x) - 1/tan(c))*tan(a)**2*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)) + 2*log(tan(b*x) - 1/tan(c))*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/(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(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))*tan(a)*tan(c)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 371 vs. \(2 (39) = 78\).

Time = 0.22 (sec) , antiderivative size = 371, normalized size of antiderivative = 9.51 \[ \int \tan (a+b x) \tan (c+b x) \, dx=-\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 (2 \, b x\right ) - \sin \left (2 \, a\right ), \cos \left (2 \, b x\right ) + \cos \left (2 \, a\right )\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 (2 \, b x\right ) - \sin \left (2 \, c\right ), \cos \left (2 \, b x\right ) + \cos \left (2 \, c\right )\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 (2 \, b x\right )^{2} + 2 \, \cos \left (2 \, b x\right ) \cos \left (2 \, a\right ) + \cos \left (2 \, a\right )^{2} + \sin \left (2 \, b x\right )^{2} - 2 \, \sin \left (2 \, b x\right ) \sin \left (2 \, a\right ) + \sin \left (2 \, a\right )^{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 (2 \, b x\right )^{2} + 2 \, \cos \left (2 \, b x\right ) \cos \left (2 \, c\right ) + \cos \left (2 \, c\right )^{2} + \sin \left (2 \, b x\right )^{2} - 2 \, \sin \left (2 \, b x\right ) \sin \left (2 \, c\right ) + \sin \left (2 \, c\right )^{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} \]

[In]

integrate(tan(b*x+a)*tan(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(2*b*x) - sin(2*a), cos(2*b*x) + cos(2*a))
- (cos(2*a)^2 - cos(2*c)^2 + sin(2*a)^2 - sin(2*c)^2)*arctan2(sin(2*b*x) - sin(2*c), cos(2*b*x) + cos(2*c)) -
(cos(2*c)*sin(2*a) - cos(2*a)*sin(2*c))*log(cos(2*b*x)^2 + 2*cos(2*b*x)*cos(2*a) + cos(2*a)^2 + sin(2*b*x)^2 -
2*sin(2*b*x)*sin(2*a) + sin(2*a)^2) + (cos(2*c)*sin(2*a) - cos(2*a)*sin(2*c))*log(cos(2*b*x)^2 + 2*cos(2*b*x)
*cos(2*c) + cos(2*c)^2 + sin(2*b*x)^2 - 2*sin(2*b*x)*sin(2*c) + sin(2*c)^2))/(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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (39) = 78\).

Time = 0.29 (sec) , antiderivative size = 242, normalized size of antiderivative = 6.21 \[ \int \tan (a+b x) \tan (c+b x) \, dx=-\frac {2 \, b x + \frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{3} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right )^{3} + 4 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )\right )} \log \left ({\left | 2 \, \tan \left (b x\right ) \tan \left (\frac {1}{2} \, a\right ) + \tan \left (\frac {1}{2} \, a\right )^{2} - 1 \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{3} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )} - \frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, c\right )\right )} \log \left ({\left | 2 \, \tan \left (b x\right ) \tan \left (\frac {1}{2} \, c\right ) + \tan \left (\frac {1}{2} \, c\right )^{2} - 1 \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) - \tan \left (\frac {1}{2} \, c\right )^{2}}}{2 \, b} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 31.89 (sec) , antiderivative size = 207, normalized size of antiderivative = 5.31 \[ \int \tan (a+b x) \tan (c+b x) \, dx=-\frac {\frac {x}{2}+x\,\left ({\sin \left (a-c\right )}^2-\frac {1}{2}\right )}{{\sin \left (a-c\right )}^2}-\frac {\frac {\sin \left (2\,a-2\,c\right )\,\ln \left ({\sin \left (2\,a-2\,c\right )}^2\,2{}\mathrm {i}-{\sin \left (a+b\,x\right )}^2\,2{}\mathrm {i}+{\sin \left (3\,a-2\,c+b\,x\right )}^2\,2{}\mathrm {i}+\sin \left (4\,a-4\,c\right )+\sin \left (6\,a-4\,c+2\,b\,x\right )-\sin \left (2\,a+2\,b\,x\right )\right )}{2}-\frac {\sin \left (2\,a-2\,c\right )\,\ln \left ({\sin \left (2\,a-2\,c\right )}^2\,2{}\mathrm {i}-{\sin \left (c+b\,x\right )}^2\,2{}\mathrm {i}+{\sin \left (2\,a-c+b\,x\right )}^2\,2{}\mathrm {i}+\sin \left (4\,a-4\,c\right )+\sin \left (4\,a-2\,c+2\,b\,x\right )-\sin \left (2\,c+2\,b\,x\right )\right )}{2}}{b\,{\sin \left (a-c\right )}^2} \]

[In]

int(tan(a + b*x)*tan(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)