\(\int (c+d x) \csc ^2(a+b x) \sec ^2(a+b x) \, dx\) [275]

Optimal result

Integrand size = 22, antiderivative size = 35 \[ \int (c+d x) \csc ^2(a+b x) \sec ^2(a+b x) \, dx=-\frac {2 (c+d x) \cot (2 a+2 b x)}{b}+\frac {d \log (\sin (2 a+2 b x))}{b^2} \]

[Out]

-2*(d*x+c)*cot(2*b*x+2*a)/b+d*ln(sin(2*b*x+2*a))/b^2

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4504, 4269, 3556} \[ \int (c+d x) \csc ^2(a+b x) \sec ^2(a+b x) \, dx=\frac {d \log (\sin (2 a+2 b x))}{b^2}-\frac {2 (c+d x) \cot (2 a+2 b x)}{b} \]

[In]

Int[(c + d*x)*Csc[a + b*x]^2*Sec[a + b*x]^2,x]

[Out]

(-2*(c + d*x)*Cot[2*a + 2*b*x])/b + (d*Log[Sin[2*a + 2*b*x]])/b^2

Rule 3556

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

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4504

Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dist[
2^n, Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]

Rubi steps \begin{align*} \text {integral}& = 4 \int (c+d x) \csc ^2(2 a+2 b x) \, dx \\ & = -\frac {2 (c+d x) \cot (2 a+2 b x)}{b}+\frac {(2 d) \int \cot (2 a+2 b x) \, dx}{b} \\ & = -\frac {2 (c+d x) \cot (2 a+2 b x)}{b}+\frac {d \log (\sin (2 a+2 b x))}{b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int (c+d x) \csc ^2(a+b x) \sec ^2(a+b x) \, dx=\frac {-2 b (c+d x) \cot (2 (a+b x))+d \log (\sin (2 (a+b x)))}{b^2} \]

[In]

Integrate[(c + d*x)*Csc[a + b*x]^2*Sec[a + b*x]^2,x]

[Out]

(-2*b*(c + d*x)*Cot[2*(a + b*x)] + d*Log[Sin[2*(a + b*x)]])/b^2

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.52 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.06

method	result	size
risch	\(-\frac {4 i d x}{b}-\frac {4 i d a}{b^{2}}-\frac {4 i \left (d x +c \right )}{b \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}+\frac {d \ln \left ({\mathrm e}^{4 i \left (x b +a \right )}-1\right )}{b^{2}}\)	\(72\)
parallelrisch	\(\frac {-4 \ln \left (\sec \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right ) d \cos \left (x b +a \right )+2 \ln \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )-1\right ) d \cos \left (x b +a \right )+2 \ln \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1\right ) d \cos \left (x b +a \right )+2 \ln \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )\right ) d \cos \left (x b +a \right )-b \cos \left (2 x b +2 a \right ) \csc \left (\frac {a}{2}+\frac {x b}{2}\right ) \sec \left (\frac {a}{2}+\frac {x b}{2}\right ) \left (d x +c \right )}{2 b^{2} \cos \left (x b +a \right )}\)	\(132\)
norman	\(\frac {\frac {c}{2 b}-\frac {3 c \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{b}+\frac {c \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{2 b}+\frac {d x}{2 b}-\frac {3 d x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{b}+\frac {d x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{2 b}}{\tan \left (\frac {a}{2}+\frac {x b}{2}\right ) \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}-1\right )}+\frac {d \ln \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )\right )}{b^{2}}+\frac {d \ln \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )-1\right )}{b^{2}}+\frac {d \ln \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1\right )}{b^{2}}-\frac {2 d \ln \left (1+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right )}{b^{2}}\)	\(182\)

[In]

int((d*x+c)*csc(b*x+a)^2*sec(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

-4*I*d/b*x-4*I*d/b^2*a-4*I*(d*x+c)/b/(exp(2*I*(b*x+a))+1)/(exp(2*I*(b*x+a))-1)+d/b^2*ln(exp(4*I*(b*x+a))-1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (35) = 70\).

Time = 0.26 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.14 \[ \int (c+d x) \csc ^2(a+b x) \sec ^2(a+b x) \, dx=\frac {d \cos \left (b x + a\right ) \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + b d x - 2 \, {\left (b d x + b c\right )} \cos \left (b x + a\right )^{2} + b c}{b^{2} \cos \left (b x + a\right ) \sin \left (b x + a\right )} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral((c + d*x)*csc(a + b*x)**2*sec(a + b*x)**2, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (35) = 70\).

Time = 0.32 (sec) , antiderivative size = 308, normalized size of antiderivative = 8.80 \[ \int (c+d x) \csc ^2(a+b x) \sec ^2(a+b x) \, dx=-\frac {2 \, c {\left (\frac {1}{\tan \left (b x + a\right )} - \tan \left (b x + a\right )\right )} - \frac {2 \, a d {\left (\frac {1}{\tan \left (b x + a\right )} - \tan \left (b x + a\right )\right )}}{b} - \frac {{\left ({\left (\cos \left (4 \, b x + 4 \, a\right )^{2} + \sin \left (4 \, b x + 4 \, a\right )^{2} - 2 \, \cos \left (4 \, b x + 4 \, a\right ) + 1\right )} \log \left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) + {\left (\cos \left (4 \, b x + 4 \, a\right )^{2} + \sin \left (4 \, b x + 4 \, a\right )^{2} - 2 \, \cos \left (4 \, b x + 4 \, a\right ) + 1\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right ) + {\left (\cos \left (4 \, b x + 4 \, a\right )^{2} + \sin \left (4 \, b x + 4 \, a\right )^{2} - 2 \, \cos \left (4 \, b x + 4 \, a\right ) + 1\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right ) - 8 \, {\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right )\right )} d}{{\left (\cos \left (4 \, b x + 4 \, a\right )^{2} + \sin \left (4 \, b x + 4 \, a\right )^{2} - 2 \, \cos \left (4 \, b x + 4 \, a\right ) + 1\right )} b}}{2 \, b} \]

[In]

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

[Out]

-1/2*(2*c*(1/tan(b*x + a) - tan(b*x + a)) - 2*a*d*(1/tan(b*x + a) - tan(b*x + a))/b - ((cos(4*b*x + 4*a)^2 + s
in(4*b*x + 4*a)^2 - 2*cos(4*b*x + 4*a) + 1)*log(cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) +
1) + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 - 2*cos(4*b*x + 4*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 +
2*cos(b*x + a) + 1) + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 - 2*cos(4*b*x + 4*a) + 1)*log(cos(b*x + a)^2 +
sin(b*x + a)^2 - 2*cos(b*x + a) + 1) - 8*(b*x + a)*sin(4*b*x + 4*a))*d/((cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a
)^2 - 2*cos(4*b*x + 4*a) + 1)*b))/b

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 10271 vs. \(2 (35) = 70\).

Time = 2.84 (sec) , antiderivative size = 10271, normalized size of antiderivative = 293.46 \[ \int (c+d x) \csc ^2(a+b x) \sec ^2(a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)*csc(b*x+a)^2*sec(b*x+a)^2,x, algorithm="giac")

[Out]

1/2*(b*d*x*tan(1/2*b*x)^4*tan(1/2*a)^4 + b*c*tan(1/2*b*x)^4*tan(1/2*a)^4 - 6*b*d*x*tan(1/2*b*x)^4*tan(1/2*a)^2
- 16*b*d*x*tan(1/2*b*x)^3*tan(1/2*a)^3 + d*log(64*(tan(1/2*b*x)^8*tan(1/2*a)^6 + 2*tan(1/2*b*x)^7*tan(1/2*a)^
7 + tan(1/2*b*x)^6*tan(1/2*a)^8 - 2*tan(1/2*b*x)^8*tan(1/2*a)^4 - 14*tan(1/2*b*x)^7*tan(1/2*a)^5 - 24*tan(1/2*
b*x)^6*tan(1/2*a)^6 - 14*tan(1/2*b*x)^5*tan(1/2*a)^7 - 2*tan(1/2*b*x)^4*tan(1/2*a)^8 + tan(1/2*b*x)^8*tan(1/2*
a)^2 + 14*tan(1/2*b*x)^7*tan(1/2*a)^3 + 62*tan(1/2*b*x)^6*tan(1/2*a)^4 + 98*tan(1/2*b*x)^5*tan(1/2*a)^5 + 62*t
an(1/2*b*x)^4*tan(1/2*a)^6 + 14*tan(1/2*b*x)^3*tan(1/2*a)^7 + tan(1/2*b*x)^2*tan(1/2*a)^8 - 2*tan(1/2*b*x)^7*t
an(1/2*a) - 24*tan(1/2*b*x)^6*tan(1/2*a)^2 - 98*tan(1/2*b*x)^5*tan(1/2*a)^3 - 152*tan(1/2*b*x)^4*tan(1/2*a)^4
- 98*tan(1/2*b*x)^3*tan(1/2*a)^5 - 24*tan(1/2*b*x)^2*tan(1/2*a)^6 - 2*tan(1/2*b*x)*tan(1/2*a)^7 + tan(1/2*b*x)
^6 + 14*tan(1/2*b*x)^5*tan(1/2*a) + 62*tan(1/2*b*x)^4*tan(1/2*a)^2 + 98*tan(1/2*b*x)^3*tan(1/2*a)^3 + 62*tan(1
/2*b*x)^2*tan(1/2*a)^4 + 14*tan(1/2*b*x)*tan(1/2*a)^5 + tan(1/2*a)^6 - 2*tan(1/2*b*x)^4 - 14*tan(1/2*b*x)^3*ta
n(1/2*a) - 24*tan(1/2*b*x)^2*tan(1/2*a)^2 - 14*tan(1/2*b*x)*tan(1/2*a)^3 - 2*tan(1/2*a)^4 + tan(1/2*b*x)^2 + 2
*tan(1/2*b*x)*tan(1/2*a) + tan(1/2*a)^2)/(tan(1/2*b*x)^8*tan(1/2*a)^8 + 4*tan(1/2*b*x)^8*tan(1/2*a)^6 + 4*tan(
1/2*b*x)^6*tan(1/2*a)^8 + 6*tan(1/2*b*x)^8*tan(1/2*a)^4 + 16*tan(1/2*b*x)^6*tan(1/2*a)^6 + 6*tan(1/2*b*x)^4*ta
n(1/2*a)^8 + 4*tan(1/2*b*x)^8*tan(1/2*a)^2 + 24*tan(1/2*b*x)^6*tan(1/2*a)^4 + 24*tan(1/2*b*x)^4*tan(1/2*a)^6 +
4*tan(1/2*b*x)^2*tan(1/2*a)^8 + tan(1/2*b*x)^8 + 16*tan(1/2*b*x)^6*tan(1/2*a)^2 + 36*tan(1/2*b*x)^4*tan(1/2*a
)^4 + 16*tan(1/2*b*x)^2*tan(1/2*a)^6 + tan(1/2*a)^8 + 4*tan(1/2*b*x)^6 + 24*tan(1/2*b*x)^4*tan(1/2*a)^2 + 24*t
an(1/2*b*x)^2*tan(1/2*a)^4 + 4*tan(1/2*a)^6 + 6*tan(1/2*b*x)^4 + 16*tan(1/2*b*x)^2*tan(1/2*a)^2 + 6*tan(1/2*a)
^4 + 4*tan(1/2*b*x)^2 + 4*tan(1/2*a)^2 + 1))*tan(1/2*b*x)^4*tan(1/2*a)^3 - 6*b*d*x*tan(1/2*b*x)^2*tan(1/2*a)^4
+ d*log(64*(tan(1/2*b*x)^8*tan(1/2*a)^6 + 2*tan(1/2*b*x)^7*tan(1/2*a)^7 + tan(1/2*b*x)^6*tan(1/2*a)^8 - 2*tan
(1/2*b*x)^8*tan(1/2*a)^4 - 14*tan(1/2*b*x)^7*tan(1/2*a)^5 - 24*tan(1/2*b*x)^6*tan(1/2*a)^6 - 14*tan(1/2*b*x)^5
*tan(1/2*a)^7 - 2*tan(1/2*b*x)^4*tan(1/2*a)^8 + tan(1/2*b*x)^8*tan(1/2*a)^2 + 14*tan(1/2*b*x)^7*tan(1/2*a)^3 +
62*tan(1/2*b*x)^6*tan(1/2*a)^4 + 98*tan(1/2*b*x)^5*tan(1/2*a)^5 + 62*tan(1/2*b*x)^4*tan(1/2*a)^6 + 14*tan(1/2
*b*x)^3*tan(1/2*a)^7 + tan(1/2*b*x)^2*tan(1/2*a)^8 - 2*tan(1/2*b*x)^7*tan(1/2*a) - 24*tan(1/2*b*x)^6*tan(1/2*a
)^2 - 98*tan(1/2*b*x)^5*tan(1/2*a)^3 - 152*tan(1/2*b*x)^4*tan(1/2*a)^4 - 98*tan(1/2*b*x)^3*tan(1/2*a)^5 - 24*t
an(1/2*b*x)^2*tan(1/2*a)^6 - 2*tan(1/2*b*x)*tan(1/2*a)^7 + tan(1/2*b*x)^6 + 14*tan(1/2*b*x)^5*tan(1/2*a) + 62*
tan(1/2*b*x)^4*tan(1/2*a)^2 + 98*tan(1/2*b*x)^3*tan(1/2*a)^3 + 62*tan(1/2*b*x)^2*tan(1/2*a)^4 + 14*tan(1/2*b*x
)*tan(1/2*a)^5 + tan(1/2*a)^6 - 2*tan(1/2*b*x)^4 - 14*tan(1/2*b*x)^3*tan(1/2*a) - 24*tan(1/2*b*x)^2*tan(1/2*a)
^2 - 14*tan(1/2*b*x)*tan(1/2*a)^3 - 2*tan(1/2*a)^4 + tan(1/2*b*x)^2 + 2*tan(1/2*b*x)*tan(1/2*a) + tan(1/2*a)^2
)/(tan(1/2*b*x)^8*tan(1/2*a)^8 + 4*tan(1/2*b*x)^8*tan(1/2*a)^6 + 4*tan(1/2*b*x)^6*tan(1/2*a)^8 + 6*tan(1/2*b*x
)^8*tan(1/2*a)^4 + 16*tan(1/2*b*x)^6*tan(1/2*a)^6 + 6*tan(1/2*b*x)^4*tan(1/2*a)^8 + 4*tan(1/2*b*x)^8*tan(1/2*a
)^2 + 24*tan(1/2*b*x)^6*tan(1/2*a)^4 + 24*tan(1/2*b*x)^4*tan(1/2*a)^6 + 4*tan(1/2*b*x)^2*tan(1/2*a)^8 + tan(1/
2*b*x)^8 + 16*tan(1/2*b*x)^6*tan(1/2*a)^2 + 36*tan(1/2*b*x)^4*tan(1/2*a)^4 + 16*tan(1/2*b*x)^2*tan(1/2*a)^6 +
tan(1/2*a)^8 + 4*tan(1/2*b*x)^6 + 24*tan(1/2*b*x)^4*tan(1/2*a)^2 + 24*tan(1/2*b*x)^2*tan(1/2*a)^4 + 4*tan(1/2*
a)^6 + 6*tan(1/2*b*x)^4 + 16*tan(1/2*b*x)^2*tan(1/2*a)^2 + 6*tan(1/2*a)^4 + 4*tan(1/2*b*x)^2 + 4*tan(1/2*a)^2
+ 1))*tan(1/2*b*x)^3*tan(1/2*a)^4 - 6*b*c*tan(1/2*b*x)^4*tan(1/2*a)^2 - 16*b*c*tan(1/2*b*x)^3*tan(1/2*a)^3 - 6
*b*c*tan(1/2*b*x)^2*tan(1/2*a)^4 + b*d*x*tan(1/2*b*x)^4 + 16*b*d*x*tan(1/2*b*x)^3*tan(1/2*a) - d*log(64*(tan(1
/2*b*x)^8*tan(1/2*a)^6 + 2*tan(1/2*b*x)^7*tan(1/2*a)^7 + tan(1/2*b*x)^6*tan(1/2*a)^8 - 2*tan(1/2*b*x)^8*tan(1/
2*a)^4 - 14*tan(1/2*b*x)^7*tan(1/2*a)^5 - 24*tan(1/2*b*x)^6*tan(1/2*a)^6 - 14*tan(1/2*b*x)^5*tan(1/2*a)^7 - 2*
tan(1/2*b*x)^4*tan(1/2*a)^8 + tan(1/2*b*x)^8*tan(1/2*a)^2 + 14*tan(1/2*b*x)^7*tan(1/2*a)^3 + 62*tan(1/2*b*x)^6
*tan(1/2*a)^4 + 98*tan(1/2*b*x)^5*tan(1/2*a)^5 + 62*tan(1/2*b*x)^4*tan(1/2*a)^6 + 14*tan(1/2*b*x)^3*tan(1/2*a)
^7 + tan(1/2*b*x)^2*tan(1/2*a)^8 - 2*tan(1/2*b*x)^7*tan(1/2*a) - 24*tan(1/2*b*x)^6*tan(1/2*a)^2 - 98*tan(1/2*b
*x)^5*tan(1/2*a)^3 - 152*tan(1/2*b*x)^4*tan(1/2*a)^4 - 98*tan(1/2*b*x)^3*tan(1/2*a)^5 - 24*tan(1/2*b*x)^2*tan(
1/2*a)^6 - 2*tan(1/2*b*x)*tan(1/2*a)^7 + tan(1/2*b*x)^6 + 14*tan(1/2*b*x)^5*tan(1/2*a) + 62*tan(1/2*b*x)^4*tan
(1/2*a)^2 + 98*tan(1/2*b*x)^3*tan(1/2*a)^3 + 62*tan(1/2*b*x)^2*tan(1/2*a)^4 + 14*tan(1/2*b*x)*tan(1/2*a)^5 + t
an(1/2*a)^6 - 2*tan(1/2*b*x)^4 - 14*tan(1/2*b*x)^3*tan(1/2*a) - 24*tan(1/2*b*x)^2*tan(1/2*a)^2 - 14*tan(1/2*b*
x)*tan(1/2*a)^3 - 2*tan(1/2*a)^4 + tan(1/2*b*x)^2 + 2*tan(1/2*b*x)*tan(1/2*a) + tan(1/2*a)^2)/(tan(1/2*b*x)^8*
tan(1/2*a)^8 + 4*tan(1/2*b*x)^8*tan(1/2*a)^6 + 4*tan(1/2*b*x)^6*tan(1/2*a)^8 + 6*tan(1/2*b*x)^8*tan(1/2*a)^4 +
16*tan(1/2*b*x)^6*tan(1/2*a)^6 + 6*tan(1/2*b*x)^4*tan(1/2*a)^8 + 4*tan(1/2*b*x)^8*tan(1/2*a)^2 + 24*tan(1/2*b
*x)^6*tan(1/2*a)^4 + 24*tan(1/2*b*x)^4*tan(1/2*a)^6 + 4*tan(1/2*b*x)^2*tan(1/2*a)^8 + tan(1/2*b*x)^8 + 16*tan(
1/2*b*x)^6*tan(1/2*a)^2 + 36*tan(1/2*b*x)^4*tan(1/2*a)^4 + 16*tan(1/2*b*x)^2*tan(1/2*a)^6 + tan(1/2*a)^8 + 4*t
an(1/2*b*x)^6 + 24*tan(1/2*b*x)^4*tan(1/2*a)^2 + 24*tan(1/2*b*x)^2*tan(1/2*a)^4 + 4*tan(1/2*a)^6 + 6*tan(1/2*b
*x)^4 + 16*tan(1/2*b*x)^2*tan(1/2*a)^2 + 6*tan(1/2*a)^4 + 4*tan(1/2*b*x)^2 + 4*tan(1/2*a)^2 + 1))*tan(1/2*b*x)
^4*tan(1/2*a) + 36*b*d*x*tan(1/2*b*x)^2*tan(1/2*a)^2 - 6*d*log(64*(tan(1/2*b*x)^8*tan(1/2*a)^6 + 2*tan(1/2*b*x
)^7*tan(1/2*a)^7 + tan(1/2*b*x)^6*tan(1/2*a)^8 - 2*tan(1/2*b*x)^8*tan(1/2*a)^4 - 14*tan(1/2*b*x)^7*tan(1/2*a)^
5 - 24*tan(1/2*b*x)^6*tan(1/2*a)^6 - 14*tan(1/2*b*x)^5*tan(1/2*a)^7 - 2*tan(1/2*b*x)^4*tan(1/2*a)^8 + tan(1/2*
b*x)^8*tan(1/2*a)^2 + 14*tan(1/2*b*x)^7*tan(1/2*a)^3 + 62*tan(1/2*b*x)^6*tan(1/2*a)^4 + 98*tan(1/2*b*x)^5*tan(
1/2*a)^5 + 62*tan(1/2*b*x)^4*tan(1/2*a)^6 + 14*tan(1/2*b*x)^3*tan(1/2*a)^7 + tan(1/2*b*x)^2*tan(1/2*a)^8 - 2*t
an(1/2*b*x)^7*tan(1/2*a) - 24*tan(1/2*b*x)^6*tan(1/2*a)^2 - 98*tan(1/2*b*x)^5*tan(1/2*a)^3 - 152*tan(1/2*b*x)^
4*tan(1/2*a)^4 - 98*tan(1/2*b*x)^3*tan(1/2*a)^5 - 24*tan(1/2*b*x)^2*tan(1/2*a)^6 - 2*tan(1/2*b*x)*tan(1/2*a)^7
+ tan(1/2*b*x)^6 + 14*tan(1/2*b*x)^5*tan(1/2*a) + 62*tan(1/2*b*x)^4*tan(1/2*a)^2 + 98*tan(1/2*b*x)^3*tan(1/2*
a)^3 + 62*tan(1/2*b*x)^2*tan(1/2*a)^4 + 14*tan(1/2*b*x)*tan(1/2*a)^5 + tan(1/2*a)^6 - 2*tan(1/2*b*x)^4 - 14*ta
n(1/2*b*x)^3*tan(1/2*a) - 24*tan(1/2*b*x)^2*tan(1/2*a)^2 - 14*tan(1/2*b*x)*tan(1/2*a)^3 - 2*tan(1/2*a)^4 + tan
(1/2*b*x)^2 + 2*tan(1/2*b*x)*tan(1/2*a) + tan(1/2*a)^2)/(tan(1/2*b*x)^8*tan(1/2*a)^8 + 4*tan(1/2*b*x)^8*tan(1/
2*a)^6 + 4*tan(1/2*b*x)^6*tan(1/2*a)^8 + 6*tan(1/2*b*x)^8*tan(1/2*a)^4 + 16*tan(1/2*b*x)^6*tan(1/2*a)^6 + 6*ta
n(1/2*b*x)^4*tan(1/2*a)^8 + 4*tan(1/2*b*x)^8*tan(1/2*a)^2 + 24*tan(1/2*b*x)^6*tan(1/2*a)^4 + 24*tan(1/2*b*x)^4
*tan(1/2*a)^6 + 4*tan(1/2*b*x)^2*tan(1/2*a)^8 + tan(1/2*b*x)^8 + 16*tan(1/2*b*x)^6*tan(1/2*a)^2 + 36*tan(1/2*b
*x)^4*tan(1/2*a)^4 + 16*tan(1/2*b*x)^2*tan(1/2*a)^6 + tan(1/2*a)^8 + 4*tan(1/2*b*x)^6 + 24*tan(1/2*b*x)^4*tan(
1/2*a)^2 + 24*tan(1/2*b*x)^2*tan(1/2*a)^4 + 4*tan(1/2*a)^6 + 6*tan(1/2*b*x)^4 + 16*tan(1/2*b*x)^2*tan(1/2*a)^2
+ 6*tan(1/2*a)^4 + 4*tan(1/2*b*x)^2 + 4*tan(1/2*a)^2 + 1))*tan(1/2*b*x)^3*tan(1/2*a)^2 + 16*b*d*x*tan(1/2*b*x
)*tan(1/2*a)^3 - 6*d*log(64*(tan(1/2*b*x)^8*tan(1/2*a)^6 + 2*tan(1/2*b*x)^7*tan(1/2*a)^7 + tan(1/2*b*x)^6*tan(
1/2*a)^8 - 2*tan(1/2*b*x)^8*tan(1/2*a)^4 - 14*tan(1/2*b*x)^7*tan(1/2*a)^5 - 24*tan(1/2*b*x)^6*tan(1/2*a)^6 - 1
4*tan(1/2*b*x)^5*tan(1/2*a)^7 - 2*tan(1/2*b*x)^4*tan(1/2*a)^8 + tan(1/2*b*x)^8*tan(1/2*a)^2 + 14*tan(1/2*b*x)^
7*tan(1/2*a)^3 + 62*tan(1/2*b*x)^6*tan(1/2*a)^4 + 98*tan(1/2*b*x)^5*tan(1/2*a)^5 + 62*tan(1/2*b*x)^4*tan(1/2*a
)^6 + 14*tan(1/2*b*x)^3*tan(1/2*a)^7 + tan(1/2*b*x)^2*tan(1/2*a)^8 - 2*tan(1/2*b*x)^7*tan(1/2*a) - 24*tan(1/2*
b*x)^6*tan(1/2*a)^2 - 98*tan(1/2*b*x)^5*tan(1/2*a)^3 - 152*tan(1/2*b*x)^4*tan(1/2*a)^4 - 98*tan(1/2*b*x)^3*tan
(1/2*a)^5 - 24*tan(1/2*b*x)^2*tan(1/2*a)^6 - 2*tan(1/2*b*x)*tan(1/2*a)^7 + tan(1/2*b*x)^6 + 14*tan(1/2*b*x)^5*
tan(1/2*a) + 62*tan(1/2*b*x)^4*tan(1/2*a)^2 + 98*tan(1/2*b*x)^3*tan(1/2*a)^3 + 62*tan(1/2*b*x)^2*tan(1/2*a)^4
+ 14*tan(1/2*b*x)*tan(1/2*a)^5 + tan(1/2*a)^6 - 2*tan(1/2*b*x)^4 - 14*tan(1/2*b*x)^3*tan(1/2*a) - 24*tan(1/2*b
*x)^2*tan(1/2*a)^2 - 14*tan(1/2*b*x)*tan(1/2*a)^3 - 2*tan(1/2*a)^4 + tan(1/2*b*x)^2 + 2*tan(1/2*b*x)*tan(1/2*a
) + tan(1/2*a)^2)/(tan(1/2*b*x)^8*tan(1/2*a)^8 + 4*tan(1/2*b*x)^8*tan(1/2*a)^6 + 4*tan(1/2*b*x)^6*tan(1/2*a)^8
+ 6*tan(1/2*b*x)^8*tan(1/2*a)^4 + 16*tan(1/2*b*x)^6*tan(1/2*a)^6 + 6*tan(1/2*b*x)^4*tan(1/2*a)^8 + 4*tan(1/2*
b*x)^8*tan(1/2*a)^2 + 24*tan(1/2*b*x)^6*tan(1/2*a)^4 + 24*tan(1/2*b*x)^4*tan(1/2*a)^6 + 4*tan(1/2*b*x)^2*tan(1
/2*a)^8 + tan(1/2*b*x)^8 + 16*tan(1/2*b*x)^6*tan(1/2*a)^2 + 36*tan(1/2*b*x)^4*tan(1/2*a)^4 + 16*tan(1/2*b*x)^2
*tan(1/2*a)^6 + tan(1/2*a)^8 + 4*tan(1/2*b*x)^6 + 24*tan(1/2*b*x)^4*tan(1/2*a)^2 + 24*tan(1/2*b*x)^2*tan(1/2*a
)^4 + 4*tan(1/2*a)^6 + 6*tan(1/2*b*x)^4 + 16*tan(1/2*b*x)^2*tan(1/2*a)^2 + 6*tan(1/2*a)^4 + 4*tan(1/2*b*x)^2 +
4*tan(1/2*a)^2 + 1))*tan(1/2*b*x)^2*tan(1/2*a)^3 + b*d*x*tan(1/2*a)^4 - d*log(64*(tan(1/2*b*x)^8*tan(1/2*a)^6
+ 2*tan(1/2*b*x)^7*tan(1/2*a)^7 + tan(1/2*b*x)^6*tan(1/2*a)^8 - 2*tan(1/2*b*x)^8*tan(1/2*a)^4 - 14*tan(1/2*b*
x)^7*tan(1/2*a)^5 - 24*tan(1/2*b*x)^6*tan(1/2*a)^6 - 14*tan(1/2*b*x)^5*tan(1/2*a)^7 - 2*tan(1/2*b*x)^4*tan(1/2
*a)^8 + tan(1/2*b*x)^8*tan(1/2*a)^2 + 14*tan(1/2*b*x)^7*tan(1/2*a)^3 + 62*tan(1/2*b*x)^6*tan(1/2*a)^4 + 98*tan
(1/2*b*x)^5*tan(1/2*a)^5 + 62*tan(1/2*b*x)^4*tan(1/2*a)^6 + 14*tan(1/2*b*x)^3*tan(1/2*a)^7 + tan(1/2*b*x)^2*ta
n(1/2*a)^8 - 2*tan(1/2*b*x)^7*tan(1/2*a) - 24*tan(1/2*b*x)^6*tan(1/2*a)^2 - 98*tan(1/2*b*x)^5*tan(1/2*a)^3 - 1
52*tan(1/2*b*x)^4*tan(1/2*a)^4 - 98*tan(1/2*b*x)^3*tan(1/2*a)^5 - 24*tan(1/2*b*x)^2*tan(1/2*a)^6 - 2*tan(1/2*b
*x)*tan(1/2*a)^7 + tan(1/2*b*x)^6 + 14*tan(1/2*b*x)^5*tan(1/2*a) + 62*tan(1/2*b*x)^4*tan(1/2*a)^2 + 98*tan(1/2
*b*x)^3*tan(1/2*a)^3 + 62*tan(1/2*b*x)^2*tan(1/2*a)^4 + 14*tan(1/2*b*x)*tan(1/2*a)^5 + tan(1/2*a)^6 - 2*tan(1/
2*b*x)^4 - 14*tan(1/2*b*x)^3*tan(1/2*a) - 24*tan(1/2*b*x)^2*tan(1/2*a)^2 - 14*tan(1/2*b*x)*tan(1/2*a)^3 - 2*ta
n(1/2*a)^4 + tan(1/2*b*x)^2 + 2*tan(1/2*b*x)*tan(1/2*a) + tan(1/2*a)^2)/(tan(1/2*b*x)^8*tan(1/2*a)^8 + 4*tan(1
/2*b*x)^8*tan(1/2*a)^6 + 4*tan(1/2*b*x)^6*tan(1/2*a)^8 + 6*tan(1/2*b*x)^8*tan(1/2*a)^4 + 16*tan(1/2*b*x)^6*tan
(1/2*a)^6 + 6*tan(1/2*b*x)^4*tan(1/2*a)^8 + 4*tan(1/2*b*x)^8*tan(1/2*a)^2 + 24*tan(1/2*b*x)^6*tan(1/2*a)^4 + 2
4*tan(1/2*b*x)^4*tan(1/2*a)^6 + 4*tan(1/2*b*x)^2*tan(1/2*a)^8 + tan(1/2*b*x)^8 + 16*tan(1/2*b*x)^6*tan(1/2*a)^
2 + 36*tan(1/2*b*x)^4*tan(1/2*a)^4 + 16*tan(1/2*b*x)^2*tan(1/2*a)^6 + tan(1/2*a)^8 + 4*tan(1/2*b*x)^6 + 24*tan
(1/2*b*x)^4*tan(1/2*a)^2 + 24*tan(1/2*b*x)^2*tan(1/2*a)^4 + 4*tan(1/2*a)^6 + 6*tan(1/2*b*x)^4 + 16*tan(1/2*b*x
)^2*tan(1/2*a)^2 + 6*tan(1/2*a)^4 + 4*tan(1/2*b*x)^2 + 4*tan(1/2*a)^2 + 1))*tan(1/2*b*x)*tan(1/2*a)^4 + b*c*ta
n(1/2*b*x)^4 + 16*b*c*tan(1/2*b*x)^3*tan(1/2*a) + 36*b*c*tan(1/2*b*x)^2*tan(1/2*a)^2 + 16*b*c*tan(1/2*b*x)*tan
(1/2*a)^3 + b*c*tan(1/2*a)^4 - 6*b*d*x*tan(1/2*b*x)^2 + d*log(64*(tan(1/2*b*x)^8*tan(1/2*a)^6 + 2*tan(1/2*b*x)
^7*tan(1/2*a)^7 + tan(1/2*b*x)^6*tan(1/2*a)^8 - 2*tan(1/2*b*x)^8*tan(1/2*a)^4 - 14*tan(1/2*b*x)^7*tan(1/2*a)^5
- 24*tan(1/2*b*x)^6*tan(1/2*a)^6 - 14*tan(1/2*b*x)^5*tan(1/2*a)^7 - 2*tan(1/2*b*x)^4*tan(1/2*a)^8 + tan(1/2*b
*x)^8*tan(1/2*a)^2 + 14*tan(1/2*b*x)^7*tan(1/2*a)^3 + 62*tan(1/2*b*x)^6*tan(1/2*a)^4 + 98*tan(1/2*b*x)^5*tan(1
/2*a)^5 + 62*tan(1/2*b*x)^4*tan(1/2*a)^6 + 14*tan(1/2*b*x)^3*tan(1/2*a)^7 + tan(1/2*b*x)^2*tan(1/2*a)^8 - 2*ta
n(1/2*b*x)^7*tan(1/2*a) - 24*tan(1/2*b*x)^6*tan(1/2*a)^2 - 98*tan(1/2*b*x)^5*tan(1/2*a)^3 - 152*tan(1/2*b*x)^4
*tan(1/2*a)^4 - 98*tan(1/2*b*x)^3*tan(1/2*a)^5 - 24*tan(1/2*b*x)^2*tan(1/2*a)^6 - 2*tan(1/2*b*x)*tan(1/2*a)^7
+ tan(1/2*b*x)^6 + 14*tan(1/2*b*x)^5*tan(1/2*a) + 62*tan(1/2*b*x)^4*tan(1/2*a)^2 + 98*tan(1/2*b*x)^3*tan(1/2*a
)^3 + 62*tan(1/2*b*x)^2*tan(1/2*a)^4 + 14*tan(1/2*b*x)*tan(1/2*a)^5 + tan(1/2*a)^6 - 2*tan(1/2*b*x)^4 - 14*tan
(1/2*b*x)^3*tan(1/2*a) - 24*tan(1/2*b*x)^2*tan(1/2*a)^2 - 14*tan(1/2*b*x)*tan(1/2*a)^3 - 2*tan(1/2*a)^4 + tan(
1/2*b*x)^2 + 2*tan(1/2*b*x)*tan(1/2*a) + tan(1/2*a)^2)/(tan(1/2*b*x)^8*tan(1/2*a)^8 + 4*tan(1/2*b*x)^8*tan(1/2
*a)^6 + 4*tan(1/2*b*x)^6*tan(1/2*a)^8 + 6*tan(1/2*b*x)^8*tan(1/2*a)^4 + 16*tan(1/2*b*x)^6*tan(1/2*a)^6 + 6*tan
(1/2*b*x)^4*tan(1/2*a)^8 + 4*tan(1/2*b*x)^8*tan(1/2*a)^2 + 24*tan(1/2*b*x)^6*tan(1/2*a)^4 + 24*tan(1/2*b*x)^4*
tan(1/2*a)^6 + 4*tan(1/2*b*x)^2*tan(1/2*a)^8 + tan(1/2*b*x)^8 + 16*tan(1/2*b*x)^6*tan(1/2*a)^2 + 36*tan(1/2*b*
x)^4*tan(1/2*a)^4 + 16*tan(1/2*b*x)^2*tan(1/2*a)^6 + tan(1/2*a)^8 + 4*tan(1/2*b*x)^6 + 24*tan(1/2*b*x)^4*tan(1
/2*a)^2 + 24*tan(1/2*b*x)^2*tan(1/2*a)^4 + 4*tan(1/2*a)^6 + 6*tan(1/2*b*x)^4 + 16*tan(1/2*b*x)^2*tan(1/2*a)^2
+ 6*tan(1/2*a)^4 + 4*tan(1/2*b*x)^2 + 4*tan(1/2*a)^2 + 1))*tan(1/2*b*x)^3 - 16*b*d*x*tan(1/2*b*x)*tan(1/2*a) +
6*d*log(64*(tan(1/2*b*x)^8*tan(1/2*a)^6 + 2*tan(1/2*b*x)^7*tan(1/2*a)^7 + tan(1/2*b*x)^6*tan(1/2*a)^8 - 2*tan
(1/2*b*x)^8*tan(1/2*a)^4 - 14*tan(1/2*b*x)^7*tan(1/2*a)^5 - 24*tan(1/2*b*x)^6*tan(1/2*a)^6 - 14*tan(1/2*b*x)^5
*tan(1/2*a)^7 - 2*tan(1/2*b*x)^4*tan(1/2*a)^8 + tan(1/2*b*x)^8*tan(1/2*a)^2 + 14*tan(1/2*b*x)^7*tan(1/2*a)^3 +
62*tan(1/2*b*x)^6*tan(1/2*a)^4 + 98*tan(1/2*b*x)^5*tan(1/2*a)^5 + 62*tan(1/2*b*x)^4*tan(1/2*a)^6 + 14*tan(1/2
*b*x)^3*tan(1/2*a)^7 + tan(1/2*b*x)^2*tan(1/2*a)^8 - 2*tan(1/2*b*x)^7*tan(1/2*a) - 24*tan(1/2*b*x)^6*tan(1/2*a
)^2 - 98*tan(1/2*b*x)^5*tan(1/2*a)^3 - 152*tan(1/2*b*x)^4*tan(1/2*a)^4 - 98*tan(1/2*b*x)^3*tan(1/2*a)^5 - 24*t
an(1/2*b*x)^2*tan(1/2*a)^6 - 2*tan(1/2*b*x)*tan(1/2*a)^7 + tan(1/2*b*x)^6 + 14*tan(1/2*b*x)^5*tan(1/2*a) + 62*
tan(1/2*b*x)^4*tan(1/2*a)^2 + 98*tan(1/2*b*x)^3*tan(1/2*a)^3 + 62*tan(1/2*b*x)^2*tan(1/2*a)^4 + 14*tan(1/2*b*x
)*tan(1/2*a)^5 + tan(1/2*a)^6 - 2*tan(1/2*b*x)^4 - 14*tan(1/2*b*x)^3*tan(1/2*a) - 24*tan(1/2*b*x)^2*tan(1/2*a)
^2 - 14*tan(1/2*b*x)*tan(1/2*a)^3 - 2*tan(1/2*a)^4 + tan(1/2*b*x)^2 + 2*tan(1/2*b*x)*tan(1/2*a) + tan(1/2*a)^2
)/(tan(1/2*b*x)^8*tan(1/2*a)^8 + 4*tan(1/2*b*x)^8*tan(1/2*a)^6 + 4*tan(1/2*b*x)^6*tan(1/2*a)^8 + 6*tan(1/2*b*x
)^8*tan(1/2*a)^4 + 16*tan(1/2*b*x)^6*tan(1/2*a)^6 + 6*tan(1/2*b*x)^4*tan(1/2*a)^8 + 4*tan(1/2*b*x)^8*tan(1/2*a
)^2 + 24*tan(1/2*b*x)^6*tan(1/2*a)^4 + 24*tan(1/2*b*x)^4*tan(1/2*a)^6 + 4*tan(1/2*b*x)^2*tan(1/2*a)^8 + tan(1/
2*b*x)^8 + 16*tan(1/2*b*x)^6*tan(1/2*a)^2 + 36*tan(1/2*b*x)^4*tan(1/2*a)^4 + 16*tan(1/2*b*x)^2*tan(1/2*a)^6 +
tan(1/2*a)^8 + 4*tan(1/2*b*x)^6 + 24*tan(1/2*b*x)^4*tan(1/2*a)^2 + 24*tan(1/2*b*x)^2*tan(1/2*a)^4 + 4*tan(1/2*
a)^6 + 6*tan(1/2*b*x)^4 + 16*tan(1/2*b*x)^2*tan(1/2*a)^2 + 6*tan(1/2*a)^4 + 4*tan(1/2*b*x)^2 + 4*tan(1/2*a)^2
+ 1))*tan(1/2*b*x)^2*tan(1/2*a) - 6*b*d*x*tan(1/2*a)^2 + 6*d*log(64*(tan(1/2*b*x)^8*tan(1/2*a)^6 + 2*tan(1/2*b
*x)^7*tan(1/2*a)^7 + tan(1/2*b*x)^6*tan(1/2*a)^8 - 2*tan(1/2*b*x)^8*tan(1/2*a)^4 - 14*tan(1/2*b*x)^7*tan(1/2*a
)^5 - 24*tan(1/2*b*x)^6*tan(1/2*a)^6 - 14*tan(1/2*b*x)^5*tan(1/2*a)^7 - 2*tan(1/2*b*x)^4*tan(1/2*a)^8 + tan(1/
2*b*x)^8*tan(1/2*a)^2 + 14*tan(1/2*b*x)^7*tan(1/2*a)^3 + 62*tan(1/2*b*x)^6*tan(1/2*a)^4 + 98*tan(1/2*b*x)^5*ta
n(1/2*a)^5 + 62*tan(1/2*b*x)^4*tan(1/2*a)^6 + 14*tan(1/2*b*x)^3*tan(1/2*a)^7 + tan(1/2*b*x)^2*tan(1/2*a)^8 - 2
*tan(1/2*b*x)^7*tan(1/2*a) - 24*tan(1/2*b*x)^6*tan(1/2*a)^2 - 98*tan(1/2*b*x)^5*tan(1/2*a)^3 - 152*tan(1/2*b*x
)^4*tan(1/2*a)^4 - 98*tan(1/2*b*x)^3*tan(1/2*a)^5 - 24*tan(1/2*b*x)^2*tan(1/2*a)^6 - 2*tan(1/2*b*x)*tan(1/2*a)
^7 + tan(1/2*b*x)^6 + 14*tan(1/2*b*x)^5*tan(1/2*a) + 62*tan(1/2*b*x)^4*tan(1/2*a)^2 + 98*tan(1/2*b*x)^3*tan(1/
2*a)^3 + 62*tan(1/2*b*x)^2*tan(1/2*a)^4 + 14*tan(1/2*b*x)*tan(1/2*a)^5 + tan(1/2*a)^6 - 2*tan(1/2*b*x)^4 - 14*
tan(1/2*b*x)^3*tan(1/2*a) - 24*tan(1/2*b*x)^2*tan(1/2*a)^2 - 14*tan(1/2*b*x)*tan(1/2*a)^3 - 2*tan(1/2*a)^4 + t
an(1/2*b*x)^2 + 2*tan(1/2*b*x)*tan(1/2*a) + tan(1/2*a)^2)/(tan(1/2*b*x)^8*tan(1/2*a)^8 + 4*tan(1/2*b*x)^8*tan(
1/2*a)^6 + 4*tan(1/2*b*x)^6*tan(1/2*a)^8 + 6*tan(1/2*b*x)^8*tan(1/2*a)^4 + 16*tan(1/2*b*x)^6*tan(1/2*a)^6 + 6*
tan(1/2*b*x)^4*tan(1/2*a)^8 + 4*tan(1/2*b*x)^8*tan(1/2*a)^2 + 24*tan(1/2*b*x)^6*tan(1/2*a)^4 + 24*tan(1/2*b*x)
^4*tan(1/2*a)^6 + 4*tan(1/2*b*x)^2*tan(1/2*a)^8 + tan(1/2*b*x)^8 + 16*tan(1/2*b*x)^6*tan(1/2*a)^2 + 36*tan(1/2
*b*x)^4*tan(1/2*a)^4 + 16*tan(1/2*b*x)^2*tan(1/2*a)^6 + tan(1/2*a)^8 + 4*tan(1/2*b*x)^6 + 24*tan(1/2*b*x)^4*ta
n(1/2*a)^2 + 24*tan(1/2*b*x)^2*tan(1/2*a)^4 + 4*tan(1/2*a)^6 + 6*tan(1/2*b*x)^4 + 16*tan(1/2*b*x)^2*tan(1/2*a)
^2 + 6*tan(1/2*a)^4 + 4*tan(1/2*b*x)^2 + 4*tan(1/2*a)^2 + 1))*tan(1/2*b*x)*tan(1/2*a)^2 + d*log(64*(tan(1/2*b*
x)^8*tan(1/2*a)^6 + 2*tan(1/2*b*x)^7*tan(1/2*a)^7 + tan(1/2*b*x)^6*tan(1/2*a)^8 - 2*tan(1/2*b*x)^8*tan(1/2*a)^
4 - 14*tan(1/2*b*x)^7*tan(1/2*a)^5 - 24*tan(1/2*b*x)^6*tan(1/2*a)^6 - 14*tan(1/2*b*x)^5*tan(1/2*a)^7 - 2*tan(1
/2*b*x)^4*tan(1/2*a)^8 + tan(1/2*b*x)^8*tan(1/2*a)^2 + 14*tan(1/2*b*x)^7*tan(1/2*a)^3 + 62*tan(1/2*b*x)^6*tan(
1/2*a)^4 + 98*tan(1/2*b*x)^5*tan(1/2*a)^5 + 62*tan(1/2*b*x)^4*tan(1/2*a)^6 + 14*tan(1/2*b*x)^3*tan(1/2*a)^7 +
tan(1/2*b*x)^2*tan(1/2*a)^8 - 2*tan(1/2*b*x)^7*tan(1/2*a) - 24*tan(1/2*b*x)^6*tan(1/2*a)^2 - 98*tan(1/2*b*x)^5
*tan(1/2*a)^3 - 152*tan(1/2*b*x)^4*tan(1/2*a)^4 - 98*tan(1/2*b*x)^3*tan(1/2*a)^5 - 24*tan(1/2*b*x)^2*tan(1/2*a
)^6 - 2*tan(1/2*b*x)*tan(1/2*a)^7 + tan(1/2*b*x)^6 + 14*tan(1/2*b*x)^5*tan(1/2*a) + 62*tan(1/2*b*x)^4*tan(1/2*
a)^2 + 98*tan(1/2*b*x)^3*tan(1/2*a)^3 + 62*tan(1/2*b*x)^2*tan(1/2*a)^4 + 14*tan(1/2*b*x)*tan(1/2*a)^5 + tan(1/
2*a)^6 - 2*tan(1/2*b*x)^4 - 14*tan(1/2*b*x)^3*tan(1/2*a) - 24*tan(1/2*b*x)^2*tan(1/2*a)^2 - 14*tan(1/2*b*x)*ta
n(1/2*a)^3 - 2*tan(1/2*a)^4 + tan(1/2*b*x)^2 + 2*tan(1/2*b*x)*tan(1/2*a) + tan(1/2*a)^2)/(tan(1/2*b*x)^8*tan(1
/2*a)^8 + 4*tan(1/2*b*x)^8*tan(1/2*a)^6 + 4*tan(1/2*b*x)^6*tan(1/2*a)^8 + 6*tan(1/2*b*x)^8*tan(1/2*a)^4 + 16*t
an(1/2*b*x)^6*tan(1/2*a)^6 + 6*tan(1/2*b*x)^4*tan(1/2*a)^8 + 4*tan(1/2*b*x)^8*tan(1/2*a)^2 + 24*tan(1/2*b*x)^6
*tan(1/2*a)^4 + 24*tan(1/2*b*x)^4*tan(1/2*a)^6 + 4*tan(1/2*b*x)^2*tan(1/2*a)^8 + tan(1/2*b*x)^8 + 16*tan(1/2*b
*x)^6*tan(1/2*a)^2 + 36*tan(1/2*b*x)^4*tan(1/2*a)^4 + 16*tan(1/2*b*x)^2*tan(1/2*a)^6 + tan(1/2*a)^8 + 4*tan(1/
2*b*x)^6 + 24*tan(1/2*b*x)^4*tan(1/2*a)^2 + 24*tan(1/2*b*x)^2*tan(1/2*a)^4 + 4*tan(1/2*a)^6 + 6*tan(1/2*b*x)^4
+ 16*tan(1/2*b*x)^2*tan(1/2*a)^2 + 6*tan(1/2*a)^4 + 4*tan(1/2*b*x)^2 + 4*tan(1/2*a)^2 + 1))*tan(1/2*a)^3 - 6*
b*c*tan(1/2*b*x)^2 - 16*b*c*tan(1/2*b*x)*tan(1/2*a) - 6*b*c*tan(1/2*a)^2 + b*d*x - d*log(64*(tan(1/2*b*x)^8*ta
n(1/2*a)^6 + 2*tan(1/2*b*x)^7*tan(1/2*a)^7 + tan(1/2*b*x)^6*tan(1/2*a)^8 - 2*tan(1/2*b*x)^8*tan(1/2*a)^4 - 14*
tan(1/2*b*x)^7*tan(1/2*a)^5 - 24*tan(1/2*b*x)^6*tan(1/2*a)^6 - 14*tan(1/2*b*x)^5*tan(1/2*a)^7 - 2*tan(1/2*b*x)
^4*tan(1/2*a)^8 + tan(1/2*b*x)^8*tan(1/2*a)^2 + 14*tan(1/2*b*x)^7*tan(1/2*a)^3 + 62*tan(1/2*b*x)^6*tan(1/2*a)^
4 + 98*tan(1/2*b*x)^5*tan(1/2*a)^5 + 62*tan(1/2*b*x)^4*tan(1/2*a)^6 + 14*tan(1/2*b*x)^3*tan(1/2*a)^7 + tan(1/2
*b*x)^2*tan(1/2*a)^8 - 2*tan(1/2*b*x)^7*tan(1/2*a) - 24*tan(1/2*b*x)^6*tan(1/2*a)^2 - 98*tan(1/2*b*x)^5*tan(1/
2*a)^3 - 152*tan(1/2*b*x)^4*tan(1/2*a)^4 - 98*tan(1/2*b*x)^3*tan(1/2*a)^5 - 24*tan(1/2*b*x)^2*tan(1/2*a)^6 - 2
*tan(1/2*b*x)*tan(1/2*a)^7 + tan(1/2*b*x)^6 + 14*tan(1/2*b*x)^5*tan(1/2*a) + 62*tan(1/2*b*x)^4*tan(1/2*a)^2 +
98*tan(1/2*b*x)^3*tan(1/2*a)^3 + 62*tan(1/2*b*x)^2*tan(1/2*a)^4 + 14*tan(1/2*b*x)*tan(1/2*a)^5 + tan(1/2*a)^6
- 2*tan(1/2*b*x)^4 - 14*tan(1/2*b*x)^3*tan(1/2*a) - 24*tan(1/2*b*x)^2*tan(1/2*a)^2 - 14*tan(1/2*b*x)*tan(1/2*a
)^3 - 2*tan(1/2*a)^4 + tan(1/2*b*x)^2 + 2*tan(1/2*b*x)*tan(1/2*a) + tan(1/2*a)^2)/(tan(1/2*b*x)^8*tan(1/2*a)^8
+ 4*tan(1/2*b*x)^8*tan(1/2*a)^6 + 4*tan(1/2*b*x)^6*tan(1/2*a)^8 + 6*tan(1/2*b*x)^8*tan(1/2*a)^4 + 16*tan(1/2*
b*x)^6*tan(1/2*a)^6 + 6*tan(1/2*b*x)^4*tan(1/2*a)^8 + 4*tan(1/2*b*x)^8*tan(1/2*a)^2 + 24*tan(1/2*b*x)^6*tan(1/
2*a)^4 + 24*tan(1/2*b*x)^4*tan(1/2*a)^6 + 4*tan(1/2*b*x)^2*tan(1/2*a)^8 + tan(1/2*b*x)^8 + 16*tan(1/2*b*x)^6*t
an(1/2*a)^2 + 36*tan(1/2*b*x)^4*tan(1/2*a)^4 + 16*tan(1/2*b*x)^2*tan(1/2*a)^6 + tan(1/2*a)^8 + 4*tan(1/2*b*x)^
6 + 24*tan(1/2*b*x)^4*tan(1/2*a)^2 + 24*tan(1/2*b*x)^2*tan(1/2*a)^4 + 4*tan(1/2*a)^6 + 6*tan(1/2*b*x)^4 + 16*t
an(1/2*b*x)^2*tan(1/2*a)^2 + 6*tan(1/2*a)^4 + 4*tan(1/2*b*x)^2 + 4*tan(1/2*a)^2 + 1))*tan(1/2*b*x) - d*log(64*
(tan(1/2*b*x)^8*tan(1/2*a)^6 + 2*tan(1/2*b*x)^7*tan(1/2*a)^7 + tan(1/2*b*x)^6*tan(1/2*a)^8 - 2*tan(1/2*b*x)^8*
tan(1/2*a)^4 - 14*tan(1/2*b*x)^7*tan(1/2*a)^5 - 24*tan(1/2*b*x)^6*tan(1/2*a)^6 - 14*tan(1/2*b*x)^5*tan(1/2*a)^
7 - 2*tan(1/2*b*x)^4*tan(1/2*a)^8 + tan(1/2*b*x)^8*tan(1/2*a)^2 + 14*tan(1/2*b*x)^7*tan(1/2*a)^3 + 62*tan(1/2*
b*x)^6*tan(1/2*a)^4 + 98*tan(1/2*b*x)^5*tan(1/2*a)^5 + 62*tan(1/2*b*x)^4*tan(1/2*a)^6 + 14*tan(1/2*b*x)^3*tan(
1/2*a)^7 + tan(1/2*b*x)^2*tan(1/2*a)^8 - 2*tan(1/2*b*x)^7*tan(1/2*a) - 24*tan(1/2*b*x)^6*tan(1/2*a)^2 - 98*tan
(1/2*b*x)^5*tan(1/2*a)^3 - 152*tan(1/2*b*x)^4*tan(1/2*a)^4 - 98*tan(1/2*b*x)^3*tan(1/2*a)^5 - 24*tan(1/2*b*x)^
2*tan(1/2*a)^6 - 2*tan(1/2*b*x)*tan(1/2*a)^7 + tan(1/2*b*x)^6 + 14*tan(1/2*b*x)^5*tan(1/2*a) + 62*tan(1/2*b*x)
^4*tan(1/2*a)^2 + 98*tan(1/2*b*x)^3*tan(1/2*a)^3 + 62*tan(1/2*b*x)^2*tan(1/2*a)^4 + 14*tan(1/2*b*x)*tan(1/2*a)
^5 + tan(1/2*a)^6 - 2*tan(1/2*b*x)^4 - 14*tan(1/2*b*x)^3*tan(1/2*a) - 24*tan(1/2*b*x)^2*tan(1/2*a)^2 - 14*tan(
1/2*b*x)*tan(1/2*a)^3 - 2*tan(1/2*a)^4 + tan(1/2*b*x)^2 + 2*tan(1/2*b*x)*tan(1/2*a) + tan(1/2*a)^2)/(tan(1/2*b
*x)^8*tan(1/2*a)^8 + 4*tan(1/2*b*x)^8*tan(1/2*a)^6 + 4*tan(1/2*b*x)^6*tan(1/2*a)^8 + 6*tan(1/2*b*x)^8*tan(1/2*
a)^4 + 16*tan(1/2*b*x)^6*tan(1/2*a)^6 + 6*tan(1/2*b*x)^4*tan(1/2*a)^8 + 4*tan(1/2*b*x)^8*tan(1/2*a)^2 + 24*tan
(1/2*b*x)^6*tan(1/2*a)^4 + 24*tan(1/2*b*x)^4*tan(1/2*a)^6 + 4*tan(1/2*b*x)^2*tan(1/2*a)^8 + tan(1/2*b*x)^8 + 1
6*tan(1/2*b*x)^6*tan(1/2*a)^2 + 36*tan(1/2*b*x)^4*tan(1/2*a)^4 + 16*tan(1/2*b*x)^2*tan(1/2*a)^6 + tan(1/2*a)^8
+ 4*tan(1/2*b*x)^6 + 24*tan(1/2*b*x)^4*tan(1/2*a)^2 + 24*tan(1/2*b*x)^2*tan(1/2*a)^4 + 4*tan(1/2*a)^6 + 6*tan
(1/2*b*x)^4 + 16*tan(1/2*b*x)^2*tan(1/2*a)^2 + 6*tan(1/2*a)^4 + 4*tan(1/2*b*x)^2 + 4*tan(1/2*a)^2 + 1))*tan(1/
2*a) + b*c)/(b^2*tan(1/2*b*x)^4*tan(1/2*a)^3 + b^2*tan(1/2*b*x)^3*tan(1/2*a)^4 - b^2*tan(1/2*b*x)^4*tan(1/2*a)
- 6*b^2*tan(1/2*b*x)^3*tan(1/2*a)^2 - 6*b^2*tan(1/2*b*x)^2*tan(1/2*a)^3 - b^2*tan(1/2*b*x)*tan(1/2*a)^4 + b^2
*tan(1/2*b*x)^3 + 6*b^2*tan(1/2*b*x)^2*tan(1/2*a) + 6*b^2*tan(1/2*b*x)*tan(1/2*a)^2 + b^2*tan(1/2*a)^3 - b^2*t
an(1/2*b*x) - b^2*tan(1/2*a))

Mupad [B] (verification not implemented)

Time = 27.60 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.57 \[ \int (c+d x) \csc ^2(a+b x) \sec ^2(a+b x) \, dx=\frac {d\,\ln \left ({\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,4{}\mathrm {i}}-1\right )}{b^2}-\frac {\left (c+d\,x\right )\,4{}\mathrm {i}}{b\,\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}-1\right )}-\frac {d\,x\,4{}\mathrm {i}}{b} \]

[In]

int((c + d*x)/(cos(a + b*x)^2*sin(a + b*x)^2),x)

[Out]

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