3.279 \(\int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx\)
Optimal. Leaf size=137 \[ \frac {b (A b-a B)}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {x \left (a^2 (-B)+2 a A b+b^2 B\right )}{\left (a^2+b^2\right )^2}+\frac {A \log (\sin (c+d x))}{a^2 d}-\frac {b \left (-2 a^3 B+3 a^2 A b+A b^3\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^2 d \left (a^2+b^2\right )^2} \]
[Out]
-(2*A*a*b-B*a^2+B*b^2)*x/(a^2+b^2)^2+A*ln(sin(d*x+c))/a^2/d-b*(3*A*a^2*b+A*b^3-2*B*a^3)*ln(a*cos(d*x+c)+b*sin(
d*x+c))/a^2/(a^2+b^2)^2/d+b*(A*b-B*a)/a/(a^2+b^2)/d/(a+b*tan(d*x+c))
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Rubi [A] time = 0.32, antiderivative size = 137, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used =
{3609, 3651, 3530, 3475} \[ \frac {b (A b-a B)}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {b \left (3 a^2 A b-2 a^3 B+A b^3\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^2 d \left (a^2+b^2\right )^2}-\frac {x \left (a^2 (-B)+2 a A b+b^2 B\right )}{\left (a^2+b^2\right )^2}+\frac {A \log (\sin (c+d x))}{a^2 d} \]
Antiderivative was successfully verified.
[In]
Int[(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]
[Out]
-(((2*a*A*b - a^2*B + b^2*B)*x)/(a^2 + b^2)^2) + (A*Log[Sin[c + d*x]])/(a^2*d) - (b*(3*a^2*A*b + A*b^3 - 2*a^3
*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^2*(a^2 + b^2)^2*d) + (b*(A*b - a*B))/(a*(a^2 + b^2)*d*(a + b*Tan[
c + d*x]))
Rule 3475
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]
Rule 3530
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c*Log[Re
moveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]
Rule 3609
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n
+ 1))/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e +
f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(
m + n + 2)) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 2)*Tan[e + f*x]^2, x], x
], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& LtQ[m, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ
[a, 0])))
Rule 3651
Int[((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2)/(((a_) + (b_.)*tan[(e_.) + (f_.)
*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[((a*(A*c - c*C + B*d) + b*(B*c - A*d + C*d
))*x)/((a^2 + b^2)*(c^2 + d^2)), x] + (Dist[(A*b^2 - a*b*B + a^2*C)/((b*c - a*d)*(a^2 + b^2)), Int[(b - a*Tan[
e + f*x])/(a + b*Tan[e + f*x]), x], x] - Dist[(c^2*C - B*c*d + A*d^2)/((b*c - a*d)*(c^2 + d^2)), Int[(d - c*Ta
n[e + f*x])/(c + d*Tan[e + f*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ
[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Rubi steps
\begin {align*} \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx &=\frac {b (A b-a B)}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {\cot (c+d x) \left (A \left (a^2+b^2\right )-a (A b-a B) \tan (c+d x)+b (A b-a B) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{a \left (a^2+b^2\right )}\\ &=-\frac {\left (2 a A b-a^2 B+b^2 B\right ) x}{\left (a^2+b^2\right )^2}+\frac {b (A b-a B)}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {A \int \cot (c+d x) \, dx}{a^2}-\frac {\left (b \left (3 a^2 A b+A b^3-2 a^3 B\right )\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^2 \left (a^2+b^2\right )^2}\\ &=-\frac {\left (2 a A b-a^2 B+b^2 B\right ) x}{\left (a^2+b^2\right )^2}+\frac {A \log (\sin (c+d x))}{a^2 d}-\frac {b \left (3 a^2 A b+A b^3-2 a^3 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^2 \left (a^2+b^2\right )^2 d}+\frac {b (A b-a B)}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [C] time = 0.85, size = 183, normalized size = 1.34 \[ \frac {\frac {A \left (a^2+b^2\right ) \log (\tan (c+d x))}{a}+\frac {b \left (2 a^3 B-3 a^2 A b-A b^3\right ) \log (a+b \tan (c+d x))}{a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{a+b \tan (c+d x)}-\frac {a (a-i b) (A+i B) \log (-\tan (c+d x)+i)}{2 (a+i b)}-\frac {a (a+i b) (A-i B) \log (\tan (c+d x)+i)}{2 (a-i b)}}{a d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]
[Out]
(-1/2*(a*(a - I*b)*(A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b) + (A*(a^2 + b^2)*Log[Tan[c + d*x]])/a - (a*(a +
I*b)*(A - I*B)*Log[I + Tan[c + d*x]])/(2*(a - I*b)) + (b*(-3*a^2*A*b - A*b^3 + 2*a^3*B)*Log[a + b*Tan[c + d*x]
])/(a*(a^2 + b^2)) + (b*(A*b - a*B))/(a + b*Tan[c + d*x]))/(a*(a^2 + b^2)*d)
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fricas [B] time = 0.78, size = 323, normalized size = 2.36 \[ -\frac {2 \, B a^{2} b^{3} - 2 \, A a b^{4} - 2 \, {\left (B a^{5} - 2 \, A a^{4} b - B a^{3} b^{2}\right )} d x - {\left (A a^{5} + 2 \, A a^{3} b^{2} + A a b^{4} + {\left (A a^{4} b + 2 \, A a^{2} b^{3} + A b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (2 \, B a^{4} b - 3 \, A a^{3} b^{2} - A a b^{4} + {\left (2 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3} - A b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (B a^{3} b^{2} - A a^{2} b^{3} + {\left (B a^{4} b - 2 \, A a^{3} b^{2} - B a^{2} b^{3}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d \tan \left (d x + c\right ) + {\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^2,x, algorithm="fricas")
[Out]
-1/2*(2*B*a^2*b^3 - 2*A*a*b^4 - 2*(B*a^5 - 2*A*a^4*b - B*a^3*b^2)*d*x - (A*a^5 + 2*A*a^3*b^2 + A*a*b^4 + (A*a^
4*b + 2*A*a^2*b^3 + A*b^5)*tan(d*x + c))*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) - (2*B*a^4*b - 3*A*a^3*b^2 -
A*a*b^4 + (2*B*a^3*b^2 - 3*A*a^2*b^3 - A*b^5)*tan(d*x + c))*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^
2)/(tan(d*x + c)^2 + 1)) - 2*(B*a^3*b^2 - A*a^2*b^3 + (B*a^4*b - 2*A*a^3*b^2 - B*a^2*b^3)*d*x)*tan(d*x + c))/(
(a^6*b + 2*a^4*b^3 + a^2*b^5)*d*tan(d*x + c) + (a^7 + 2*a^5*b^2 + a^3*b^4)*d)
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giac [B] time = 0.87, size = 279, normalized size = 2.04 \[ \frac {\frac {2 \, {\left (B a^{2} - 2 \, A a b - B b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {{\left (A a^{2} + 2 \, B a b - A b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (2 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3} - A b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}} + \frac {2 \, A \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{2}} - \frac {2 \, {\left (2 \, B a^{3} b^{2} \tan \left (d x + c\right ) - 3 \, A a^{2} b^{3} \tan \left (d x + c\right ) - A b^{5} \tan \left (d x + c\right ) + 3 \, B a^{4} b - 4 \, A a^{3} b^{2} + B a^{2} b^{3} - 2 \, A a b^{4}\right )}}{{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^2,x, algorithm="giac")
[Out]
1/2*(2*(B*a^2 - 2*A*a*b - B*b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) - (A*a^2 + 2*B*a*b - A*b^2)*log(tan(d*x + c
)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) + 2*(2*B*a^3*b^2 - 3*A*a^2*b^3 - A*b^5)*log(abs(b*tan(d*x + c) + a))/(a^6*b +
2*a^4*b^3 + a^2*b^5) + 2*A*log(abs(tan(d*x + c)))/a^2 - 2*(2*B*a^3*b^2*tan(d*x + c) - 3*A*a^2*b^3*tan(d*x + c
) - A*b^5*tan(d*x + c) + 3*B*a^4*b - 4*A*a^3*b^2 + B*a^2*b^3 - 2*A*a*b^4)/((a^6 + 2*a^4*b^2 + a^2*b^4)*(b*tan(
d*x + c) + a)))/d
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maple [B] time = 0.72, size = 325, normalized size = 2.37 \[ -\frac {3 \ln \left (a +b \tan \left (d x +c \right )\right ) A \,b^{2}}{d \left (a^{2}+b^{2}\right )^{2}}-\frac {b^{4} \ln \left (a +b \tan \left (d x +c \right )\right ) A}{d \,a^{2} \left (a^{2}+b^{2}\right )^{2}}+\frac {2 \ln \left (a +b \tan \left (d x +c \right )\right ) B a b}{d \left (a^{2}+b^{2}\right )^{2}}+\frac {b^{2} A}{d a \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}-\frac {b B}{d \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}+\frac {A \ln \left (\tan \left (d x +c \right )\right )}{a^{2} d}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2} A}{2 d \left (a^{2}+b^{2}\right )^{2}}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) A \,b^{2}}{2 d \left (a^{2}+b^{2}\right )^{2}}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) B a b}{d \left (a^{2}+b^{2}\right )^{2}}-\frac {2 A \arctan \left (\tan \left (d x +c \right )\right ) a b}{d \left (a^{2}+b^{2}\right )^{2}}+\frac {B \arctan \left (\tan \left (d x +c \right )\right ) a^{2}}{d \left (a^{2}+b^{2}\right )^{2}}-\frac {B \arctan \left (\tan \left (d x +c \right )\right ) b^{2}}{d \left (a^{2}+b^{2}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(d*x+c)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^2,x)
[Out]
-3/d/(a^2+b^2)^2*ln(a+b*tan(d*x+c))*A*b^2-1/d*b^4/a^2/(a^2+b^2)^2*ln(a+b*tan(d*x+c))*A+2/d/(a^2+b^2)^2*ln(a+b*
tan(d*x+c))*B*a*b+1/d*b^2/a/(a^2+b^2)/(a+b*tan(d*x+c))*A-1/d*b/(a^2+b^2)/(a+b*tan(d*x+c))*B+1/a^2/d*A*ln(tan(d
*x+c))-1/2/d/(a^2+b^2)^2*ln(1+tan(d*x+c)^2)*a^2*A+1/2/d/(a^2+b^2)^2*ln(1+tan(d*x+c)^2)*A*b^2-1/d/(a^2+b^2)^2*l
n(1+tan(d*x+c)^2)*B*a*b-2/d/(a^2+b^2)^2*A*arctan(tan(d*x+c))*a*b+1/d/(a^2+b^2)^2*B*arctan(tan(d*x+c))*a^2-1/d/
(a^2+b^2)^2*B*arctan(tan(d*x+c))*b^2
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maxima [A] time = 0.88, size = 208, normalized size = 1.52 \[ \frac {\frac {2 \, {\left (B a^{2} - 2 \, A a b - B b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (2 \, B a^{3} b - 3 \, A a^{2} b^{2} - A b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}} - \frac {{\left (A a^{2} + 2 \, B a b - A b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (B a b - A b^{2}\right )}}{a^{4} + a^{2} b^{2} + {\left (a^{3} b + a b^{3}\right )} \tan \left (d x + c\right )} + \frac {2 \, A \log \left (\tan \left (d x + c\right )\right )}{a^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^2,x, algorithm="maxima")
[Out]
1/2*(2*(B*a^2 - 2*A*a*b - B*b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) + 2*(2*B*a^3*b - 3*A*a^2*b^2 - A*b^4)*log(b
*tan(d*x + c) + a)/(a^6 + 2*a^4*b^2 + a^2*b^4) - (A*a^2 + 2*B*a*b - A*b^2)*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^
2*b^2 + b^4) - 2*(B*a*b - A*b^2)/(a^4 + a^2*b^2 + (a^3*b + a*b^3)*tan(d*x + c)) + 2*A*log(tan(d*x + c))/a^2)/d
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mupad [B] time = 8.00, size = 180, normalized size = 1.31 \[ \frac {A\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )}{2\,d\,\left (a^2+a\,b\,2{}\mathrm {i}-b^2\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (a^2\,1{}\mathrm {i}+2\,a\,b-b^2\,1{}\mathrm {i}\right )}+\frac {A\,b^2-B\,a\,b}{a\,d\,\left (a^2+b^2\right )\,\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}-\frac {b\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-2\,B\,a^3+3\,A\,a^2\,b+A\,b^3\right )}{a^2\,d\,{\left (a^2+b^2\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((cot(c + d*x)*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^2,x)
[Out]
(A*log(tan(c + d*x)))/(a^2*d) - (log(tan(c + d*x) - 1i)*(A + B*1i))/(2*d*(a*b*2i + a^2 - b^2)) - (log(tan(c +
d*x) + 1i)*(A*1i + B))/(2*d*(2*a*b + a^2*1i - b^2*1i)) + (A*b^2 - B*a*b)/(a*d*(a^2 + b^2)*(a + b*tan(c + d*x))
) - (b*log(a + b*tan(c + d*x))*(A*b^3 - 2*B*a^3 + 3*A*a^2*b))/(a^2*d*(a^2 + b^2)^2)
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sympy [A] time = 4.15, size = 4447, normalized size = 32.46 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**2,x)
[Out]
Piecewise((zoo*x*(A + B*tan(c))*cot(c)/tan(c)**2, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((-A*log(tan(c + d*x)**2 +
1)/(2*d) + A*log(tan(c + d*x))/d + B*x)/a**2, Eq(b, 0)), ((A*log(tan(c + d*x)**2 + 1)/(2*d) - A*log(tan(c + d*
x))/d - A/(2*d*tan(c + d*x)**2) - B*x - B/(d*tan(c + d*x)))/b**2, Eq(a, 0)), (3*I*A*d*x*tan(c + d*x)**2/(4*b**
2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 6*A*d*x*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8
*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 3*I*A*d*x/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d)
+ 2*A*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d
) - 4*I*A*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d
) - 2*A*log(tan(c + d*x)**2 + 1)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 4*A*log(tan
(c + d*x))*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 8*I*A*log(tan(c +
d*x))*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 4*A*log(tan(c + d*x))/(4
*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 3*I*A*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 -
8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 4*A/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - B
*d*x*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 2*I*B*d*x*tan(c + d*x)/
(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + B*d*x/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d
*tan(c + d*x) - 4*b**2*d) - B*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 2
*I*B/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d), Eq(a, -I*b)), (-3*I*A*d*x*tan(c + d*x)**
2/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 6*A*d*x*tan(c + d*x)/(4*b**2*d*tan(c + d*x
)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 3*I*A*d*x/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4
*b**2*d) + 2*A*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) -
4*b**2*d) + 4*I*A*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) -
4*b**2*d) - 2*A*log(tan(c + d*x)**2 + 1)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 4*A
*log(tan(c + d*x))*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 8*I*A*log
(tan(c + d*x))*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 4*A*log(tan(c +
d*x))/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 3*I*A*tan(c + d*x)/(4*b**2*d*tan(c + d
*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 4*A/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**
2*d) - B*d*x*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 2*I*B*d*x*tan(c
+ d*x)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + B*d*x/(4*b**2*d*tan(c + d*x)**2 + 8*
I*b**2*d*tan(c + d*x) - 4*b**2*d) - B*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**
2*d) - 2*I*B/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d), Eq(a, I*b)), (x*(A + B*tan(c))*c
ot(c)/(a + b*tan(c))**2, Eq(d, 0)), (-A*a**5*log(tan(c + d*x)**2 + 1)/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*
a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) + 2*A*a**5*log(tan(c +
d*x))/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**
2*b**5*d*tan(c + d*x)) - 4*A*a**4*b*d*x/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*ta
n(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) - A*a**4*b*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(2*a
**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*t
an(c + d*x)) + 2*A*a**4*b*log(tan(c + d*x))*tan(c + d*x)/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d +
4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) - 4*A*a**3*b**2*d*x*tan(c + d*x)/(2*
a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*
tan(c + d*x)) - 6*A*a**3*b**2*log(a/b + tan(c + d*x))/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*
a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) + A*a**3*b**2*log(tan(c + d*x)**2 + 1)/
(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5
*d*tan(c + d*x)) + 4*A*a**3*b**2*log(tan(c + d*x))/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**
4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) + 2*A*a**3*b**2/(2*a**7*d + 2*a**6*b*d*tan
(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) - 6*A*a**
2*b**3*log(a/b + tan(c + d*x))*tan(c + d*x)/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*
d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) + A*a**2*b**3*log(tan(c + d*x)**2 + 1)*tan(c + d*
x)/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b
**5*d*tan(c + d*x)) + 4*A*a**2*b**3*log(tan(c + d*x))*tan(c + d*x)/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**
5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) - 2*A*a*b**4*log(a/b + tan
(c + d*x))/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d +
2*a**2*b**5*d*tan(c + d*x)) + 2*A*a*b**4*log(tan(c + d*x))/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d
+ 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) + 2*A*a*b**4/(2*a**7*d + 2*a**6*b*
d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) - 2*
A*b**5*log(a/b + tan(c + d*x))*tan(c + d*x)/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*
d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) + 2*A*b**5*log(tan(c + d*x))*tan(c + d*x)/(2*a**7
*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(
c + d*x)) + 2*B*a**5*d*x/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*
a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) + 2*B*a**4*b*d*x*tan(c + d*x)/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) +
4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) + 4*B*a**4*b*log(a/b
+ tan(c + d*x))/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4
*d + 2*a**2*b**5*d*tan(c + d*x)) - 2*B*a**4*b*log(tan(c + d*x)**2 + 1)/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4
*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) - 2*B*a**4*b/(2*a**7*d
+ 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(c
+ d*x)) - 2*B*a**3*b**2*d*x/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) +
2*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) + 4*B*a**3*b**2*log(a/b + tan(c + d*x))*tan(c + d*x)/(2*a**7*d +
2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d
*x)) - 2*B*a**3*b**2*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d
+ 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) - 2*B*a**2*b**3*d*x*tan(c + d*x)/(
2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*
d*tan(c + d*x)) - 2*B*a**2*b**3/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*
x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)), True))
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