3.476 \(\int \frac{\cot ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx\)

Optimal. Leaf size=189 \[ \frac{b^2 \left (2 a^2+3 b^2\right )}{a^3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac{\left (a^2-3 b^2\right ) \log (\sin (c+d x))}{a^4 d}-\frac{b^4 \left (5 a^2+3 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )^2}+\frac{2 a b x}{\left (a^2+b^2\right )^2}+\frac{3 b \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))}-\frac{\cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))} \]

[Out]

(2*a*b*x)/(a^2 + b^2)^2 - ((a^2 - 3*b^2)*Log[Sin[c + d*x]])/(a^4*d) - (b^4*(5*a^2 + 3*b^2)*Log[a*Cos[c + d*x]
+ b*Sin[c + d*x]])/(a^4*(a^2 + b^2)^2*d) + (b^2*(2*a^2 + 3*b^2))/(a^3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])) + (3
*b*Cot[c + d*x])/(2*a^2*d*(a + b*Tan[c + d*x])) - Cot[c + d*x]^2/(2*a*d*(a + b*Tan[c + d*x]))

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Rubi [A]  time = 0.57283, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3569, 3649, 3650, 3651, 3530, 3475} \[ \frac{b^2 \left (2 a^2+3 b^2\right )}{a^3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac{\left (a^2-3 b^2\right ) \log (\sin (c+d x))}{a^4 d}-\frac{b^4 \left (5 a^2+3 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )^2}+\frac{2 a b x}{\left (a^2+b^2\right )^2}+\frac{3 b \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))}-\frac{\cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^3/(a + b*Tan[c + d*x])^2,x]

[Out]

(2*a*b*x)/(a^2 + b^2)^2 - ((a^2 - 3*b^2)*Log[Sin[c + d*x]])/(a^4*d) - (b^4*(5*a^2 + 3*b^2)*Log[a*Cos[c + d*x]
+ b*Sin[c + d*x]])/(a^4*(a^2 + b^2)^2*d) + (b^2*(2*a^2 + 3*b^2))/(a^3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])) + (3
*b*Cot[c + d*x])/(2*a^2*d*(a + b*Tan[c + d*x])) - Cot[c + d*x]^2/(2*a*d*(a + b*Tan[c + d*x]))

Rule 3569

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b^2*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d)), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] &&
NeQ[a, 0])))

Rule 3649

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[((A*b^2 - a*(b*B - a*C))*(a + b*T
an[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[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3650

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*t
an[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[((A*b^2 + a^2*C)*(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[A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2)) - a*C*(b*c*(m + 1)
+ a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - b*C)*Tan[e + f*x] - d*(A*b^2 + a^2*C)*(m + n + 2)*Tan[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^
2, 0] && LtQ[m, -1] &&  !(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]

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 3475

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

Rubi steps

\begin{align*} \int \frac{\cot ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx &=-\frac{\cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))}-\frac{\int \frac{\cot ^2(c+d x) \left (3 b+2 a \tan (c+d x)+3 b \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 a}\\ &=\frac{3 b \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))}-\frac{\cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))}+\frac{\int \frac{\cot (c+d x) \left (-2 \left (a^2-3 b^2\right )+6 b^2 \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 a^2}\\ &=\frac{b^2 \left (2 a^2+3 b^2\right )}{a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac{3 b \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))}-\frac{\cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))}+\frac{\int \frac{\cot (c+d x) \left (-2 \left (a^2-3 b^2\right ) \left (a^2+b^2\right )+2 a^3 b \tan (c+d x)+2 b^2 \left (2 a^2+3 b^2\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 a^3 \left (a^2+b^2\right )}\\ &=\frac{2 a b x}{\left (a^2+b^2\right )^2}+\frac{b^2 \left (2 a^2+3 b^2\right )}{a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac{3 b \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))}-\frac{\cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))}-\frac{\left (a^2-3 b^2\right ) \int \cot (c+d x) \, dx}{a^4}-\frac{\left (b^4 \left (5 a^2+3 b^2\right )\right ) \int \frac{b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^4 \left (a^2+b^2\right )^2}\\ &=\frac{2 a b x}{\left (a^2+b^2\right )^2}-\frac{\left (a^2-3 b^2\right ) \log (\sin (c+d x))}{a^4 d}-\frac{b^4 \left (5 a^2+3 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right )^2 d}+\frac{b^2 \left (2 a^2+3 b^2\right )}{a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac{3 b \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))}-\frac{\cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))}\\ \end{align*}

Mathematica [C]  time = 1.17794, size = 146, normalized size = 0.77 \[ -\frac{\frac{2 b^5}{a^4 \left (a^2+b^2\right ) (a \cot (c+d x)+b)}+\frac{2 b^4 \left (5 a^2+3 b^2\right ) \log (a \cot (c+d x)+b)}{a^4 \left (a^2+b^2\right )^2}-\frac{4 b \cot (c+d x)}{a^3}+\frac{\cot ^2(c+d x)}{a^2}-\frac{\log (-\cot (c+d x)+i)}{(a-i b)^2}-\frac{\log (\cot (c+d x)+i)}{(a+i b)^2}}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^3/(a + b*Tan[c + d*x])^2,x]

[Out]

-((-4*b*Cot[c + d*x])/a^3 + Cot[c + d*x]^2/a^2 + (2*b^5)/(a^4*(a^2 + b^2)*(b + a*Cot[c + d*x])) - Log[I - Cot[
c + d*x]]/(a - I*b)^2 - Log[I + Cot[c + d*x]]/(a + I*b)^2 + (2*b^4*(5*a^2 + 3*b^2)*Log[b + a*Cot[c + d*x]])/(a
^4*(a^2 + b^2)^2))/(2*d)

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Maple [A]  time = 0.09, size = 240, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){a}^{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){b}^{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}+2\,{\frac{ab\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{1}{2\,{a}^{2}d \left ( \tan \left ( dx+c \right ) \right ) ^{2}}}-{\frac{\ln \left ( \tan \left ( dx+c \right ) \right ) }{{a}^{2}d}}+3\,{\frac{\ln \left ( \tan \left ( dx+c \right ) \right ){b}^{2}}{d{a}^{4}}}+2\,{\frac{b}{d{a}^{3}\tan \left ( dx+c \right ) }}+{\frac{{b}^{4}}{d \left ({a}^{2}+{b}^{2} \right ){a}^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-5\,{\frac{{b}^{4}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{2}{a}^{2}}}-3\,{\frac{{b}^{6}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{2}{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^3/(a+b*tan(d*x+c))^2,x)

[Out]

1/2/d/(a^2+b^2)^2*ln(1+tan(d*x+c)^2)*a^2-1/2/d/(a^2+b^2)^2*ln(1+tan(d*x+c)^2)*b^2+2/d/(a^2+b^2)^2*a*b*arctan(t
an(d*x+c))-1/2/d/a^2/tan(d*x+c)^2-1/d/a^2*ln(tan(d*x+c))+3/d/a^4*ln(tan(d*x+c))*b^2+2/d/a^3*b/tan(d*x+c)+1/d*b
^4/(a^2+b^2)/a^3/(a+b*tan(d*x+c))-5/d*b^4/(a^2+b^2)^2/a^2*ln(a+b*tan(d*x+c))-3/d*b^6/(a^2+b^2)^2/a^4*ln(a+b*ta
n(d*x+c))

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Maxima [A]  time = 1.59519, size = 324, normalized size = 1.71 \begin{align*} \frac{\frac{4 \,{\left (d x + c\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac{2 \,{\left (5 \, a^{2} b^{4} + 3 \, b^{6}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4}} + \frac{{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac{a^{4} + a^{2} b^{2} - 2 \,{\left (2 \, a^{2} b^{2} + 3 \, b^{4}\right )} \tan \left (d x + c\right )^{2} - 3 \,{\left (a^{3} b + a b^{3}\right )} \tan \left (d x + c\right )}{{\left (a^{5} b + a^{3} b^{3}\right )} \tan \left (d x + c\right )^{3} +{\left (a^{6} + a^{4} b^{2}\right )} \tan \left (d x + c\right )^{2}} - \frac{2 \,{\left (a^{2} - 3 \, b^{2}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{4}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3/(a+b*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

1/2*(4*(d*x + c)*a*b/(a^4 + 2*a^2*b^2 + b^4) - 2*(5*a^2*b^4 + 3*b^6)*log(b*tan(d*x + c) + a)/(a^8 + 2*a^6*b^2
+ a^4*b^4) + (a^2 - b^2)*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) - (a^4 + a^2*b^2 - 2*(2*a^2*b^2 + 3*b
^4)*tan(d*x + c)^2 - 3*(a^3*b + a*b^3)*tan(d*x + c))/((a^5*b + a^3*b^3)*tan(d*x + c)^3 + (a^6 + a^4*b^2)*tan(d
*x + c)^2) - 2*(a^2 - 3*b^2)*log(tan(d*x + c))/a^4)/d

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Fricas [B]  time = 2.35913, size = 834, normalized size = 4.41 \begin{align*} -\frac{a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4} -{\left (4 \, a^{5} b^{2} d x - a^{6} b - 2 \, a^{4} b^{3} - 3 \, a^{2} b^{5}\right )} \tan \left (d x + c\right )^{3} -{\left (4 \, a^{6} b d x - a^{7} + 2 \, a^{5} b^{2} + 7 \, a^{3} b^{4} + 6 \, a b^{6}\right )} \tan \left (d x + c\right )^{2} +{\left ({\left (a^{6} b - a^{4} b^{3} - 5 \, a^{2} b^{5} - 3 \, b^{7}\right )} \tan \left (d x + c\right )^{3} +{\left (a^{7} - a^{5} b^{2} - 5 \, a^{3} b^{4} - 3 \, a b^{6}\right )} \tan \left (d x + c\right )^{2}\right )} \log \left (\frac{\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) +{\left ({\left (5 \, a^{2} b^{5} + 3 \, b^{7}\right )} \tan \left (d x + c\right )^{3} +{\left (5 \, a^{3} b^{4} + 3 \, a b^{6}\right )} \tan \left (d x + c\right )^{2}\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 ) - 3 \,{\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \tan \left (d x + c\right )}{2 \,{\left ({\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} d \tan \left (d x + c\right )^{3} +{\left (a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} d \tan \left (d x + c\right )^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3/(a+b*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/2*(a^7 + 2*a^5*b^2 + a^3*b^4 - (4*a^5*b^2*d*x - a^6*b - 2*a^4*b^3 - 3*a^2*b^5)*tan(d*x + c)^3 - (4*a^6*b*d*
x - a^7 + 2*a^5*b^2 + 7*a^3*b^4 + 6*a*b^6)*tan(d*x + c)^2 + ((a^6*b - a^4*b^3 - 5*a^2*b^5 - 3*b^7)*tan(d*x + c
)^3 + (a^7 - a^5*b^2 - 5*a^3*b^4 - 3*a*b^6)*tan(d*x + c)^2)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) + ((5*a^2
*b^5 + 3*b^7)*tan(d*x + c)^3 + (5*a^3*b^4 + 3*a*b^6)*tan(d*x + c)^2)*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x +
c) + a^2)/(tan(d*x + c)^2 + 1)) - 3*(a^6*b + 2*a^4*b^3 + a^2*b^5)*tan(d*x + c))/((a^8*b + 2*a^6*b^3 + a^4*b^5
)*d*tan(d*x + c)^3 + (a^9 + 2*a^7*b^2 + a^5*b^4)*d*tan(d*x + c)^2)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**3/(a+b*tan(d*x+c))**2,x)

[Out]

Exception raised: AttributeError

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Giac [A]  time = 1.29361, size = 367, normalized size = 1.94 \begin{align*} \frac{\frac{4 \,{\left (d x + c\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac{{\left (a^{2} - 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 (5 \, a^{2} b^{5} + 3 \, b^{7}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}} + \frac{2 \,{\left (5 \, a^{2} b^{5} \tan \left (d x + c\right ) + 3 \, b^{7} \tan \left (d x + c\right ) + 6 \, a^{3} b^{4} + 4 \, a b^{6}\right )}}{{\left (a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4}\right )}{\left (b \tan \left (d x + c\right ) + a\right )}} - \frac{2 \,{\left (a^{2} - 3 \, b^{2}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4}} + \frac{3 \, a^{2} \tan \left (d x + c\right )^{2} - 9 \, b^{2} \tan \left (d x + c\right )^{2} + 4 \, a b \tan \left (d x + c\right ) - a^{2}}{a^{4} \tan \left (d x + c\right )^{2}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3/(a+b*tan(d*x+c))^2,x, algorithm="giac")

[Out]

1/2*(4*(d*x + c)*a*b/(a^4 + 2*a^2*b^2 + b^4) + (a^2 - b^2)*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) - 2
*(5*a^2*b^5 + 3*b^7)*log(abs(b*tan(d*x + c) + a))/(a^8*b + 2*a^6*b^3 + a^4*b^5) + 2*(5*a^2*b^5*tan(d*x + c) +
3*b^7*tan(d*x + c) + 6*a^3*b^4 + 4*a*b^6)/((a^8 + 2*a^6*b^2 + a^4*b^4)*(b*tan(d*x + c) + a)) - 2*(a^2 - 3*b^2)
*log(abs(tan(d*x + c)))/a^4 + (3*a^2*tan(d*x + c)^2 - 9*b^2*tan(d*x + c)^2 + 4*a*b*tan(d*x + c) - a^2)/(a^4*ta
n(d*x + c)^2))/d