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

Optimal. Leaf size=114 \[ \frac {a^2 \left (a^2+3 b^2\right ) \log (a+b \tan (c+d x))}{b^2 d \left (a^2+b^2\right )^2}+\frac {\left (a^2-b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^2}-\frac {2 a b x}{\left (a^2+b^2\right )^2}+\frac {a^3}{b^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))} \]

[Out]

-2*a*b*x/(a^2+b^2)^2+(a^2-b^2)*ln(cos(d*x+c))/(a^2+b^2)^2/d+a^2*(a^2+3*b^2)*ln(a+b*tan(d*x+c))/b^2/(a^2+b^2)^2
/d+a^3/b^2/(a^2+b^2)/d/(a+b*tan(d*x+c))

________________________________________________________________________________________

Rubi [A]  time = 0.16, antiderivative size = 121, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3565, 3626, 3617, 31, 3475} \[ -\frac {a^2 \tan (c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {a^2 \left (a^2+3 b^2\right ) \log (a+b \tan (c+d x))}{b^2 d \left (a^2+b^2\right )^2}+\frac {\left (a^2-b^2\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^2}-\frac {2 a b x}{\left (a^2+b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*b*x)/(a^2 + b^2)^2 + ((a^2 - b^2)*Log[Cos[c + d*x]])/((a^2 + b^2)^2*d) + (a^2*(a^2 + 3*b^2)*Log[a + b*Ta
n[c + d*x]])/(b^2*(a^2 + b^2)^2*d) - (a^2*Tan[c + d*x])/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, 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 3565

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[((b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

Rule 3617

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[
A/(b*f), Subst[Int[(a + x)^m, x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A, C]

Rule 3626

Int[((A_) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2)/((a_.) + (b_.)*tan[(e_.) + (f_.)*
(x_)]), x_Symbol] :> Simp[((a*A + b*B - a*C)*x)/(a^2 + b^2), x] + (Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 + b^2), I
nt[(1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x], x] - Dist[(A*b - a*B - b*C)/(a^2 + b^2), Int[Tan[e + f*x], x
], x]) /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && NeQ[a^2 + b^2, 0] && NeQ[A*b - a
*B - b*C, 0]

Rubi steps

\begin {align*} \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx &=-\frac {a^2 \tan (c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {a^2-a b \tan (c+d x)+\left (a^2+b^2\right ) \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b \left (a^2+b^2\right )}\\ &=-\frac {2 a b x}{\left (a^2+b^2\right )^2}-\frac {a^2 \tan (c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {\left (a^2-b^2\right ) \int \tan (c+d x) \, dx}{\left (a^2+b^2\right )^2}+\frac {\left (a^2 \left (a^2+3 b^2\right )\right ) \int \frac {1+\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b \left (a^2+b^2\right )^2}\\ &=-\frac {2 a b x}{\left (a^2+b^2\right )^2}+\frac {\left (a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {a^2 \tan (c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\left (a^2 \left (a^2+3 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^2 \left (a^2+b^2\right )^2 d}\\ &=-\frac {2 a b x}{\left (a^2+b^2\right )^2}+\frac {\left (a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {a^2 \left (a^2+3 b^2\right ) \log (a+b \tan (c+d x))}{b^2 \left (a^2+b^2\right )^2 d}-\frac {a^2 \tan (c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 1.07, size = 251, normalized size = 2.20 \[ \frac {-2 i a^2 \left (a^2+3 b^2\right ) \tan ^{-1}(\tan (c+d x)) (a+b \tan (c+d x))+a \left (-2 \left (a^2+b^2\right )^2 \log (\cos (c+d x))+a \left (a \left (a^2+3 b^2\right ) \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right )+2 (2 b+i a) (a+i b)^2 (c+d x)\right )\right )+b \tan (c+d x) \left (-2 \left (a^2+b^2\right )^2 \log (\cos (c+d x))+a \left (a \left (a^2+3 b^2\right ) \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right )+2 i \left (a^3 (c+d x+i)+a b^2 (3 c+3 d x+i)+2 i b^3 (c+d x)\right )\right )\right )}{2 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.54, size = 226, normalized size = 1.98 \[ -\frac {4 \, a^{2} b^{3} d x - 2 \, a^{3} b^{2} - {\left (a^{5} + 3 \, a^{3} b^{2} + {\left (a^{4} b + 3 \, a^{2} b^{3}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \, {\left (2 \, a b^{4} d x + a^{4} b\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} d \tan \left (d x + c\right ) + {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 3.04, size = 181, normalized size = 1.59 \[ -\frac {\frac {4 \, {\left (d x + c\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (a^{4} + 3 \, a^{2} b^{2}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (a^{4} \tan \left (d x + c\right ) + 3 \, a^{2} b^{2} \tan \left (d x + c\right ) + 2 \, a^{3} b\right )}}{{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(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*(a^4 + 3*a^2*b^2)*log(abs(b*tan(d*x + c) + a))/(a^4*b^2 + 2*a^2*b^4 + b^6) + 2*(a^4*tan(d*x + c) + 3*a^2*b^2
*tan(d*x + c) + 2*a^3*b)/((a^4*b + 2*a^2*b^3 + b^5)*(b*tan(d*x + c) + a)))/d

________________________________________________________________________________________

maple [A]  time = 0.21, size = 170, normalized size = 1.49 \[ \frac {a^{4} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \,b^{2} \left (a^{2}+b^{2}\right )^{2}}+\frac {3 a^{2} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{2}}+\frac {a^{3}}{b^{2} \left (a^{2}+b^{2}\right ) d \left (a +b \tan \left (d x +c \right )\right )}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2}}{2 d \left (a^{2}+b^{2}\right )^{2}}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) b^{2}}{2 d \left (a^{2}+b^{2}\right )^{2}}-\frac {2 a b \arctan \left (\tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.82, size = 155, normalized size = 1.36 \[ \frac {\frac {2 \, a^{3}}{a^{3} b^{2} + a b^{4} + {\left (a^{2} b^{3} + b^{5}\right )} \tan \left (d x + c\right )} - \frac {4 \, {\left (d x + c\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (a^{4} + 3 \, a^{2} b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}} - \frac {{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 4.25, size = 137, normalized size = 1.20 \[ \frac {a^3}{b^2\,d\,\left (a^2+b^2\right )\,\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^2+a\,b\,2{}\mathrm {i}-b^2\right )}+\frac {a^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^2+3\,b^2\right )}{b^2\,d\,{\left (a^2+b^2\right )}^2}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^2\,1{}\mathrm {i}+2\,a\,b-b^2\,1{}\mathrm {i}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3/(b^2*d*(a^2 + b^2)*(a + b*tan(c + d*x))) - (log(tan(c + d*x) + 1i)*1i)/(2*d*(2*a*b + a^2*1i - b^2*1i)) - l
og(tan(c + d*x) - 1i)/(2*d*(a*b*2i + a^2 - b^2)) + (a^2*log(a + b*tan(c + d*x))*(a^2 + 3*b^2))/(b^2*d*(a^2 + b
^2)^2)

________________________________________________________________________________________

sympy [A]  time = 2.08, size = 1992, normalized size = 17.47 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x*tan(c), Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((-log(tan(c + d*x)**2 + 1)/(2*d) + tan(c + d*x)**2/
(2*d))/a**2, Eq(b, 0)), (3*I*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*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*d*x/(4*b**2*d*ta
n(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 2*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*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*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) - 5*I*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/(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*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*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*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*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*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*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) + 5*I*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/(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*tan(c)**3/(a + b*tan(c))**2, Eq(d, 0)), (2*a**5*log(a/b + tan(c +
d*x))/(2*a**5*b**2*d + 2*a**4*b**3*d*tan(c + d*x) + 4*a**3*b**4*d + 4*a**2*b**5*d*tan(c + d*x) + 2*a*b**6*d +
2*b**7*d*tan(c + d*x)) + 2*a**5/(2*a**5*b**2*d + 2*a**4*b**3*d*tan(c + d*x) + 4*a**3*b**4*d + 4*a**2*b**5*d*t
an(c + d*x) + 2*a*b**6*d + 2*b**7*d*tan(c + d*x)) + 2*a**4*b*log(a/b + tan(c + d*x))*tan(c + d*x)/(2*a**5*b**2
*d + 2*a**4*b**3*d*tan(c + d*x) + 4*a**3*b**4*d + 4*a**2*b**5*d*tan(c + d*x) + 2*a*b**6*d + 2*b**7*d*tan(c + d
*x)) + 6*a**3*b**2*log(a/b + tan(c + d*x))/(2*a**5*b**2*d + 2*a**4*b**3*d*tan(c + d*x) + 4*a**3*b**4*d + 4*a**
2*b**5*d*tan(c + d*x) + 2*a*b**6*d + 2*b**7*d*tan(c + d*x)) - a**3*b**2*log(tan(c + d*x)**2 + 1)/(2*a**5*b**2*
d + 2*a**4*b**3*d*tan(c + d*x) + 4*a**3*b**4*d + 4*a**2*b**5*d*tan(c + d*x) + 2*a*b**6*d + 2*b**7*d*tan(c + d*
x)) + 2*a**3*b**2/(2*a**5*b**2*d + 2*a**4*b**3*d*tan(c + d*x) + 4*a**3*b**4*d + 4*a**2*b**5*d*tan(c + d*x) + 2
*a*b**6*d + 2*b**7*d*tan(c + d*x)) - 4*a**2*b**3*d*x/(2*a**5*b**2*d + 2*a**4*b**3*d*tan(c + d*x) + 4*a**3*b**4
*d + 4*a**2*b**5*d*tan(c + d*x) + 2*a*b**6*d + 2*b**7*d*tan(c + d*x)) + 6*a**2*b**3*log(a/b + tan(c + d*x))*ta
n(c + d*x)/(2*a**5*b**2*d + 2*a**4*b**3*d*tan(c + d*x) + 4*a**3*b**4*d + 4*a**2*b**5*d*tan(c + d*x) + 2*a*b**6
*d + 2*b**7*d*tan(c + d*x)) - a**2*b**3*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(2*a**5*b**2*d + 2*a**4*b**3*d*t
an(c + d*x) + 4*a**3*b**4*d + 4*a**2*b**5*d*tan(c + d*x) + 2*a*b**6*d + 2*b**7*d*tan(c + d*x)) - 4*a*b**4*d*x*
tan(c + d*x)/(2*a**5*b**2*d + 2*a**4*b**3*d*tan(c + d*x) + 4*a**3*b**4*d + 4*a**2*b**5*d*tan(c + d*x) + 2*a*b*
*6*d + 2*b**7*d*tan(c + d*x)) + a*b**4*log(tan(c + d*x)**2 + 1)/(2*a**5*b**2*d + 2*a**4*b**3*d*tan(c + d*x) +
4*a**3*b**4*d + 4*a**2*b**5*d*tan(c + d*x) + 2*a*b**6*d + 2*b**7*d*tan(c + d*x)) + b**5*log(tan(c + d*x)**2 +
1)*tan(c + d*x)/(2*a**5*b**2*d + 2*a**4*b**3*d*tan(c + d*x) + 4*a**3*b**4*d + 4*a**2*b**5*d*tan(c + d*x) + 2*a
*b**6*d + 2*b**7*d*tan(c + d*x)), True))

________________________________________________________________________________________