3.231 \(\int \frac {\tan ^4(e+f x)}{(a+b \tan ^2(e+f x))^2} \, dx\)

Optimal. Leaf size=95 \[ \frac {\sqrt {a} (a-3 b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{2 b^{3/2} f (a-b)^2}-\frac {a \tan (e+f x)}{2 b f (a-b) \left (a+b \tan ^2(e+f x)\right )}+\frac {x}{(a-b)^2} \]

[Out]

x/(a-b)^2+1/2*(a-3*b)*arctan(b^(1/2)*tan(f*x+e)/a^(1/2))*a^(1/2)/(a-b)^2/b^(3/2)/f-1/2*a*tan(f*x+e)/(a-b)/b/f/
(a+b*tan(f*x+e)^2)

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Rubi [A]  time = 0.12, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3670, 470, 522, 203, 205} \[ \frac {\sqrt {a} (a-3 b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{2 b^{3/2} f (a-b)^2}-\frac {a \tan (e+f x)}{2 b f (a-b) \left (a+b \tan ^2(e+f x)\right )}+\frac {x}{(a-b)^2} \]

Antiderivative was successfully verified.

[In]

Int[Tan[e + f*x]^4/(a + b*Tan[e + f*x]^2)^2,x]

[Out]

x/(a - b)^2 + (Sqrt[a]*(a - 3*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*(a - b)^2*b^(3/2)*f) - (a*Tan[e +
f*x])/(2*(a - b)*b*f*(a + b*Tan[e + f*x]^2))

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(a*e^(2
*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*n*(b*c - a*d)*(p + 1)), x] + Dist[e^(2
*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) +
(a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 3670

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff)/f, Subst[Int[(((d*ff*x)/c)^m*(a + b*(ff*x)^n)^p)/(c^
2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rubi steps

\begin {align*} \int \frac {\tan ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4}{\left (1+x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {a \tan (e+f x)}{2 (a-b) b f \left (a+b \tan ^2(e+f x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {a+(a-2 b) x^2}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{2 (a-b) b f}\\ &=-\frac {a \tan (e+f x)}{2 (a-b) b f \left (a+b \tan ^2(e+f x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{(a-b)^2 f}+\frac {(a (a-3 b)) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tan (e+f x)\right )}{2 (a-b)^2 b f}\\ &=\frac {x}{(a-b)^2}+\frac {\sqrt {a} (a-3 b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{2 (a-b)^2 b^{3/2} f}-\frac {a \tan (e+f x)}{2 (a-b) b f \left (a+b \tan ^2(e+f x)\right )}\\ \end {align*}

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Mathematica [A]  time = 0.82, size = 94, normalized size = 0.99 \[ \frac {\frac {\sqrt {a} (a-3 b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{b^{3/2}}-\frac {a (a-b) \sin (2 (e+f x))}{b ((a-b) \cos (2 (e+f x))+a+b)}+2 (e+f x)}{2 f (a-b)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[e + f*x]^4/(a + b*Tan[e + f*x]^2)^2,x]

[Out]

(2*(e + f*x) + (Sqrt[a]*(a - 3*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/b^(3/2) - (a*(a - b)*Sin[2*(e + f*x)
])/(b*(a + b + (a - b)*Cos[2*(e + f*x)])))/(2*(a - b)^2*f)

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fricas [A]  time = 0.49, size = 381, normalized size = 4.01 \[ \left [\frac {8 \, b^{2} f x \tan \left (f x + e\right )^{2} + 8 \, a b f x - {\left ({\left (a b - 3 \, b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} - 3 \, a b\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b^{2} \tan \left (f x + e\right )^{4} - 6 \, a b \tan \left (f x + e\right )^{2} + a^{2} - 4 \, {\left (b^{2} \tan \left (f x + e\right )^{3} - a b \tan \left (f x + e\right )\right )} \sqrt {-\frac {a}{b}}}{b^{2} \tan \left (f x + e\right )^{4} + 2 \, a b \tan \left (f x + e\right )^{2} + a^{2}}\right ) - 4 \, {\left (a^{2} - a b\right )} \tan \left (f x + e\right )}{8 \, {\left ({\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} f\right )}}, \frac {4 \, b^{2} f x \tan \left (f x + e\right )^{2} + 4 \, a b f x + {\left ({\left (a b - 3 \, b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} - 3 \, a b\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {{\left (b \tan \left (f x + e\right )^{2} - a\right )} \sqrt {\frac {a}{b}}}{2 \, a \tan \left (f x + e\right )}\right ) - 2 \, {\left (a^{2} - a b\right )} \tan \left (f x + e\right )}{4 \, {\left ({\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} f\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^4/(a+b*tan(f*x+e)^2)^2,x, algorithm="fricas")

[Out]

[1/8*(8*b^2*f*x*tan(f*x + e)^2 + 8*a*b*f*x - ((a*b - 3*b^2)*tan(f*x + e)^2 + a^2 - 3*a*b)*sqrt(-a/b)*log((b^2*
tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 - 4*(b^2*tan(f*x + e)^3 - a*b*tan(f*x + e))*sqrt(-a/b))/(b^2*tan(f
*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)) - 4*(a^2 - a*b)*tan(f*x + e))/((a^2*b^2 - 2*a*b^3 + b^4)*f*tan(f*x +
e)^2 + (a^3*b - 2*a^2*b^2 + a*b^3)*f), 1/4*(4*b^2*f*x*tan(f*x + e)^2 + 4*a*b*f*x + ((a*b - 3*b^2)*tan(f*x + e)
^2 + a^2 - 3*a*b)*sqrt(a/b)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(a/b)/(a*tan(f*x + e))) - 2*(a^2 - a*b)*tan(
f*x + e))/((a^2*b^2 - 2*a*b^3 + b^4)*f*tan(f*x + e)^2 + (a^3*b - 2*a^2*b^2 + a*b^3)*f)]

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giac [A]  time = 3.70, size = 127, normalized size = 1.34 \[ \frac {\frac {{\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (b) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )\right )} {\left (a^{2} - 3 \, a b\right )}}{{\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} \sqrt {a b}} + \frac {2 \, {\left (f x + e\right )}}{a^{2} - 2 \, a b + b^{2}} - \frac {a \tan \left (f x + e\right )}{{\left (b \tan \left (f x + e\right )^{2} + a\right )} {\left (a b - b^{2}\right )}}}{2 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^4/(a+b*tan(f*x+e)^2)^2,x, algorithm="giac")

[Out]

1/2*((pi*floor((f*x + e)/pi + 1/2)*sgn(b) + arctan(b*tan(f*x + e)/sqrt(a*b)))*(a^2 - 3*a*b)/((a^2*b - 2*a*b^2
+ b^3)*sqrt(a*b)) + 2*(f*x + e)/(a^2 - 2*a*b + b^2) - a*tan(f*x + e)/((b*tan(f*x + e)^2 + a)*(a*b - b^2)))/f

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maple [A]  time = 0.20, size = 160, normalized size = 1.68 \[ -\frac {a^{2} \tan \left (f x +e \right )}{2 f b \left (a -b \right )^{2} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )}+\frac {a \tan \left (f x +e \right )}{2 f \left (a -b \right )^{2} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )}+\frac {a^{2} \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {a b}}\right )}{2 f b \left (a -b \right )^{2} \sqrt {a b}}-\frac {3 a \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {a b}}\right )}{2 f \left (a -b \right )^{2} \sqrt {a b}}+\frac {\arctan \left (\tan \left (f x +e \right )\right )}{f \left (a -b \right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(f*x+e)^4/(a+b*tan(f*x+e)^2)^2,x)

[Out]

-1/2/f*a^2/b/(a-b)^2*tan(f*x+e)/(a+b*tan(f*x+e)^2)+1/2/f*a/(a-b)^2*tan(f*x+e)/(a+b*tan(f*x+e)^2)+1/2/f*a^2/b/(
a-b)^2/(a*b)^(1/2)*arctan(tan(f*x+e)*b/(a*b)^(1/2))-3/2/f*a/(a-b)^2/(a*b)^(1/2)*arctan(tan(f*x+e)*b/(a*b)^(1/2
))+1/f/(a-b)^2*arctan(tan(f*x+e))

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maxima [A]  time = 0.75, size = 114, normalized size = 1.20 \[ -\frac {\frac {a \tan \left (f x + e\right )}{a^{2} b - a b^{2} + {\left (a b^{2} - b^{3}\right )} \tan \left (f x + e\right )^{2}} - \frac {{\left (a^{2} - 3 \, a b\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )}{{\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} \sqrt {a b}} - \frac {2 \, {\left (f x + e\right )}}{a^{2} - 2 \, a b + b^{2}}}{2 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^4/(a+b*tan(f*x+e)^2)^2,x, algorithm="maxima")

[Out]

-1/2*(a*tan(f*x + e)/(a^2*b - a*b^2 + (a*b^2 - b^3)*tan(f*x + e)^2) - (a^2 - 3*a*b)*arctan(b*tan(f*x + e)/sqrt
(a*b))/((a^2*b - 2*a*b^2 + b^3)*sqrt(a*b)) - 2*(f*x + e)/(a^2 - 2*a*b + b^2))/f

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mupad [B]  time = 13.52, size = 2358, normalized size = 24.82 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(e + f*x)^4/(a + b*tan(e + f*x)^2)^2,x)

[Out]

(2*atan((((((2*a*b^6 - 8*a^2*b^5 + 12*a^3*b^4 - 8*a^4*b^3 + 2*a^5*b^2)*1i)/(3*a*b^3 + a^3*b - b^4 - 3*a^2*b^2)
- (tan(e + f*x)*(16*b^8 - 48*a*b^7 + 32*a^2*b^6 + 32*a^3*b^5 - 48*a^4*b^4 + 16*a^5*b^3))/(2*(a^2*b - 2*a*b^2
+ b^3)*(2*a^2 - 4*a*b + 2*b^2)))/(2*a^2 - 4*a*b + 2*b^2) + (tan(e + f*x)*(a^4 - 6*a^3*b + 4*b^4 + 9*a^2*b^2))/
(2*(a^2*b - 2*a*b^2 + b^3)))/(2*a^2 - 4*a*b + 2*b^2) - ((((2*a*b^6 - 8*a^2*b^5 + 12*a^3*b^4 - 8*a^4*b^3 + 2*a^
5*b^2)*1i)/(3*a*b^3 + a^3*b - b^4 - 3*a^2*b^2) + (tan(e + f*x)*(16*b^8 - 48*a*b^7 + 32*a^2*b^6 + 32*a^3*b^5 -
48*a^4*b^4 + 16*a^5*b^3))/(2*(a^2*b - 2*a*b^2 + b^3)*(2*a^2 - 4*a*b + 2*b^2)))/(2*a^2 - 4*a*b + 2*b^2) - (tan(
e + f*x)*(a^4 - 6*a^3*b + 4*b^4 + 9*a^2*b^2))/(2*(a^2*b - 2*a*b^2 + b^3)))/(2*a^2 - 4*a*b + 2*b^2))/((((((2*a*
b^6 - 8*a^2*b^5 + 12*a^3*b^4 - 8*a^4*b^3 + 2*a^5*b^2)*1i)/(3*a*b^3 + a^3*b - b^4 - 3*a^2*b^2) - (tan(e + f*x)*
(16*b^8 - 48*a*b^7 + 32*a^2*b^6 + 32*a^3*b^5 - 48*a^4*b^4 + 16*a^5*b^3))/(2*(a^2*b - 2*a*b^2 + b^3)*(2*a^2 - 4
*a*b + 2*b^2)))*1i)/(2*a^2 - 4*a*b + 2*b^2) + (tan(e + f*x)*(a^4 - 6*a^3*b + 4*b^4 + 9*a^2*b^2)*1i)/(2*(a^2*b
- 2*a*b^2 + b^3)))/(2*a^2 - 4*a*b + 2*b^2) + (((((2*a*b^6 - 8*a^2*b^5 + 12*a^3*b^4 - 8*a^4*b^3 + 2*a^5*b^2)*1i
)/(3*a*b^3 + a^3*b - b^4 - 3*a^2*b^2) + (tan(e + f*x)*(16*b^8 - 48*a*b^7 + 32*a^2*b^6 + 32*a^3*b^5 - 48*a^4*b^
4 + 16*a^5*b^3))/(2*(a^2*b - 2*a*b^2 + b^3)*(2*a^2 - 4*a*b + 2*b^2)))*1i)/(2*a^2 - 4*a*b + 2*b^2) - (tan(e + f
*x)*(a^4 - 6*a^3*b + 4*b^4 + 9*a^2*b^2)*1i)/(2*(a^2*b - 2*a*b^2 + b^3)))/(2*a^2 - 4*a*b + 2*b^2) + (3*a*b^2 -
(5*a^2*b)/2 + a^3/2)/(3*a*b^3 + a^3*b - b^4 - 3*a^2*b^2))))/(f*(2*a^2 - 4*a*b + 2*b^2)) + (atan(((((tan(e + f*
x)*(a^4 - 6*a^3*b + 4*b^4 + 9*a^2*b^2))/(2*(a^2*b - 2*a*b^2 + b^3)) - (((2*a*b^6 - 8*a^2*b^5 + 12*a^3*b^4 - 8*
a^4*b^3 + 2*a^5*b^2)/(3*a*b^3 + a^3*b - b^4 - 3*a^2*b^2) - (tan(e + f*x)*(a - 3*b)*(-a*b^3)^(1/2)*(16*b^8 - 48
*a*b^7 + 32*a^2*b^6 + 32*a^3*b^5 - 48*a^4*b^4 + 16*a^5*b^3))/(8*(a^2*b - 2*a*b^2 + b^3)*(b^5 - 2*a*b^4 + a^2*b
^3)))*(a - 3*b)*(-a*b^3)^(1/2))/(4*(b^5 - 2*a*b^4 + a^2*b^3)))*(a - 3*b)*(-a*b^3)^(1/2)*1i)/(4*(b^5 - 2*a*b^4
+ a^2*b^3)) + (((tan(e + f*x)*(a^4 - 6*a^3*b + 4*b^4 + 9*a^2*b^2))/(2*(a^2*b - 2*a*b^2 + b^3)) + (((2*a*b^6 -
8*a^2*b^5 + 12*a^3*b^4 - 8*a^4*b^3 + 2*a^5*b^2)/(3*a*b^3 + a^3*b - b^4 - 3*a^2*b^2) + (tan(e + f*x)*(a - 3*b)*
(-a*b^3)^(1/2)*(16*b^8 - 48*a*b^7 + 32*a^2*b^6 + 32*a^3*b^5 - 48*a^4*b^4 + 16*a^5*b^3))/(8*(a^2*b - 2*a*b^2 +
b^3)*(b^5 - 2*a*b^4 + a^2*b^3)))*(a - 3*b)*(-a*b^3)^(1/2))/(4*(b^5 - 2*a*b^4 + a^2*b^3)))*(a - 3*b)*(-a*b^3)^(
1/2)*1i)/(4*(b^5 - 2*a*b^4 + a^2*b^3)))/((3*a*b^2 - (5*a^2*b)/2 + a^3/2)/(3*a*b^3 + a^3*b - b^4 - 3*a^2*b^2) -
(((tan(e + f*x)*(a^4 - 6*a^3*b + 4*b^4 + 9*a^2*b^2))/(2*(a^2*b - 2*a*b^2 + b^3)) - (((2*a*b^6 - 8*a^2*b^5 + 1
2*a^3*b^4 - 8*a^4*b^3 + 2*a^5*b^2)/(3*a*b^3 + a^3*b - b^4 - 3*a^2*b^2) - (tan(e + f*x)*(a - 3*b)*(-a*b^3)^(1/2
)*(16*b^8 - 48*a*b^7 + 32*a^2*b^6 + 32*a^3*b^5 - 48*a^4*b^4 + 16*a^5*b^3))/(8*(a^2*b - 2*a*b^2 + b^3)*(b^5 - 2
*a*b^4 + a^2*b^3)))*(a - 3*b)*(-a*b^3)^(1/2))/(4*(b^5 - 2*a*b^4 + a^2*b^3)))*(a - 3*b)*(-a*b^3)^(1/2))/(4*(b^5
- 2*a*b^4 + a^2*b^3)) + (((tan(e + f*x)*(a^4 - 6*a^3*b + 4*b^4 + 9*a^2*b^2))/(2*(a^2*b - 2*a*b^2 + b^3)) + ((
(2*a*b^6 - 8*a^2*b^5 + 12*a^3*b^4 - 8*a^4*b^3 + 2*a^5*b^2)/(3*a*b^3 + a^3*b - b^4 - 3*a^2*b^2) + (tan(e + f*x)
*(a - 3*b)*(-a*b^3)^(1/2)*(16*b^8 - 48*a*b^7 + 32*a^2*b^6 + 32*a^3*b^5 - 48*a^4*b^4 + 16*a^5*b^3))/(8*(a^2*b -
2*a*b^2 + b^3)*(b^5 - 2*a*b^4 + a^2*b^3)))*(a - 3*b)*(-a*b^3)^(1/2))/(4*(b^5 - 2*a*b^4 + a^2*b^3)))*(a - 3*b)
*(-a*b^3)^(1/2))/(4*(b^5 - 2*a*b^4 + a^2*b^3))))*(a - 3*b)*(-a*b^3)^(1/2)*1i)/(2*f*(b^5 - 2*a*b^4 + a^2*b^3))
- (a*tan(e + f*x))/(2*b*f*(a + b*tan(e + f*x)^2)*(a - b))

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sympy [A]  time = 28.76, size = 2416, normalized size = 25.43 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)**4/(a+b*tan(f*x+e)**2)**2,x)

[Out]

Piecewise((zoo*x, Eq(a, 0) & Eq(b, 0) & Eq(f, 0)), (x/b**2, Eq(a, 0)), (3*f*x*tan(e + f*x)**4/(8*b**2*f*tan(e
+ f*x)**4 + 16*b**2*f*tan(e + f*x)**2 + 8*b**2*f) + 6*f*x*tan(e + f*x)**2/(8*b**2*f*tan(e + f*x)**4 + 16*b**2*
f*tan(e + f*x)**2 + 8*b**2*f) + 3*f*x/(8*b**2*f*tan(e + f*x)**4 + 16*b**2*f*tan(e + f*x)**2 + 8*b**2*f) - 5*ta
n(e + f*x)**3/(8*b**2*f*tan(e + f*x)**4 + 16*b**2*f*tan(e + f*x)**2 + 8*b**2*f) - 3*tan(e + f*x)/(8*b**2*f*tan
(e + f*x)**4 + 16*b**2*f*tan(e + f*x)**2 + 8*b**2*f), Eq(a, b)), (x*tan(e)**4/(a + b*tan(e)**2)**2, Eq(f, 0)),
((x + tan(e + f*x)**3/(3*f) - tan(e + f*x)/f)/a**2, Eq(b, 0)), (-2*I*a**(5/2)*b*sqrt(1/b)*tan(e + f*x)/(4*I*a
**(7/2)*b**2*f*sqrt(1/b) + 4*I*a**(5/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 8*I*a**(5/2)*b**3*f*sqrt(1/b) - 8*I
*a**(3/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 4*I*a**(3/2)*b**4*f*sqrt(1/b) + 4*I*sqrt(a)*b**5*f*sqrt(1/b)*tan(
e + f*x)**2) + 4*I*a**(3/2)*b**2*f*x*sqrt(1/b)/(4*I*a**(7/2)*b**2*f*sqrt(1/b) + 4*I*a**(5/2)*b**3*f*sqrt(1/b)*
tan(e + f*x)**2 - 8*I*a**(5/2)*b**3*f*sqrt(1/b) - 8*I*a**(3/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 4*I*a**(3/2)
*b**4*f*sqrt(1/b) + 4*I*sqrt(a)*b**5*f*sqrt(1/b)*tan(e + f*x)**2) + 2*I*a**(3/2)*b**2*sqrt(1/b)*tan(e + f*x)/(
4*I*a**(7/2)*b**2*f*sqrt(1/b) + 4*I*a**(5/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 8*I*a**(5/2)*b**3*f*sqrt(1/b)
- 8*I*a**(3/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 4*I*a**(3/2)*b**4*f*sqrt(1/b) + 4*I*sqrt(a)*b**5*f*sqrt(1/b)
*tan(e + f*x)**2) + 4*I*sqrt(a)*b**3*f*x*sqrt(1/b)*tan(e + f*x)**2/(4*I*a**(7/2)*b**2*f*sqrt(1/b) + 4*I*a**(5/
2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 8*I*a**(5/2)*b**3*f*sqrt(1/b) - 8*I*a**(3/2)*b**4*f*sqrt(1/b)*tan(e + f*
x)**2 + 4*I*a**(3/2)*b**4*f*sqrt(1/b) + 4*I*sqrt(a)*b**5*f*sqrt(1/b)*tan(e + f*x)**2) + a**3*log(-I*sqrt(a)*sq
rt(1/b) + tan(e + f*x))/(4*I*a**(7/2)*b**2*f*sqrt(1/b) + 4*I*a**(5/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 8*I*a
**(5/2)*b**3*f*sqrt(1/b) - 8*I*a**(3/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 4*I*a**(3/2)*b**4*f*sqrt(1/b) + 4*I
*sqrt(a)*b**5*f*sqrt(1/b)*tan(e + f*x)**2) - a**3*log(I*sqrt(a)*sqrt(1/b) + tan(e + f*x))/(4*I*a**(7/2)*b**2*f
*sqrt(1/b) + 4*I*a**(5/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 8*I*a**(5/2)*b**3*f*sqrt(1/b) - 8*I*a**(3/2)*b**4
*f*sqrt(1/b)*tan(e + f*x)**2 + 4*I*a**(3/2)*b**4*f*sqrt(1/b) + 4*I*sqrt(a)*b**5*f*sqrt(1/b)*tan(e + f*x)**2) +
a**2*b*log(-I*sqrt(a)*sqrt(1/b) + tan(e + f*x))*tan(e + f*x)**2/(4*I*a**(7/2)*b**2*f*sqrt(1/b) + 4*I*a**(5/2)
*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 8*I*a**(5/2)*b**3*f*sqrt(1/b) - 8*I*a**(3/2)*b**4*f*sqrt(1/b)*tan(e + f*x)
**2 + 4*I*a**(3/2)*b**4*f*sqrt(1/b) + 4*I*sqrt(a)*b**5*f*sqrt(1/b)*tan(e + f*x)**2) - 3*a**2*b*log(-I*sqrt(a)*
sqrt(1/b) + tan(e + f*x))/(4*I*a**(7/2)*b**2*f*sqrt(1/b) + 4*I*a**(5/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 8*I
*a**(5/2)*b**3*f*sqrt(1/b) - 8*I*a**(3/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 4*I*a**(3/2)*b**4*f*sqrt(1/b) + 4
*I*sqrt(a)*b**5*f*sqrt(1/b)*tan(e + f*x)**2) - a**2*b*log(I*sqrt(a)*sqrt(1/b) + tan(e + f*x))*tan(e + f*x)**2/
(4*I*a**(7/2)*b**2*f*sqrt(1/b) + 4*I*a**(5/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 8*I*a**(5/2)*b**3*f*sqrt(1/b)
- 8*I*a**(3/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 4*I*a**(3/2)*b**4*f*sqrt(1/b) + 4*I*sqrt(a)*b**5*f*sqrt(1/b
)*tan(e + f*x)**2) + 3*a**2*b*log(I*sqrt(a)*sqrt(1/b) + tan(e + f*x))/(4*I*a**(7/2)*b**2*f*sqrt(1/b) + 4*I*a**
(5/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 8*I*a**(5/2)*b**3*f*sqrt(1/b) - 8*I*a**(3/2)*b**4*f*sqrt(1/b)*tan(e +
f*x)**2 + 4*I*a**(3/2)*b**4*f*sqrt(1/b) + 4*I*sqrt(a)*b**5*f*sqrt(1/b)*tan(e + f*x)**2) - 3*a*b**2*log(-I*sqr
t(a)*sqrt(1/b) + tan(e + f*x))*tan(e + f*x)**2/(4*I*a**(7/2)*b**2*f*sqrt(1/b) + 4*I*a**(5/2)*b**3*f*sqrt(1/b)*
tan(e + f*x)**2 - 8*I*a**(5/2)*b**3*f*sqrt(1/b) - 8*I*a**(3/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 4*I*a**(3/2)
*b**4*f*sqrt(1/b) + 4*I*sqrt(a)*b**5*f*sqrt(1/b)*tan(e + f*x)**2) + 3*a*b**2*log(I*sqrt(a)*sqrt(1/b) + tan(e +
f*x))*tan(e + f*x)**2/(4*I*a**(7/2)*b**2*f*sqrt(1/b) + 4*I*a**(5/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 8*I*a*
*(5/2)*b**3*f*sqrt(1/b) - 8*I*a**(3/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 4*I*a**(3/2)*b**4*f*sqrt(1/b) + 4*I*
sqrt(a)*b**5*f*sqrt(1/b)*tan(e + f*x)**2), True))

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