3.244 \(\int \frac {\tan ^4(e+f x)}{(a+b \tan ^2(e+f x))^3} \, dx\)

Optimal. Leaf size=145 \[ \frac {\left (a^2-6 a b-3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{8 \sqrt {a} b^{3/2} f (a-b)^3}+\frac {(a-5 b) \tan (e+f x)}{8 b f (a-b)^2 \left (a+b \tan ^2(e+f x)\right )}-\frac {a \tan (e+f x)}{4 b f (a-b) \left (a+b \tan ^2(e+f x)\right )^2}+\frac {x}{(a-b)^3} \]

[Out]

x/(a-b)^3+1/8*(a^2-6*a*b-3*b^2)*arctan(b^(1/2)*tan(f*x+e)/a^(1/2))/(a-b)^3/b^(3/2)/f/a^(1/2)-1/4*a*tan(f*x+e)/
(a-b)/b/f/(a+b*tan(f*x+e)^2)^2+1/8*(a-5*b)*tan(f*x+e)/(a-b)^2/b/f/(a+b*tan(f*x+e)^2)

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Rubi [A]  time = 0.18, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3670, 470, 527, 522, 203, 205} \[ \frac {\left (a^2-6 a b-3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{8 \sqrt {a} b^{3/2} f (a-b)^3}+\frac {(a-5 b) \tan (e+f x)}{8 b f (a-b)^2 \left (a+b \tan ^2(e+f x)\right )}-\frac {a \tan (e+f x)}{4 b f (a-b) \left (a+b \tan ^2(e+f x)\right )^2}+\frac {x}{(a-b)^3} \]

Antiderivative was successfully verified.

[In]

Int[Tan[e + f*x]^4/(a + b*Tan[e + f*x]^2)^3,x]

[Out]

x/(a - b)^3 + ((a^2 - 6*a*b - 3*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*Sqrt[a]*(a - b)^3*b^(3/2)*f) -
(a*Tan[e + f*x])/(4*(a - b)*b*f*(a + b*Tan[e + f*x]^2)^2) + ((a - 5*b)*Tan[e + f*x])/(8*(a - b)^2*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 527

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

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 )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4}{\left (1+x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {a \tan (e+f x)}{4 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {a+(a-4 b) x^2}{\left (1+x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{4 (a-b) b f}\\ &=-\frac {a \tan (e+f x)}{4 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^2}+\frac {(a-5 b) \tan (e+f x)}{8 (a-b)^2 b f \left (a+b \tan ^2(e+f x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {a (a+3 b)+a (a-5 b) x^2}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{8 a (a-b)^2 b f}\\ &=-\frac {a \tan (e+f x)}{4 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^2}+\frac {(a-5 b) \tan (e+f x)}{8 (a-b)^2 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)^3 f}+\frac {\left (a^2-6 a b-3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tan (e+f x)\right )}{8 (a-b)^3 b f}\\ &=\frac {x}{(a-b)^3}+\frac {\left (a^2-6 a b-3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{8 \sqrt {a} (a-b)^3 b^{3/2} f}-\frac {a \tan (e+f x)}{4 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^2}+\frac {(a-5 b) \tan (e+f x)}{8 (a-b)^2 b f \left (a+b \tan ^2(e+f x)\right )}\\ \end {align*}

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Mathematica [A]  time = 2.09, size = 136, normalized size = 0.94 \[ \frac {-\frac {(a-b) \sin (2 (e+f x)) \left (\left (a^2+4 a b-5 b^2\right ) \cos (2 (e+f x))+a^2+2 a b+5 b^2\right )}{b ((a-b) \cos (2 (e+f x))+a+b)^2}+\frac {\left (a^2-6 a b-3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}+8 (e+f x)}{8 f (a-b)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[e + f*x]^4/(a + b*Tan[e + f*x]^2)^3,x]

[Out]

(8*(e + f*x) + ((a^2 - 6*a*b - 3*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(Sqrt[a]*b^(3/2)) - ((a - b)*(a^
2 + 2*a*b + 5*b^2 + (a^2 + 4*a*b - 5*b^2)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(b*(a + b + (a - b)*Cos[2*(e + f
*x)])^2))/(8*(a - b)^3*f)

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fricas [B]  time = 0.50, size = 749, normalized size = 5.17 \[ \left [\frac {32 \, a b^{4} f x \tan \left (f x + e\right )^{4} + 64 \, a^{2} b^{3} f x \tan \left (f x + e\right )^{2} + 32 \, a^{3} b^{2} f x + 4 \, {\left (a^{3} b^{2} - 6 \, a^{2} b^{3} + 5 \, a b^{4}\right )} \tan \left (f x + e\right )^{3} - {\left ({\left (a^{2} b^{2} - 6 \, a b^{3} - 3 \, b^{4}\right )} \tan \left (f x + e\right )^{4} + a^{4} - 6 \, a^{3} b - 3 \, a^{2} b^{2} + 2 \, {\left (a^{3} b - 6 \, a^{2} b^{2} - 3 \, a b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {-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 \tan \left (f x + e\right )^{3} - a \tan \left (f x + e\right )\right )} \sqrt {-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^{4} b + 2 \, a^{3} b^{2} - 3 \, a^{2} b^{3}\right )} \tan \left (f x + e\right )}{32 \, {\left ({\left (a^{4} b^{4} - 3 \, a^{3} b^{5} + 3 \, a^{2} b^{6} - a b^{7}\right )} f \tan \left (f x + e\right )^{4} + 2 \, {\left (a^{5} b^{3} - 3 \, a^{4} b^{4} + 3 \, a^{3} b^{5} - a^{2} b^{6}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5}\right )} f\right )}}, \frac {16 \, a b^{4} f x \tan \left (f x + e\right )^{4} + 32 \, a^{2} b^{3} f x \tan \left (f x + e\right )^{2} + 16 \, a^{3} b^{2} f x + 2 \, {\left (a^{3} b^{2} - 6 \, a^{2} b^{3} + 5 \, a b^{4}\right )} \tan \left (f x + e\right )^{3} + {\left ({\left (a^{2} b^{2} - 6 \, a b^{3} - 3 \, b^{4}\right )} \tan \left (f x + e\right )^{4} + a^{4} - 6 \, a^{3} b - 3 \, a^{2} b^{2} + 2 \, {\left (a^{3} b - 6 \, a^{2} b^{2} - 3 \, a b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {a b} \arctan \left (\frac {{\left (b \tan \left (f x + e\right )^{2} - a\right )} \sqrt {a b}}{2 \, a b \tan \left (f x + e\right )}\right ) - 2 \, {\left (a^{4} b + 2 \, a^{3} b^{2} - 3 \, a^{2} b^{3}\right )} \tan \left (f x + e\right )}{16 \, {\left ({\left (a^{4} b^{4} - 3 \, a^{3} b^{5} + 3 \, a^{2} b^{6} - a b^{7}\right )} f \tan \left (f x + e\right )^{4} + 2 \, {\left (a^{5} b^{3} - 3 \, a^{4} b^{4} + 3 \, a^{3} b^{5} - a^{2} b^{6}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5}\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)^3,x, algorithm="fricas")

[Out]

[1/32*(32*a*b^4*f*x*tan(f*x + e)^4 + 64*a^2*b^3*f*x*tan(f*x + e)^2 + 32*a^3*b^2*f*x + 4*(a^3*b^2 - 6*a^2*b^3 +
5*a*b^4)*tan(f*x + e)^3 - ((a^2*b^2 - 6*a*b^3 - 3*b^4)*tan(f*x + e)^4 + a^4 - 6*a^3*b - 3*a^2*b^2 + 2*(a^3*b
- 6*a^2*b^2 - 3*a*b^3)*tan(f*x + e)^2)*sqrt(-a*b)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 - 4*(b*
tan(f*x + e)^3 - a*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^4*b + 2
*a^3*b^2 - 3*a^2*b^3)*tan(f*x + e))/((a^4*b^4 - 3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*f*tan(f*x + e)^4 + 2*(a^5*b^3 -
3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*f*tan(f*x + e)^2 + (a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)*f), 1/16*(16*
a*b^4*f*x*tan(f*x + e)^4 + 32*a^2*b^3*f*x*tan(f*x + e)^2 + 16*a^3*b^2*f*x + 2*(a^3*b^2 - 6*a^2*b^3 + 5*a*b^4)*
tan(f*x + e)^3 + ((a^2*b^2 - 6*a*b^3 - 3*b^4)*tan(f*x + e)^4 + a^4 - 6*a^3*b - 3*a^2*b^2 + 2*(a^3*b - 6*a^2*b^
2 - 3*a*b^3)*tan(f*x + e)^2)*sqrt(a*b)*arctan(1/2*(b*tan(f*x + e)^2 - a)*sqrt(a*b)/(a*b*tan(f*x + e))) - 2*(a^
4*b + 2*a^3*b^2 - 3*a^2*b^3)*tan(f*x + e))/((a^4*b^4 - 3*a^3*b^5 + 3*a^2*b^6 - a*b^7)*f*tan(f*x + e)^4 + 2*(a^
5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*f*tan(f*x + e)^2 + (a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5)*f)]

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giac [A]  time = 4.02, size = 199, normalized size = 1.37 \[ \frac {\frac {{\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (b) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )\right )} {\left (a^{2} - 6 \, a b - 3 \, b^{2}\right )}}{{\left (a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right )} \sqrt {a b}} + \frac {8 \, {\left (f x + e\right )}}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {a b \tan \left (f x + e\right )^{3} - 5 \, b^{2} \tan \left (f x + e\right )^{3} - a^{2} \tan \left (f x + e\right ) - 3 \, a b \tan \left (f x + e\right )}{{\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{2}}}{8 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^4/(a+b*tan(f*x+e)^2)^3,x, algorithm="giac")

[Out]

1/8*((pi*floor((f*x + e)/pi + 1/2)*sgn(b) + arctan(b*tan(f*x + e)/sqrt(a*b)))*(a^2 - 6*a*b - 3*b^2)/((a^3*b -
3*a^2*b^2 + 3*a*b^3 - b^4)*sqrt(a*b)) + 8*(f*x + e)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + (a*b*tan(f*x + e)^3 - 5*
b^2*tan(f*x + e)^3 - a^2*tan(f*x + e) - 3*a*b*tan(f*x + e))/((a^2*b - 2*a*b^2 + b^3)*(b*tan(f*x + e)^2 + a)^2)
)/f

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maple [B]  time = 0.21, size = 338, normalized size = 2.33 \[ \frac {a^{2} \left (\tan ^{3}\left (f x +e \right )\right )}{8 f \left (a -b \right )^{3} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {3 a b \left (\tan ^{3}\left (f x +e \right )\right )}{4 f \left (a -b \right )^{3} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {5 \left (\tan ^{3}\left (f x +e \right )\right ) b^{2}}{8 f \left (a -b \right )^{3} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {a^{3} \tan \left (f x +e \right )}{8 f \left (a -b \right )^{3} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2} b}-\frac {a^{2} \tan \left (f x +e \right )}{4 f \left (a -b \right )^{3} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {3 a b \tan \left (f x +e \right )}{8 f \left (a -b \right )^{3} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {a^{2} \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {a b}}\right )}{8 f \left (a -b \right )^{3} b \sqrt {a b}}-\frac {3 a \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {a b}}\right )}{4 f \left (a -b \right )^{3} \sqrt {a b}}-\frac {3 b \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {a b}}\right )}{8 f \left (a -b \right )^{3} \sqrt {a b}}+\frac {\arctan \left (\tan \left (f x +e \right )\right )}{f \left (a -b \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(f*x+e)^4/(a+b*tan(f*x+e)^2)^3,x)

[Out]

1/8/f*a^2/(a-b)^3/(a+b*tan(f*x+e)^2)^2*tan(f*x+e)^3-3/4/f*a/(a-b)^3/(a+b*tan(f*x+e)^2)^2*b*tan(f*x+e)^3+5/8/f/
(a-b)^3/(a+b*tan(f*x+e)^2)^2*tan(f*x+e)^3*b^2-1/8/f*a^3/(a-b)^3/(a+b*tan(f*x+e)^2)^2/b*tan(f*x+e)-1/4/f*a^2/(a
-b)^3/(a+b*tan(f*x+e)^2)^2*tan(f*x+e)+3/8/f/(a-b)^3/(a+b*tan(f*x+e)^2)^2*a*b*tan(f*x+e)+1/8/f*a^2/(a-b)^3/b/(a
*b)^(1/2)*arctan(tan(f*x+e)*b/(a*b)^(1/2))-3/4/f*a/(a-b)^3/(a*b)^(1/2)*arctan(tan(f*x+e)*b/(a*b)^(1/2))-3/8/f/
(a-b)^3*b/(a*b)^(1/2)*arctan(tan(f*x+e)*b/(a*b)^(1/2))+1/f/(a-b)^3*arctan(tan(f*x+e))

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maxima [A]  time = 0.88, size = 212, normalized size = 1.46 \[ \frac {\frac {{\left (a^{2} - 6 \, a b - 3 \, b^{2}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )}{{\left (a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right )} \sqrt {a b}} + \frac {{\left (a b - 5 \, b^{2}\right )} \tan \left (f x + e\right )^{3} - {\left (a^{2} + 3 \, a b\right )} \tan \left (f x + e\right )}{a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} \tan \left (f x + e\right )^{4} + 2 \, {\left (a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right )} \tan \left (f x + e\right )^{2}} + \frac {8 \, {\left (f x + e\right )}}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}}}{8 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^4/(a+b*tan(f*x+e)^2)^3,x, algorithm="maxima")

[Out]

1/8*((a^2 - 6*a*b - 3*b^2)*arctan(b*tan(f*x + e)/sqrt(a*b))/((a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*sqrt(a*b)) +
((a*b - 5*b^2)*tan(f*x + e)^3 - (a^2 + 3*a*b)*tan(f*x + e))/(a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^2*b^3 - 2*a*b^4
+ b^5)*tan(f*x + e)^4 + 2*(a^3*b^2 - 2*a^2*b^3 + a*b^4)*tan(f*x + e)^2) + 8*(f*x + e)/(a^3 - 3*a^2*b + 3*a*b^2
- b^3))/f

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mupad [B]  time = 15.49, size = 3667, normalized size = 25.29 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(e + f*x)^4/(a + b*tan(e + f*x)^2)^3,x)

[Out]

((tan(e + f*x)^3*(a - 5*b))/(8*(a^2 - 2*a*b + b^2)) - (a*tan(e + f*x)*(a + 3*b))/(8*(a^2*b - 2*a*b^2 + b^3)))/
(f*(a^2 + b^2*tan(e + f*x)^4 + 2*a*b*tan(e + f*x)^2)) - (2*atan((((((544*a*b^8 - 96*b^9 - 1248*a^2*b^7 + 1440*
a^3*b^6 - 800*a^4*b^5 + 96*a^5*b^4 + 96*a^6*b^3 - 32*a^7*b^2)/(64*(a^6*b - 6*a*b^6 + b^7 + 15*a^2*b^5 - 20*a^3
*b^4 + 15*a^4*b^3 - 6*a^5*b^2)) - (tan(e + f*x)*(1280*a*b^9 - 256*b^10 - 2304*a^2*b^8 + 1280*a^3*b^7 + 1280*a^
4*b^6 - 2304*a^5*b^5 + 1280*a^6*b^4 - 256*a^7*b^3)*1i)/(32*(6*a*b^2 - 6*a^2*b + 2*a^3 - 2*b^3)*(a^4*b - 4*a*b^
4 + b^5 + 6*a^2*b^3 - 4*a^3*b^2)))*1i)/(6*a*b^2 - 6*a^2*b + 2*a^3 - 2*b^3) + (tan(e + f*x)*(36*a*b^3 - 12*a^3*
b + a^4 + 73*b^4 + 30*a^2*b^2))/(32*(a^4*b - 4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a^3*b^2)))/(6*a*b^2 - 6*a^2*b + 2*a
^3 - 2*b^3) - ((((544*a*b^8 - 96*b^9 - 1248*a^2*b^7 + 1440*a^3*b^6 - 800*a^4*b^5 + 96*a^5*b^4 + 96*a^6*b^3 - 3
2*a^7*b^2)/(64*(a^6*b - 6*a*b^6 + b^7 + 15*a^2*b^5 - 20*a^3*b^4 + 15*a^4*b^3 - 6*a^5*b^2)) + (tan(e + f*x)*(12
80*a*b^9 - 256*b^10 - 2304*a^2*b^8 + 1280*a^3*b^7 + 1280*a^4*b^6 - 2304*a^5*b^5 + 1280*a^6*b^4 - 256*a^7*b^3)*
1i)/(32*(6*a*b^2 - 6*a^2*b + 2*a^3 - 2*b^3)*(a^4*b - 4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a^3*b^2)))*1i)/(6*a*b^2 - 6
*a^2*b + 2*a^3 - 2*b^3) - (tan(e + f*x)*(36*a*b^3 - 12*a^3*b + a^4 + 73*b^4 + 30*a^2*b^2))/(32*(a^4*b - 4*a*b^
4 + b^5 + 6*a^2*b^3 - 4*a^3*b^2)))/(6*a*b^2 - 6*a^2*b + 2*a^3 - 2*b^3))/((((((544*a*b^8 - 96*b^9 - 1248*a^2*b^
7 + 1440*a^3*b^6 - 800*a^4*b^5 + 96*a^5*b^4 + 96*a^6*b^3 - 32*a^7*b^2)/(64*(a^6*b - 6*a*b^6 + b^7 + 15*a^2*b^5
- 20*a^3*b^4 + 15*a^4*b^3 - 6*a^5*b^2)) - (tan(e + f*x)*(1280*a*b^9 - 256*b^10 - 2304*a^2*b^8 + 1280*a^3*b^7
+ 1280*a^4*b^6 - 2304*a^5*b^5 + 1280*a^6*b^4 - 256*a^7*b^3)*1i)/(32*(6*a*b^2 - 6*a^2*b + 2*a^3 - 2*b^3)*(a^4*b
- 4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a^3*b^2)))*1i)/(6*a*b^2 - 6*a^2*b + 2*a^3 - 2*b^3) + (tan(e + f*x)*(36*a*b^3
- 12*a^3*b + a^4 + 73*b^4 + 30*a^2*b^2))/(32*(a^4*b - 4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a^3*b^2)))*1i)/(6*a*b^2 -
6*a^2*b + 2*a^3 - 2*b^3) - (27*a*b^2 - 11*a^2*b + a^3 + 15*b^3)/(32*(a^6*b - 6*a*b^6 + b^7 + 15*a^2*b^5 - 20*a
^3*b^4 + 15*a^4*b^3 - 6*a^5*b^2)) + (((((544*a*b^8 - 96*b^9 - 1248*a^2*b^7 + 1440*a^3*b^6 - 800*a^4*b^5 + 96*a
^5*b^4 + 96*a^6*b^3 - 32*a^7*b^2)/(64*(a^6*b - 6*a*b^6 + b^7 + 15*a^2*b^5 - 20*a^3*b^4 + 15*a^4*b^3 - 6*a^5*b^
2)) + (tan(e + f*x)*(1280*a*b^9 - 256*b^10 - 2304*a^2*b^8 + 1280*a^3*b^7 + 1280*a^4*b^6 - 2304*a^5*b^5 + 1280*
a^6*b^4 - 256*a^7*b^3)*1i)/(32*(6*a*b^2 - 6*a^2*b + 2*a^3 - 2*b^3)*(a^4*b - 4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a^3*
b^2)))*1i)/(6*a*b^2 - 6*a^2*b + 2*a^3 - 2*b^3) - (tan(e + f*x)*(36*a*b^3 - 12*a^3*b + a^4 + 73*b^4 + 30*a^2*b^
2))/(32*(a^4*b - 4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a^3*b^2)))*1i)/(6*a*b^2 - 6*a^2*b + 2*a^3 - 2*b^3))))/(f*(6*a*b
^2 - 6*a^2*b + 2*a^3 - 2*b^3)) + (atan((((-a*b^3)^(1/2)*((tan(e + f*x)*(36*a*b^3 - 12*a^3*b + a^4 + 73*b^4 + 3
0*a^2*b^2))/(32*(a^4*b - 4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a^3*b^2)) + (((544*a*b^8 - 96*b^9 - 1248*a^2*b^7 + 1440
*a^3*b^6 - 800*a^4*b^5 + 96*a^5*b^4 + 96*a^6*b^3 - 32*a^7*b^2)/(64*(a^6*b - 6*a*b^6 + b^7 + 15*a^2*b^5 - 20*a^
3*b^4 + 15*a^4*b^3 - 6*a^5*b^2)) - (tan(e + f*x)*(-a*b^3)^(1/2)*(6*a*b - a^2 + 3*b^2)*(1280*a*b^9 - 256*b^10 -
2304*a^2*b^8 + 1280*a^3*b^7 + 1280*a^4*b^6 - 2304*a^5*b^5 + 1280*a^6*b^4 - 256*a^7*b^3))/(512*(a*b^6 - 3*a^2*
b^5 + 3*a^3*b^4 - a^4*b^3)*(a^4*b - 4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a^3*b^2)))*(-a*b^3)^(1/2)*(6*a*b - a^2 + 3*b
^2))/(16*(a*b^6 - 3*a^2*b^5 + 3*a^3*b^4 - a^4*b^3)))*(6*a*b - a^2 + 3*b^2)*1i)/(16*(a*b^6 - 3*a^2*b^5 + 3*a^3*
b^4 - a^4*b^3)) + ((-a*b^3)^(1/2)*((tan(e + f*x)*(36*a*b^3 - 12*a^3*b + a^4 + 73*b^4 + 30*a^2*b^2))/(32*(a^4*b
- 4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a^3*b^2)) - (((544*a*b^8 - 96*b^9 - 1248*a^2*b^7 + 1440*a^3*b^6 - 800*a^4*b^5
+ 96*a^5*b^4 + 96*a^6*b^3 - 32*a^7*b^2)/(64*(a^6*b - 6*a*b^6 + b^7 + 15*a^2*b^5 - 20*a^3*b^4 + 15*a^4*b^3 - 6
*a^5*b^2)) + (tan(e + f*x)*(-a*b^3)^(1/2)*(6*a*b - a^2 + 3*b^2)*(1280*a*b^9 - 256*b^10 - 2304*a^2*b^8 + 1280*a
^3*b^7 + 1280*a^4*b^6 - 2304*a^5*b^5 + 1280*a^6*b^4 - 256*a^7*b^3))/(512*(a*b^6 - 3*a^2*b^5 + 3*a^3*b^4 - a^4*
b^3)*(a^4*b - 4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a^3*b^2)))*(-a*b^3)^(1/2)*(6*a*b - a^2 + 3*b^2))/(16*(a*b^6 - 3*a^
2*b^5 + 3*a^3*b^4 - a^4*b^3)))*(6*a*b - a^2 + 3*b^2)*1i)/(16*(a*b^6 - 3*a^2*b^5 + 3*a^3*b^4 - a^4*b^3)))/((27*
a*b^2 - 11*a^2*b + a^3 + 15*b^3)/(32*(a^6*b - 6*a*b^6 + b^7 + 15*a^2*b^5 - 20*a^3*b^4 + 15*a^4*b^3 - 6*a^5*b^2
)) - ((-a*b^3)^(1/2)*((tan(e + f*x)*(36*a*b^3 - 12*a^3*b + a^4 + 73*b^4 + 30*a^2*b^2))/(32*(a^4*b - 4*a*b^4 +
b^5 + 6*a^2*b^3 - 4*a^3*b^2)) + (((544*a*b^8 - 96*b^9 - 1248*a^2*b^7 + 1440*a^3*b^6 - 800*a^4*b^5 + 96*a^5*b^4
+ 96*a^6*b^3 - 32*a^7*b^2)/(64*(a^6*b - 6*a*b^6 + b^7 + 15*a^2*b^5 - 20*a^3*b^4 + 15*a^4*b^3 - 6*a^5*b^2)) -
(tan(e + f*x)*(-a*b^3)^(1/2)*(6*a*b - a^2 + 3*b^2)*(1280*a*b^9 - 256*b^10 - 2304*a^2*b^8 + 1280*a^3*b^7 + 1280
*a^4*b^6 - 2304*a^5*b^5 + 1280*a^6*b^4 - 256*a^7*b^3))/(512*(a*b^6 - 3*a^2*b^5 + 3*a^3*b^4 - a^4*b^3)*(a^4*b -
4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a^3*b^2)))*(-a*b^3)^(1/2)*(6*a*b - a^2 + 3*b^2))/(16*(a*b^6 - 3*a^2*b^5 + 3*a^3
*b^4 - a^4*b^3)))*(6*a*b - a^2 + 3*b^2))/(16*(a*b^6 - 3*a^2*b^5 + 3*a^3*b^4 - a^4*b^3)) + ((-a*b^3)^(1/2)*((ta
n(e + f*x)*(36*a*b^3 - 12*a^3*b + a^4 + 73*b^4 + 30*a^2*b^2))/(32*(a^4*b - 4*a*b^4 + b^5 + 6*a^2*b^3 - 4*a^3*b
^2)) - (((544*a*b^8 - 96*b^9 - 1248*a^2*b^7 + 1440*a^3*b^6 - 800*a^4*b^5 + 96*a^5*b^4 + 96*a^6*b^3 - 32*a^7*b^
2)/(64*(a^6*b - 6*a*b^6 + b^7 + 15*a^2*b^5 - 20*a^3*b^4 + 15*a^4*b^3 - 6*a^5*b^2)) + (tan(e + f*x)*(-a*b^3)^(1
/2)*(6*a*b - a^2 + 3*b^2)*(1280*a*b^9 - 256*b^10 - 2304*a^2*b^8 + 1280*a^3*b^7 + 1280*a^4*b^6 - 2304*a^5*b^5 +
1280*a^6*b^4 - 256*a^7*b^3))/(512*(a*b^6 - 3*a^2*b^5 + 3*a^3*b^4 - a^4*b^3)*(a^4*b - 4*a*b^4 + b^5 + 6*a^2*b^
3 - 4*a^3*b^2)))*(-a*b^3)^(1/2)*(6*a*b - a^2 + 3*b^2))/(16*(a*b^6 - 3*a^2*b^5 + 3*a^3*b^4 - a^4*b^3)))*(6*a*b
- a^2 + 3*b^2))/(16*(a*b^6 - 3*a^2*b^5 + 3*a^3*b^4 - a^4*b^3))))*(-a*b^3)^(1/2)*(6*a*b - a^2 + 3*b^2)*1i)/(8*f
*(a*b^6 - 3*a^2*b^5 + 3*a^3*b^4 - a^4*b^3))

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sympy [A]  time = 140.86, size = 9811, normalized size = 67.66 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)**4/(a+b*tan(f*x+e)**2)**3,x)

[Out]

Piecewise((zoo*x/tan(e)**2, Eq(a, 0) & Eq(b, 0) & Eq(f, 0)), ((-x - 1/(f*tan(e + f*x)))/b**3, Eq(a, 0)), (3*f*
x*tan(e + f*x)**6/(48*b**3*f*tan(e + f*x)**6 + 144*b**3*f*tan(e + f*x)**4 + 144*b**3*f*tan(e + f*x)**2 + 48*b*
*3*f) + 9*f*x*tan(e + f*x)**4/(48*b**3*f*tan(e + f*x)**6 + 144*b**3*f*tan(e + f*x)**4 + 144*b**3*f*tan(e + f*x
)**2 + 48*b**3*f) + 9*f*x*tan(e + f*x)**2/(48*b**3*f*tan(e + f*x)**6 + 144*b**3*f*tan(e + f*x)**4 + 144*b**3*f
*tan(e + f*x)**2 + 48*b**3*f) + 3*f*x/(48*b**3*f*tan(e + f*x)**6 + 144*b**3*f*tan(e + f*x)**4 + 144*b**3*f*tan
(e + f*x)**2 + 48*b**3*f) + 3*tan(e + f*x)**5/(48*b**3*f*tan(e + f*x)**6 + 144*b**3*f*tan(e + f*x)**4 + 144*b*
*3*f*tan(e + f*x)**2 + 48*b**3*f) - 8*tan(e + f*x)**3/(48*b**3*f*tan(e + f*x)**6 + 144*b**3*f*tan(e + f*x)**4
+ 144*b**3*f*tan(e + f*x)**2 + 48*b**3*f) - 3*tan(e + f*x)/(48*b**3*f*tan(e + f*x)**6 + 144*b**3*f*tan(e + f*x
)**4 + 144*b**3*f*tan(e + f*x)**2 + 48*b**3*f), Eq(a, b)), (x*tan(e)**4/(a + b*tan(e)**2)**3, Eq(f, 0)), ((x +
tan(e + f*x)**3/(3*f) - tan(e + f*x)/f)/a**3, Eq(b, 0)), (-2*I*a**(7/2)*b*sqrt(1/b)*tan(e + f*x)/(16*I*a**(11
/2)*b**2*f*sqrt(1/b) + 32*I*a**(9/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 48*I*a**(9/2)*b**3*f*sqrt(1/b) + 16*I*
a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**4 - 96*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 48*I*a**(7/2)*b**
4*f*sqrt(1/b) - 48*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**4 + 96*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**
2 - 16*I*a**(5/2)*b**5*f*sqrt(1/b) + 48*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**4 - 32*I*a**(3/2)*b**6*f*sqr
t(1/b)*tan(e + f*x)**2 - 16*I*sqrt(a)*b**7*f*sqrt(1/b)*tan(e + f*x)**4) + 16*I*a**(5/2)*b**2*f*x*sqrt(1/b)/(16
*I*a**(11/2)*b**2*f*sqrt(1/b) + 32*I*a**(9/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 48*I*a**(9/2)*b**3*f*sqrt(1/b
) + 16*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**4 - 96*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 48*I*a**
(7/2)*b**4*f*sqrt(1/b) - 48*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**4 + 96*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e
+ f*x)**2 - 16*I*a**(5/2)*b**5*f*sqrt(1/b) + 48*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**4 - 32*I*a**(3/2)*b
**6*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*sqrt(a)*b**7*f*sqrt(1/b)*tan(e + f*x)**4) + 2*I*a**(5/2)*b**2*sqrt(1/b)
*tan(e + f*x)**3/(16*I*a**(11/2)*b**2*f*sqrt(1/b) + 32*I*a**(9/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 48*I*a**(
9/2)*b**3*f*sqrt(1/b) + 16*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**4 - 96*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e
+ f*x)**2 + 48*I*a**(7/2)*b**4*f*sqrt(1/b) - 48*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**4 + 96*I*a**(5/2)*b*
*5*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*a**(5/2)*b**5*f*sqrt(1/b) + 48*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)*
*4 - 32*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*sqrt(a)*b**7*f*sqrt(1/b)*tan(e + f*x)**4) - 4*I*a**
(5/2)*b**2*sqrt(1/b)*tan(e + f*x)/(16*I*a**(11/2)*b**2*f*sqrt(1/b) + 32*I*a**(9/2)*b**3*f*sqrt(1/b)*tan(e + f*
x)**2 - 48*I*a**(9/2)*b**3*f*sqrt(1/b) + 16*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**4 - 96*I*a**(7/2)*b**4*f
*sqrt(1/b)*tan(e + f*x)**2 + 48*I*a**(7/2)*b**4*f*sqrt(1/b) - 48*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**4 +
96*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*a**(5/2)*b**5*f*sqrt(1/b) + 48*I*a**(3/2)*b**6*f*sqrt(1
/b)*tan(e + f*x)**4 - 32*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*sqrt(a)*b**7*f*sqrt(1/b)*tan(e + f
*x)**4) + 32*I*a**(3/2)*b**3*f*x*sqrt(1/b)*tan(e + f*x)**2/(16*I*a**(11/2)*b**2*f*sqrt(1/b) + 32*I*a**(9/2)*b*
*3*f*sqrt(1/b)*tan(e + f*x)**2 - 48*I*a**(9/2)*b**3*f*sqrt(1/b) + 16*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)*
*4 - 96*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 48*I*a**(7/2)*b**4*f*sqrt(1/b) - 48*I*a**(5/2)*b**5*f*sq
rt(1/b)*tan(e + f*x)**4 + 96*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*a**(5/2)*b**5*f*sqrt(1/b) + 48
*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**4 - 32*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*sqrt(a)*b
**7*f*sqrt(1/b)*tan(e + f*x)**4) - 12*I*a**(3/2)*b**3*sqrt(1/b)*tan(e + f*x)**3/(16*I*a**(11/2)*b**2*f*sqrt(1/
b) + 32*I*a**(9/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 48*I*a**(9/2)*b**3*f*sqrt(1/b) + 16*I*a**(7/2)*b**4*f*sq
rt(1/b)*tan(e + f*x)**4 - 96*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 48*I*a**(7/2)*b**4*f*sqrt(1/b) - 48
*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**4 + 96*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*a**(5/2)*
b**5*f*sqrt(1/b) + 48*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**4 - 32*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x
)**2 - 16*I*sqrt(a)*b**7*f*sqrt(1/b)*tan(e + f*x)**4) + 6*I*a**(3/2)*b**3*sqrt(1/b)*tan(e + f*x)/(16*I*a**(11/
2)*b**2*f*sqrt(1/b) + 32*I*a**(9/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 48*I*a**(9/2)*b**3*f*sqrt(1/b) + 16*I*a
**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**4 - 96*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 48*I*a**(7/2)*b**4
*f*sqrt(1/b) - 48*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**4 + 96*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**2
- 16*I*a**(5/2)*b**5*f*sqrt(1/b) + 48*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**4 - 32*I*a**(3/2)*b**6*f*sqrt
(1/b)*tan(e + f*x)**2 - 16*I*sqrt(a)*b**7*f*sqrt(1/b)*tan(e + f*x)**4) + 16*I*sqrt(a)*b**4*f*x*sqrt(1/b)*tan(e
+ f*x)**4/(16*I*a**(11/2)*b**2*f*sqrt(1/b) + 32*I*a**(9/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 48*I*a**(9/2)*b
**3*f*sqrt(1/b) + 16*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**4 - 96*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)
**2 + 48*I*a**(7/2)*b**4*f*sqrt(1/b) - 48*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**4 + 96*I*a**(5/2)*b**5*f*s
qrt(1/b)*tan(e + f*x)**2 - 16*I*a**(5/2)*b**5*f*sqrt(1/b) + 48*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**4 - 3
2*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*sqrt(a)*b**7*f*sqrt(1/b)*tan(e + f*x)**4) + 10*I*sqrt(a)*
b**4*sqrt(1/b)*tan(e + f*x)**3/(16*I*a**(11/2)*b**2*f*sqrt(1/b) + 32*I*a**(9/2)*b**3*f*sqrt(1/b)*tan(e + f*x)*
*2 - 48*I*a**(9/2)*b**3*f*sqrt(1/b) + 16*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**4 - 96*I*a**(7/2)*b**4*f*sq
rt(1/b)*tan(e + f*x)**2 + 48*I*a**(7/2)*b**4*f*sqrt(1/b) - 48*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**4 + 96
*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*a**(5/2)*b**5*f*sqrt(1/b) + 48*I*a**(3/2)*b**6*f*sqrt(1/b)
*tan(e + f*x)**4 - 32*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*sqrt(a)*b**7*f*sqrt(1/b)*tan(e + f*x)
**4) + a**4*log(-I*sqrt(a)*sqrt(1/b) + tan(e + f*x))/(16*I*a**(11/2)*b**2*f*sqrt(1/b) + 32*I*a**(9/2)*b**3*f*s
qrt(1/b)*tan(e + f*x)**2 - 48*I*a**(9/2)*b**3*f*sqrt(1/b) + 16*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**4 - 9
6*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 48*I*a**(7/2)*b**4*f*sqrt(1/b) - 48*I*a**(5/2)*b**5*f*sqrt(1/b
)*tan(e + f*x)**4 + 96*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*a**(5/2)*b**5*f*sqrt(1/b) + 48*I*a**
(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**4 - 32*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*sqrt(a)*b**7*f*
sqrt(1/b)*tan(e + f*x)**4) - a**4*log(I*sqrt(a)*sqrt(1/b) + tan(e + f*x))/(16*I*a**(11/2)*b**2*f*sqrt(1/b) + 3
2*I*a**(9/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 48*I*a**(9/2)*b**3*f*sqrt(1/b) + 16*I*a**(7/2)*b**4*f*sqrt(1/b
)*tan(e + f*x)**4 - 96*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 48*I*a**(7/2)*b**4*f*sqrt(1/b) - 48*I*a**
(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**4 + 96*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*a**(5/2)*b**5*f
*sqrt(1/b) + 48*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**4 - 32*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**2 -
16*I*sqrt(a)*b**7*f*sqrt(1/b)*tan(e + f*x)**4) + 2*a**3*b*log(-I*sqrt(a)*sqrt(1/b) + tan(e + f*x))*tan(e + f*
x)**2/(16*I*a**(11/2)*b**2*f*sqrt(1/b) + 32*I*a**(9/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 48*I*a**(9/2)*b**3*f
*sqrt(1/b) + 16*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**4 - 96*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 +
48*I*a**(7/2)*b**4*f*sqrt(1/b) - 48*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**4 + 96*I*a**(5/2)*b**5*f*sqrt(1
/b)*tan(e + f*x)**2 - 16*I*a**(5/2)*b**5*f*sqrt(1/b) + 48*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**4 - 32*I*a
**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*sqrt(a)*b**7*f*sqrt(1/b)*tan(e + f*x)**4) - 6*a**3*b*log(-I*sq
rt(a)*sqrt(1/b) + tan(e + f*x))/(16*I*a**(11/2)*b**2*f*sqrt(1/b) + 32*I*a**(9/2)*b**3*f*sqrt(1/b)*tan(e + f*x)
**2 - 48*I*a**(9/2)*b**3*f*sqrt(1/b) + 16*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**4 - 96*I*a**(7/2)*b**4*f*s
qrt(1/b)*tan(e + f*x)**2 + 48*I*a**(7/2)*b**4*f*sqrt(1/b) - 48*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**4 + 9
6*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*a**(5/2)*b**5*f*sqrt(1/b) + 48*I*a**(3/2)*b**6*f*sqrt(1/b
)*tan(e + f*x)**4 - 32*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*sqrt(a)*b**7*f*sqrt(1/b)*tan(e + f*x
)**4) - 2*a**3*b*log(I*sqrt(a)*sqrt(1/b) + tan(e + f*x))*tan(e + f*x)**2/(16*I*a**(11/2)*b**2*f*sqrt(1/b) + 32
*I*a**(9/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 48*I*a**(9/2)*b**3*f*sqrt(1/b) + 16*I*a**(7/2)*b**4*f*sqrt(1/b)
*tan(e + f*x)**4 - 96*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 48*I*a**(7/2)*b**4*f*sqrt(1/b) - 48*I*a**(
5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**4 + 96*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*a**(5/2)*b**5*f*
sqrt(1/b) + 48*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**4 - 32*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**2 -
16*I*sqrt(a)*b**7*f*sqrt(1/b)*tan(e + f*x)**4) + 6*a**3*b*log(I*sqrt(a)*sqrt(1/b) + tan(e + f*x))/(16*I*a**(11
/2)*b**2*f*sqrt(1/b) + 32*I*a**(9/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 48*I*a**(9/2)*b**3*f*sqrt(1/b) + 16*I*
a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**4 - 96*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 48*I*a**(7/2)*b**
4*f*sqrt(1/b) - 48*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**4 + 96*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**
2 - 16*I*a**(5/2)*b**5*f*sqrt(1/b) + 48*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**4 - 32*I*a**(3/2)*b**6*f*sqr
t(1/b)*tan(e + f*x)**2 - 16*I*sqrt(a)*b**7*f*sqrt(1/b)*tan(e + f*x)**4) + a**2*b**2*log(-I*sqrt(a)*sqrt(1/b) +
tan(e + f*x))*tan(e + f*x)**4/(16*I*a**(11/2)*b**2*f*sqrt(1/b) + 32*I*a**(9/2)*b**3*f*sqrt(1/b)*tan(e + f*x)*
*2 - 48*I*a**(9/2)*b**3*f*sqrt(1/b) + 16*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**4 - 96*I*a**(7/2)*b**4*f*sq
rt(1/b)*tan(e + f*x)**2 + 48*I*a**(7/2)*b**4*f*sqrt(1/b) - 48*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**4 + 96
*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*a**(5/2)*b**5*f*sqrt(1/b) + 48*I*a**(3/2)*b**6*f*sqrt(1/b)
*tan(e + f*x)**4 - 32*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*sqrt(a)*b**7*f*sqrt(1/b)*tan(e + f*x)
**4) - 12*a**2*b**2*log(-I*sqrt(a)*sqrt(1/b) + tan(e + f*x))*tan(e + f*x)**2/(16*I*a**(11/2)*b**2*f*sqrt(1/b)
+ 32*I*a**(9/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 48*I*a**(9/2)*b**3*f*sqrt(1/b) + 16*I*a**(7/2)*b**4*f*sqrt(
1/b)*tan(e + f*x)**4 - 96*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 48*I*a**(7/2)*b**4*f*sqrt(1/b) - 48*I*
a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**4 + 96*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*a**(5/2)*b**
5*f*sqrt(1/b) + 48*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**4 - 32*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**
2 - 16*I*sqrt(a)*b**7*f*sqrt(1/b)*tan(e + f*x)**4) - 3*a**2*b**2*log(-I*sqrt(a)*sqrt(1/b) + tan(e + f*x))/(16*
I*a**(11/2)*b**2*f*sqrt(1/b) + 32*I*a**(9/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 48*I*a**(9/2)*b**3*f*sqrt(1/b)
+ 16*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**4 - 96*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 48*I*a**(
7/2)*b**4*f*sqrt(1/b) - 48*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**4 + 96*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e
+ f*x)**2 - 16*I*a**(5/2)*b**5*f*sqrt(1/b) + 48*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**4 - 32*I*a**(3/2)*b*
*6*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*sqrt(a)*b**7*f*sqrt(1/b)*tan(e + f*x)**4) - a**2*b**2*log(I*sqrt(a)*sqrt
(1/b) + tan(e + f*x))*tan(e + f*x)**4/(16*I*a**(11/2)*b**2*f*sqrt(1/b) + 32*I*a**(9/2)*b**3*f*sqrt(1/b)*tan(e
+ f*x)**2 - 48*I*a**(9/2)*b**3*f*sqrt(1/b) + 16*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**4 - 96*I*a**(7/2)*b*
*4*f*sqrt(1/b)*tan(e + f*x)**2 + 48*I*a**(7/2)*b**4*f*sqrt(1/b) - 48*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)*
*4 + 96*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*a**(5/2)*b**5*f*sqrt(1/b) + 48*I*a**(3/2)*b**6*f*sq
rt(1/b)*tan(e + f*x)**4 - 32*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*sqrt(a)*b**7*f*sqrt(1/b)*tan(e
+ f*x)**4) + 12*a**2*b**2*log(I*sqrt(a)*sqrt(1/b) + tan(e + f*x))*tan(e + f*x)**2/(16*I*a**(11/2)*b**2*f*sqrt
(1/b) + 32*I*a**(9/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 48*I*a**(9/2)*b**3*f*sqrt(1/b) + 16*I*a**(7/2)*b**4*f
*sqrt(1/b)*tan(e + f*x)**4 - 96*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 48*I*a**(7/2)*b**4*f*sqrt(1/b) -
48*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**4 + 96*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*a**(5/
2)*b**5*f*sqrt(1/b) + 48*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**4 - 32*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e +
f*x)**2 - 16*I*sqrt(a)*b**7*f*sqrt(1/b)*tan(e + f*x)**4) + 3*a**2*b**2*log(I*sqrt(a)*sqrt(1/b) + tan(e + f*x))
/(16*I*a**(11/2)*b**2*f*sqrt(1/b) + 32*I*a**(9/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 48*I*a**(9/2)*b**3*f*sqrt
(1/b) + 16*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**4 - 96*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 48*I
*a**(7/2)*b**4*f*sqrt(1/b) - 48*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**4 + 96*I*a**(5/2)*b**5*f*sqrt(1/b)*t
an(e + f*x)**2 - 16*I*a**(5/2)*b**5*f*sqrt(1/b) + 48*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**4 - 32*I*a**(3/
2)*b**6*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*sqrt(a)*b**7*f*sqrt(1/b)*tan(e + f*x)**4) - 6*a*b**3*log(-I*sqrt(a)
*sqrt(1/b) + tan(e + f*x))*tan(e + f*x)**4/(16*I*a**(11/2)*b**2*f*sqrt(1/b) + 32*I*a**(9/2)*b**3*f*sqrt(1/b)*t
an(e + f*x)**2 - 48*I*a**(9/2)*b**3*f*sqrt(1/b) + 16*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**4 - 96*I*a**(7/
2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 48*I*a**(7/2)*b**4*f*sqrt(1/b) - 48*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e +
f*x)**4 + 96*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*a**(5/2)*b**5*f*sqrt(1/b) + 48*I*a**(3/2)*b**6
*f*sqrt(1/b)*tan(e + f*x)**4 - 32*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*sqrt(a)*b**7*f*sqrt(1/b)*
tan(e + f*x)**4) - 6*a*b**3*log(-I*sqrt(a)*sqrt(1/b) + tan(e + f*x))*tan(e + f*x)**2/(16*I*a**(11/2)*b**2*f*sq
rt(1/b) + 32*I*a**(9/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 48*I*a**(9/2)*b**3*f*sqrt(1/b) + 16*I*a**(7/2)*b**4
*f*sqrt(1/b)*tan(e + f*x)**4 - 96*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 48*I*a**(7/2)*b**4*f*sqrt(1/b)
- 48*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**4 + 96*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*a**(
5/2)*b**5*f*sqrt(1/b) + 48*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**4 - 32*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e
+ f*x)**2 - 16*I*sqrt(a)*b**7*f*sqrt(1/b)*tan(e + f*x)**4) + 6*a*b**3*log(I*sqrt(a)*sqrt(1/b) + tan(e + f*x))*
tan(e + f*x)**4/(16*I*a**(11/2)*b**2*f*sqrt(1/b) + 32*I*a**(9/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 48*I*a**(9
/2)*b**3*f*sqrt(1/b) + 16*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**4 - 96*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e +
f*x)**2 + 48*I*a**(7/2)*b**4*f*sqrt(1/b) - 48*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**4 + 96*I*a**(5/2)*b**
5*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*a**(5/2)*b**5*f*sqrt(1/b) + 48*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**
4 - 32*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*sqrt(a)*b**7*f*sqrt(1/b)*tan(e + f*x)**4) + 6*a*b**3
*log(I*sqrt(a)*sqrt(1/b) + tan(e + f*x))*tan(e + f*x)**2/(16*I*a**(11/2)*b**2*f*sqrt(1/b) + 32*I*a**(9/2)*b**3
*f*sqrt(1/b)*tan(e + f*x)**2 - 48*I*a**(9/2)*b**3*f*sqrt(1/b) + 16*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**4
- 96*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 48*I*a**(7/2)*b**4*f*sqrt(1/b) - 48*I*a**(5/2)*b**5*f*sqrt
(1/b)*tan(e + f*x)**4 + 96*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*a**(5/2)*b**5*f*sqrt(1/b) + 48*I
*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**4 - 32*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*sqrt(a)*b**
7*f*sqrt(1/b)*tan(e + f*x)**4) - 3*b**4*log(-I*sqrt(a)*sqrt(1/b) + tan(e + f*x))*tan(e + f*x)**4/(16*I*a**(11/
2)*b**2*f*sqrt(1/b) + 32*I*a**(9/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 - 48*I*a**(9/2)*b**3*f*sqrt(1/b) + 16*I*a
**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**4 - 96*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**2 + 48*I*a**(7/2)*b**4
*f*sqrt(1/b) - 48*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**4 + 96*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**2
- 16*I*a**(5/2)*b**5*f*sqrt(1/b) + 48*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**4 - 32*I*a**(3/2)*b**6*f*sqrt
(1/b)*tan(e + f*x)**2 - 16*I*sqrt(a)*b**7*f*sqrt(1/b)*tan(e + f*x)**4) + 3*b**4*log(I*sqrt(a)*sqrt(1/b) + tan(
e + f*x))*tan(e + f*x)**4/(16*I*a**(11/2)*b**2*f*sqrt(1/b) + 32*I*a**(9/2)*b**3*f*sqrt(1/b)*tan(e + f*x)**2 -
48*I*a**(9/2)*b**3*f*sqrt(1/b) + 16*I*a**(7/2)*b**4*f*sqrt(1/b)*tan(e + f*x)**4 - 96*I*a**(7/2)*b**4*f*sqrt(1/
b)*tan(e + f*x)**2 + 48*I*a**(7/2)*b**4*f*sqrt(1/b) - 48*I*a**(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**4 + 96*I*a*
*(5/2)*b**5*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*a**(5/2)*b**5*f*sqrt(1/b) + 48*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(
e + f*x)**4 - 32*I*a**(3/2)*b**6*f*sqrt(1/b)*tan(e + f*x)**2 - 16*I*sqrt(a)*b**7*f*sqrt(1/b)*tan(e + f*x)**4),
True))

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