3.236 \(\int \frac {\cot ^6(e+f x)}{(a+b \tan ^2(e+f x))^2} \, dx\)

Optimal. Leaf size=218 \[ \frac {b^{7/2} (9 a-7 b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{2 a^{9/2} f (a-b)^2}-\frac {(2 a-7 b) \cot ^5(e+f x)}{10 a^2 f (a-b)}+\frac {\left (2 a^2+2 a b-7 b^2\right ) \cot ^3(e+f x)}{6 a^3 f (a-b)}-\frac {\left (2 a^3+2 a^2 b+2 a b^2-7 b^3\right ) \cot (e+f x)}{2 a^4 f (a-b)}-\frac {b \cot ^5(e+f x)}{2 a f (a-b) \left (a+b \tan ^2(e+f x)\right )}-\frac {x}{(a-b)^2} \]

[Out]

-x/(a-b)^2+1/2*(9*a-7*b)*b^(7/2)*arctan(b^(1/2)*tan(f*x+e)/a^(1/2))/a^(9/2)/(a-b)^2/f-1/2*(2*a^3+2*a^2*b+2*a*b
^2-7*b^3)*cot(f*x+e)/a^4/(a-b)/f+1/6*(2*a^2+2*a*b-7*b^2)*cot(f*x+e)^3/a^3/(a-b)/f-1/10*(2*a-7*b)*cot(f*x+e)^5/
a^2/(a-b)/f-1/2*b*cot(f*x+e)^5/a/(a-b)/f/(a+b*tan(f*x+e)^2)

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Rubi [A]  time = 0.34, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3670, 472, 583, 522, 203, 205} \[ \frac {b^{7/2} (9 a-7 b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{2 a^{9/2} f (a-b)^2}+\frac {\left (2 a^2+2 a b-7 b^2\right ) \cot ^3(e+f x)}{6 a^3 f (a-b)}-\frac {\left (2 a^2 b+2 a^3+2 a b^2-7 b^3\right ) \cot (e+f x)}{2 a^4 f (a-b)}-\frac {(2 a-7 b) \cot ^5(e+f x)}{10 a^2 f (a-b)}-\frac {b \cot ^5(e+f x)}{2 a f (a-b) \left (a+b \tan ^2(e+f x)\right )}-\frac {x}{(a-b)^2} \]

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^6/(a + b*Tan[e + f*x]^2)^2,x]

[Out]

-(x/(a - b)^2) + ((9*a - 7*b)*b^(7/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*a^(9/2)*(a - b)^2*f) - ((2*a^
3 + 2*a^2*b + 2*a*b^2 - 7*b^3)*Cot[e + f*x])/(2*a^4*(a - b)*f) + ((2*a^2 + 2*a*b - 7*b^2)*Cot[e + f*x]^3)/(6*a
^3*(a - b)*f) - ((2*a - 7*b)*Cot[e + f*x]^5)/(10*a^2*(a - b)*f) - (b*Cot[e + f*x]^5)/(2*a*(a - 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 472

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*(e*x
)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*e*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d)*(
p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n*(
p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p
, -1] && 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 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -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 {\cot ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^6 \left (1+x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {b \cot ^5(e+f x)}{2 a (a-b) f \left (a+b \tan ^2(e+f x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {2 a-7 b-7 b x^2}{x^6 \left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{2 a (a-b) f}\\ &=-\frac {(2 a-7 b) \cot ^5(e+f x)}{10 a^2 (a-b) f}-\frac {b \cot ^5(e+f x)}{2 a (a-b) f \left (a+b \tan ^2(e+f x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {5 \left (2 a^2+2 a b-7 b^2\right )+5 (2 a-7 b) b x^2}{x^4 \left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{10 a^2 (a-b) f}\\ &=\frac {\left (2 a^2+2 a b-7 b^2\right ) \cot ^3(e+f x)}{6 a^3 (a-b) f}-\frac {(2 a-7 b) \cot ^5(e+f x)}{10 a^2 (a-b) f}-\frac {b \cot ^5(e+f x)}{2 a (a-b) f \left (a+b \tan ^2(e+f x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {15 \left (2 a^3+2 a^2 b+2 a b^2-7 b^3\right )+15 b \left (2 a^2+2 a b-7 b^2\right ) x^2}{x^2 \left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{30 a^3 (a-b) f}\\ &=-\frac {\left (2 a^3+2 a^2 b+2 a b^2-7 b^3\right ) \cot (e+f x)}{2 a^4 (a-b) f}+\frac {\left (2 a^2+2 a b-7 b^2\right ) \cot ^3(e+f x)}{6 a^3 (a-b) f}-\frac {(2 a-7 b) \cot ^5(e+f x)}{10 a^2 (a-b) f}-\frac {b \cot ^5(e+f x)}{2 a (a-b) f \left (a+b \tan ^2(e+f x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {15 \left (2 a^4+2 a^3 b+2 a^2 b^2+2 a b^3-7 b^4\right )+15 b \left (2 a^3+2 a^2 b+2 a b^2-7 b^3\right ) x^2}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{30 a^4 (a-b) f}\\ &=-\frac {\left (2 a^3+2 a^2 b+2 a b^2-7 b^3\right ) \cot (e+f x)}{2 a^4 (a-b) f}+\frac {\left (2 a^2+2 a b-7 b^2\right ) \cot ^3(e+f x)}{6 a^3 (a-b) f}-\frac {(2 a-7 b) \cot ^5(e+f x)}{10 a^2 (a-b) f}-\frac {b \cot ^5(e+f x)}{2 a (a-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 {\left ((9 a-7 b) b^4\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tan (e+f x)\right )}{2 a^4 (a-b)^2 f}\\ &=-\frac {x}{(a-b)^2}+\frac {(9 a-7 b) b^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{2 a^{9/2} (a-b)^2 f}-\frac {\left (2 a^3+2 a^2 b+2 a b^2-7 b^3\right ) \cot (e+f x)}{2 a^4 (a-b) f}+\frac {\left (2 a^2+2 a b-7 b^2\right ) \cot ^3(e+f x)}{6 a^3 (a-b) f}-\frac {(2 a-7 b) \cot ^5(e+f x)}{10 a^2 (a-b) f}-\frac {b \cot ^5(e+f x)}{2 a (a-b) f \left (a+b \tan ^2(e+f x)\right )}\\ \end {align*}

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Mathematica [A]  time = 6.28, size = 231, normalized size = 1.06 \[ \frac {b^{7/2} (9 a-7 b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{2 a^{9/2} f (a-b)^2}+\frac {b^4 \sin (2 (e+f x))}{2 a^4 f (a-b) (a \cos (2 (e+f x))+a-b \cos (2 (e+f x))+b)}+\frac {\csc ^3(e+f x) (11 a \cos (e+f x)+10 b \cos (e+f x))}{15 a^3 f}-\frac {\cot (e+f x) \csc ^4(e+f x)}{5 a^2 f}+\frac {\csc (e+f x) \left (-23 a^2 \cos (e+f x)-40 a b \cos (e+f x)-45 b^2 \cos (e+f x)\right )}{15 a^4 f}-\frac {e+f x}{f (a-b)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[e + f*x]^6/(a + b*Tan[e + f*x]^2)^2,x]

[Out]

-((e + f*x)/((a - b)^2*f)) + ((9*a - 7*b)*b^(7/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*a^(9/2)*(a - b)^2
*f) + ((-23*a^2*Cos[e + f*x] - 40*a*b*Cos[e + f*x] - 45*b^2*Cos[e + f*x])*Csc[e + f*x])/(15*a^4*f) + ((11*a*Co
s[e + f*x] + 10*b*Cos[e + f*x])*Csc[e + f*x]^3)/(15*a^3*f) - (Cot[e + f*x]*Csc[e + f*x]^4)/(5*a^2*f) + (b^4*Si
n[2*(e + f*x)])/(2*a^4*(a - b)*f*(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)]))

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fricas [A]  time = 0.54, size = 672, normalized size = 3.08 \[ \left [-\frac {120 \, a^{4} b f x \tan \left (f x + e\right )^{7} + 120 \, a^{5} f x \tan \left (f x + e\right )^{5} + 60 \, {\left (2 \, a^{4} b - 9 \, a b^{4} + 7 \, b^{5}\right )} \tan \left (f x + e\right )^{6} + 24 \, a^{5} - 48 \, a^{4} b + 24 \, a^{3} b^{2} + 40 \, {\left (3 \, a^{5} - a^{4} b - 9 \, a^{2} b^{3} + 7 \, a b^{4}\right )} \tan \left (f x + e\right )^{4} - 8 \, {\left (5 \, a^{5} - 3 \, a^{4} b - 9 \, a^{3} b^{2} + 7 \, a^{2} b^{3}\right )} \tan \left (f x + e\right )^{2} + 15 \, {\left ({\left (9 \, a b^{4} - 7 \, b^{5}\right )} \tan \left (f x + e\right )^{7} + {\left (9 \, a^{2} b^{3} - 7 \, a b^{4}\right )} \tan \left (f x + e\right )^{5}\right )} \sqrt {-\frac {b}{a}} \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 (a b \tan \left (f x + e\right )^{3} - a^{2} \tan \left (f x + e\right )\right )} \sqrt {-\frac {b}{a}}}{b^{2} \tan \left (f x + e\right )^{4} + 2 \, a b \tan \left (f x + e\right )^{2} + a^{2}}\right )}{120 \, {\left ({\left (a^{6} b - 2 \, a^{5} b^{2} + a^{4} b^{3}\right )} f \tan \left (f x + e\right )^{7} + {\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} f \tan \left (f x + e\right )^{5}\right )}}, -\frac {60 \, a^{4} b f x \tan \left (f x + e\right )^{7} + 60 \, a^{5} f x \tan \left (f x + e\right )^{5} + 30 \, {\left (2 \, a^{4} b - 9 \, a b^{4} + 7 \, b^{5}\right )} \tan \left (f x + e\right )^{6} + 12 \, a^{5} - 24 \, a^{4} b + 12 \, a^{3} b^{2} + 20 \, {\left (3 \, a^{5} - a^{4} b - 9 \, a^{2} b^{3} + 7 \, a b^{4}\right )} \tan \left (f x + e\right )^{4} - 4 \, {\left (5 \, a^{5} - 3 \, a^{4} b - 9 \, a^{3} b^{2} + 7 \, a^{2} b^{3}\right )} \tan \left (f x + e\right )^{2} - 15 \, {\left ({\left (9 \, a b^{4} - 7 \, b^{5}\right )} \tan \left (f x + e\right )^{7} + {\left (9 \, a^{2} b^{3} - 7 \, a b^{4}\right )} \tan \left (f x + e\right )^{5}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {{\left (b \tan \left (f x + e\right )^{2} - a\right )} \sqrt {\frac {b}{a}}}{2 \, b \tan \left (f x + e\right )}\right )}{60 \, {\left ({\left (a^{6} b - 2 \, a^{5} b^{2} + a^{4} b^{3}\right )} f \tan \left (f x + e\right )^{7} + {\left (a^{7} - 2 \, a^{6} b + a^{5} b^{2}\right )} f \tan \left (f x + e\right )^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^6/(a+b*tan(f*x+e)^2)^2,x, algorithm="fricas")

[Out]

[-1/120*(120*a^4*b*f*x*tan(f*x + e)^7 + 120*a^5*f*x*tan(f*x + e)^5 + 60*(2*a^4*b - 9*a*b^4 + 7*b^5)*tan(f*x +
e)^6 + 24*a^5 - 48*a^4*b + 24*a^3*b^2 + 40*(3*a^5 - a^4*b - 9*a^2*b^3 + 7*a*b^4)*tan(f*x + e)^4 - 8*(5*a^5 - 3
*a^4*b - 9*a^3*b^2 + 7*a^2*b^3)*tan(f*x + e)^2 + 15*((9*a*b^4 - 7*b^5)*tan(f*x + e)^7 + (9*a^2*b^3 - 7*a*b^4)*
tan(f*x + e)^5)*sqrt(-b/a)*log((b^2*tan(f*x + e)^4 - 6*a*b*tan(f*x + e)^2 + a^2 - 4*(a*b*tan(f*x + e)^3 - a^2*
tan(f*x + e))*sqrt(-b/a))/(b^2*tan(f*x + e)^4 + 2*a*b*tan(f*x + e)^2 + a^2)))/((a^6*b - 2*a^5*b^2 + a^4*b^3)*f
*tan(f*x + e)^7 + (a^7 - 2*a^6*b + a^5*b^2)*f*tan(f*x + e)^5), -1/60*(60*a^4*b*f*x*tan(f*x + e)^7 + 60*a^5*f*x
*tan(f*x + e)^5 + 30*(2*a^4*b - 9*a*b^4 + 7*b^5)*tan(f*x + e)^6 + 12*a^5 - 24*a^4*b + 12*a^3*b^2 + 20*(3*a^5 -
a^4*b - 9*a^2*b^3 + 7*a*b^4)*tan(f*x + e)^4 - 4*(5*a^5 - 3*a^4*b - 9*a^3*b^2 + 7*a^2*b^3)*tan(f*x + e)^2 - 15
*((9*a*b^4 - 7*b^5)*tan(f*x + e)^7 + (9*a^2*b^3 - 7*a*b^4)*tan(f*x + e)^5)*sqrt(b/a)*arctan(1/2*(b*tan(f*x + e
)^2 - a)*sqrt(b/a)/(b*tan(f*x + e))))/((a^6*b - 2*a^5*b^2 + a^4*b^3)*f*tan(f*x + e)^7 + (a^7 - 2*a^6*b + a^5*b
^2)*f*tan(f*x + e)^5)]

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giac [A]  time = 7.35, size = 225, normalized size = 1.03 \[ \frac {\frac {15 \, b^{4} \tan \left (f x + e\right )}{{\left (a^{5} - a^{4} b\right )} {\left (b \tan \left (f x + e\right )^{2} + a\right )}} + \frac {15 \, {\left (9 \, a b^{4} - 7 \, b^{5}\right )} {\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (b) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )\right )}}{{\left (a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right )} \sqrt {a b}} - \frac {30 \, {\left (f x + e\right )}}{a^{2} - 2 \, a b + b^{2}} - \frac {2 \, {\left (15 \, a^{2} \tan \left (f x + e\right )^{4} + 30 \, a b \tan \left (f x + e\right )^{4} + 45 \, b^{2} \tan \left (f x + e\right )^{4} - 5 \, a^{2} \tan \left (f x + e\right )^{2} - 10 \, a b \tan \left (f x + e\right )^{2} + 3 \, a^{2}\right )}}{a^{4} \tan \left (f x + e\right )^{5}}}{30 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^6/(a+b*tan(f*x+e)^2)^2,x, algorithm="giac")

[Out]

1/30*(15*b^4*tan(f*x + e)/((a^5 - a^4*b)*(b*tan(f*x + e)^2 + a)) + 15*(9*a*b^4 - 7*b^5)*(pi*floor((f*x + e)/pi
+ 1/2)*sgn(b) + arctan(b*tan(f*x + e)/sqrt(a*b)))/((a^6 - 2*a^5*b + a^4*b^2)*sqrt(a*b)) - 30*(f*x + e)/(a^2 -
2*a*b + b^2) - 2*(15*a^2*tan(f*x + e)^4 + 30*a*b*tan(f*x + e)^4 + 45*b^2*tan(f*x + e)^4 - 5*a^2*tan(f*x + e)^
2 - 10*a*b*tan(f*x + e)^2 + 3*a^2)/(a^4*tan(f*x + e)^5))/f

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maple [A]  time = 0.94, size = 272, normalized size = 1.25 \[ \frac {b^{4} \tan \left (f x +e \right )}{2 f \,a^{3} \left (a -b \right )^{2} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )}-\frac {b^{5} \tan \left (f x +e \right )}{2 f \,a^{4} \left (a -b \right )^{2} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )}+\frac {9 b^{4} \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {a b}}\right )}{2 f \,a^{3} \left (a -b \right )^{2} \sqrt {a b}}-\frac {7 b^{5} \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {a b}}\right )}{2 f \,a^{4} \left (a -b \right )^{2} \sqrt {a b}}-\frac {1}{5 f \,a^{2} \tan \left (f x +e \right )^{5}}+\frac {1}{3 f \,a^{2} \tan \left (f x +e \right )^{3}}+\frac {2 b}{3 f \,a^{3} \tan \left (f x +e \right )^{3}}-\frac {1}{f \,a^{2} \tan \left (f x +e \right )}-\frac {2 b}{f \,a^{3} \tan \left (f x +e \right )}-\frac {3 b^{2}}{f \,a^{4} \tan \left (f x +e \right )}-\frac {\arctan \left (\tan \left (f x +e \right )\right )}{f \left (a -b \right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^6/(a+b*tan(f*x+e)^2)^2,x)

[Out]

1/2/f*b^4/a^3/(a-b)^2*tan(f*x+e)/(a+b*tan(f*x+e)^2)-1/2/f*b^5/a^4/(a-b)^2*tan(f*x+e)/(a+b*tan(f*x+e)^2)+9/2/f*
b^4/a^3/(a-b)^2/(a*b)^(1/2)*arctan(tan(f*x+e)*b/(a*b)^(1/2))-7/2/f*b^5/a^4/(a-b)^2/(a*b)^(1/2)*arctan(tan(f*x+
e)*b/(a*b)^(1/2))-1/5/f/a^2/tan(f*x+e)^5+1/3/f/a^2/tan(f*x+e)^3+2/3/f/a^3/tan(f*x+e)^3*b-1/f/a^2/tan(f*x+e)-2/
f/a^3/tan(f*x+e)*b-3/f/a^4/tan(f*x+e)*b^2-1/f/(a-b)^2*arctan(tan(f*x+e))

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maxima [A]  time = 0.75, size = 239, normalized size = 1.10 \[ \frac {\frac {15 \, {\left (9 \, a b^{4} - 7 \, b^{5}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )}{{\left (a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right )} \sqrt {a b}} - \frac {15 \, {\left (2 \, a^{3} b + 2 \, a^{2} b^{2} + 2 \, a b^{3} - 7 \, b^{4}\right )} \tan \left (f x + e\right )^{6} + 10 \, {\left (3 \, a^{4} + 2 \, a^{3} b + 2 \, a^{2} b^{2} - 7 \, a b^{3}\right )} \tan \left (f x + e\right )^{4} + 6 \, a^{4} - 6 \, a^{3} b - 2 \, {\left (5 \, a^{4} + 2 \, a^{3} b - 7 \, a^{2} b^{2}\right )} \tan \left (f x + e\right )^{2}}{{\left (a^{5} b - a^{4} b^{2}\right )} \tan \left (f x + e\right )^{7} + {\left (a^{6} - a^{5} b\right )} \tan \left (f x + e\right )^{5}} - \frac {30 \, {\left (f x + e\right )}}{a^{2} - 2 \, a b + b^{2}}}{30 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^6/(a+b*tan(f*x+e)^2)^2,x, algorithm="maxima")

[Out]

1/30*(15*(9*a*b^4 - 7*b^5)*arctan(b*tan(f*x + e)/sqrt(a*b))/((a^6 - 2*a^5*b + a^4*b^2)*sqrt(a*b)) - (15*(2*a^3
*b + 2*a^2*b^2 + 2*a*b^3 - 7*b^4)*tan(f*x + e)^6 + 10*(3*a^4 + 2*a^3*b + 2*a^2*b^2 - 7*a*b^3)*tan(f*x + e)^4 +
6*a^4 - 6*a^3*b - 2*(5*a^4 + 2*a^3*b - 7*a^2*b^2)*tan(f*x + e)^2)/((a^5*b - a^4*b^2)*tan(f*x + e)^7 + (a^6 -
a^5*b)*tan(f*x + e)^5) - 30*(f*x + e)/(a^2 - 2*a*b + b^2))/f

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mupad [B]  time = 16.00, size = 3030, normalized size = 13.90 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(e + f*x)^6/(a + b*tan(e + f*x)^2)^2,x)

[Out]

- (1/(5*a) + (tan(e + f*x)^4*(5*a*b + 3*a^2 + 7*b^2))/(3*a^3) - (tan(e + f*x)^2*(5*a + 7*b))/(15*a^2) + (tan(e
+ f*x)^6*(2*a*b^3 + 2*a^3*b - 7*b^4 + 2*a^2*b^2))/(2*a^4*(a - b)))/(f*(a*tan(e + f*x)^5 + b*tan(e + f*x)^7))
- (2*atan(((tan(e + f*x)*(784*a^12*b^14 - 4368*a^13*b^13 + 9696*a^14*b^12 - 10720*a^15*b^11 + 5904*a^16*b^10 -
1296*a^17*b^9 + 64*a^20*b^6 - 192*a^21*b^5 + 192*a^22*b^4 - 64*a^23*b^3) + ((2816*a^17*b^11 - 448*a^16*b^12 -
7360*a^18*b^10 + 10240*a^19*b^9 - 8000*a^20*b^8 + 3200*a^21*b^7 + 64*a^22*b^6 - 1280*a^23*b^5 + 1280*a^24*b^4
- 640*a^25*b^3 + 128*a^26*b^2 + (tan(e + f*x)*(256*a^20*b^10 - 1536*a^21*b^9 + 3584*a^22*b^8 - 3584*a^23*b^7
+ 3584*a^25*b^5 - 3584*a^26*b^4 + 1536*a^27*b^3 - 256*a^28*b^2)*1i)/(2*a^2 - 4*a*b + 2*b^2))*1i)/(2*a^2 - 4*a*
b + 2*b^2))/(2*a^2 - 4*a*b + 2*b^2) + (tan(e + f*x)*(784*a^12*b^14 - 4368*a^13*b^13 + 9696*a^14*b^12 - 10720*a
^15*b^11 + 5904*a^16*b^10 - 1296*a^17*b^9 + 64*a^20*b^6 - 192*a^21*b^5 + 192*a^22*b^4 - 64*a^23*b^3) + ((448*a
^16*b^12 - 2816*a^17*b^11 + 7360*a^18*b^10 - 10240*a^19*b^9 + 8000*a^20*b^8 - 3200*a^21*b^7 - 64*a^22*b^6 + 12
80*a^23*b^5 - 1280*a^24*b^4 + 640*a^25*b^3 - 128*a^26*b^2 + (tan(e + f*x)*(256*a^20*b^10 - 1536*a^21*b^9 + 358
4*a^22*b^8 - 3584*a^23*b^7 + 3584*a^25*b^5 - 3584*a^26*b^4 + 1536*a^27*b^3 - 256*a^28*b^2)*1i)/(2*a^2 - 4*a*b
+ 2*b^2))*1i)/(2*a^2 - 4*a*b + 2*b^2))/(2*a^2 - 4*a*b + 2*b^2))/(((tan(e + f*x)*(784*a^12*b^14 - 4368*a^13*b^1
3 + 9696*a^14*b^12 - 10720*a^15*b^11 + 5904*a^16*b^10 - 1296*a^17*b^9 + 64*a^20*b^6 - 192*a^21*b^5 + 192*a^22*
b^4 - 64*a^23*b^3) + ((448*a^16*b^12 - 2816*a^17*b^11 + 7360*a^18*b^10 - 10240*a^19*b^9 + 8000*a^20*b^8 - 3200
*a^21*b^7 - 64*a^22*b^6 + 1280*a^23*b^5 - 1280*a^24*b^4 + 640*a^25*b^3 - 128*a^26*b^2 + (tan(e + f*x)*(256*a^2
0*b^10 - 1536*a^21*b^9 + 3584*a^22*b^8 - 3584*a^23*b^7 + 3584*a^25*b^5 - 3584*a^26*b^4 + 1536*a^27*b^3 - 256*a
^28*b^2)*1i)/(2*a^2 - 4*a*b + 2*b^2))*1i)/(2*a^2 - 4*a*b + 2*b^2))*1i)/(2*a^2 - 4*a*b + 2*b^2) - ((tan(e + f*x
)*(784*a^12*b^14 - 4368*a^13*b^13 + 9696*a^14*b^12 - 10720*a^15*b^11 + 5904*a^16*b^10 - 1296*a^17*b^9 + 64*a^2
0*b^6 - 192*a^21*b^5 + 192*a^22*b^4 - 64*a^23*b^3) + ((2816*a^17*b^11 - 448*a^16*b^12 - 7360*a^18*b^10 + 10240
*a^19*b^9 - 8000*a^20*b^8 + 3200*a^21*b^7 + 64*a^22*b^6 - 1280*a^23*b^5 + 1280*a^24*b^4 - 640*a^25*b^3 + 128*a
^26*b^2 + (tan(e + f*x)*(256*a^20*b^10 - 1536*a^21*b^9 + 3584*a^22*b^8 - 3584*a^23*b^7 + 3584*a^25*b^5 - 3584*
a^26*b^4 + 1536*a^27*b^3 - 256*a^28*b^2)*1i)/(2*a^2 - 4*a*b + 2*b^2))*1i)/(2*a^2 - 4*a*b + 2*b^2))*1i)/(2*a^2
- 4*a*b + 2*b^2) + 784*a^12*b^12 - 2800*a^13*b^11 + 3312*a^14*b^10 - 1296*a^15*b^9 + 224*a^16*b^8 - 512*a^17*b
^7 + 288*a^18*b^6)))/(f*(2*a^2 - 4*a*b + 2*b^2)) - (atan((((tan(e + f*x)*(784*a^12*b^14 - 4368*a^13*b^13 + 969
6*a^14*b^12 - 10720*a^15*b^11 + 5904*a^16*b^10 - 1296*a^17*b^9 + 64*a^20*b^6 - 192*a^21*b^5 + 192*a^22*b^4 - 6
4*a^23*b^3) + ((9*a - 7*b)*(-a^9*b^7)^(1/2)*(2816*a^17*b^11 - 448*a^16*b^12 - 7360*a^18*b^10 + 10240*a^19*b^9
- 8000*a^20*b^8 + 3200*a^21*b^7 + 64*a^22*b^6 - 1280*a^23*b^5 + 1280*a^24*b^4 - 640*a^25*b^3 + 128*a^26*b^2 +
(tan(e + f*x)*(9*a - 7*b)*(-a^9*b^7)^(1/2)*(256*a^20*b^10 - 1536*a^21*b^9 + 3584*a^22*b^8 - 3584*a^23*b^7 + 35
84*a^25*b^5 - 3584*a^26*b^4 + 1536*a^27*b^3 - 256*a^28*b^2))/(4*(a^11 - 2*a^10*b + a^9*b^2))))/(4*(a^11 - 2*a^
10*b + a^9*b^2)))*(9*a - 7*b)*(-a^9*b^7)^(1/2)*1i)/(4*(a^11 - 2*a^10*b + a^9*b^2)) + ((tan(e + f*x)*(784*a^12*
b^14 - 4368*a^13*b^13 + 9696*a^14*b^12 - 10720*a^15*b^11 + 5904*a^16*b^10 - 1296*a^17*b^9 + 64*a^20*b^6 - 192*
a^21*b^5 + 192*a^22*b^4 - 64*a^23*b^3) + ((9*a - 7*b)*(-a^9*b^7)^(1/2)*(448*a^16*b^12 - 2816*a^17*b^11 + 7360*
a^18*b^10 - 10240*a^19*b^9 + 8000*a^20*b^8 - 3200*a^21*b^7 - 64*a^22*b^6 + 1280*a^23*b^5 - 1280*a^24*b^4 + 640
*a^25*b^3 - 128*a^26*b^2 + (tan(e + f*x)*(9*a - 7*b)*(-a^9*b^7)^(1/2)*(256*a^20*b^10 - 1536*a^21*b^9 + 3584*a^
22*b^8 - 3584*a^23*b^7 + 3584*a^25*b^5 - 3584*a^26*b^4 + 1536*a^27*b^3 - 256*a^28*b^2))/(4*(a^11 - 2*a^10*b +
a^9*b^2))))/(4*(a^11 - 2*a^10*b + a^9*b^2)))*(9*a - 7*b)*(-a^9*b^7)^(1/2)*1i)/(4*(a^11 - 2*a^10*b + a^9*b^2)))
/(784*a^12*b^12 - 2800*a^13*b^11 + 3312*a^14*b^10 - 1296*a^15*b^9 + 224*a^16*b^8 - 512*a^17*b^7 + 288*a^18*b^6
- ((tan(e + f*x)*(784*a^12*b^14 - 4368*a^13*b^13 + 9696*a^14*b^12 - 10720*a^15*b^11 + 5904*a^16*b^10 - 1296*a
^17*b^9 + 64*a^20*b^6 - 192*a^21*b^5 + 192*a^22*b^4 - 64*a^23*b^3) + ((9*a - 7*b)*(-a^9*b^7)^(1/2)*(2816*a^17*
b^11 - 448*a^16*b^12 - 7360*a^18*b^10 + 10240*a^19*b^9 - 8000*a^20*b^8 + 3200*a^21*b^7 + 64*a^22*b^6 - 1280*a^
23*b^5 + 1280*a^24*b^4 - 640*a^25*b^3 + 128*a^26*b^2 + (tan(e + f*x)*(9*a - 7*b)*(-a^9*b^7)^(1/2)*(256*a^20*b^
10 - 1536*a^21*b^9 + 3584*a^22*b^8 - 3584*a^23*b^7 + 3584*a^25*b^5 - 3584*a^26*b^4 + 1536*a^27*b^3 - 256*a^28*
b^2))/(4*(a^11 - 2*a^10*b + a^9*b^2))))/(4*(a^11 - 2*a^10*b + a^9*b^2)))*(9*a - 7*b)*(-a^9*b^7)^(1/2))/(4*(a^1
1 - 2*a^10*b + a^9*b^2)) + ((tan(e + f*x)*(784*a^12*b^14 - 4368*a^13*b^13 + 9696*a^14*b^12 - 10720*a^15*b^11 +
5904*a^16*b^10 - 1296*a^17*b^9 + 64*a^20*b^6 - 192*a^21*b^5 + 192*a^22*b^4 - 64*a^23*b^3) + ((9*a - 7*b)*(-a^
9*b^7)^(1/2)*(448*a^16*b^12 - 2816*a^17*b^11 + 7360*a^18*b^10 - 10240*a^19*b^9 + 8000*a^20*b^8 - 3200*a^21*b^7
- 64*a^22*b^6 + 1280*a^23*b^5 - 1280*a^24*b^4 + 640*a^25*b^3 - 128*a^26*b^2 + (tan(e + f*x)*(9*a - 7*b)*(-a^9
*b^7)^(1/2)*(256*a^20*b^10 - 1536*a^21*b^9 + 3584*a^22*b^8 - 3584*a^23*b^7 + 3584*a^25*b^5 - 3584*a^26*b^4 + 1
536*a^27*b^3 - 256*a^28*b^2))/(4*(a^11 - 2*a^10*b + a^9*b^2))))/(4*(a^11 - 2*a^10*b + a^9*b^2)))*(9*a - 7*b)*(
-a^9*b^7)^(1/2))/(4*(a^11 - 2*a^10*b + a^9*b^2))))*(9*a - 7*b)*(-a^9*b^7)^(1/2)*1i)/(2*f*(a^11 - 2*a^10*b + a^
9*b^2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**6/(a+b*tan(f*x+e)**2)**2,x)

[Out]

Timed out

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