3.3.43 \(\int \frac {\tan ^6(e+f x)}{(a+b \tan ^2(e+f x))^3} \, dx\) [243]

Optimal. Leaf size=153 \[ -\frac {x}{(a-b)^3}+\frac {\sqrt {a} \left (3 a^2-10 a b+15 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{8 (a-b)^3 b^{5/2} f}-\frac {a \tan ^3(e+f x)}{4 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {a (3 a-7 b) \tan (e+f x)}{8 (a-b)^2 b^2 f \left (a+b \tan ^2(e+f x)\right )} \]

[Out]

-x/(a-b)^3+1/8*(3*a^2-10*a*b+15*b^2)*arctan(b^(1/2)*tan(f*x+e)/a^(1/2))*a^(1/2)/(a-b)^3/b^(5/2)/f-1/4*a*tan(f*
x+e)^3/(a-b)/b/f/(a+b*tan(f*x+e)^2)^2-1/8*a*(3*a-7*b)*tan(f*x+e)/(a-b)^2/b^2/f/(a+b*tan(f*x+e)^2)

________________________________________________________________________________________

Rubi [A]
time = 0.16, antiderivative size = 153, 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 = {3751, 481, 592, 536, 209, 211} \begin {gather*} \frac {\sqrt {a} \left (3 a^2-10 a b+15 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{8 b^{5/2} f (a-b)^3}-\frac {a (3 a-7 b) \tan (e+f x)}{8 b^2 f (a-b)^2 \left (a+b \tan ^2(e+f x)\right )}-\frac {a \tan ^3(e+f x)}{4 b f (a-b) \left (a+b \tan ^2(e+f x)\right )^2}-\frac {x}{(a-b)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tan[e + f*x]^6/(a + b*Tan[e + f*x]^2)^3,x]

[Out]

-(x/(a - b)^3) + (Sqrt[a]*(3*a^2 - 10*a*b + 15*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*(a - b)^3*b^(5/
2)*f) - (a*Tan[e + f*x]^3)/(4*(a - b)*b*f*(a + b*Tan[e + f*x]^2)^2) - (a*(3*a - 7*b)*Tan[e + f*x])/(8*(a - b)^
2*b^2*f*(a + b*Tan[e + f*x]^2))

Rule 209

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

Rule 211

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

Rule 481

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 536

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 592

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Simp[g^(n - 1)*(b*e - a*f)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c -
a*d)*(p + 1))), x] - Dist[g^n/(b*n*(b*c - a*d)*(p + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*S
imp[c*(b*e - a*f)*(m - n + 1) + (d*(b*e - a*f)*(m + n*q + 1) - b*n*(c*f - d*e)*(p + 1))*x^n, x], x], x] /; Fre
eQ[{a, b, c, d, e, f, g, q}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, 0]

Rule 3751

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 ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {x^6}{\left (1+x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {a \tan ^3(e+f x)}{4 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^2}+\frac {\text {Subst}\left (\int \frac {x^2 \left (3 a+(3 a-4 b) x^2\right )}{\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 ^3(e+f x)}{4 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {a (3 a-7 b) \tan (e+f x)}{8 (a-b)^2 b^2 f \left (a+b \tan ^2(e+f x)\right )}-\frac {\text {Subst}\left (\int \frac {-a (3 a-7 b)+\left (-3 a^2+7 a b-8 b^2\right ) x^2}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{8 (a-b)^2 b^2 f}\\ &=-\frac {a \tan ^3(e+f x)}{4 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {a (3 a-7 b) \tan (e+f x)}{8 (a-b)^2 b^2 f \left (a+b \tan ^2(e+f x)\right )}-\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{(a-b)^3 f}+\frac {\left (a \left (3 a^2-10 a b+15 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tan (e+f x)\right )}{8 (a-b)^3 b^2 f}\\ &=-\frac {x}{(a-b)^3}+\frac {\sqrt {a} \left (3 a^2-10 a b+15 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{8 (a-b)^3 b^{5/2} f}-\frac {a \tan ^3(e+f x)}{4 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {a (3 a-7 b) \tan (e+f x)}{8 (a-b)^2 b^2 f \left (a+b \tan ^2(e+f x)\right )}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.59, size = 142, normalized size = 0.93 \begin {gather*} \frac {-8 (e+f x)+\frac {\sqrt {a} \left (3 a^2-10 a b+15 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{b^{5/2}}-\frac {a (a-b) \left (3 a^2-2 a b-9 b^2+3 \left (a^2-4 a b+3 b^2\right ) \cos (2 (e+f x))\right ) \sin (2 (e+f x))}{b^2 (a+b+(a-b) \cos (2 (e+f x)))^2}}{8 (a-b)^3 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tan[e + f*x]^6/(a + b*Tan[e + f*x]^2)^3,x]

[Out]

(-8*(e + f*x) + (Sqrt[a]*(3*a^2 - 10*a*b + 15*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/b^(5/2) - (a*(a - b
)*(3*a^2 - 2*a*b - 9*b^2 + 3*(a^2 - 4*a*b + 3*b^2)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)])/(b^2*(a + b + (a - b)*C
os[2*(e + f*x)])^2))/(8*(a - b)^3*f)

________________________________________________________________________________________

Maple [A]
time = 0.35, size = 142, normalized size = 0.93

method	result	size
derivativedivides	\(\frac {-\frac {\arctan \left (\tan \left (f x +e \right )\right )}{\left (a -b \right )^{3}}+\frac {a \left (\frac {-\frac {\left (5 a^{2}-14 a b +9 b^{2}\right ) \left (\tan ^{3}\left (f x +e \right )\right )}{8 b}-\frac {a \left (3 a^{2}-10 a b +7 b^{2}\right ) \tan \left (f x +e \right )}{8 b^{2}}}{\left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {\left (3 a^{2}-10 a b +15 b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {a b}}\right )}{8 b^{2} \sqrt {a b}}\right )}{\left (a -b \right )^{3}}}{f}\)	\(142\)
default	\(\frac {-\frac {\arctan \left (\tan \left (f x +e \right )\right )}{\left (a -b \right )^{3}}+\frac {a \left (\frac {-\frac {\left (5 a^{2}-14 a b +9 b^{2}\right ) \left (\tan ^{3}\left (f x +e \right )\right )}{8 b}-\frac {a \left (3 a^{2}-10 a b +7 b^{2}\right ) \tan \left (f x +e \right )}{8 b^{2}}}{\left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {\left (3 a^{2}-10 a b +15 b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {a b}}\right )}{8 b^{2} \sqrt {a b}}\right )}{\left (a -b \right )^{3}}}{f}\)	\(142\)
risch	\(-\frac {x}{a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}}-\frac {i \left (3 a^{3} {\mathrm e}^{6 i \left (f x +e \right )}-13 a^{2} b \,{\mathrm e}^{6 i \left (f x +e \right )}+a \,b^{2} {\mathrm e}^{6 i \left (f x +e \right )}+9 b^{3} {\mathrm e}^{6 i \left (f x +e \right )}+9 a^{3} {\mathrm e}^{4 i \left (f x +e \right )}-21 a^{2} b \,{\mathrm e}^{4 i \left (f x +e \right )}-9 a \,b^{2} {\mathrm e}^{4 i \left (f x +e \right )}-27 b^{3} {\mathrm e}^{4 i \left (f x +e \right )}+9 a^{3} {\mathrm e}^{2 i \left (f x +e \right )}-23 a^{2} b \,{\mathrm e}^{2 i \left (f x +e \right )}-13 a \,b^{2} {\mathrm e}^{2 i \left (f x +e \right )}+27 b^{3} {\mathrm e}^{2 i \left (f x +e \right )}+3 a^{3}-15 a^{2} b +21 a \,b^{2}-9 b^{3}\right ) a}{4 \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}-b \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+2 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a -b \right )^{2} f \left (a -b \right )^{3} b^{2}}-\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-a b}+a +b}{a -b}\right ) a^{2}}{16 b^{3} \left (a -b \right )^{3} f}+\frac {5 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-a b}+a +b}{a -b}\right ) a}{8 b^{2} \left (a -b \right )^{3} f}-\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-a b}+a +b}{a -b}\right )}{16 b \left (a -b \right )^{3} f}+\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-a b}-a -b}{a -b}\right ) a^{2}}{16 b^{3} \left (a -b \right )^{3} f}-\frac {5 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-a b}-a -b}{a -b}\right ) a}{8 b^{2} \left (a -b \right )^{3} f}+\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-a b}-a -b}{a -b}\right )}{16 b \left (a -b \right )^{3} f}\)	\(630\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(f*x+e)^6/(a+b*tan(f*x+e)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/f*(-1/(a-b)^3*arctan(tan(f*x+e))+a/(a-b)^3*((-1/8*(5*a^2-14*a*b+9*b^2)/b*tan(f*x+e)^3-1/8*a*(3*a^2-10*a*b+7*
b^2)/b^2*tan(f*x+e))/(a+b*tan(f*x+e)^2)^2+1/8*(3*a^2-10*a*b+15*b^2)/b^2/(a*b)^(1/2)*arctan(b*tan(f*x+e)/(a*b)^
(1/2))))

________________________________________________________________________________________

Maxima [A]
time = 0.50, size = 235, normalized size = 1.54 \begin {gather*} \frac {\frac {{\left (3 \, a^{3} - 10 \, a^{2} b + 15 \, a b^{2}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )}{{\left (a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right )} \sqrt {a b}} - \frac {{\left (5 \, a^{2} b - 9 \, a b^{2}\right )} \tan \left (f x + e\right )^{3} + {\left (3 \, a^{3} - 7 \, a^{2} b\right )} \tan \left (f x + e\right )}{a^{4} b^{2} - 2 \, a^{3} b^{3} + a^{2} b^{4} + {\left (a^{2} b^{4} - 2 \, a b^{5} + b^{6}\right )} \tan \left (f x + e\right )^{4} + 2 \, {\left (a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5}\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^6/(a+b*tan(f*x+e)^2)^3,x, algorithm="maxima")

[Out]

1/8*((3*a^3 - 10*a^2*b + 15*a*b^2)*arctan(b*tan(f*x + e)/sqrt(a*b))/((a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*sqr
t(a*b)) - ((5*a^2*b - 9*a*b^2)*tan(f*x + e)^3 + (3*a^3 - 7*a^2*b)*tan(f*x + e))/(a^4*b^2 - 2*a^3*b^3 + a^2*b^4
+ (a^2*b^4 - 2*a*b^5 + b^6)*tan(f*x + e)^4 + 2*(a^3*b^3 - 2*a^2*b^4 + a*b^5)*tan(f*x + e)^2) - 8*(f*x + e)/(a
^3 - 3*a^2*b + 3*a*b^2 - b^3))/f

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (144) = 288\).
time = 3.72, size = 767, normalized size = 5.01 \begin {gather*} \left [-\frac {32 \, b^{4} f x \tan \left (f x + e\right )^{4} + 64 \, a b^{3} f x \tan \left (f x + e\right )^{2} + 32 \, a^{2} b^{2} f x + 4 \, {\left (5 \, a^{3} b - 14 \, a^{2} b^{2} + 9 \, a b^{3}\right )} \tan \left (f x + e\right )^{3} + {\left ({\left (3 \, a^{2} b^{2} - 10 \, a b^{3} + 15 \, b^{4}\right )} \tan \left (f x + e\right )^{4} + 3 \, a^{4} - 10 \, a^{3} b + 15 \, a^{2} b^{2} + 2 \, {\left (3 \, a^{3} b - 10 \, a^{2} b^{2} + 15 \, a b^{3}\right )} \tan \left (f x + e\right )^{2}\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 (3 \, a^{4} - 10 \, a^{3} b + 7 \, a^{2} b^{2}\right )} \tan \left (f x + e\right )}{32 \, {\left ({\left (a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right )} f \tan \left (f x + e\right )^{4} + 2 \, {\left (a^{4} b^{3} - 3 \, a^{3} b^{4} + 3 \, a^{2} b^{5} - a b^{6}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - a^{2} b^{5}\right )} f\right )}}, -\frac {16 \, b^{4} f x \tan \left (f x + e\right )^{4} + 32 \, a b^{3} f x \tan \left (f x + e\right )^{2} + 16 \, a^{2} b^{2} f x + 2 \, {\left (5 \, a^{3} b - 14 \, a^{2} b^{2} + 9 \, a b^{3}\right )} \tan \left (f x + e\right )^{3} - {\left ({\left (3 \, a^{2} b^{2} - 10 \, a b^{3} + 15 \, b^{4}\right )} \tan \left (f x + e\right )^{4} + 3 \, a^{4} - 10 \, a^{3} b + 15 \, a^{2} b^{2} + 2 \, {\left (3 \, a^{3} b - 10 \, a^{2} b^{2} + 15 \, a b^{3}\right )} \tan \left (f x + e\right )^{2}\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 (3 \, a^{4} - 10 \, a^{3} b + 7 \, a^{2} b^{2}\right )} \tan \left (f x + e\right )}{16 \, {\left ({\left (a^{3} b^{4} - 3 \, a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right )} f \tan \left (f x + e\right )^{4} + 2 \, {\left (a^{4} b^{3} - 3 \, a^{3} b^{4} + 3 \, a^{2} b^{5} - a b^{6}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - a^{2} b^{5}\right )} f\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^6/(a+b*tan(f*x+e)^2)^3,x, algorithm="fricas")

[Out]

[-1/32*(32*b^4*f*x*tan(f*x + e)^4 + 64*a*b^3*f*x*tan(f*x + e)^2 + 32*a^2*b^2*f*x + 4*(5*a^3*b - 14*a^2*b^2 + 9
*a*b^3)*tan(f*x + e)^3 + ((3*a^2*b^2 - 10*a*b^3 + 15*b^4)*tan(f*x + e)^4 + 3*a^4 - 10*a^3*b + 15*a^2*b^2 + 2*(
3*a^3*b - 10*a^2*b^2 + 15*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^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*(3*a^4 - 10*a^3*b + 7*a^2*b^2)*tan(f*x + e))/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*f*tan(f*x + e)^4 + 2*(
a^4*b^3 - 3*a^3*b^4 + 3*a^2*b^5 - a*b^6)*f*tan(f*x + e)^2 + (a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*f), -1
/16*(16*b^4*f*x*tan(f*x + e)^4 + 32*a*b^3*f*x*tan(f*x + e)^2 + 16*a^2*b^2*f*x + 2*(5*a^3*b - 14*a^2*b^2 + 9*a*
b^3)*tan(f*x + e)^3 - ((3*a^2*b^2 - 10*a*b^3 + 15*b^4)*tan(f*x + e)^4 + 3*a^4 - 10*a^3*b + 15*a^2*b^2 + 2*(3*a
^3*b - 10*a^2*b^2 + 15*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*tan(f*x
+ e))) + 2*(3*a^4 - 10*a^3*b + 7*a^2*b^2)*tan(f*x + e))/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*f*tan(f*x + e)
^4 + 2*(a^4*b^3 - 3*a^3*b^4 + 3*a^2*b^5 - a*b^6)*f*tan(f*x + e)^2 + (a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5
)*f)]

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 8974 vs. \(2 (133) = 266\).
time = 77.53, size = 8974, normalized size = 58.65 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)**6/(a+b*tan(f*x+e)**2)**3,x)

[Out]

Piecewise((zoo*x, Eq(a, 0) & Eq(b, 0) & Eq(f, 0)), ((-x + tan(e + f*x)**5/(5*f) - tan(e + f*x)**3/(3*f) + tan(
e + f*x)/f)/a**3, Eq(b, 0)), (x/b**3, Eq(a, 0)), (15*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) + 45*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) + 45*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) + 15*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) - 33*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) - 40*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) - 15*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), E
q(a, b)), (x*tan(e)**6/(a + b*tan(e)**2)**3, Eq(f, 0)), (3*a**5*log(-sqrt(-a/b) + tan(e + f*x))/(16*a**5*b**3*
f*sqrt(-a/b) + 32*a**4*b**4*f*sqrt(-a/b)*tan(e + f*x)**2 - 48*a**4*b**4*f*sqrt(-a/b) + 16*a**3*b**5*f*sqrt(-a/
b)*tan(e + f*x)**4 - 96*a**3*b**5*f*sqrt(-a/b)*tan(e + f*x)**2 + 48*a**3*b**5*f*sqrt(-a/b) - 48*a**2*b**6*f*sq
rt(-a/b)*tan(e + f*x)**4 + 96*a**2*b**6*f*sqrt(-a/b)*tan(e + f*x)**2 - 16*a**2*b**6*f*sqrt(-a/b) + 48*a*b**7*f
*sqrt(-a/b)*tan(e + f*x)**4 - 32*a*b**7*f*sqrt(-a/b)*tan(e + f*x)**2 - 16*b**8*f*sqrt(-a/b)*tan(e + f*x)**4) -
3*a**5*log(sqrt(-a/b) + tan(e + f*x))/(16*a**5*b**3*f*sqrt(-a/b) + 32*a**4*b**4*f*sqrt(-a/b)*tan(e + f*x)**2
- 48*a**4*b**4*f*sqrt(-a/b) + 16*a**3*b**5*f*sqrt(-a/b)*tan(e + f*x)**4 - 96*a**3*b**5*f*sqrt(-a/b)*tan(e + f*
x)**2 + 48*a**3*b**5*f*sqrt(-a/b) - 48*a**2*b**6*f*sqrt(-a/b)*tan(e + f*x)**4 + 96*a**2*b**6*f*sqrt(-a/b)*tan(
e + f*x)**2 - 16*a**2*b**6*f*sqrt(-a/b) + 48*a*b**7*f*sqrt(-a/b)*tan(e + f*x)**4 - 32*a*b**7*f*sqrt(-a/b)*tan(
e + f*x)**2 - 16*b**8*f*sqrt(-a/b)*tan(e + f*x)**4) - 6*a**4*b*sqrt(-a/b)*tan(e + f*x)/(16*a**5*b**3*f*sqrt(-a
/b) + 32*a**4*b**4*f*sqrt(-a/b)*tan(e + f*x)**2 - 48*a**4*b**4*f*sqrt(-a/b) + 16*a**3*b**5*f*sqrt(-a/b)*tan(e
+ f*x)**4 - 96*a**3*b**5*f*sqrt(-a/b)*tan(e + f*x)**2 + 48*a**3*b**5*f*sqrt(-a/b) - 48*a**2*b**6*f*sqrt(-a/b)*
tan(e + f*x)**4 + 96*a**2*b**6*f*sqrt(-a/b)*tan(e + f*x)**2 - 16*a**2*b**6*f*sqrt(-a/b) + 48*a*b**7*f*sqrt(-a/
b)*tan(e + f*x)**4 - 32*a*b**7*f*sqrt(-a/b)*tan(e + f*x)**2 - 16*b**8*f*sqrt(-a/b)*tan(e + f*x)**4) + 6*a**4*b
*log(-sqrt(-a/b) + tan(e + f*x))*tan(e + f*x)**2/(16*a**5*b**3*f*sqrt(-a/b) + 32*a**4*b**4*f*sqrt(-a/b)*tan(e
+ f*x)**2 - 48*a**4*b**4*f*sqrt(-a/b) + 16*a**3*b**5*f*sqrt(-a/b)*tan(e + f*x)**4 - 96*a**3*b**5*f*sqrt(-a/b)*
tan(e + f*x)**2 + 48*a**3*b**5*f*sqrt(-a/b) - 48*a**2*b**6*f*sqrt(-a/b)*tan(e + f*x)**4 + 96*a**2*b**6*f*sqrt(
-a/b)*tan(e + f*x)**2 - 16*a**2*b**6*f*sqrt(-a/b) + 48*a*b**7*f*sqrt(-a/b)*tan(e + f*x)**4 - 32*a*b**7*f*sqrt(
-a/b)*tan(e + f*x)**2 - 16*b**8*f*sqrt(-a/b)*tan(e + f*x)**4) - 10*a**4*b*log(-sqrt(-a/b) + tan(e + f*x))/(16*
a**5*b**3*f*sqrt(-a/b) + 32*a**4*b**4*f*sqrt(-a/b)*tan(e + f*x)**2 - 48*a**4*b**4*f*sqrt(-a/b) + 16*a**3*b**5*
f*sqrt(-a/b)*tan(e + f*x)**4 - 96*a**3*b**5*f*sqrt(-a/b)*tan(e + f*x)**2 + 48*a**3*b**5*f*sqrt(-a/b) - 48*a**2
*b**6*f*sqrt(-a/b)*tan(e + f*x)**4 + 96*a**2*b**6*f*sqrt(-a/b)*tan(e + f*x)**2 - 16*a**2*b**6*f*sqrt(-a/b) + 4
8*a*b**7*f*sqrt(-a/b)*tan(e + f*x)**4 - 32*a*b**7*f*sqrt(-a/b)*tan(e + f*x)**2 - 16*b**8*f*sqrt(-a/b)*tan(e +
f*x)**4) - 6*a**4*b*log(sqrt(-a/b) + tan(e + f*x))*tan(e + f*x)**2/(16*a**5*b**3*f*sqrt(-a/b) + 32*a**4*b**4*f
*sqrt(-a/b)*tan(e + f*x)**2 - 48*a**4*b**4*f*sqrt(-a/b) + 16*a**3*b**5*f*sqrt(-a/b)*tan(e + f*x)**4 - 96*a**3*
b**5*f*sqrt(-a/b)*tan(e + f*x)**2 + 48*a**3*b**5*f*sqrt(-a/b) - 48*a**2*b**6*f*sqrt(-a/b)*tan(e + f*x)**4 + 96
*a**2*b**6*f*sqrt(-a/b)*tan(e + f*x)**2 - 16*a**2*b**6*f*sqrt(-a/b) + 48*a*b**7*f*sqrt(-a/b)*tan(e + f*x)**4 -
32*a*b**7*f*sqrt(-a/b)*tan(e + f*x)**2 - 16*b**8*f*sqrt(-a/b)*tan(e + f*x)**4) + 10*a**4*b*log(sqrt(-a/b) + t
an(e + f*x))/(16*a**5*b**3*f*sqrt(-a/b) + 32*a**4*b**4*f*sqrt(-a/b)*tan(e + f*x)**2 - 48*a**4*b**4*f*sqrt(-a/b
) + 16*a**3*b**5*f*sqrt(-a/b)*tan(e + f*x)**4 - 96*a**3*b**5*f*sqrt(-a/b)*tan(e + f*x)**2 + 48*a**3*b**5*f*sqr
t(-a/b) - 48*a**2*b**6*f*sqrt(-a/b)*tan(e + f*x)**4 + 96*a**2*b**6*f*sqrt(-a/b)*tan(e + f*x)**2 - 16*a**2*b**6
*f*sqrt(-a/b) + 48*a*b**7*f*sqrt(-a/b)*tan(e + f*x)**4 - 32*a*b**7*f*sqrt(-a/b)*tan(e + f*x)**2 - 16*b**8*f*sq
rt(-a/b)*tan(e + f*x)**4) - 10*a**3*b**2*sqrt(-a/b)*tan(e + f*x)**3/(16*a**5*b**3*f*sqrt(-a/b) + 32*a**4*b**4*
f*sqrt(-a/b)*tan(e + f*x)**2 - 48*a**4*b**4*f*sqrt(-a/b) + 16*a**3*b**5*f*sqrt(-a/b)*tan(e + f*x)**4 - 96*a**3
*b**5*f*sqrt(-a/b)*tan(e + f*x)**2 + 48*a**3*b**5*f*sqrt(-a/b) - 48*a**2*b**6*f*sqrt(-a/b)*tan(e + f*x)**4 + 9
6*a**2*b**6*f*sqrt(-a/b)*tan(e + f*x)**2 - 16*a**2*b**6*f*sqrt(-a/b) + 48*a*b**7*f*sqrt(-a/b)*tan(e + f*x)**4
- 32*a*b**7*f*sqrt(-a/b)*tan(e + f*x)**2 - 16*b...

________________________________________________________________________________________

Giac [A]
time = 2.50, size = 215, normalized size = 1.41 \begin {gather*} \frac {\frac {{\left (3 \, a^{3} - 10 \, a^{2} b + 15 \, a b^{2}\right )} {\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )\right )}}{{\left (a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right )} \sqrt {a b}} - \frac {8 \, {\left (f x + e\right )}}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {5 \, a^{2} b \tan \left (f x + e\right )^{3} - 9 \, a b^{2} \tan \left (f x + e\right )^{3} + 3 \, a^{3} \tan \left (f x + e\right ) - 7 \, a^{2} b \tan \left (f x + e\right )}{{\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{2}}}{8 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^6/(a+b*tan(f*x+e)^2)^3,x, algorithm="giac")

[Out]

1/8*((3*a^3 - 10*a^2*b + 15*a*b^2)*(pi*floor((f*x + e)/pi + 1/2)*sgn(b) + arctan(b*tan(f*x + e)/sqrt(a*b)))/((
a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*sqrt(a*b)) - 8*(f*x + e)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) - (5*a^2*b*tan(f
*x + e)^3 - 9*a*b^2*tan(f*x + e)^3 + 3*a^3*tan(f*x + e) - 7*a^2*b*tan(f*x + e))/((a^2*b^2 - 2*a*b^3 + b^4)*(b*
tan(f*x + e)^2 + a)^2))/f

________________________________________________________________________________________

Mupad [B]
time = 15.52, size = 2500, normalized size = 16.34 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(e + f*x)^6/(a + b*tan(e + f*x)^2)^3,x)

[Out]

((tan(e + f*x)^3*(9*a*b - 5*a^2))/(8*(a^2*b - 2*a*b^2 + b^3)) + (a*tan(e + f*x)*(7*a*b - 3*a^2))/(8*b*(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((((((224*a*b^10 - 1440*a^2*b
^9 + 3936*a^3*b^8 - 5920*a^4*b^7 + 5280*a^5*b^6 - 2784*a^6*b^5 + 800*a^7*b^4 - 96*a^8*b^3)/(64*(b^9 - 6*a*b^8
+ 15*a^2*b^7 - 20*a^3*b^6 + 15*a^4*b^5 - 6*a^5*b^4 + a^6*b^3)) - (tan(e + f*x)*(1280*a*b^11 - 256*b^12 - 2304*
a^2*b^10 + 1280*a^3*b^9 + 1280*a^4*b^8 - 2304*a^5*b^7 + 1280*a^6*b^6 - 256*a^7*b^5)*1i)/(32*(6*a*b^2 - 6*a^2*b
+ 2*a^3 - 2*b^3)*(b^7 - 4*a*b^6 + 6*a^2*b^5 - 4*a^3*b^4 + a^4*b^3)))*1i)/(6*a*b^2 - 6*a^2*b + 2*a^3 - 2*b^3)
+ (tan(e + f*x)*(9*a^6 - 60*a^5*b + 64*b^6 + 225*a^2*b^4 - 300*a^3*b^3 + 190*a^4*b^2))/(32*(b^7 - 4*a*b^6 + 6*
a^2*b^5 - 4*a^3*b^4 + a^4*b^3)))/(6*a*b^2 - 6*a^2*b + 2*a^3 - 2*b^3) - ((((224*a*b^10 - 1440*a^2*b^9 + 3936*a^
3*b^8 - 5920*a^4*b^7 + 5280*a^5*b^6 - 2784*a^6*b^5 + 800*a^7*b^4 - 96*a^8*b^3)/(64*(b^9 - 6*a*b^8 + 15*a^2*b^7
- 20*a^3*b^6 + 15*a^4*b^5 - 6*a^5*b^4 + a^6*b^3)) + (tan(e + f*x)*(1280*a*b^11 - 256*b^12 - 2304*a^2*b^10 + 1
280*a^3*b^9 + 1280*a^4*b^8 - 2304*a^5*b^7 + 1280*a^6*b^6 - 256*a^7*b^5)*1i)/(32*(6*a*b^2 - 6*a^2*b + 2*a^3 - 2
*b^3)*(b^7 - 4*a*b^6 + 6*a^2*b^5 - 4*a^3*b^4 + a^4*b^3)))*1i)/(6*a*b^2 - 6*a^2*b + 2*a^3 - 2*b^3) - (tan(e + f
*x)*(9*a^6 - 60*a^5*b + 64*b^6 + 225*a^2*b^4 - 300*a^3*b^3 + 190*a^4*b^2))/(32*(b^7 - 4*a*b^6 + 6*a^2*b^5 - 4*
a^3*b^4 + a^4*b^3)))/(6*a*b^2 - 6*a^2*b + 2*a^3 - 2*b^3))/((120*a*b^4 - 51*a^4*b + 9*a^5 - 185*a^2*b^3 + 139*a
^3*b^2)/(32*(b^9 - 6*a*b^8 + 15*a^2*b^7 - 20*a^3*b^6 + 15*a^4*b^5 - 6*a^5*b^4 + a^6*b^3)) + (((((224*a*b^10 -
1440*a^2*b^9 + 3936*a^3*b^8 - 5920*a^4*b^7 + 5280*a^5*b^6 - 2784*a^6*b^5 + 800*a^7*b^4 - 96*a^8*b^3)/(64*(b^9
- 6*a*b^8 + 15*a^2*b^7 - 20*a^3*b^6 + 15*a^4*b^5 - 6*a^5*b^4 + a^6*b^3)) - (tan(e + f*x)*(1280*a*b^11 - 256*b^
12 - 2304*a^2*b^10 + 1280*a^3*b^9 + 1280*a^4*b^8 - 2304*a^5*b^7 + 1280*a^6*b^6 - 256*a^7*b^5)*1i)/(32*(6*a*b^2
- 6*a^2*b + 2*a^3 - 2*b^3)*(b^7 - 4*a*b^6 + 6*a^2*b^5 - 4*a^3*b^4 + a^4*b^3)))*1i)/(6*a*b^2 - 6*a^2*b + 2*a^3
- 2*b^3) + (tan(e + f*x)*(9*a^6 - 60*a^5*b + 64*b^6 + 225*a^2*b^4 - 300*a^3*b^3 + 190*a^4*b^2))/(32*(b^7 - 4*
a*b^6 + 6*a^2*b^5 - 4*a^3*b^4 + a^4*b^3)))*1i)/(6*a*b^2 - 6*a^2*b + 2*a^3 - 2*b^3) + (((((224*a*b^10 - 1440*a^
2*b^9 + 3936*a^3*b^8 - 5920*a^4*b^7 + 5280*a^5*b^6 - 2784*a^6*b^5 + 800*a^7*b^4 - 96*a^8*b^3)/(64*(b^9 - 6*a*b
^8 + 15*a^2*b^7 - 20*a^3*b^6 + 15*a^4*b^5 - 6*a^5*b^4 + a^6*b^3)) + (tan(e + f*x)*(1280*a*b^11 - 256*b^12 - 23
04*a^2*b^10 + 1280*a^3*b^9 + 1280*a^4*b^8 - 2304*a^5*b^7 + 1280*a^6*b^6 - 256*a^7*b^5)*1i)/(32*(6*a*b^2 - 6*a^
2*b + 2*a^3 - 2*b^3)*(b^7 - 4*a*b^6 + 6*a^2*b^5 - 4*a^3*b^4 + a^4*b^3)))*1i)/(6*a*b^2 - 6*a^2*b + 2*a^3 - 2*b^
3) - (tan(e + f*x)*(9*a^6 - 60*a^5*b + 64*b^6 + 225*a^2*b^4 - 300*a^3*b^3 + 190*a^4*b^2))/(32*(b^7 - 4*a*b^6 +
6*a^2*b^5 - 4*a^3*b^4 + a^4*b^3)))*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^5)^(1/2)*((tan(e + f*x)*(9*a^6 - 60*a^5*b + 64*b^6 + 225*a^2*b^4 - 300*a^3*b^3 + 190*
a^4*b^2))/(32*(b^7 - 4*a*b^6 + 6*a^2*b^5 - 4*a^3*b^4 + a^4*b^3)) + ((-a*b^5)^(1/2)*((224*a*b^10 - 1440*a^2*b^9
+ 3936*a^3*b^8 - 5920*a^4*b^7 + 5280*a^5*b^6 - 2784*a^6*b^5 + 800*a^7*b^4 - 96*a^8*b^3)/(64*(b^9 - 6*a*b^8 +
15*a^2*b^7 - 20*a^3*b^6 + 15*a^4*b^5 - 6*a^5*b^4 + a^6*b^3)) - (tan(e + f*x)*(-a*b^5)^(1/2)*(3*a^2 - 10*a*b +
15*b^2)*(1280*a*b^11 - 256*b^12 - 2304*a^2*b^10 + 1280*a^3*b^9 + 1280*a^4*b^8 - 2304*a^5*b^7 + 1280*a^6*b^6 -
256*a^7*b^5))/(512*(3*a*b^7 - b^8 - 3*a^2*b^6 + a^3*b^5)*(b^7 - 4*a*b^6 + 6*a^2*b^5 - 4*a^3*b^4 + a^4*b^3)))*(
3*a^2 - 10*a*b + 15*b^2))/(16*(3*a*b^7 - b^8 - 3*a^2*b^6 + a^3*b^5)))*(3*a^2 - 10*a*b + 15*b^2)*1i)/(16*(3*a*b
^7 - b^8 - 3*a^2*b^6 + a^3*b^5)) + ((-a*b^5)^(1/2)*((tan(e + f*x)*(9*a^6 - 60*a^5*b + 64*b^6 + 225*a^2*b^4 - 3
00*a^3*b^3 + 190*a^4*b^2))/(32*(b^7 - 4*a*b^6 + 6*a^2*b^5 - 4*a^3*b^4 + a^4*b^3)) - ((-a*b^5)^(1/2)*((224*a*b^
10 - 1440*a^2*b^9 + 3936*a^3*b^8 - 5920*a^4*b^7 + 5280*a^5*b^6 - 2784*a^6*b^5 + 800*a^7*b^4 - 96*a^8*b^3)/(64*
(b^9 - 6*a*b^8 + 15*a^2*b^7 - 20*a^3*b^6 + 15*a^4*b^5 - 6*a^5*b^4 + a^6*b^3)) + (tan(e + f*x)*(-a*b^5)^(1/2)*(
3*a^2 - 10*a*b + 15*b^2)*(1280*a*b^11 - 256*b^12 - 2304*a^2*b^10 + 1280*a^3*b^9 + 1280*a^4*b^8 - 2304*a^5*b^7
+ 1280*a^6*b^6 - 256*a^7*b^5))/(512*(3*a*b^7 - b^8 - 3*a^2*b^6 + a^3*b^5)*(b^7 - 4*a*b^6 + 6*a^2*b^5 - 4*a^3*b
^4 + a^4*b^3)))*(3*a^2 - 10*a*b + 15*b^2))/(16*(3*a*b^7 - b^8 - 3*a^2*b^6 + a^3*b^5)))*(3*a^2 - 10*a*b + 15*b^
2)*1i)/(16*(3*a*b^7 - b^8 - 3*a^2*b^6 + a^3*b^5)))/((120*a*b^4 - 51*a^4*b + 9*a^5 - 185*a^2*b^3 + 139*a^3*b^2)
/(32*(b^9 - 6*a*b^8 + 15*a^2*b^7 - 20*a^3*b^6 + 15*a^4*b^5 - 6*a^5*b^4 + a^6*b^3)) + ((-a*b^5)^(1/2)*((tan(e +
f*x)*(9*a^6 - 60*a^5*b + 64*b^6 + 225*a^2*b^4 - 300*a^3*b^3 + 190*a^4*b^2))/(32*(b^7 - 4*a*b^6 + 6*a^2*b^5 -
4*a^3*b^4 + a^4*b^3)) + ((-a*b^5)^(1/2)*((224*a*b^10 - 1440*a^2*b^9 + 3936*a^3*b^8 - 5920*a^4*b^7 + 5280*a^5*b
^6 - 2784*a^6*b^5 + 800*a^7*b^4 - 96*a^8*b^3)/(...

________________________________________________________________________________________