\(\int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx\) [467]

Optimal result

Integrand size = 21, antiderivative size = 239 \[ \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}-\frac {2 a b \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {2 a^5 \left (2 a^2+3 b^2\right ) \log (a+b \tan (c+d x))}{b^5 \left (a^2+b^2\right )^2 d}+\frac {\left (4 a^4+2 a^2 b^2-b^4\right ) \tan (c+d x)}{b^4 \left (a^2+b^2\right ) d}-\frac {a \left (2 a^2+b^2\right ) \tan ^2(c+d x)}{b^3 \left (a^2+b^2\right ) d}+\frac {\left (4 a^2+b^2\right ) \tan ^3(c+d x)}{3 b^2 \left (a^2+b^2\right ) d}-\frac {a^2 \tan ^4(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3646, 3728, 3707, 3698, 31, 3556} \[ \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {a^2 \tan ^4(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {\left (4 a^2+b^2\right ) \tan ^3(c+d x)}{3 b^2 d \left (a^2+b^2\right )}-\frac {2 a b \log (\cos (c+d x))}{d \left (a^2+b^2\right )^2}-\frac {x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^2}-\frac {a \left (2 a^2+b^2\right ) \tan ^2(c+d x)}{b^3 d \left (a^2+b^2\right )}-\frac {2 a^5 \left (2 a^2+3 b^2\right ) \log (a+b \tan (c+d x))}{b^5 d \left (a^2+b^2\right )^2}+\frac {\left (4 a^4+2 a^2 b^2-b^4\right ) \tan (c+d x)}{b^4 d \left (a^2+b^2\right )} \]

[In]

Int[Tan[c + d*x]^6/(a + b*Tan[c + d*x])^2,x]

[Out]

-(((a^2 - b^2)*x)/(a^2 + b^2)^2) - (2*a*b*Log[Cos[c + d*x]])/((a^2 + b^2)^2*d) - (2*a^5*(2*a^2 + 3*b^2)*Log[a
+ b*Tan[c + d*x]])/(b^5*(a^2 + b^2)^2*d) + ((4*a^4 + 2*a^2*b^2 - b^4)*Tan[c + d*x])/(b^4*(a^2 + b^2)*d) - (a*(
2*a^2 + b^2)*Tan[c + d*x]^2)/(b^3*(a^2 + b^2)*d) + ((4*a^2 + b^2)*Tan[c + d*x]^3)/(3*b^2*(a^2 + b^2)*d) - (a^2
*Tan[c + d*x]^4)/(b*(a^2 + b^2)*d*(a + b*Tan[c + d*x]))

Rule 31

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

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3646

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

Rule 3698

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

Rule 3707

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

Rule 3728

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*
tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d
*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c +
d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f
*x] - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}
, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !Intege
rQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \tan ^4(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {\tan ^3(c+d x) \left (4 a^2-a b \tan (c+d x)+\left (4 a^2+b^2\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{b \left (a^2+b^2\right )} \\ & = \frac {\left (4 a^2+b^2\right ) \tan ^3(c+d x)}{3 b^2 \left (a^2+b^2\right ) d}-\frac {a^2 \tan ^4(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {\tan ^2(c+d x) \left (-3 a \left (4 a^2+b^2\right )-3 b^3 \tan (c+d x)-6 a \left (2 a^2+b^2\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{3 b^2 \left (a^2+b^2\right )} \\ & = -\frac {a \left (2 a^2+b^2\right ) \tan ^2(c+d x)}{b^3 \left (a^2+b^2\right ) d}+\frac {\left (4 a^2+b^2\right ) \tan ^3(c+d x)}{3 b^2 \left (a^2+b^2\right ) d}-\frac {a^2 \tan ^4(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {\tan (c+d x) \left (12 a^2 \left (2 a^2+b^2\right )+6 a b^3 \tan (c+d x)+6 \left (4 a^4+2 a^2 b^2-b^4\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{6 b^3 \left (a^2+b^2\right )} \\ & = \frac {\left (4 a^4+2 a^2 b^2-b^4\right ) \tan (c+d x)}{b^4 \left (a^2+b^2\right ) d}-\frac {a \left (2 a^2+b^2\right ) \tan ^2(c+d x)}{b^3 \left (a^2+b^2\right ) d}+\frac {\left (4 a^2+b^2\right ) \tan ^3(c+d x)}{3 b^2 \left (a^2+b^2\right ) d}-\frac {a^2 \tan ^4(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {-6 a \left (4 a^4+2 a^2 b^2-b^4\right )+6 b^5 \tan (c+d x)-12 a \left (2 a^4+a^2 b^2-b^4\right ) \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{6 b^4 \left (a^2+b^2\right )} \\ & = -\frac {\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}+\frac {\left (4 a^4+2 a^2 b^2-b^4\right ) \tan (c+d x)}{b^4 \left (a^2+b^2\right ) d}-\frac {a \left (2 a^2+b^2\right ) \tan ^2(c+d x)}{b^3 \left (a^2+b^2\right ) d}+\frac {\left (4 a^2+b^2\right ) \tan ^3(c+d x)}{3 b^2 \left (a^2+b^2\right ) d}-\frac {a^2 \tan ^4(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {(2 a b) \int \tan (c+d x) \, dx}{\left (a^2+b^2\right )^2}-\frac {\left (2 a^5 \left (2 a^2+3 b^2\right )\right ) \int \frac {1+\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b^4 \left (a^2+b^2\right )^2} \\ & = -\frac {\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}-\frac {2 a b \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {\left (4 a^4+2 a^2 b^2-b^4\right ) \tan (c+d x)}{b^4 \left (a^2+b^2\right ) d}-\frac {a \left (2 a^2+b^2\right ) \tan ^2(c+d x)}{b^3 \left (a^2+b^2\right ) d}+\frac {\left (4 a^2+b^2\right ) \tan ^3(c+d x)}{3 b^2 \left (a^2+b^2\right ) d}-\frac {a^2 \tan ^4(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {\left (2 a^5 \left (2 a^2+3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^5 \left (a^2+b^2\right )^2 d} \\ & = -\frac {\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}-\frac {2 a b \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {2 a^5 \left (2 a^2+3 b^2\right ) \log (a+b \tan (c+d x))}{b^5 \left (a^2+b^2\right )^2 d}+\frac {\left (4 a^4+2 a^2 b^2-b^4\right ) \tan (c+d x)}{b^4 \left (a^2+b^2\right ) d}-\frac {a \left (2 a^2+b^2\right ) \tan ^2(c+d x)}{b^3 \left (a^2+b^2\right ) d}+\frac {\left (4 a^2+b^2\right ) \tan ^3(c+d x)}{3 b^2 \left (a^2+b^2\right ) d}-\frac {a^2 \tan ^4(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.25 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.01 \[ \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\tan ^4(c+d x)}{3 b d (a+b \tan (c+d x))}+\frac {-\frac {2 a \tan ^3(c+d x)}{b d (a+b \tan (c+d x))}+\frac {\frac {3 i b^2 \log (i-\tan (c+d x))}{(a+i b)^2 d}-\frac {3 i b^2 \log (i+\tan (c+d x))}{(a-i b)^2 d}-\frac {12 a^5 \left (2 a^2+3 b^2\right ) \log (a+b \tan (c+d x))}{b^3 \left (a^2+b^2\right )^2 d}-\frac {6 \left (1-\frac {2 a^2}{b^2}\right ) \tan (c+d x)}{d}-\frac {6 a^4 \left (2 a^2+b^2\right )}{b^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}}{2 b}}{3 b} \]

[In]

Integrate[Tan[c + d*x]^6/(a + b*Tan[c + d*x])^2,x]

[Out]

Tan[c + d*x]^4/(3*b*d*(a + b*Tan[c + d*x])) + ((-2*a*Tan[c + d*x]^3)/(b*d*(a + b*Tan[c + d*x])) + (((3*I)*b^2*
Log[I - Tan[c + d*x]])/((a + I*b)^2*d) - ((3*I)*b^2*Log[I + Tan[c + d*x]])/((a - I*b)^2*d) - (12*a^5*(2*a^2 +
3*b^2)*Log[a + b*Tan[c + d*x]])/(b^3*(a^2 + b^2)^2*d) - (6*(1 - (2*a^2)/b^2)*Tan[c + d*x])/d - (6*a^4*(2*a^2 +
b^2))/(b^3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])))/(2*b))/(3*b)

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.70

method	result	size
derivativedivides	\(\frac {\frac {\frac {b^{2} \left (\tan ^{3}\left (d x +c \right )\right )}{3}-a b \left (\tan ^{2}\left (d x +c \right )\right )+3 \tan \left (d x +c \right ) a^{2}-b^{2} \tan \left (d x +c \right )}{b^{4}}+\frac {a b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+\left (-a^{2}+b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {a^{6}}{b^{5} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}-\frac {2 a^{5} \left (2 a^{2}+3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{5} \left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(168\)
default	\(\frac {\frac {\frac {b^{2} \left (\tan ^{3}\left (d x +c \right )\right )}{3}-a b \left (\tan ^{2}\left (d x +c \right )\right )+3 \tan \left (d x +c \right ) a^{2}-b^{2} \tan \left (d x +c \right )}{b^{4}}+\frac {a b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+\left (-a^{2}+b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {a^{6}}{b^{5} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}-\frac {2 a^{5} \left (2 a^{2}+3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{5} \left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(168\)
norman	\(\frac {\frac {\left (2 a^{2}-b^{2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{b^{3} d}+\frac {\tan ^{4}\left (d x +c \right )}{3 b d}-\frac {2 a \left (\tan ^{3}\left (d x +c \right )\right )}{3 b^{2} d}-\frac {\left (a^{2}-b^{2}\right ) a x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {b \left (a^{2}-b^{2}\right ) x \tan \left (d x +c \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {\left (4 a^{5}+2 a^{3} b^{2}-a \,b^{4}\right ) a}{d \,b^{5} \left (a^{2}+b^{2}\right )}}{a +b \tan \left (d x +c \right )}+\frac {a b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 a^{5} \left (2 a^{2}+3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b^{5} d}\)	\(263\)
parallelrisch	\(\frac {3 b^{6} a^{2}-12 a^{8}-18 a^{6} b^{2}-3 a^{4} b^{4}+2 \left (\tan ^{4}\left (d x +c \right )\right ) a^{2} b^{6}+\left (\tan ^{4}\left (d x +c \right )\right ) b^{8}-3 \left (\tan ^{2}\left (d x +c \right )\right ) b^{8}-12 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{8}-2 \left (\tan ^{3}\left (d x +c \right )\right ) a^{5} b^{3}-4 \left (\tan ^{3}\left (d x +c \right )\right ) a^{3} b^{5}-2 \left (\tan ^{3}\left (d x +c \right )\right ) a \,b^{7}+6 \left (\tan ^{2}\left (d x +c \right )\right ) a^{6} b^{2}+9 \left (\tan ^{2}\left (d x +c \right )\right ) a^{4} b^{4}+3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2} b^{6}+\left (\tan ^{4}\left (d x +c \right )\right ) a^{4} b^{4}-18 \ln \left (a +b \tan \left (d x +c \right )\right ) a^{6} b^{2}+3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right ) a \,b^{7}-12 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{7} b -18 \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{5} b^{3}+3 x \tan \left (d x +c \right ) b^{8} d -3 x \,a^{3} b^{5} d +3 x a \,b^{7} d -3 x \tan \left (d x +c \right ) a^{2} b^{6} d}{3 \left (a +b \tan \left (d x +c \right )\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b^{5} d}\)	\(374\)
risch	\(\frac {x}{2 i a b -a^{2}+b^{2}}-\frac {8 i a^{3} x}{b^{5}}-\frac {8 i a^{3} c}{b^{5} d}+\frac {4 i a x}{b^{3}}+\frac {4 i a c}{b^{3} d}+\frac {8 i a^{7} x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b^{5}}+\frac {8 i a^{7} c}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b^{5} d}+\frac {12 i a^{5} x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b^{3}}+\frac {12 i a^{5} c}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b^{3} d}-\frac {2 i \left (-12 i a^{3} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-12 i a^{5} b \,{\mathrm e}^{2 i \left (d x +c \right )}+6 i a \,b^{5} {\mathrm e}^{6 i \left (d x +c \right )}+12 a^{6} {\mathrm e}^{6 i \left (d x +c \right )}+6 a^{4} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+3 a^{2} b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+6 b^{6} {\mathrm e}^{6 i \left (d x +c \right )}-6 i a^{3} b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+2 i a^{3} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+14 i a \,b^{5} {\mathrm e}^{2 i \left (d x +c \right )}+36 a^{6} {\mathrm e}^{4 i \left (d x +c \right )}+18 a^{4} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-9 a^{2} b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-12 i a^{5} b \,{\mathrm e}^{6 i \left (d x +c \right )}-24 i a^{5} b \,{\mathrm e}^{4 i \left (d x +c \right )}+12 i a \,b^{5} {\mathrm e}^{4 i \left (d x +c \right )}+36 a^{6} {\mathrm e}^{2 i \left (d x +c \right )}+26 a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-3 a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-2 b^{6} {\mathrm e}^{2 i \left (d x +c \right )}+12 a^{6}+14 a^{4} b^{2}+a^{2} b^{4}-4 b^{6}\right )}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3} \left (-i a +b \right ) \left (i a +b \right )^{2} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right ) d \,b^{4}}+\frac {4 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{5} d}-\frac {2 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{3} d}-\frac {4 a^{7} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b^{5} d}-\frac {6 a^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b^{3} d}\)	\(754\)

[In]

int(tan(d*x+c)^6/(a+b*tan(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.62 \[ \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {6 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - {\left (a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )^{4} + 2 \, {\left (a^{5} b^{3} + 2 \, a^{3} b^{5} + a b^{7}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{3} b^{5} - a b^{7}\right )} d x - 3 \, {\left (2 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - b^{8}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (2 \, a^{8} + 3 \, a^{6} b^{2} + {\left (2 \, a^{7} b + 3 \, a^{5} b^{3}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 3 \, {\left (2 \, a^{8} + 3 \, a^{6} b^{2} - a^{2} b^{6} + {\left (2 \, a^{7} b + 3 \, a^{5} b^{3} - a b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 3 \, {\left (4 \, a^{7} b + 4 \, a^{5} b^{3} - a^{3} b^{5} - 2 \, a b^{7} - {\left (a^{2} b^{6} - b^{8}\right )} d x\right )} \tan \left (d x + c\right )}{3 \, {\left ({\left (a^{4} b^{6} + 2 \, a^{2} b^{8} + b^{10}\right )} d \tan \left (d x + c\right ) + {\left (a^{5} b^{5} + 2 \, a^{3} b^{7} + a b^{9}\right )} d\right )}} \]

[In]

integrate(tan(d*x+c)^6/(a+b*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/3*(6*a^6*b^2 + 6*a^4*b^4 + 3*a^2*b^6 - (a^4*b^4 + 2*a^2*b^6 + b^8)*tan(d*x + c)^4 + 2*(a^5*b^3 + 2*a^3*b^5
+ a*b^7)*tan(d*x + c)^3 + 3*(a^3*b^5 - a*b^7)*d*x - 3*(2*a^6*b^2 + 3*a^4*b^4 - b^8)*tan(d*x + c)^2 + 3*(2*a^8
+ 3*a^6*b^2 + (2*a^7*b + 3*a^5*b^3)*tan(d*x + c))*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x
+ c)^2 + 1)) - 3*(2*a^8 + 3*a^6*b^2 - a^2*b^6 + (2*a^7*b + 3*a^5*b^3 - a*b^7)*tan(d*x + c))*log(1/(tan(d*x +
c)^2 + 1)) - 3*(4*a^7*b + 4*a^5*b^3 - a^3*b^5 - 2*a*b^7 - (a^2*b^6 - b^8)*d*x)*tan(d*x + c))/((a^4*b^6 + 2*a^2
*b^8 + b^10)*d*tan(d*x + c) + (a^5*b^5 + 2*a^3*b^7 + a*b^9)*d)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.54 (sec) , antiderivative size = 3279, normalized size of antiderivative = 13.72 \[ \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

integrate(tan(d*x+c)**6/(a+b*tan(d*x+c))**2,x)

[Out]

Piecewise((zoo*x*tan(c)**4, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((-x + tan(c + d*x)**5/(5*d) - tan(c + d*x)**3/(3
*d) + tan(c + d*x)/d)/a**2, Eq(b, 0)), (75*d*x*tan(c + d*x)**2/(12*b**2*d*tan(c + d*x)**2 - 24*I*b**2*d*tan(c
+ d*x) - 12*b**2*d) - 150*I*d*x*tan(c + d*x)/(12*b**2*d*tan(c + d*x)**2 - 24*I*b**2*d*tan(c + d*x) - 12*b**2*d
) - 75*d*x/(12*b**2*d*tan(c + d*x)**2 - 24*I*b**2*d*tan(c + d*x) - 12*b**2*d) - 36*I*log(tan(c + d*x)**2 + 1)*
tan(c + d*x)**2/(12*b**2*d*tan(c + d*x)**2 - 24*I*b**2*d*tan(c + d*x) - 12*b**2*d) - 72*log(tan(c + d*x)**2 +
1)*tan(c + d*x)/(12*b**2*d*tan(c + d*x)**2 - 24*I*b**2*d*tan(c + d*x) - 12*b**2*d) + 36*I*log(tan(c + d*x)**2
+ 1)/(12*b**2*d*tan(c + d*x)**2 - 24*I*b**2*d*tan(c + d*x) - 12*b**2*d) + 4*tan(c + d*x)**5/(12*b**2*d*tan(c +
d*x)**2 - 24*I*b**2*d*tan(c + d*x) - 12*b**2*d) + 4*I*tan(c + d*x)**4/(12*b**2*d*tan(c + d*x)**2 - 24*I*b**2*
d*tan(c + d*x) - 12*b**2*d) - 28*tan(c + d*x)**3/(12*b**2*d*tan(c + d*x)**2 - 24*I*b**2*d*tan(c + d*x) - 12*b*
*2*d) - 153*tan(c + d*x)/(12*b**2*d*tan(c + d*x)**2 - 24*I*b**2*d*tan(c + d*x) - 12*b**2*d) + 114*I/(12*b**2*d
*tan(c + d*x)**2 - 24*I*b**2*d*tan(c + d*x) - 12*b**2*d), Eq(a, -I*b)), (75*d*x*tan(c + d*x)**2/(12*b**2*d*tan
(c + d*x)**2 + 24*I*b**2*d*tan(c + d*x) - 12*b**2*d) + 150*I*d*x*tan(c + d*x)/(12*b**2*d*tan(c + d*x)**2 + 24*
I*b**2*d*tan(c + d*x) - 12*b**2*d) - 75*d*x/(12*b**2*d*tan(c + d*x)**2 + 24*I*b**2*d*tan(c + d*x) - 12*b**2*d)
+ 36*I*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(12*b**2*d*tan(c + d*x)**2 + 24*I*b**2*d*tan(c + d*x) - 12*b*
*2*d) - 72*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(12*b**2*d*tan(c + d*x)**2 + 24*I*b**2*d*tan(c + d*x) - 12*b*
*2*d) - 36*I*log(tan(c + d*x)**2 + 1)/(12*b**2*d*tan(c + d*x)**2 + 24*I*b**2*d*tan(c + d*x) - 12*b**2*d) + 4*t
an(c + d*x)**5/(12*b**2*d*tan(c + d*x)**2 + 24*I*b**2*d*tan(c + d*x) - 12*b**2*d) - 4*I*tan(c + d*x)**4/(12*b*
*2*d*tan(c + d*x)**2 + 24*I*b**2*d*tan(c + d*x) - 12*b**2*d) - 28*tan(c + d*x)**3/(12*b**2*d*tan(c + d*x)**2 +
24*I*b**2*d*tan(c + d*x) - 12*b**2*d) - 153*tan(c + d*x)/(12*b**2*d*tan(c + d*x)**2 + 24*I*b**2*d*tan(c + d*x
) - 12*b**2*d) - 114*I/(12*b**2*d*tan(c + d*x)**2 + 24*I*b**2*d*tan(c + d*x) - 12*b**2*d), Eq(a, I*b)), (x*tan
(c)**6/(a + b*tan(c))**2, Eq(d, 0)), (-12*a**8*log(a/b + tan(c + d*x))/(3*a**5*b**5*d + 3*a**4*b**6*d*tan(c +
d*x) + 6*a**3*b**7*d + 6*a**2*b**8*d*tan(c + d*x) + 3*a*b**9*d + 3*b**10*d*tan(c + d*x)) - 12*a**8/(3*a**5*b**
5*d + 3*a**4*b**6*d*tan(c + d*x) + 6*a**3*b**7*d + 6*a**2*b**8*d*tan(c + d*x) + 3*a*b**9*d + 3*b**10*d*tan(c +
d*x)) - 12*a**7*b*log(a/b + tan(c + d*x))*tan(c + d*x)/(3*a**5*b**5*d + 3*a**4*b**6*d*tan(c + d*x) + 6*a**3*b
**7*d + 6*a**2*b**8*d*tan(c + d*x) + 3*a*b**9*d + 3*b**10*d*tan(c + d*x)) - 18*a**6*b**2*log(a/b + tan(c + d*x
))/(3*a**5*b**5*d + 3*a**4*b**6*d*tan(c + d*x) + 6*a**3*b**7*d + 6*a**2*b**8*d*tan(c + d*x) + 3*a*b**9*d + 3*b
**10*d*tan(c + d*x)) + 6*a**6*b**2*tan(c + d*x)**2/(3*a**5*b**5*d + 3*a**4*b**6*d*tan(c + d*x) + 6*a**3*b**7*d
+ 6*a**2*b**8*d*tan(c + d*x) + 3*a*b**9*d + 3*b**10*d*tan(c + d*x)) - 18*a**6*b**2/(3*a**5*b**5*d + 3*a**4*b*
*6*d*tan(c + d*x) + 6*a**3*b**7*d + 6*a**2*b**8*d*tan(c + d*x) + 3*a*b**9*d + 3*b**10*d*tan(c + d*x)) - 18*a**
5*b**3*log(a/b + tan(c + d*x))*tan(c + d*x)/(3*a**5*b**5*d + 3*a**4*b**6*d*tan(c + d*x) + 6*a**3*b**7*d + 6*a*
*2*b**8*d*tan(c + d*x) + 3*a*b**9*d + 3*b**10*d*tan(c + d*x)) - 2*a**5*b**3*tan(c + d*x)**3/(3*a**5*b**5*d + 3
*a**4*b**6*d*tan(c + d*x) + 6*a**3*b**7*d + 6*a**2*b**8*d*tan(c + d*x) + 3*a*b**9*d + 3*b**10*d*tan(c + d*x))
+ a**4*b**4*tan(c + d*x)**4/(3*a**5*b**5*d + 3*a**4*b**6*d*tan(c + d*x) + 6*a**3*b**7*d + 6*a**2*b**8*d*tan(c
+ d*x) + 3*a*b**9*d + 3*b**10*d*tan(c + d*x)) + 9*a**4*b**4*tan(c + d*x)**2/(3*a**5*b**5*d + 3*a**4*b**6*d*tan
(c + d*x) + 6*a**3*b**7*d + 6*a**2*b**8*d*tan(c + d*x) + 3*a*b**9*d + 3*b**10*d*tan(c + d*x)) - 3*a**4*b**4/(3
*a**5*b**5*d + 3*a**4*b**6*d*tan(c + d*x) + 6*a**3*b**7*d + 6*a**2*b**8*d*tan(c + d*x) + 3*a*b**9*d + 3*b**10*
d*tan(c + d*x)) - 3*a**3*b**5*d*x/(3*a**5*b**5*d + 3*a**4*b**6*d*tan(c + d*x) + 6*a**3*b**7*d + 6*a**2*b**8*d*
tan(c + d*x) + 3*a*b**9*d + 3*b**10*d*tan(c + d*x)) - 4*a**3*b**5*tan(c + d*x)**3/(3*a**5*b**5*d + 3*a**4*b**6
*d*tan(c + d*x) + 6*a**3*b**7*d + 6*a**2*b**8*d*tan(c + d*x) + 3*a*b**9*d + 3*b**10*d*tan(c + d*x)) - 3*a**2*b
**6*d*x*tan(c + d*x)/(3*a**5*b**5*d + 3*a**4*b**6*d*tan(c + d*x) + 6*a**3*b**7*d + 6*a**2*b**8*d*tan(c + d*x)
+ 3*a*b**9*d + 3*b**10*d*tan(c + d*x)) + 3*a**2*b**6*log(tan(c + d*x)**2 + 1)/(3*a**5*b**5*d + 3*a**4*b**6*d*t
an(c + d*x) + 6*a**3*b**7*d + 6*a**2*b**8*d*tan(c + d*x) + 3*a*b**9*d + 3*b**10*d*tan(c + d*x)) + 2*a**2*b**6*
tan(c + d*x)**4/(3*a**5*b**5*d + 3*a**4*b**6*d*tan(c + d*x) + 6*a**3*b**7*d + 6*a**2*b**8*d*tan(c + d*x) + 3*a
*b**9*d + 3*b**10*d*tan(c + d*x)) + 3*a**2*b**6/(3*a**5*b**5*d + 3*a**4*b**6*d*tan(c + d*x) + 6*a**3*b**7*d +
6*a**2*b**8*d*tan(c + d*x) + 3*a*b**9*d + 3*b**10*d*tan(c + d*x)) + 3*a*b**7*d*x/(3*a**5*b**5*d + 3*a**4*b**6*
d*tan(c + d*x) + 6*a**3*b**7*d + 6*a**2*b**8*d*tan(c + d*x) + 3*a*b**9*d + 3*b**10*d*tan(c + d*x)) + 3*a*b**7*
log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(3*a**5*b**5*d + 3*a**4*b**6*d*tan(c + d*x) + 6*a**3*b**7*d + 6*a**2*b**
8*d*tan(c + d*x) + 3*a*b**9*d + 3*b**10*d*tan(c + d*x)) - 2*a*b**7*tan(c + d*x)**3/(3*a**5*b**5*d + 3*a**4*b**
6*d*tan(c + d*x) + 6*a**3*b**7*d + 6*a**2*b**8*d*tan(c + d*x) + 3*a*b**9*d + 3*b**10*d*tan(c + d*x)) + 3*b**8*
d*x*tan(c + d*x)/(3*a**5*b**5*d + 3*a**4*b**6*d*tan(c + d*x) + 6*a**3*b**7*d + 6*a**2*b**8*d*tan(c + d*x) + 3*
a*b**9*d + 3*b**10*d*tan(c + d*x)) + b**8*tan(c + d*x)**4/(3*a**5*b**5*d + 3*a**4*b**6*d*tan(c + d*x) + 6*a**3
*b**7*d + 6*a**2*b**8*d*tan(c + d*x) + 3*a*b**9*d + 3*b**10*d*tan(c + d*x)) - 3*b**8*tan(c + d*x)**2/(3*a**5*b
**5*d + 3*a**4*b**6*d*tan(c + d*x) + 6*a**3*b**7*d + 6*a**2*b**8*d*tan(c + d*x) + 3*a*b**9*d + 3*b**10*d*tan(c
+ d*x)), True))

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.86 \[ \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {3 \, a^{6}}{a^{3} b^{5} + a b^{7} + {\left (a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )} - \frac {3 \, a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {3 \, {\left (a^{2} - b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {6 \, {\left (2 \, a^{7} + 3 \, a^{5} b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} b^{5} + 2 \, a^{2} b^{7} + b^{9}} - \frac {b^{2} \tan \left (d x + c\right )^{3} - 3 \, a b \tan \left (d x + c\right )^{2} + 3 \, {\left (3 \, a^{2} - b^{2}\right )} \tan \left (d x + c\right )}{b^{4}}}{3 \, d} \]

[In]

integrate(tan(d*x+c)^6/(a+b*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/3*(3*a^6/(a^3*b^5 + a*b^7 + (a^2*b^6 + b^8)*tan(d*x + c)) - 3*a*b*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^2*b^2
+ b^4) + 3*(a^2 - b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) + 6*(2*a^7 + 3*a^5*b^2)*log(b*tan(d*x + c) + a)/(a^4*
b^5 + 2*a^2*b^7 + b^9) - (b^2*tan(d*x + c)^3 - 3*a*b*tan(d*x + c)^2 + 3*(3*a^2 - b^2)*tan(d*x + c))/b^4)/d

Giac [A] (verification not implemented)

none

Time = 2.16 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.05 \[ \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {3 \, a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {3 \, {\left (a^{2} - b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {6 \, {\left (2 \, a^{7} + 3 \, a^{5} b^{2}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b^{5} + 2 \, a^{2} b^{7} + b^{9}} + \frac {3 \, {\left (4 \, a^{7} b \tan \left (d x + c\right ) + 6 \, a^{5} b^{3} \tan \left (d x + c\right ) + 3 \, a^{8} + 5 \, a^{6} b^{2}\right )}}{{\left (a^{4} b^{5} + 2 \, a^{2} b^{7} + b^{9}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}} + \frac {b^{4} \tan \left (d x + c\right )^{3} - 3 \, a b^{3} \tan \left (d x + c\right )^{2} + 9 \, a^{2} b^{2} \tan \left (d x + c\right ) - 3 \, b^{4} \tan \left (d x + c\right )}{b^{6}}}{3 \, d} \]

[In]

integrate(tan(d*x+c)^6/(a+b*tan(d*x+c))^2,x, algorithm="giac")

[Out]

1/3*(3*a*b*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) - 3*(a^2 - b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) -
6*(2*a^7 + 3*a^5*b^2)*log(abs(b*tan(d*x + c) + a))/(a^4*b^5 + 2*a^2*b^7 + b^9) + 3*(4*a^7*b*tan(d*x + c) + 6*
a^5*b^3*tan(d*x + c) + 3*a^8 + 5*a^6*b^2)/((a^4*b^5 + 2*a^2*b^7 + b^9)*(b*tan(d*x + c) + a)) + (b^4*tan(d*x +
c)^3 - 3*a*b^3*tan(d*x + c)^2 + 9*a^2*b^2*tan(d*x + c) - 3*b^4*tan(d*x + c))/b^6)/d

Mupad [B] (verification not implemented)

Time = 5.83 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.89 \[ \int \frac {\tan ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (-a^2\,1{}\mathrm {i}+2\,a\,b+b^2\,1{}\mathrm {i}\right )}-\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {4\,a^3}{b^5}-\frac {2\,a}{b^3}+\frac {2\,a\,b}{{\left (a^2+b^2\right )}^2}\right )}{d}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3}{3\,b^2\,d}-\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {a^2+b^2}{b^4}-\frac {4\,a^2}{b^4}\right )}{d}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{b^3\,d}-\frac {a^6}{b\,d\,\left (\mathrm {tan}\left (c+d\,x\right )\,b^5+a\,b^4\right )\,\left (a^2+b^2\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-a^2+a\,b\,2{}\mathrm {i}+b^2\right )} \]

[In]

int(tan(c + d*x)^6/(a + b*tan(c + d*x))^2,x)

[Out]

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