\(\int \frac {\tan ^6(c+d x)}{a+b \tan (c+d x)} \, dx\) [456]

Optimal result

Integrand size = 21, antiderivative size = 154 \[ \int \frac {\tan ^6(c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {a x}{a^2+b^2}-\frac {b \log (\cos (c+d x))}{\left (a^2+b^2\right ) d}+\frac {a^6 \log (a+b \tan (c+d x))}{b^5 \left (a^2+b^2\right ) d}-\frac {a \left (a^2-b^2\right ) \tan (c+d x)}{b^4 d}+\frac {\left (a^2-b^2\right ) \tan ^2(c+d x)}{2 b^3 d}-\frac {a \tan ^3(c+d x)}{3 b^2 d}+\frac {\tan ^4(c+d x)}{4 b d} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.19 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.08 \[ \int \frac {\tan ^6(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {6 \left (b^5 (i a+b) \log (i-\tan (c+d x))+b^5 (-i a+b) \log (i+\tan (c+d x))+2 a^6 \log (a+b \tan (c+d x))\right )-12 a b \left (a^4-b^4\right ) \tan (c+d x)+6 b^2 \left (a^4-b^4\right ) \tan ^2(c+d x)-4 a b^3 \left (a^2+b^2\right ) \tan ^3(c+d x)+3 b^4 \left (a^2+b^2\right ) \tan ^4(c+d x)}{12 b^5 \left (a^2+b^2\right ) d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.39 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.18, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {3042, 4049, 27, 3042, 4130, 27, 3042, 4131, 27, 3042, 4130, 25, 3042, 4110, 3042, 3956, 4100, 16}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\tan ^6(c+d x)}{a+b \tan (c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\tan (c+d x)^6}{a+b \tan (c+d x)}dx\)

\(\Big \downarrow \) 4049

\(\displaystyle \frac {\int -\frac {4 \tan ^3(c+d x) \left (a \tan ^2(c+d x)+b \tan (c+d x)+a\right )}{a+b \tan (c+d x)}dx}{4 b}+\frac {\tan ^4(c+d x)}{4 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\tan ^4(c+d x)}{4 b d}-\frac {\int \frac {\tan ^3(c+d x) \left (a \tan ^2(c+d x)+b \tan (c+d x)+a\right )}{a+b \tan (c+d x)}dx}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\tan ^4(c+d x)}{4 b d}-\frac {\int \frac {\tan (c+d x)^3 \left (a \tan (c+d x)^2+b \tan (c+d x)+a\right )}{a+b \tan (c+d x)}dx}{b}\)

\(\Big \downarrow \) 4130

\(\displaystyle \frac {\tan ^4(c+d x)}{4 b d}-\frac {\frac {\int -\frac {3 \tan ^2(c+d x) \left (a^2+\left (a^2-b^2\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)}dx}{3 b}+\frac {a \tan ^3(c+d x)}{3 b d}}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\tan ^4(c+d x)}{4 b d}-\frac {\frac {a \tan ^3(c+d x)}{3 b d}-\frac {\int \frac {\tan ^2(c+d x) \left (a^2+\left (a^2-b^2\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)}dx}{b}}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\tan ^4(c+d x)}{4 b d}-\frac {\frac {a \tan ^3(c+d x)}{3 b d}-\frac {\int \frac {\tan (c+d x)^2 \left (a^2+\left (a^2-b^2\right ) \tan (c+d x)^2\right )}{a+b \tan (c+d x)}dx}{b}}{b}\)

\(\Big \downarrow \) 4131

\(\displaystyle \frac {\tan ^4(c+d x)}{4 b d}-\frac {\frac {a \tan ^3(c+d x)}{3 b d}-\frac {\frac {\int -\frac {2 \tan (c+d x) \left (-\tan (c+d x) b^3+a \left (a^2-b^2\right ) \tan ^2(c+d x)+a \left (a^2-b^2\right )\right )}{a+b \tan (c+d x)}dx}{2 b}+\frac {\left (a^2-b^2\right ) \tan ^2(c+d x)}{2 b d}}{b}}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\tan ^4(c+d x)}{4 b d}-\frac {\frac {a \tan ^3(c+d x)}{3 b d}-\frac {\frac {\left (a^2-b^2\right ) \tan ^2(c+d x)}{2 b d}-\frac {\int \frac {\tan (c+d x) \left (-\tan (c+d x) b^3+a \left (a^2-b^2\right ) \tan ^2(c+d x)+a \left (a^2-b^2\right )\right )}{a+b \tan (c+d x)}dx}{b}}{b}}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\tan ^4(c+d x)}{4 b d}-\frac {\frac {a \tan ^3(c+d x)}{3 b d}-\frac {\frac {\left (a^2-b^2\right ) \tan ^2(c+d x)}{2 b d}-\frac {\int \frac {\tan (c+d x) \left (-\tan (c+d x) b^3+a \left (a^2-b^2\right ) \tan (c+d x)^2+a \left (a^2-b^2\right )\right )}{a+b \tan (c+d x)}dx}{b}}{b}}{b}\)

\(\Big \downarrow \) 4130

\(\displaystyle \frac {\tan ^4(c+d x)}{4 b d}-\frac {\frac {a \tan ^3(c+d x)}{3 b d}-\frac {\frac {\left (a^2-b^2\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {\int -\frac {\left (a^2-b^2\right ) a^2+\left (a^4-b^2 a^2+b^4\right ) \tan ^2(c+d x)}{a+b \tan (c+d x)}dx}{b}+\frac {a \left (a^2-b^2\right ) \tan (c+d x)}{b d}}{b}}{b}}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\tan ^4(c+d x)}{4 b d}-\frac {\frac {a \tan ^3(c+d x)}{3 b d}-\frac {\frac {\left (a^2-b^2\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {a \left (a^2-b^2\right ) \tan (c+d x)}{b d}-\frac {\int \frac {\left (a^2-b^2\right ) a^2+\left (a^4-b^2 a^2+b^4\right ) \tan ^2(c+d x)}{a+b \tan (c+d x)}dx}{b}}{b}}{b}}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\tan ^4(c+d x)}{4 b d}-\frac {\frac {a \tan ^3(c+d x)}{3 b d}-\frac {\frac {\left (a^2-b^2\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {a \left (a^2-b^2\right ) \tan (c+d x)}{b d}-\frac {\int \frac {\left (a^2-b^2\right ) a^2+\left (a^4-b^2 a^2+b^4\right ) \tan (c+d x)^2}{a+b \tan (c+d x)}dx}{b}}{b}}{b}}{b}\)

\(\Big \downarrow \) 4110

\(\displaystyle \frac {\tan ^4(c+d x)}{4 b d}-\frac {\frac {a \tan ^3(c+d x)}{3 b d}-\frac {\frac {\left (a^2-b^2\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {a \left (a^2-b^2\right ) \tan (c+d x)}{b d}-\frac {\frac {b^5 \int \tan (c+d x)dx}{a^2+b^2}+\frac {a^6 \int \frac {\tan ^2(c+d x)+1}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {a b^4 x}{a^2+b^2}}{b}}{b}}{b}}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\tan ^4(c+d x)}{4 b d}-\frac {\frac {a \tan ^3(c+d x)}{3 b d}-\frac {\frac {\left (a^2-b^2\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {a \left (a^2-b^2\right ) \tan (c+d x)}{b d}-\frac {\frac {b^5 \int \tan (c+d x)dx}{a^2+b^2}+\frac {a^6 \int \frac {\tan (c+d x)^2+1}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {a b^4 x}{a^2+b^2}}{b}}{b}}{b}}{b}\)

\(\Big \downarrow \) 3956

\(\displaystyle \frac {\tan ^4(c+d x)}{4 b d}-\frac {\frac {a \tan ^3(c+d x)}{3 b d}-\frac {\frac {\left (a^2-b^2\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {a \left (a^2-b^2\right ) \tan (c+d x)}{b d}-\frac {\frac {a^6 \int \frac {\tan (c+d x)^2+1}{a+b \tan (c+d x)}dx}{a^2+b^2}-\frac {b^5 \log (\cos (c+d x))}{d \left (a^2+b^2\right )}-\frac {a b^4 x}{a^2+b^2}}{b}}{b}}{b}}{b}\)

\(\Big \downarrow \) 4100

\(\displaystyle \frac {\tan ^4(c+d x)}{4 b d}-\frac {\frac {a \tan ^3(c+d x)}{3 b d}-\frac {\frac {\left (a^2-b^2\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {a \left (a^2-b^2\right ) \tan (c+d x)}{b d}-\frac {\frac {a^6 \int \frac {1}{a+b \tan (c+d x)}d(b \tan (c+d x))}{b d \left (a^2+b^2\right )}-\frac {b^5 \log (\cos (c+d x))}{d \left (a^2+b^2\right )}-\frac {a b^4 x}{a^2+b^2}}{b}}{b}}{b}}{b}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {\tan ^4(c+d x)}{4 b d}-\frac {\frac {a \tan ^3(c+d x)}{3 b d}-\frac {\frac {\left (a^2-b^2\right ) \tan ^2(c+d x)}{2 b d}-\frac {\frac {a \left (a^2-b^2\right ) \tan (c+d x)}{b d}-\frac {-\frac {b^5 \log (\cos (c+d x))}{d \left (a^2+b^2\right )}-\frac {a b^4 x}{a^2+b^2}+\frac {a^6 \log (a+b \tan (c+d x))}{b d \left (a^2+b^2\right )}}{b}}{b}}{b}}{b}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 16

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

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3956

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

rule 4049

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c 
+ d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Simp[1/(d*(m + n - 1)) 
 Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n 
- 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[ 
e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2, x], x], 
x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2 
, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || I 
ntegerQ[m]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])) 
)
 

rule 4100

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*tan[(e_.) + 
 (f_.)*(x_)]^2), x_Symbol] :> Simp[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 4110

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

rule 4130

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] + Simp[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] && ( !IntegerQ[m] || (EqQ[ 
c, 0] && NeQ[a, 0])))
 

rule 4131

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) 
+ (f_.)*(x_)])^(n_)*((A_.) + (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] + Simp[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 
- b*C)*(m + n + 1)*Tan[e + f*x] - C*m*(b*c - a*d)*Tan[e + f*x]^2, x], x], x 
] /; FreeQ[{a, b, c, d, e, f, A, 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] && ( !IntegerQ 
[m] || (EqQ[c, 0] && NeQ[a, 0])))
 

Maple [A] (verified)

Time = 0.69 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(\frac {-\frac {-\frac {b^{3} \tan \left (d x +c \right )^{4}}{4}+\frac {a \,b^{2} \tan \left (d x +c \right )^{3}}{3}-\frac {\left (a^{2}-b^{2}\right ) \tan \left (d x +c \right )^{2} b}{2}+a \left (a^{2}-b^{2}\right ) \tan \left (d x +c \right )}{b^{4}}+\frac {a^{6} \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{5} \left (a^{2}+b^{2}\right )}+\frac {\frac {b \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}-a \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}}{d}\)	\(138\)
default	\(\frac {-\frac {-\frac {b^{3} \tan \left (d x +c \right )^{4}}{4}+\frac {a \,b^{2} \tan \left (d x +c \right )^{3}}{3}-\frac {\left (a^{2}-b^{2}\right ) \tan \left (d x +c \right )^{2} b}{2}+a \left (a^{2}-b^{2}\right ) \tan \left (d x +c \right )}{b^{4}}+\frac {a^{6} \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{5} \left (a^{2}+b^{2}\right )}+\frac {\frac {b \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}-a \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}}{d}\)	\(138\)
norman	\(-\frac {a x}{a^{2}+b^{2}}+\frac {\tan \left (d x +c \right )^{4}}{4 b d}-\frac {a \tan \left (d x +c \right )^{3}}{3 b^{2} d}+\frac {\left (a^{2}-b^{2}\right ) \tan \left (d x +c \right )^{2}}{2 b^{3} d}-\frac {a \left (a^{2}-b^{2}\right ) \tan \left (d x +c \right )}{b^{4} d}+\frac {a^{6} \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{5} \left (a^{2}+b^{2}\right ) d}+\frac {b \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{2}+b^{2}\right )}\)	\(153\)
parallelrisch	\(\frac {3 \tan \left (d x +c \right )^{4} a^{2} b^{4}+3 \tan \left (d x +c \right )^{4} b^{6}-4 \tan \left (d x +c \right )^{3} a^{3} b^{3}-4 \tan \left (d x +c \right )^{3} a \,b^{5}-12 a x \,b^{5} d +6 \tan \left (d x +c \right )^{2} a^{4} b^{2}-6 \tan \left (d x +c \right )^{2} b^{6}+6 b^{6} \ln \left (1+\tan \left (d x +c \right )^{2}\right )+12 a^{6} \ln \left (a +b \tan \left (d x +c \right )\right )-12 \tan \left (d x +c \right ) a^{5} b +12 \tan \left (d x +c \right ) a \,b^{5}}{12 \left (a^{2}+b^{2}\right ) b^{5} d}\)	\(171\)
risch	\(\frac {x}{i b -a}+\frac {2 i a^{4} x}{b^{5}}+\frac {2 i a^{4} c}{b^{5} d}-\frac {2 i x \,a^{2}}{b^{3}}-\frac {2 i a^{2} c}{b^{3} d}+\frac {2 i x}{b}+\frac {2 i c}{b d}-\frac {2 i a^{6} x}{\left (a^{2}+b^{2}\right ) b^{5}}-\frac {2 i a^{6} c}{\left (a^{2}+b^{2}\right ) b^{5} d}-\frac {2 \left (3 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-6 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-3 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+6 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+9 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-12 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-6 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+6 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+9 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-10 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-3 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+6 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+3 i a^{3}-4 i a \,b^{2}\right )}{3 b^{4} d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{4}}{b^{5} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{2}}{b^{3} d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b d}+\frac {a^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{2}+b^{2}\right ) b^{5} d}\)	\(446\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.18 \[ \int \frac {\tan ^6(c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {12 \, a b^{5} d x - 6 \, a^{6} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 3 \, {\left (a^{2} b^{4} + b^{6}\right )} \tan \left (d x + c\right )^{4} + 4 \, {\left (a^{3} b^{3} + a b^{5}\right )} \tan \left (d x + c\right )^{3} - 6 \, {\left (a^{4} b^{2} - b^{6}\right )} \tan \left (d x + c\right )^{2} + 6 \, {\left (a^{6} + b^{6}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 12 \, {\left (a^{5} b - a b^{5}\right )} \tan \left (d x + c\right )}{12 \, {\left (a^{2} b^{5} + b^{7}\right )} d} \] Input:

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

Output:

-1/12*(12*a*b^5*d*x - 6*a^6*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + 
 a^2)/(tan(d*x + c)^2 + 1)) - 3*(a^2*b^4 + b^6)*tan(d*x + c)^4 + 4*(a^3*b^ 
3 + a*b^5)*tan(d*x + c)^3 - 6*(a^4*b^2 - b^6)*tan(d*x + c)^2 + 6*(a^6 + b^ 
6)*log(1/(tan(d*x + c)^2 + 1)) + 12*(a^5*b - a*b^5)*tan(d*x + c))/((a^2*b^ 
5 + b^7)*d)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.15 (sec) , antiderivative size = 947, normalized size of antiderivative = 6.15 \[ \int \frac {\tan ^6(c+d x)}{a+b \tan (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((zoo*x*tan(c)**5, 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, Eq(b, 0)), (3 
0*I*d*x*tan(c + d*x)/(12*b*d*tan(c + d*x) - 12*I*b*d) + 30*d*x/(12*b*d*tan 
(c + d*x) - 12*I*b*d) + 18*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(12*b*d*t 
an(c + d*x) - 12*I*b*d) - 18*I*log(tan(c + d*x)**2 + 1)/(12*b*d*tan(c + d* 
x) - 12*I*b*d) + 3*tan(c + d*x)**5/(12*b*d*tan(c + d*x) - 12*I*b*d) + I*ta 
n(c + d*x)**4/(12*b*d*tan(c + d*x) - 12*I*b*d) - 8*tan(c + d*x)**3/(12*b*d 
*tan(c + d*x) - 12*I*b*d) - 12*I*tan(c + d*x)**2/(12*b*d*tan(c + d*x) - 12 
*I*b*d) - 30*I/(12*b*d*tan(c + d*x) - 12*I*b*d), Eq(a, -I*b)), (-30*I*d*x* 
tan(c + d*x)/(12*b*d*tan(c + d*x) + 12*I*b*d) + 30*d*x/(12*b*d*tan(c + d*x 
) + 12*I*b*d) + 18*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(12*b*d*tan(c + d 
*x) + 12*I*b*d) + 18*I*log(tan(c + d*x)**2 + 1)/(12*b*d*tan(c + d*x) + 12* 
I*b*d) + 3*tan(c + d*x)**5/(12*b*d*tan(c + d*x) + 12*I*b*d) - I*tan(c + d* 
x)**4/(12*b*d*tan(c + d*x) + 12*I*b*d) - 8*tan(c + d*x)**3/(12*b*d*tan(c + 
 d*x) + 12*I*b*d) + 12*I*tan(c + d*x)**2/(12*b*d*tan(c + d*x) + 12*I*b*d) 
+ 30*I/(12*b*d*tan(c + d*x) + 12*I*b*d), Eq(a, I*b)), (x*tan(c)**6/(a + b* 
tan(c)), Eq(d, 0)), (12*a**6*log(a/b + tan(c + d*x))/(12*a**2*b**5*d + 12* 
b**7*d) - 12*a**5*b*tan(c + d*x)/(12*a**2*b**5*d + 12*b**7*d) + 6*a**4*b** 
2*tan(c + d*x)**2/(12*a**2*b**5*d + 12*b**7*d) - 4*a**3*b**3*tan(c + d*x)* 
*3/(12*a**2*b**5*d + 12*b**7*d) + 3*a**2*b**4*tan(c + d*x)**4/(12*a**2*...
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.95 \[ \int \frac {\tan ^6(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {\frac {12 \, a^{6} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{2} b^{5} + b^{7}} - \frac {12 \, {\left (d x + c\right )} a}{a^{2} + b^{2}} + \frac {6 \, b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac {3 \, b^{3} \tan \left (d x + c\right )^{4} - 4 \, a b^{2} \tan \left (d x + c\right )^{3} + 6 \, {\left (a^{2} b - b^{3}\right )} \tan \left (d x + c\right )^{2} - 12 \, {\left (a^{3} - a b^{2}\right )} \tan \left (d x + c\right )}{b^{4}}}{12 \, d} \] Input:

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

Output:

1/12*(12*a^6*log(b*tan(d*x + c) + a)/(a^2*b^5 + b^7) - 12*(d*x + c)*a/(a^2 
 + b^2) + 6*b*log(tan(d*x + c)^2 + 1)/(a^2 + b^2) + (3*b^3*tan(d*x + c)^4 
- 4*a*b^2*tan(d*x + c)^3 + 6*(a^2*b - b^3)*tan(d*x + c)^2 - 12*(a^3 - a*b^ 
2)*tan(d*x + c))/b^4)/d
 

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.20 \[ \int \frac {\tan ^6(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {a^{6} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{2} b^{5} d + b^{7} d} - \frac {{\left (d x + c\right )} a}{a^{2} d + b^{2} d} + \frac {b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, {\left (a^{2} d + b^{2} d\right )}} + \frac {3 \, b^{3} d^{3} \tan \left (d x + c\right )^{4} - 4 \, a b^{2} d^{3} \tan \left (d x + c\right )^{3} + 6 \, a^{2} b d^{3} \tan \left (d x + c\right )^{2} - 6 \, b^{3} d^{3} \tan \left (d x + c\right )^{2} - 12 \, a^{3} d^{3} \tan \left (d x + c\right ) + 12 \, a b^{2} d^{3} \tan \left (d x + c\right )}{12 \, b^{4} d^{4}} \] Input:

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

Output:

a^6*log(abs(b*tan(d*x + c) + a))/(a^2*b^5*d + b^7*d) - (d*x + c)*a/(a^2*d 
+ b^2*d) + 1/2*b*log(tan(d*x + c)^2 + 1)/(a^2*d + b^2*d) + 1/12*(3*b^3*d^3 
*tan(d*x + c)^4 - 4*a*b^2*d^3*tan(d*x + c)^3 + 6*a^2*b*d^3*tan(d*x + c)^2 
- 6*b^3*d^3*tan(d*x + c)^2 - 12*a^3*d^3*tan(d*x + c) + 12*a*b^2*d^3*tan(d* 
x + c))/(b^4*d^4)
 

Mupad [B] (verification not implemented)

Time = 1.44 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.07 \[ \int \frac {\tan ^6(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (b+a\,1{}\mathrm {i}\right )}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {1}{2\,b}-\frac {a^2}{2\,b^3}\right )}{d}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4}{4\,b\,d}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3\,b^2\,d}+\frac {a^6\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}{d\,\left (a^2\,b^5+b^7\right )}+\frac {a\,\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {1}{b}-\frac {a^2}{b^3}\right )}{b\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a+b\,1{}\mathrm {i}\right )} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.10 \[ \int \frac {\tan ^6(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {6 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) b^{6}+12 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) a^{6}+3 \tan \left (d x +c \right )^{4} a^{2} b^{4}+3 \tan \left (d x +c \right )^{4} b^{6}-4 \tan \left (d x +c \right )^{3} a^{3} b^{3}-4 \tan \left (d x +c \right )^{3} a \,b^{5}+6 \tan \left (d x +c \right )^{2} a^{4} b^{2}-6 \tan \left (d x +c \right )^{2} b^{6}-12 \tan \left (d x +c \right ) a^{5} b +12 \tan \left (d x +c \right ) a \,b^{5}-12 a \,b^{5} d x}{12 b^{5} d \left (a^{2}+b^{2}\right )} \] Input:

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

Output:

(6*log(tan(c + d*x)**2 + 1)*b**6 + 12*log(tan(c + d*x)*b + a)*a**6 + 3*tan 
(c + d*x)**4*a**2*b**4 + 3*tan(c + d*x)**4*b**6 - 4*tan(c + d*x)**3*a**3*b 
**3 - 4*tan(c + d*x)**3*a*b**5 + 6*tan(c + d*x)**2*a**4*b**2 - 6*tan(c + d 
*x)**2*b**6 - 12*tan(c + d*x)*a**5*b + 12*tan(c + d*x)*a*b**5 - 12*a*b**5* 
d*x)/(12*b**5*d*(a**2 + b**2))