\(\int (a+a \sec (c+d x))^3 \tan ^6(c+d x) \, dx\) [47]

Optimal result

Integrand size = 21, antiderivative size = 237 \[ \int (a+a \sec (c+d x))^3 \tan ^6(c+d x) \, dx=-a^3 x-\frac {125 a^3 \text {arctanh}(\sin (c+d x))}{128 d}+\frac {a^3 \tan (c+d x)}{d}+\frac {115 a^3 \sec (c+d x) \tan (c+d x)}{128 d}+\frac {5 a^3 \sec ^3(c+d x) \tan (c+d x)}{64 d}-\frac {a^3 \tan ^3(c+d x)}{3 d}-\frac {5 a^3 \sec (c+d x) \tan ^3(c+d x)}{8 d}-\frac {5 a^3 \sec ^3(c+d x) \tan ^3(c+d x)}{48 d}+\frac {a^3 \tan ^5(c+d x)}{5 d}+\frac {a^3 \sec (c+d x) \tan ^5(c+d x)}{2 d}+\frac {a^3 \sec ^3(c+d x) \tan ^5(c+d x)}{8 d}+\frac {3 a^3 \tan ^7(c+d x)}{7 d} \] Output:

-a^3*x-125/128*a^3*arctanh(sin(d*x+c))/d+a^3*tan(d*x+c)/d+115/128*a^3*sec( 
d*x+c)*tan(d*x+c)/d+5/64*a^3*sec(d*x+c)^3*tan(d*x+c)/d-1/3*a^3*tan(d*x+c)^ 
3/d-5/8*a^3*sec(d*x+c)*tan(d*x+c)^3/d-5/48*a^3*sec(d*x+c)^3*tan(d*x+c)^3/d 
+1/5*a^3*tan(d*x+c)^5/d+1/2*a^3*sec(d*x+c)*tan(d*x+c)^5/d+1/8*a^3*sec(d*x+ 
c)^3*tan(d*x+c)^5/d+3/7*a^3*tan(d*x+c)^7/d
 

Mathematica [A] (verified)

Time = 1.10 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.23 \[ \int (a+a \sec (c+d x))^3 \tan ^6(c+d x) \, dx=-\frac {a^3 \arctan (\tan (c+d x))}{d}-\frac {125 a^3 \text {arctanh}(\sin (c+d x))}{128 d}+\frac {a^3 \tan (c+d x)}{d}-\frac {125 a^3 \sec (c+d x) \tan (c+d x)}{128 d}-\frac {125 a^3 \sec ^3(c+d x) \tan (c+d x)}{192 d}+\frac {119 a^3 \sec ^5(c+d x) \tan (c+d x)}{48 d}+\frac {a^3 \sec ^7(c+d x) \tan (c+d x)}{8 d}-\frac {a^3 \tan ^3(c+d x)}{3 d}-\frac {5 a^3 \sec ^3(c+d x) \tan ^3(c+d x)}{d}-\frac {a^3 \sec ^5(c+d x) \tan ^3(c+d x)}{3 d}+\frac {a^3 \tan ^5(c+d x)}{5 d}+\frac {3 a^3 \sec (c+d x) \tan ^5(c+d x)}{d}+\frac {a^3 \sec ^3(c+d x) \tan ^5(c+d x)}{3 d}+\frac {3 a^3 \tan ^7(c+d x)}{7 d} \] Input:

Integrate[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^6,x]
 

Output:

-((a^3*ArcTan[Tan[c + d*x]])/d) - (125*a^3*ArcTanh[Sin[c + d*x]])/(128*d) 
+ (a^3*Tan[c + d*x])/d - (125*a^3*Sec[c + d*x]*Tan[c + d*x])/(128*d) - (12 
5*a^3*Sec[c + d*x]^3*Tan[c + d*x])/(192*d) + (119*a^3*Sec[c + d*x]^5*Tan[c 
 + d*x])/(48*d) + (a^3*Sec[c + d*x]^7*Tan[c + d*x])/(8*d) - (a^3*Tan[c + d 
*x]^3)/(3*d) - (5*a^3*Sec[c + d*x]^3*Tan[c + d*x]^3)/d - (a^3*Sec[c + d*x] 
^5*Tan[c + d*x]^3)/(3*d) + (a^3*Tan[c + d*x]^5)/(5*d) + (3*a^3*Sec[c + d*x 
]*Tan[c + d*x]^5)/d + (a^3*Sec[c + d*x]^3*Tan[c + d*x]^5)/(3*d) + (3*a^3*T 
an[c + d*x]^7)/(7*d)
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 4374, 2009}

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.

\(\displaystyle \int \tan ^6(c+d x) (a \sec (c+d x)+a)^3 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cot \left (c+d x+\frac {\pi }{2}\right )^6 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3dx\)

\(\Big \downarrow \) 4374

\(\displaystyle \int \left (a^3 \tan ^6(c+d x)+a^3 \tan ^6(c+d x) \sec ^3(c+d x)+3 a^3 \tan ^6(c+d x) \sec ^2(c+d x)+3 a^3 \tan ^6(c+d x) \sec (c+d x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {125 a^3 \text {arctanh}(\sin (c+d x))}{128 d}+\frac {3 a^3 \tan ^7(c+d x)}{7 d}+\frac {a^3 \tan ^5(c+d x)}{5 d}-\frac {a^3 \tan ^3(c+d x)}{3 d}+\frac {a^3 \tan (c+d x)}{d}+\frac {a^3 \tan ^5(c+d x) \sec ^3(c+d x)}{8 d}-\frac {5 a^3 \tan ^3(c+d x) \sec ^3(c+d x)}{48 d}+\frac {5 a^3 \tan (c+d x) \sec ^3(c+d x)}{64 d}+\frac {a^3 \tan ^5(c+d x) \sec (c+d x)}{2 d}-\frac {5 a^3 \tan ^3(c+d x) \sec (c+d x)}{8 d}+\frac {115 a^3 \tan (c+d x) \sec (c+d x)}{128 d}-a^3 x\)

Input:

Int[(a + a*Sec[c + d*x])^3*Tan[c + d*x]^6,x]
 

Output:

-(a^3*x) - (125*a^3*ArcTanh[Sin[c + d*x]])/(128*d) + (a^3*Tan[c + d*x])/d 
+ (115*a^3*Sec[c + d*x]*Tan[c + d*x])/(128*d) + (5*a^3*Sec[c + d*x]^3*Tan[ 
c + d*x])/(64*d) - (a^3*Tan[c + d*x]^3)/(3*d) - (5*a^3*Sec[c + d*x]*Tan[c 
+ d*x]^3)/(8*d) - (5*a^3*Sec[c + d*x]^3*Tan[c + d*x]^3)/(48*d) + (a^3*Tan[ 
c + d*x]^5)/(5*d) + (a^3*Sec[c + d*x]*Tan[c + d*x]^5)/(2*d) + (a^3*Sec[c + 
 d*x]^3*Tan[c + d*x]^5)/(8*d) + (3*a^3*Tan[c + d*x]^7)/(7*d)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 4374

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( 
a_))^(n_), x_Symbol] :> Int[ExpandIntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[ 
c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.76 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.96

method	result	size
risch	\(-a^{3} x -\frac {i a^{3} \left (27195 \,{\mathrm e}^{15 i \left (d x +c \right )}+65135 \,{\mathrm e}^{13 i \left (d x +c \right )}-161280 \,{\mathrm e}^{12 i \left (d x +c \right )}+63595 \,{\mathrm e}^{11 i \left (d x +c \right )}-286720 \,{\mathrm e}^{10 i \left (d x +c \right )}+133175 \,{\mathrm e}^{9 i \left (d x +c \right )}-519680 \,{\mathrm e}^{8 i \left (d x +c \right )}-133175 \,{\mathrm e}^{7 i \left (d x +c \right )}-544768 \,{\mathrm e}^{6 i \left (d x +c \right )}-63595 \,{\mathrm e}^{5 i \left (d x +c \right )}-254464 \,{\mathrm e}^{4 i \left (d x +c \right )}-65135 \,{\mathrm e}^{3 i \left (d x +c \right )}-118784 \,{\mathrm e}^{2 i \left (d x +c \right )}-27195 \,{\mathrm e}^{i \left (d x +c \right )}-14848\right )}{6720 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{8}}-\frac {125 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{128 d}+\frac {125 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d}\)	\(228\)
derivativedivides	\(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{7}}{48 \cos \left (d x +c \right )^{6}}-\frac {\sin \left (d x +c \right )^{7}}{192 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{7}}{128 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{5}}{128}+\frac {5 \sin \left (d x +c \right )^{3}}{384}+\frac {5 \sin \left (d x +c \right )}{128}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+\frac {3 a^{3} \sin \left (d x +c \right )^{7}}{7 \cos \left (d x +c \right )^{7}}+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{6 \cos \left (d x +c \right )^{6}}-\frac {\sin \left (d x +c \right )^{7}}{24 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{7}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{5}}{16}+\frac {5 \sin \left (d x +c \right )^{3}}{48}+\frac {5 \sin \left (d x +c \right )}{16}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+a^{3} \left (\frac {\tan \left (d x +c \right )^{5}}{5}-\frac {\tan \left (d x +c \right )^{3}}{3}+\tan \left (d x +c \right )-d x -c \right )}{d}\)	\(290\)
default	\(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{7}}{48 \cos \left (d x +c \right )^{6}}-\frac {\sin \left (d x +c \right )^{7}}{192 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{7}}{128 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{5}}{128}+\frac {5 \sin \left (d x +c \right )^{3}}{384}+\frac {5 \sin \left (d x +c \right )}{128}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+\frac {3 a^{3} \sin \left (d x +c \right )^{7}}{7 \cos \left (d x +c \right )^{7}}+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{6 \cos \left (d x +c \right )^{6}}-\frac {\sin \left (d x +c \right )^{7}}{24 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{7}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{5}}{16}+\frac {5 \sin \left (d x +c \right )^{3}}{48}+\frac {5 \sin \left (d x +c \right )}{16}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+a^{3} \left (\frac {\tan \left (d x +c \right )^{5}}{5}-\frac {\tan \left (d x +c \right )^{3}}{3}+\tan \left (d x +c \right )-d x -c \right )}{d}\)	\(290\)
parts	\(\frac {a^{3} \left (\frac {\tan \left (d x +c \right )^{5}}{5}-\frac {\tan \left (d x +c \right )^{3}}{3}+\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin \left (d x +c \right )^{7}}{48 \cos \left (d x +c \right )^{6}}-\frac {\sin \left (d x +c \right )^{7}}{192 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{7}}{128 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{5}}{128}+\frac {5 \sin \left (d x +c \right )^{3}}{384}+\frac {5 \sin \left (d x +c \right )}{128}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )}{d}+\frac {3 a^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{6 \cos \left (d x +c \right )^{6}}-\frac {\sin \left (d x +c \right )^{7}}{24 \cos \left (d x +c \right )^{4}}+\frac {\sin \left (d x +c \right )^{7}}{16 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{5}}{16}+\frac {5 \sin \left (d x +c \right )^{3}}{48}+\frac {5 \sin \left (d x +c \right )}{16}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {3 a^{3} \tan \left (d x +c \right )^{7}}{7 d}\)	\(292\)

Input:

int((a+a*sec(d*x+c))^3*tan(d*x+c)^6,x,method=_RETURNVERBOSE)
 

Output:

-a^3*x-1/6720*I*a^3*(27195*exp(15*I*(d*x+c))+65135*exp(13*I*(d*x+c))-16128 
0*exp(12*I*(d*x+c))+63595*exp(11*I*(d*x+c))-286720*exp(10*I*(d*x+c))+13317 
5*exp(9*I*(d*x+c))-519680*exp(8*I*(d*x+c))-133175*exp(7*I*(d*x+c))-544768* 
exp(6*I*(d*x+c))-63595*exp(5*I*(d*x+c))-254464*exp(4*I*(d*x+c))-65135*exp( 
3*I*(d*x+c))-118784*exp(2*I*(d*x+c))-27195*exp(I*(d*x+c))-14848)/d/(exp(2* 
I*(d*x+c))+1)^8-125/128*a^3/d*ln(exp(I*(d*x+c))+I)+125/128*a^3/d*ln(exp(I* 
(d*x+c))-I)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.75 \[ \int (a+a \sec (c+d x))^3 \tan ^6(c+d x) \, dx=-\frac {26880 \, a^{3} d x \cos \left (d x + c\right )^{8} + 13125 \, a^{3} \cos \left (d x + c\right )^{8} \log \left (\sin \left (d x + c\right ) + 1\right ) - 13125 \, a^{3} \cos \left (d x + c\right )^{8} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (14848 \, a^{3} \cos \left (d x + c\right )^{7} + 27195 \, a^{3} \cos \left (d x + c\right )^{6} + 7424 \, a^{3} \cos \left (d x + c\right )^{5} - 17710 \, a^{3} \cos \left (d x + c\right )^{4} - 14592 \, a^{3} \cos \left (d x + c\right )^{3} + 1960 \, a^{3} \cos \left (d x + c\right )^{2} + 5760 \, a^{3} \cos \left (d x + c\right ) + 1680 \, a^{3}\right )} \sin \left (d x + c\right )}{26880 \, d \cos \left (d x + c\right )^{8}} \] Input:

integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^6,x, algorithm="fricas")
 

Output:

-1/26880*(26880*a^3*d*x*cos(d*x + c)^8 + 13125*a^3*cos(d*x + c)^8*log(sin( 
d*x + c) + 1) - 13125*a^3*cos(d*x + c)^8*log(-sin(d*x + c) + 1) - 2*(14848 
*a^3*cos(d*x + c)^7 + 27195*a^3*cos(d*x + c)^6 + 7424*a^3*cos(d*x + c)^5 - 
 17710*a^3*cos(d*x + c)^4 - 14592*a^3*cos(d*x + c)^3 + 1960*a^3*cos(d*x + 
c)^2 + 5760*a^3*cos(d*x + c) + 1680*a^3)*sin(d*x + c))/(d*cos(d*x + c)^8)
 

Sympy [F]

\[ \int (a+a \sec (c+d x))^3 \tan ^6(c+d x) \, dx=a^{3} \left (\int 3 \tan ^{6}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 \tan ^{6}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \tan ^{6}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \tan ^{6}{\left (c + d x \right )}\, dx\right ) \] Input:

integrate((a+a*sec(d*x+c))**3*tan(d*x+c)**6,x)
 

Output:

a**3*(Integral(3*tan(c + d*x)**6*sec(c + d*x), x) + Integral(3*tan(c + d*x 
)**6*sec(c + d*x)**2, x) + Integral(tan(c + d*x)**6*sec(c + d*x)**3, x) + 
Integral(tan(c + d*x)**6, x))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.11 \[ \int (a+a \sec (c+d x))^3 \tan ^6(c+d x) \, dx=\frac {11520 \, a^{3} \tan \left (d x + c\right )^{7} + 1792 \, {\left (3 \, \tan \left (d x + c\right )^{5} - 5 \, \tan \left (d x + c\right )^{3} - 15 \, d x - 15 \, c + 15 \, \tan \left (d x + c\right )\right )} a^{3} + 35 \, a^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{7} + 73 \, \sin \left (d x + c\right )^{5} - 55 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 840 \, a^{3} {\left (\frac {2 \, {\left (33 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{26880 \, d} \] Input:

integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^6,x, algorithm="maxima")
 

Output:

1/26880*(11520*a^3*tan(d*x + c)^7 + 1792*(3*tan(d*x + c)^5 - 5*tan(d*x + c 
)^3 - 15*d*x - 15*c + 15*tan(d*x + c))*a^3 + 35*a^3*(2*(15*sin(d*x + c)^7 
+ 73*sin(d*x + c)^5 - 55*sin(d*x + c)^3 + 15*sin(d*x + c))/(sin(d*x + c)^8 
 - 4*sin(d*x + c)^6 + 6*sin(d*x + c)^4 - 4*sin(d*x + c)^2 + 1) - 15*log(si 
n(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) - 840*a^3*(2*(33*sin(d*x + c)^ 
5 - 40*sin(d*x + c)^3 + 15*sin(d*x + c))/(sin(d*x + c)^6 - 3*sin(d*x + c)^ 
4 + 3*sin(d*x + c)^2 - 1) + 15*log(sin(d*x + c) + 1) - 15*log(sin(d*x + c) 
 - 1)))/d
 

Giac [A] (verification not implemented)

Time = 0.80 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.83 \[ \int (a+a \sec (c+d x))^3 \tan ^6(c+d x) \, dx=-\frac {13440 \, {\left (d x + c\right )} a^{3} + 13125 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 13125 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (315 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} - 11375 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 79723 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 269879 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 550089 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 749973 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 212625 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 26565 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{8}}}{13440 \, d} \] Input:

integrate((a+a*sec(d*x+c))^3*tan(d*x+c)^6,x, algorithm="giac")
 

Output:

-1/13440*(13440*(d*x + c)*a^3 + 13125*a^3*log(abs(tan(1/2*d*x + 1/2*c) + 1 
)) - 13125*a^3*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*(315*a^3*tan(1/2*d*x 
 + 1/2*c)^15 - 11375*a^3*tan(1/2*d*x + 1/2*c)^13 + 79723*a^3*tan(1/2*d*x + 
 1/2*c)^11 - 269879*a^3*tan(1/2*d*x + 1/2*c)^9 + 550089*a^3*tan(1/2*d*x + 
1/2*c)^7 - 749973*a^3*tan(1/2*d*x + 1/2*c)^5 + 212625*a^3*tan(1/2*d*x + 1/ 
2*c)^3 - 26565*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^8)/d
 

Mupad [B] (verification not implemented)

Time = 13.69 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.11 \[ \int (a+a \sec (c+d x))^3 \tan ^6(c+d x) \, dx=-a^3\,x-\frac {125\,a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{64\,d}-\frac {\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}-\frac {325\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{192}+\frac {11389\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{960}-\frac {269879\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{6720}+\frac {183363\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2240}-\frac {35713\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{320}+\frac {2025\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64}-\frac {253\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \] Input:

int(tan(c + d*x)^6*(a + a/cos(c + d*x))^3,x)
 

Output:

- a^3*x - (125*a^3*atanh(tan(c/2 + (d*x)/2)))/(64*d) - ((2025*a^3*tan(c/2 
+ (d*x)/2)^3)/64 - (35713*a^3*tan(c/2 + (d*x)/2)^5)/320 + (183363*a^3*tan( 
c/2 + (d*x)/2)^7)/2240 - (269879*a^3*tan(c/2 + (d*x)/2)^9)/6720 + (11389*a 
^3*tan(c/2 + (d*x)/2)^11)/960 - (325*a^3*tan(c/2 + (d*x)/2)^13)/192 + (3*a 
^3*tan(c/2 + (d*x)/2)^15)/64 - (253*a^3*tan(c/2 + (d*x)/2))/64)/(d*(28*tan 
(c/2 + (d*x)/2)^4 - 8*tan(c/2 + (d*x)/2)^2 - 56*tan(c/2 + (d*x)/2)^6 + 70* 
tan(c/2 + (d*x)/2)^8 - 56*tan(c/2 + (d*x)/2)^10 + 28*tan(c/2 + (d*x)/2)^12 
 - 8*tan(c/2 + (d*x)/2)^14 + tan(c/2 + (d*x)/2)^16 + 1))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 597, normalized size of antiderivative = 2.52 \[ \int (a+a \sec (c+d x))^3 \tan ^6(c+d x) \, dx =\text {Too large to display} \] Input:

int((a+a*sec(d*x+c))^3*tan(d*x+c)^6,x)
 

Output:

(a**3*(5760*cos(c + d*x)*sin(c + d*x)**7 + 13125*log(tan((c + d*x)/2) - 1) 
*sin(c + d*x)**8 - 52500*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6 + 78750 
*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4 - 52500*log(tan((c + d*x)/2) - 
1)*sin(c + d*x)**2 + 13125*log(tan((c + d*x)/2) - 1) - 13125*log(tan((c + 
d*x)/2) + 1)*sin(c + d*x)**8 + 52500*log(tan((c + d*x)/2) + 1)*sin(c + d*x 
)**6 - 78750*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4 + 52500*log(tan((c 
+ d*x)/2) + 1)*sin(c + d*x)**2 - 13125*log(tan((c + d*x)/2) + 1) + 2688*si 
n(c + d*x)**8*tan(c + d*x)**5 - 4480*sin(c + d*x)**8*tan(c + d*x)**3 + 134 
40*sin(c + d*x)**8*tan(c + d*x) - 13440*sin(c + d*x)**8*d*x - 27195*sin(c 
+ d*x)**7 - 10752*sin(c + d*x)**6*tan(c + d*x)**5 + 17920*sin(c + d*x)**6* 
tan(c + d*x)**3 - 53760*sin(c + d*x)**6*tan(c + d*x) + 53760*sin(c + d*x)* 
*6*d*x + 63875*sin(c + d*x)**5 + 16128*sin(c + d*x)**4*tan(c + d*x)**5 - 2 
6880*sin(c + d*x)**4*tan(c + d*x)**3 + 80640*sin(c + d*x)**4*tan(c + d*x) 
- 80640*sin(c + d*x)**4*d*x - 48125*sin(c + d*x)**3 - 10752*sin(c + d*x)** 
2*tan(c + d*x)**5 + 17920*sin(c + d*x)**2*tan(c + d*x)**3 - 53760*sin(c + 
d*x)**2*tan(c + d*x) + 53760*sin(c + d*x)**2*d*x + 13125*sin(c + d*x) + 26 
88*tan(c + d*x)**5 - 4480*tan(c + d*x)**3 + 13440*tan(c + d*x) - 13440*d*x 
))/(13440*d*(sin(c + d*x)**8 - 4*sin(c + d*x)**6 + 6*sin(c + d*x)**4 - 4*s 
in(c + d*x)**2 + 1))