\(\int \frac {\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx\) [283]

Optimal result

Integrand size = 31, antiderivative size = 250 \[ \int \frac {\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=-\frac {\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) x}{\left (a^2+b^2\right )^3}+\frac {\left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a \left (a^2 A b^3-3 A b^5+a^5 B+3 a^3 b^2 B+6 a b^4 B\right ) \log (a+b \tan (c+d x))}{b^3 \left (a^2+b^2\right )^3 d}+\frac {a (A b-a B) \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.60 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.89 \[ \int \frac {\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {-\frac {(A+i B) \log (i-\tan (c+d x))}{(a+i b)^3}-\frac {(A-i B) \log (i+\tan (c+d x))}{(a-i b)^3}+\frac {2 a \left (a^2 A b^3-3 A b^5+a^5 B+3 a^3 b^2 B+6 a b^4 B\right ) \log (a+b \tan (c+d x))}{b^3 \left (a^2+b^2\right )^3}+\frac {a^3 (A b-a B)}{b^3 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {2 a^2 \left (-a^2 A b-3 A b^3+2 a^3 B+4 a b^2 B\right )}{b^3 \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}}{2 d} \] Input:

Integrate[(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 1.29 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.14, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {3042, 4088, 27, 3042, 4118, 3042, 4109, 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.

\(\displaystyle \int \frac {\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\tan (c+d x)^3 (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3}dx\)

\(\Big \downarrow \) 4088

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4118

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4109

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3956

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

\(\Big \downarrow \) 4100

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

\(\Big \downarrow \) 16

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

Input:

Int[(Tan[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]
 

Output:

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

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 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 4088

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

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 4109

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] + (Simp[(A*b^2 - a*b*B + a^2*C)/(a^2 + b^2)   Int[( 
1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x]), x], x] - Simp[(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 4118

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

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.97

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (-A \,a^{3}+3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-3 A \,a^{2} b +A \,b^{3}+B \,a^{3}-3 B a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {a \left (A \,a^{2} b^{3}-3 A \,b^{5}+B \,a^{5}+3 B \,a^{3} b^{2}+6 B a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} b^{3}}-\frac {a^{2} \left (A \,a^{2} b +3 A \,b^{3}-2 B \,a^{3}-4 B a \,b^{2}\right )}{b^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {a^{3} \left (A b -B a \right )}{2 b^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\)	\(242\)
default	\(\frac {\frac {\frac {\left (-A \,a^{3}+3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-3 A \,a^{2} b +A \,b^{3}+B \,a^{3}-3 B a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {a \left (A \,a^{2} b^{3}-3 A \,b^{5}+B \,a^{5}+3 B \,a^{3} b^{2}+6 B a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} b^{3}}-\frac {a^{2} \left (A \,a^{2} b +3 A \,b^{3}-2 B \,a^{3}-4 B a \,b^{2}\right )}{b^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {a^{3} \left (A b -B a \right )}{2 b^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\)	\(242\)
norman	\(\frac {-\frac {a^{2} \left (A \,a^{3} b +5 A a \,b^{3}-3 B \,a^{4}-7 B \,a^{2} b^{2}\right )}{2 d \,b^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) a^{2} x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {b^{2} \left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) x \tan \left (d x +c \right )^{2}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {a \left (A \,a^{3} b +3 A a \,b^{3}-2 B \,a^{4}-4 B \,a^{2} b^{2}\right ) \tan \left (d x +c \right )}{d \,b^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 b \left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) a x \tan \left (d x +c \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a \left (A \,a^{2} b^{3}-3 A \,b^{5}+B \,a^{5}+3 B \,a^{3} b^{2}+6 B a \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right ) d \,b^{3}}-\frac {\left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}\)	\(471\)
parallelrisch	\(\text {Expression too large to display}\)	\(1014\)
risch	\(-\frac {x B}{3 i a^{2} b -i b^{3}-a^{3}+3 a \,b^{2}}-\frac {2 i a^{3} A c}{d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}+\frac {2 i B x}{b^{3}}-\frac {6 i a^{4} B c}{b d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}-\frac {2 i a^{6} B c}{b^{3} d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}-\frac {2 i a^{6} B x}{b^{3} \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}+\frac {6 i a \,b^{2} A c}{d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}-\frac {12 i a^{2} b B x}{a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}}-\frac {12 i a^{2} b B c}{d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}-\frac {2 i \left (3 B \,a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-2 i B \,a^{5} b \,{\mathrm e}^{2 i \left (d x +c \right )}+3 i A \,a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+i A \,a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+i B \,a^{5} b -3 A \,a^{3} b^{3}+4 B \,a^{4} b^{2}+B \,a^{6} {\mathrm e}^{2 i \left (d x +c \right )}-4 i B \,a^{3} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-3 i A \,a^{2} b^{4}+4 i B \,a^{3} b^{3}+B \,a^{6}-2 A \,a^{3} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}\right )}{\left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )^{2} b^{2} \left (i b +a \right )^{2} d \left (-i b +a \right )^{3}}+\frac {6 i a \,b^{2} A x}{a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}}-\frac {i x A}{3 i a^{2} b -i b^{3}-a^{3}+3 a \,b^{2}}-\frac {2 i a^{3} A x}{a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}}-\frac {6 i a^{4} B x}{b \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}+\frac {2 i B c}{b^{3} d}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) A}{d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}-\frac {3 a \,b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) A}{d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}+\frac {a^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{b^{3} d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}+\frac {3 a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{b d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}+\frac {6 a^{2} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}-\frac {B \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{3} d}\)	\(1020\)

Input:

int(tan(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x,method=_RETURNVERBO 
SE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 666 vs. \(2 (245) = 490\).

Time = 0.20 (sec) , antiderivative size = 666, normalized size of antiderivative = 2.66 \[ \int \frac {\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {B a^{6} b^{2} + A a^{5} b^{3} + 7 \, B a^{4} b^{4} - 5 \, A a^{3} b^{5} + 2 \, {\left (B a^{5} b^{3} - 3 \, A a^{4} b^{4} - 3 \, B a^{3} b^{5} + A a^{2} b^{6}\right )} d x - {\left (3 \, B a^{6} b^{2} - A a^{5} b^{3} + 9 \, B a^{4} b^{4} - 7 \, A a^{3} b^{5} - 2 \, {\left (B a^{3} b^{5} - 3 \, A a^{2} b^{6} - 3 \, B a b^{7} + A b^{8}\right )} d x\right )} \tan \left (d x + c\right )^{2} + {\left (B a^{8} + 3 \, B a^{6} b^{2} + A a^{5} b^{3} + 6 \, B a^{4} b^{4} - 3 \, A a^{3} b^{5} + {\left (B a^{6} b^{2} + 3 \, B a^{4} b^{4} + A a^{3} b^{5} + 6 \, B a^{2} b^{6} - 3 \, A a b^{7}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (B a^{7} b + 3 \, B a^{5} b^{3} + A a^{4} b^{4} + 6 \, B a^{3} b^{5} - 3 \, A a^{2} b^{6}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (B a^{8} + 3 \, B a^{6} b^{2} + 3 \, B a^{4} b^{4} + B a^{2} b^{6} + {\left (B a^{6} b^{2} + 3 \, B a^{4} b^{4} + 3 \, B a^{2} b^{6} + B b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (B a^{7} b + 3 \, B a^{5} b^{3} + 3 \, B a^{3} b^{5} + B a b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (B a^{7} b + 3 \, B a^{5} b^{3} - 3 \, A a^{4} b^{4} - 4 \, B a^{3} b^{5} + 3 \, A a^{2} b^{6} - 2 \, {\left (B a^{4} b^{4} - 3 \, A a^{3} b^{5} - 3 \, B a^{2} b^{6} + A a b^{7}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b^{5} + 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} + b^{11}\right )} d \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b^{4} + 3 \, a^{5} b^{6} + 3 \, a^{3} b^{8} + a b^{10}\right )} d \tan \left (d x + c\right ) + {\left (a^{8} b^{3} + 3 \, a^{6} b^{5} + 3 \, a^{4} b^{7} + a^{2} b^{9}\right )} d\right )}} \] Input:

integrate(tan(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x, algorithm="f 
ricas")
 

Output:

1/2*(B*a^6*b^2 + A*a^5*b^3 + 7*B*a^4*b^4 - 5*A*a^3*b^5 + 2*(B*a^5*b^3 - 3* 
A*a^4*b^4 - 3*B*a^3*b^5 + A*a^2*b^6)*d*x - (3*B*a^6*b^2 - A*a^5*b^3 + 9*B* 
a^4*b^4 - 7*A*a^3*b^5 - 2*(B*a^3*b^5 - 3*A*a^2*b^6 - 3*B*a*b^7 + A*b^8)*d* 
x)*tan(d*x + c)^2 + (B*a^8 + 3*B*a^6*b^2 + A*a^5*b^3 + 6*B*a^4*b^4 - 3*A*a 
^3*b^5 + (B*a^6*b^2 + 3*B*a^4*b^4 + A*a^3*b^5 + 6*B*a^2*b^6 - 3*A*a*b^7)*t 
an(d*x + c)^2 + 2*(B*a^7*b + 3*B*a^5*b^3 + A*a^4*b^4 + 6*B*a^3*b^5 - 3*A*a 
^2*b^6)*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)) - (B*a^8 + 3*B*a^6*b^2 + 3*B*a^4*b^4 + B*a^2*b^6 + ( 
B*a^6*b^2 + 3*B*a^4*b^4 + 3*B*a^2*b^6 + B*b^8)*tan(d*x + c)^2 + 2*(B*a^7*b 
 + 3*B*a^5*b^3 + 3*B*a^3*b^5 + B*a*b^7)*tan(d*x + c))*log(1/(tan(d*x + c)^ 
2 + 1)) - 2*(B*a^7*b + 3*B*a^5*b^3 - 3*A*a^4*b^4 - 4*B*a^3*b^5 + 3*A*a^2*b 
^6 - 2*(B*a^4*b^4 - 3*A*a^3*b^5 - 3*B*a^2*b^6 + A*a*b^7)*d*x)*tan(d*x + c) 
)/((a^6*b^5 + 3*a^4*b^7 + 3*a^2*b^9 + b^11)*d*tan(d*x + c)^2 + 2*(a^7*b^4 
+ 3*a^5*b^6 + 3*a^3*b^8 + a*b^10)*d*tan(d*x + c) + (a^8*b^3 + 3*a^6*b^5 + 
3*a^4*b^7 + a^2*b^9)*d)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\text {Exception raised: AttributeError} \] Input:

integrate(tan(d*x+c)**3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**3,x)
 

Output:

Exception raised: AttributeError >> 'NoneType' object has no attribute 'pr 
imitive'
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.46 \[ \int \frac {\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {2 \, {\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (B a^{6} + 3 \, B a^{4} b^{2} + A a^{3} b^{3} + 6 \, B a^{2} b^{4} - 3 \, A a b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}} - \frac {{\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {3 \, B a^{6} - A a^{5} b + 7 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3} + 2 \, {\left (2 \, B a^{5} b - A a^{4} b^{2} + 4 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} \tan \left (d x + c\right )}{a^{6} b^{3} + 2 \, a^{4} b^{5} + a^{2} b^{7} + {\left (a^{4} b^{5} + 2 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{4} + 2 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )}}{2 \, d} \] Input:

integrate(tan(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x, algorithm="m 
axima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.44 \[ \int \frac {\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {{\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} {\left (d x + c\right )}}{a^{6} d + 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d + b^{6} d} - \frac {{\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, {\left (a^{6} d + 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d + b^{6} d\right )}} + \frac {{\left (B a^{6} + 3 \, B a^{4} b^{2} + A a^{3} b^{3} + 6 \, B a^{2} b^{4} - 3 \, A a b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b^{3} d + 3 \, a^{4} b^{5} d + 3 \, a^{2} b^{7} d + b^{9} d} + \frac {2 \, {\left (2 \, B a^{7} - A a^{6} b + 6 \, B a^{5} b^{2} - 4 \, A a^{4} b^{3} + 4 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5}\right )} \tan \left (d x + c\right ) + \frac {3 \, B a^{8} - A a^{7} b + 10 \, B a^{6} b^{2} - 6 \, A a^{5} b^{3} + 7 \, B a^{4} b^{4} - 5 \, A a^{3} b^{5}}{b}}{2 \, {\left (a^{2} + b^{2}\right )}^{3} {\left (b \tan \left (d x + c\right ) + a\right )}^{2} b^{2} d} \] Input:

integrate(tan(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x, algorithm="g 
iac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 4.01 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.23 \[ \int \frac {\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {3\,B\,a^6-A\,a^5\,b+7\,B\,a^4\,b^2-5\,A\,a^3\,b^3}{2\,b^3\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {a^2\,\mathrm {tan}\left (c+d\,x\right )\,\left (-2\,B\,a^3+A\,a^2\,b-4\,B\,a\,b^2+3\,A\,b^3\right )}{b^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2+2\,a\,b\,\mathrm {tan}\left (c+d\,x\right )+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}+3\,a^2\,b+a\,b^2\,3{}\mathrm {i}-b^3\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3+a^2\,b\,3{}\mathrm {i}+3\,a\,b^2-b^3\,1{}\mathrm {i}\right )}+\frac {a\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (B\,a^5+3\,B\,a^3\,b^2+A\,a^2\,b^3+6\,B\,a\,b^4-3\,A\,b^5\right )}{b^3\,d\,{\left (a^2+b^2\right )}^3} \] Input:

int((tan(c + d*x)^3*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^3,x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.11 \[ \int \frac {\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {-\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right ) a^{2} b^{3}+\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) \tan \left (d x +c \right ) b^{5}-\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) a^{3} b^{2}+\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) a \,b^{4}+2 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right ) a^{4} b +6 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) \tan \left (d x +c \right ) a^{2} b^{3}+2 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) a^{5}+6 \,\mathrm {log}\left (a +\tan \left (d x +c \right ) b \right ) a^{3} b^{2}-2 \tan \left (d x +c \right ) a^{4} b -2 \tan \left (d x +c \right ) a^{2} b^{3}-4 \tan \left (d x +c \right ) a \,b^{4} d x -4 a^{2} b^{3} d x}{2 b^{2} d \left (\tan \left (d x +c \right ) a^{4} b +2 \tan \left (d x +c \right ) a^{2} b^{3}+\tan \left (d x +c \right ) b^{5}+a^{5}+2 a^{3} b^{2}+a \,b^{4}\right )} \] Input:

int(tan(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x)
 

Output:

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