\(\int (c+d x)^4 \sec (a+b x) \tan (a+b x) \, dx\) [247]

Optimal result

Integrand size = 20, antiderivative size = 227 \[ \int (c+d x)^4 \sec (a+b x) \tan (a+b x) \, dx=\frac {8 i d (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}+\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}+\frac {24 i d^4 \operatorname {PolyLog}\left (4,-i e^{i (a+b x)}\right )}{b^5}-\frac {24 i d^4 \operatorname {PolyLog}\left (4,i e^{i (a+b x)}\right )}{b^5}+\frac {(c+d x)^4 \sec (a+b x)}{b} \] Output:

8*I*d*(d*x+c)^3*arctan(exp(I*(b*x+a)))/b^2-12*I*d^2*(d*x+c)^2*polylog(2,-I 
*exp(I*(b*x+a)))/b^3+12*I*d^2*(d*x+c)^2*polylog(2,I*exp(I*(b*x+a)))/b^3+24 
*d^3*(d*x+c)*polylog(3,-I*exp(I*(b*x+a)))/b^4-24*d^3*(d*x+c)*polylog(3,I*e 
xp(I*(b*x+a)))/b^4+24*I*d^4*polylog(4,-I*exp(I*(b*x+a)))/b^5-24*I*d^4*poly 
log(4,I*exp(I*(b*x+a)))/b^5+(d*x+c)^4*sec(b*x+a)/b
 

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.89 \[ \int (c+d x)^4 \sec (a+b x) \tan (a+b x) \, dx=-\frac {4 d \left (-2 i b^3 c^3 \arctan \left (e^{i (a+b x)}\right )+3 b^3 c^2 d x \log \left (1-i e^{i (a+b x)}\right )+3 b^3 c d^2 x^2 \log \left (1-i e^{i (a+b x)}\right )+b^3 d^3 x^3 \log \left (1-i e^{i (a+b x)}\right )-3 b^3 c^2 d x \log \left (1+i e^{i (a+b x)}\right )-3 b^3 c d^2 x^2 \log \left (1+i e^{i (a+b x)}\right )-b^3 d^3 x^3 \log \left (1+i e^{i (a+b x)}\right )+3 i b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )-3 i b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )-6 b c d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )-6 b d^3 x \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )+6 b c d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )+6 b d^3 x \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )-6 i d^3 \operatorname {PolyLog}\left (4,-i e^{i (a+b x)}\right )+6 i d^3 \operatorname {PolyLog}\left (4,i e^{i (a+b x)}\right )\right )}{b^5}+\frac {(c+d x)^4 \sec (a+b x)}{b} \] Input:

Integrate[(c + d*x)^4*Sec[a + b*x]*Tan[a + b*x],x]
 

Output:

(-4*d*((-2*I)*b^3*c^3*ArcTan[E^(I*(a + b*x))] + 3*b^3*c^2*d*x*Log[1 - I*E^ 
(I*(a + b*x))] + 3*b^3*c*d^2*x^2*Log[1 - I*E^(I*(a + b*x))] + b^3*d^3*x^3* 
Log[1 - I*E^(I*(a + b*x))] - 3*b^3*c^2*d*x*Log[1 + I*E^(I*(a + b*x))] - 3* 
b^3*c*d^2*x^2*Log[1 + I*E^(I*(a + b*x))] - b^3*d^3*x^3*Log[1 + I*E^(I*(a + 
 b*x))] + (3*I)*b^2*d*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(a + b*x))] - (3*I) 
*b^2*d*(c + d*x)^2*PolyLog[2, I*E^(I*(a + b*x))] - 6*b*c*d^2*PolyLog[3, (- 
I)*E^(I*(a + b*x))] - 6*b*d^3*x*PolyLog[3, (-I)*E^(I*(a + b*x))] + 6*b*c*d 
^2*PolyLog[3, I*E^(I*(a + b*x))] + 6*b*d^3*x*PolyLog[3, I*E^(I*(a + b*x))] 
 - (6*I)*d^3*PolyLog[4, (-I)*E^(I*(a + b*x))] + (6*I)*d^3*PolyLog[4, I*E^( 
I*(a + b*x))]))/b^5 + ((c + d*x)^4*Sec[a + b*x])/b
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {4909, 3042, 4669, 3011, 7163, 2720, 7143}

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.

\(\displaystyle \int (c+d x)^4 \tan (a+b x) \sec (a+b x) \, dx\)

\(\Big \downarrow \) 4909

\(\displaystyle \frac {(c+d x)^4 \sec (a+b x)}{b}-\frac {4 d \int (c+d x)^3 \sec (a+b x)dx}{b}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

\(\displaystyle \frac {(c+d x)^4 \sec (a+b x)}{b}-\frac {4 d \left (-\frac {3 d \int (c+d x)^2 \log \left (1-i e^{i (a+b x)}\right )dx}{b}+\frac {3 d \int (c+d x)^2 \log \left (1+i e^{i (a+b x)}\right )dx}{b}-\frac {2 i (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {(c+d x)^4 \sec (a+b x)}{b}-\frac {4 d \left (\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b}-\frac {2 i d \int (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )dx}{b}\right )}{b}-\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b}-\frac {2 i d \int (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )dx}{b}\right )}{b}-\frac {2 i (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 7163

\(\displaystyle \frac {(c+d x)^4 \sec (a+b x)}{b}-\frac {4 d \left (\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b}-\frac {2 i d \left (\frac {i d \int \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )dx}{b}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b}\right )}{b}\right )}{b}-\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b}-\frac {2 i d \left (\frac {i d \int \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )dx}{b}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b}\right )}{b}\right )}{b}-\frac {2 i (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {(c+d x)^4 \sec (a+b x)}{b}-\frac {4 d \left (\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b}-\frac {2 i d \left (\frac {d \int e^{-i (a+b x)} \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )de^{i (a+b x)}}{b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b}\right )}{b}\right )}{b}-\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b}-\frac {2 i d \left (\frac {d \int e^{-i (a+b x)} \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )de^{i (a+b x)}}{b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b}\right )}{b}\right )}{b}-\frac {2 i (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {(c+d x)^4 \sec (a+b x)}{b}-\frac {4 d \left (-\frac {2 i (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b}+\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b}-\frac {2 i d \left (\frac {d \operatorname {PolyLog}\left (4,-i e^{i (a+b x)}\right )}{b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b}\right )}{b}\right )}{b}-\frac {3 d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b}-\frac {2 i d \left (\frac {d \operatorname {PolyLog}\left (4,i e^{i (a+b x)}\right )}{b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b}\right )}{b}\right )}{b}\right )}{b}\)

Input:

Int[(c + d*x)^4*Sec[a + b*x]*Tan[a + b*x],x]
 

Output:

(-4*d*(((-2*I)*(c + d*x)^3*ArcTan[E^(I*(a + b*x))])/b + (3*d*((I*(c + d*x) 
^2*PolyLog[2, (-I)*E^(I*(a + b*x))])/b - ((2*I)*d*(((-I)*(c + d*x)*PolyLog 
[3, (-I)*E^(I*(a + b*x))])/b + (d*PolyLog[4, (-I)*E^(I*(a + b*x))])/b^2))/ 
b))/b - (3*d*((I*(c + d*x)^2*PolyLog[2, I*E^(I*(a + b*x))])/b - ((2*I)*d*( 
((-I)*(c + d*x)*PolyLog[3, I*E^(I*(a + b*x))])/b + (d*PolyLog[4, I*E^(I*(a 
 + b*x))])/b^2))/b))/b))/b + ((c + d*x)^4*Sec[a + b*x])/b
 

Defintions of rubi rules used

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

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

rule 4669

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si 
mp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], 
 x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x 
))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 4909

Int[((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.)*Tan[(a_.) + (b 
_.)*(x_)]^(p_.), x_Symbol] :> Simp[(c + d*x)^m*(Sec[a + b*x]^n/(b*n)), x] - 
 Simp[d*(m/(b*n))   Int[(c + d*x)^(m - 1)*Sec[a + b*x]^n, x], x] /; FreeQ[{ 
a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

rule 7163

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. 
)*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a 
+ b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F]))   Int[(e + f*x) 
^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c 
, d, e, f, n, p}, x] && GtQ[m, 0]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 766 vs. \(2 (202 ) = 404\).

Time = 2.48 (sec) , antiderivative size = 767, normalized size of antiderivative = 3.38

method	result	size
risch	\(\frac {12 d^{3} c \ln \left (i {\mathrm e}^{i \left (b x +a \right )}+1\right ) x^{2}}{b^{2}}-\frac {12 d^{3} a^{2} c \ln \left (i {\mathrm e}^{i \left (b x +a \right )}+1\right )}{b^{4}}-\frac {12 d^{2} c^{2} \ln \left (1-i {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}-\frac {12 d^{2} c^{2} \ln \left (1-i {\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{3}}+\frac {12 i d^{4} \operatorname {polylog}\left (2, i {\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b^{3}}-\frac {12 i d^{2} c^{2} \operatorname {polylog}\left (2, -i {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {12 i d^{4} \operatorname {polylog}\left (2, -i {\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b^{3}}+\frac {8 i d \,c^{3} \arctan \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {12 d^{3} a^{2} c \ln \left (1-i {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {12 d^{2} c^{2} \ln \left (i {\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{b^{2}}+\frac {12 d^{2} c^{2} \ln \left (i {\mathrm e}^{i \left (b x +a \right )}+1\right ) a}{b^{3}}-\frac {12 d^{3} c \ln \left (1-i {\mathrm e}^{i \left (b x +a \right )}\right ) x^{2}}{b^{2}}+\frac {24 i d^{3} c \,a^{2} \arctan \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}-\frac {24 i d^{2} c^{2} a \arctan \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {2 \,{\mathrm e}^{i \left (b x +a \right )} \left (d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}\right )}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}+\frac {4 d^{4} a^{3} \ln \left (i {\mathrm e}^{i \left (b x +a \right )}+1\right )}{b^{5}}-\frac {4 d^{4} \ln \left (1-i {\mathrm e}^{i \left (b x +a \right )}\right ) x^{3}}{b^{2}}-\frac {24 d^{3} c \operatorname {polylog}\left (3, i {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}+\frac {24 d^{3} c \operatorname {polylog}\left (3, -i {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{4}}-\frac {24 d^{4} \operatorname {polylog}\left (3, i {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{4}}-\frac {4 d^{4} a^{3} \ln \left (1-i {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{5}}+\frac {4 d^{4} \ln \left (i {\mathrm e}^{i \left (b x +a \right )}+1\right ) x^{3}}{b^{2}}+\frac {24 d^{4} \operatorname {polylog}\left (3, -i {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{4}}+\frac {24 i d^{4} \operatorname {polylog}\left (4, -i {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{5}}-\frac {24 i d^{4} \operatorname {polylog}\left (4, i {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{5}}+\frac {24 i d^{3} c \operatorname {polylog}\left (2, i {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}-\frac {24 i d^{3} c \operatorname {polylog}\left (2, -i {\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{3}}-\frac {8 i d^{4} a^{3} \arctan \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{5}}+\frac {12 i d^{2} c^{2} \operatorname {polylog}\left (2, i {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}\)	\(767\)

Input:

int((d*x+c)^4*sec(b*x+a)*tan(b*x+a),x,method=_RETURNVERBOSE)
 

Output:

24*I/b^4*d^3*c*a^2*arctan(exp(I*(b*x+a)))-24*I/b^3*d^2*c^2*a*arctan(exp(I* 
(b*x+a)))+24*I/b^3*d^3*c*polylog(2,I*exp(I*(b*x+a)))*x-24*I/b^3*d^3*c*poly 
log(2,-I*exp(I*(b*x+a)))*x+2*exp(I*(b*x+a))*(d^4*x^4+4*c*d^3*x^3+6*c^2*d^2 
*x^2+4*c^3*d*x+c^4)/b/(exp(2*I*(b*x+a))+1)+4/b^5*d^4*a^3*ln(I*exp(I*(b*x+a 
))+1)-4/b^2*d^4*ln(1-I*exp(I*(b*x+a)))*x^3-24/b^4*d^3*c*polylog(3,I*exp(I* 
(b*x+a)))+24/b^4*d^3*c*polylog(3,-I*exp(I*(b*x+a)))-24/b^4*d^4*polylog(3,I 
*exp(I*(b*x+a)))*x-4/b^5*d^4*a^3*ln(1-I*exp(I*(b*x+a)))+4/b^2*d^4*ln(I*exp 
(I*(b*x+a))+1)*x^3+24/b^4*d^4*polylog(3,-I*exp(I*(b*x+a)))*x-8*I/b^5*d^4*a 
^3*arctan(exp(I*(b*x+a)))+12*I/b^3*d^2*c^2*polylog(2,I*exp(I*(b*x+a)))-12* 
I/b^3*d^4*polylog(2,-I*exp(I*(b*x+a)))*x^2+8*I/b^2*d*c^3*arctan(exp(I*(b*x 
+a)))+12/b^4*d^3*a^2*c*ln(1-I*exp(I*(b*x+a)))+12/b^2*d^2*c^2*ln(I*exp(I*(b 
*x+a))+1)*x+12/b^3*d^2*c^2*ln(I*exp(I*(b*x+a))+1)*a-12/b^2*d^3*c*ln(1-I*ex 
p(I*(b*x+a)))*x^2+12/b^2*d^3*c*ln(I*exp(I*(b*x+a))+1)*x^2-12/b^4*d^3*a^2*c 
*ln(I*exp(I*(b*x+a))+1)-12/b^2*d^2*c^2*ln(1-I*exp(I*(b*x+a)))*x-12/b^3*d^2 
*c^2*ln(1-I*exp(I*(b*x+a)))*a+12*I/b^3*d^4*polylog(2,I*exp(I*(b*x+a)))*x^2 
-12*I/b^3*d^2*c^2*polylog(2,-I*exp(I*(b*x+a)))+24*I*d^4*polylog(4,-I*exp(I 
*(b*x+a)))/b^5-24*I*d^4*polylog(4,I*exp(I*(b*x+a)))/b^5
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1190 vs. \(2 (189) = 378\).

Time = 0.13 (sec) , antiderivative size = 1190, normalized size of antiderivative = 5.24 \[ \int (c+d x)^4 \sec (a+b x) \tan (a+b x) \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^4*sec(b*x+a)*tan(b*x+a),x, algorithm="fricas")
 

Output:

(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^4*c^3*d*x + b^4*c 
^4 - 12*I*d^4*cos(b*x + a)*polylog(4, I*cos(b*x + a) + sin(b*x + a)) - 12* 
I*d^4*cos(b*x + a)*polylog(4, I*cos(b*x + a) - sin(b*x + a)) + 12*I*d^4*co 
s(b*x + a)*polylog(4, -I*cos(b*x + a) + sin(b*x + a)) + 12*I*d^4*cos(b*x + 
 a)*polylog(4, -I*cos(b*x + a) - sin(b*x + a)) - 6*(-I*b^2*d^4*x^2 - 2*I*b 
^2*c*d^3*x - I*b^2*c^2*d^2)*cos(b*x + a)*dilog(I*cos(b*x + a) + sin(b*x + 
a)) - 6*(-I*b^2*d^4*x^2 - 2*I*b^2*c*d^3*x - I*b^2*c^2*d^2)*cos(b*x + a)*di 
log(I*cos(b*x + a) - sin(b*x + a)) - 6*(I*b^2*d^4*x^2 + 2*I*b^2*c*d^3*x + 
I*b^2*c^2*d^2)*cos(b*x + a)*dilog(-I*cos(b*x + a) + sin(b*x + a)) - 6*(I*b 
^2*d^4*x^2 + 2*I*b^2*c*d^3*x + I*b^2*c^2*d^2)*cos(b*x + a)*dilog(-I*cos(b* 
x + a) - sin(b*x + a)) - 2*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - 
a^3*d^4)*cos(b*x + a)*log(cos(b*x + a) + I*sin(b*x + a) + I) + 2*(b^3*c^3* 
d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*cos(b*x + a)*log(cos(b*x + 
a) - I*sin(b*x + a) + I) - 2*(b^3*d^4*x^3 + 3*b^3*c*d^3*x^2 + 3*b^3*c^2*d^ 
2*x + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3 + a^3*d^4)*cos(b*x + a)*log(I*cos(b* 
x + a) + sin(b*x + a) + 1) + 2*(b^3*d^4*x^3 + 3*b^3*c*d^3*x^2 + 3*b^3*c^2* 
d^2*x + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3 + a^3*d^4)*cos(b*x + a)*log(I*cos( 
b*x + a) - sin(b*x + a) + 1) - 2*(b^3*d^4*x^3 + 3*b^3*c*d^3*x^2 + 3*b^3*c^ 
2*d^2*x + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3 + a^3*d^4)*cos(b*x + a)*log(-I*c 
os(b*x + a) + sin(b*x + a) + 1) + 2*(b^3*d^4*x^3 + 3*b^3*c*d^3*x^2 + 3*...
 

Sympy [F]

\[ \int (c+d x)^4 \sec (a+b x) \tan (a+b x) \, dx=\int \left (c + d x\right )^{4} \tan {\left (a + b x \right )} \sec {\left (a + b x \right )}\, dx \] Input:

integrate((d*x+c)**4*sec(b*x+a)*tan(b*x+a),x)
 

Output:

Integral((c + d*x)**4*tan(a + b*x)*sec(a + b*x), x)
 

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2944 vs. \(2 (189) = 378\).

Time = 0.58 (sec) , antiderivative size = 2944, normalized size of antiderivative = 12.97 \[ \int (c+d x)^4 \sec (a+b x) \tan (a+b x) \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^4*sec(b*x+a)*tan(b*x+a),x, algorithm="maxima")
 

Output:

(2*(4*(b*x + a)*cos(2*b*x + 2*a)*cos(b*x + a) + 4*(b*x + a)*sin(2*b*x + 2* 
a)*sin(b*x + a) + 4*(b*x + a)*cos(b*x + a) - (cos(2*b*x + 2*a)^2 + sin(2*b 
*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 
+ 2*sin(b*x + a) + 1) + (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2 
*b*x + 2*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*sin(b*x + a) + 1) 
)*c^3*d/((cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1 
)*b) - 6*(4*(b*x + a)*cos(2*b*x + 2*a)*cos(b*x + a) + 4*(b*x + a)*sin(2*b* 
x + 2*a)*sin(b*x + a) + 4*(b*x + a)*cos(b*x + a) - (cos(2*b*x + 2*a)^2 + s 
in(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + 
 a)^2 + 2*sin(b*x + a) + 1) + (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2 
*cos(2*b*x + 2*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*sin(b*x + a 
) + 1))*a*c^2*d^2/((cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x 
+ 2*a) + 1)*b^2) + 6*(4*(b*x + a)*cos(2*b*x + 2*a)*cos(b*x + a) + 4*(b*x + 
 a)*sin(2*b*x + 2*a)*sin(b*x + a) + 4*(b*x + a)*cos(b*x + a) - (cos(2*b*x 
+ 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1)*log(cos(b*x + a)^2 
 + sin(b*x + a)^2 + 2*sin(b*x + a) + 1) + (cos(2*b*x + 2*a)^2 + sin(2*b*x 
+ 2*a)^2 + 2*cos(2*b*x + 2*a) + 1)*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2 
*sin(b*x + a) + 1))*a^2*c*d^3/((cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 
2*cos(2*b*x + 2*a) + 1)*b^3) - 2*(4*(b*x + a)*cos(2*b*x + 2*a)*cos(b*x + a 
) + 4*(b*x + a)*sin(2*b*x + 2*a)*sin(b*x + a) + 4*(b*x + a)*cos(b*x + a...
 

Giac [F]

\[ \int (c+d x)^4 \sec (a+b x) \tan (a+b x) \, dx=\int { {\left (d x + c\right )}^{4} \sec \left (b x + a\right ) \tan \left (b x + a\right ) \,d x } \] Input:

integrate((d*x+c)^4*sec(b*x+a)*tan(b*x+a),x, algorithm="giac")
 

Output:

integrate((d*x + c)^4*sec(b*x + a)*tan(b*x + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^4 \sec (a+b x) \tan (a+b x) \, dx=\int \frac {\mathrm {tan}\left (a+b\,x\right )\,{\left (c+d\,x\right )}^4}{\cos \left (a+b\,x\right )} \,d x \] Input:

int((tan(a + b*x)*(c + d*x)^4)/cos(a + b*x),x)
 

Output:

int((tan(a + b*x)*(c + d*x)^4)/cos(a + b*x), x)
 

Reduce [F]

\[ \int (c+d x)^4 \sec (a+b x) \tan (a+b x) \, dx=\frac {8 \cos \left (b x +a \right ) \left (\int \frac {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} x^{3}}{\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}d x \right ) b \,d^{4}+24 \cos \left (b x +a \right ) \left (\int \frac {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} x^{2}}{\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}d x \right ) b c \,d^{3}+24 \cos \left (b x +a \right ) \left (\int \frac {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} x}{\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}d x \right ) b \,c^{2} d^{2}+4 \cos \left (b x +a \right ) \mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right ) c^{3} d -4 \cos \left (b x +a \right ) \mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right ) c^{3} d -\cos \left (b x +a \right ) b \,c^{4}-6 \cos \left (b x +a \right ) b \,c^{2} d^{2} x^{2}-4 \cos \left (b x +a \right ) b c \,d^{3} x^{3}-\cos \left (b x +a \right ) b \,d^{4} x^{4}+b \,c^{4}+4 b \,c^{3} d x +6 b \,c^{2} d^{2} x^{2}+4 b c \,d^{3} x^{3}+b \,d^{4} x^{4}}{\cos \left (b x +a \right ) b^{2}} \] Input:

int((d*x+c)^4*sec(b*x+a)*tan(b*x+a),x)
 

Output:

(8*cos(a + b*x)*int((tan((a + b*x)/2)**2*x**3)/(tan((a + b*x)/2)**2 - 1),x 
)*b*d**4 + 24*cos(a + b*x)*int((tan((a + b*x)/2)**2*x**2)/(tan((a + b*x)/2 
)**2 - 1),x)*b*c*d**3 + 24*cos(a + b*x)*int((tan((a + b*x)/2)**2*x)/(tan(( 
a + b*x)/2)**2 - 1),x)*b*c**2*d**2 + 4*cos(a + b*x)*log(tan((a + b*x)/2) - 
 1)*c**3*d - 4*cos(a + b*x)*log(tan((a + b*x)/2) + 1)*c**3*d - cos(a + b*x 
)*b*c**4 - 6*cos(a + b*x)*b*c**2*d**2*x**2 - 4*cos(a + b*x)*b*c*d**3*x**3 
- cos(a + b*x)*b*d**4*x**4 + b*c**4 + 4*b*c**3*d*x + 6*b*c**2*d**2*x**2 + 
4*b*c*d**3*x**3 + b*d**4*x**4)/(cos(a + b*x)*b**2)