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

Optimal result

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

[Out]

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
*exp(I*(b*x+a)))/b^4+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+(d*x+c
)^4*sec(b*x+a)/b

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4494, 4266, 2611, 6744, 2320, 6724} \[ \int (c+d x)^4 \sec (a+b x) \tan (a+b x) \, dx=\frac {8 i d (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b^2}+\frac {24 i d^4 \operatorname {PolyLog}\left (4,-i e^{i (a+b x)}\right )}{b^5}-\frac {24 i d^4 \operatorname {PolyLog}\left (4,i e^{i (a+b x)}\right )}{b^5}+\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac {24 d^3 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}-\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {12 i d^2 (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}+\frac {(c+d x)^4 \sec (a+b x)}{b} \]

[In]

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

[Out]

((8*I)*d*(c + d*x)^3*ArcTan[E^(I*(a + b*x))])/b^2 - ((12*I)*d^2*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(a + b*x))])/
b^3 + ((12*I)*d^2*(c + d*x)^2*PolyLog[2, I*E^(I*(a + b*x))])/b^3 + (24*d^3*(c + d*x)*PolyLog[3, (-I)*E^(I*(a +
b*x))])/b^4 - (24*d^3*(c + d*x)*PolyLog[3, I*E^(I*(a + b*x))])/b^4 + ((24*I)*d^4*PolyLog[4, (-I)*E^(I*(a + b*
x))])/b^5 - ((24*I)*d^4*PolyLog[4, I*E^(I*(a + b*x))])/b^5 + ((c + d*x)^4*Sec[a + b*x])/b

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{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 2611

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] + Dist[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 4266

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] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[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 4494

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] - Dist[d*(m/(b*n)), Int[(c + d*x)^(m - 1)*Sec[a + b*x]^n, x], x] /; Fre
eQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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 6744

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] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^4 \sec (a+b x)}{b}-\frac {(4 d) \int (c+d x)^3 \sec (a+b x) \, dx}{b} \\ & = \frac {8 i d (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b^2}+\frac {(c+d x)^4 \sec (a+b x)}{b}+\frac {\left (12 d^2\right ) \int (c+d x)^2 \log \left (1-i e^{i (a+b x)}\right ) \, dx}{b^2}-\frac {\left (12 d^2\right ) \int (c+d x)^2 \log \left (1+i e^{i (a+b x)}\right ) \, dx}{b^2} \\ & = \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 {(c+d x)^4 \sec (a+b x)}{b}+\frac {\left (24 i d^3\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right ) \, dx}{b^3}-\frac {\left (24 i d^3\right ) \int (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right ) \, dx}{b^3} \\ & = \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 {(c+d x)^4 \sec (a+b x)}{b}-\frac {\left (24 d^4\right ) \int \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right ) \, dx}{b^4}+\frac {\left (24 d^4\right ) \int \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right ) \, dx}{b^4} \\ & = \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 {(c+d x)^4 \sec (a+b x)}{b}+\frac {\left (24 i d^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^5}-\frac {\left (24 i d^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^5} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.15 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.89 \[ \int (c+d x)^4 \sec (a+b x) \tan (a+b x) \, dx=-\frac {4 d \left (-2 i b^3 c^3 \arctan \left (e^{i (a+b x)}\right )+3 b^3 c^2 d x \log \left (1-i e^{i (a+b x)}\right )+3 b^3 c d^2 x^2 \log \left (1-i e^{i (a+b x)}\right )+b^3 d^3 x^3 \log \left (1-i e^{i (a+b x)}\right )-3 b^3 c^2 d x \log \left (1+i e^{i (a+b x)}\right )-3 b^3 c d^2 x^2 \log \left (1+i e^{i (a+b x)}\right )-b^3 d^3 x^3 \log \left (1+i e^{i (a+b x)}\right )+3 i b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )-3 i b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )-6 b c d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )-6 b d^3 x \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )+6 b c d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )+6 b d^3 x \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )-6 i d^3 \operatorname {PolyLog}\left (4,-i e^{i (a+b x)}\right )+6 i d^3 \operatorname {PolyLog}\left (4,i e^{i (a+b x)}\right )\right )}{b^5}+\frac {(c+d x)^4 \sec (a+b x)}{b} \]

[In]

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

[Out]

(-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

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 = 4.67 (sec) , antiderivative size = 767, normalized size of antiderivative = 3.38

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

[In]

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

[Out]

12*I/b^3*d^2*c^2*polylog(2,I*exp(I*(b*x+a)))-12*I/b^3*d^2*c^2*polylog(2,-I*exp(I*(b*x+a)))+8*I/b^2*d*c^3*arcta
n(exp(I*(b*x+a)))-8*I/b^5*d^4*a^3*arctan(exp(I*(b*x+a)))-12*I/b^3*d^4*polylog(2,-I*exp(I*(b*x+a)))*x^2+12*I/b^
3*d^4*polylog(2,I*exp(I*(b*x+a)))*x^2-24*I*d^4*polylog(4,I*exp(I*(b*x+a)))/b^5+24/b^4*d^3*c*polylog(3,-I*exp(I
*(b*x+a)))+4/b^5*d^4*a^3*ln(1+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)))+24/b^4*d^4*polylog(3,-I*exp(I*(b*x+a)))*x-24/b^4*d^3*c*polylog(3,I*exp(I*(b*x+a)))+4/b^2*d^4*l
n(1+I*exp(I*(b*x+a)))*x^3-4/b^2*d^4*ln(1-I*exp(I*(b*x+a)))*x^3+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*polylog(2,I*
exp(I*(b*x+a)))*x-12/b^3*d^2*c^2*ln(1-I*exp(I*(b*x+a)))*a+12/b^2*d^3*c*ln(1+I*exp(I*(b*x+a)))*x^2-12/b^2*d^3*c
*ln(1-I*exp(I*(b*x+a)))*x^2+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
/b^4*d^3*a^2*c*ln(1+I*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(1-I*exp(I*(b*x
+a)))*x+24*I*d^4*polylog(4,-I*exp(I*(b*x+a)))/b^5+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)

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 = 0.31 (sec) , antiderivative size = 1190, normalized size of antiderivative = 5.24 \[ \int (c+d x)^4 \sec (a+b x) \tan (a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

(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*
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) - s
in(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
)) - 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)*lo
g(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*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*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*c
os(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) + 12*(b*d^4*x + b*c*d^3)*cos(b*x + a)*polylog(3, I*cos(
b*x + a) + sin(b*x + a)) - 12*(b*d^4*x + b*c*d^3)*cos(b*x + a)*polylog(3, I*cos(b*x + a) - sin(b*x + a)) + 12*
(b*d^4*x + b*c*d^3)*cos(b*x + a)*polylog(3, -I*cos(b*x + a) + sin(b*x + a)) - 12*(b*d^4*x + b*c*d^3)*cos(b*x +
a)*polylog(3, -I*cos(b*x + a) - sin(b*x + a)))/(b^5*cos(b*x + a))

Sympy [F]

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

[In]

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

[Out]

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 = 0.59 (sec) , antiderivative size = 2944, normalized size of antiderivative = 12.97 \[ \int (c+d x)^4 \sec (a+b x) \tan (a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

(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 + 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*c^2*d^2/((cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*co
s(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) - (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^3*d^4/((cos(2*b*x + 2*a)
^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1)*b^4) + c^4/cos(b*x + a) - 4*a*c^3*d/(b*cos(b*x + a)) + 6*a^2
*c^2*d^2/(b^2*cos(b*x + a)) - 4*a^3*c*d^3/(b^3*cos(b*x + a)) + a^4*d^4/(b^4*cos(b*x + a)) + 2*(2*((b*x + a)^3*
d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a) + ((b*x + a)^3*d^4 +
3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*cos(2*b*x + 2*a) + (I*(b
*x + a)^3*d^4 + 3*(I*b*c*d^3 - I*a*d^4)*(b*x + a)^2 + 3*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*a^2*d^4)*(b*x + a))
*sin(2*b*x + 2*a))*arctan2(cos(b*x + a), sin(b*x + a) + 1) + 2*((b*x + a)^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a
)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a) + ((b*x + a)^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 +
3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*cos(2*b*x + 2*a) + (I*(b*x + a)^3*d^4 + 3*(I*b*c*d^3 - I*a
*d^4)*(b*x + a)^2 + 3*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*a^2*d^4)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(cos(b*x
+ a), -sin(b*x + a) + 1) + (-I*(b*x + a)^4*d^4 + 4*(-I*b*c*d^3 + I*a*d^4)*(b*x + a)^3 + 6*(-I*b^2*c^2*d^2 + 2
*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a)^2)*cos(b*x + a) + 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*d^4 + a^2*d^4
+ 2*(b*c*d^3 - a*d^4)*(b*x + a) + (b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*d^4 + a^2*d^4 + 2*(b*c*d^3 - a*d^4
)*(b*x + a))*cos(2*b*x + 2*a) + (I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*(b*x + a)^2*d^4 + I*a^2*d^4 + 2*(I*b*c*d^3
- I*a*d^4)*(b*x + a))*sin(2*b*x + 2*a))*dilog(I*e^(I*b*x + I*a)) - 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*
d^4 + a^2*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a) + (b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*d^4 + a^2*d^4 + 2*(b*
c*d^3 - a*d^4)*(b*x + a))*cos(2*b*x + 2*a) - (-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*(b*x + a)^2*d^4 - I*a^2*d^4 +
2*(-I*b*c*d^3 + I*a*d^4)*(b*x + a))*sin(2*b*x + 2*a))*dilog(-I*e^(I*b*x + I*a)) + (I*(b*x + a)^3*d^4 + 3*(I*b
*c*d^3 - I*a*d^4)*(b*x + a)^2 + 3*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*a^2*d^4)*(b*x + a) + (I*(b*x + a)^3*d^4 +
3*(I*b*c*d^3 - I*a*d^4)*(b*x + a)^2 + 3*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*a^2*d^4)*(b*x + a))*cos(2*b*x + 2*
a) - ((b*x + a)^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*s
in(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*sin(b*x + a) + 1) + (-I*(b*x + a)^3*d^4 + 3*(-I*b*c*d
^3 + I*a*d^4)*(b*x + a)^2 + 3*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a) + (-I*(b*x + a)^3*d^4 + 3
*(-I*b*c*d^3 + I*a*d^4)*(b*x + a)^2 + 3*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a))*cos(2*b*x + 2*
a) + ((b*x + a)^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*s
in(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*sin(b*x + a) + 1) - 12*(d^4*cos(2*b*x + 2*a) + I*d^4*
sin(2*b*x + 2*a) + d^4)*polylog(4, I*e^(I*b*x + I*a)) + 12*(d^4*cos(2*b*x + 2*a) + I*d^4*sin(2*b*x + 2*a) + d^
4)*polylog(4, -I*e^(I*b*x + I*a)) + 12*(I*b*c*d^3 + I*(b*x + a)*d^4 - I*a*d^4 + (I*b*c*d^3 + I*(b*x + a)*d^4 -
I*a*d^4)*cos(2*b*x + 2*a) - (b*c*d^3 + (b*x + a)*d^4 - a*d^4)*sin(2*b*x + 2*a))*polylog(3, I*e^(I*b*x + I*a))
+ 12*(-I*b*c*d^3 - I*(b*x + a)*d^4 + I*a*d^4 + (-I*b*c*d^3 - I*(b*x + a)*d^4 + I*a*d^4)*cos(2*b*x + 2*a) + (b
*c*d^3 + (b*x + a)*d^4 - a*d^4)*sin(2*b*x + 2*a))*polylog(3, -I*e^(I*b*x + I*a)) + ((b*x + a)^4*d^4 + 4*(b*c*d
^3 - a*d^4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2)*sin(b*x + a))/(-I*b^4*cos(2*b*x
+ 2*a) + b^4*sin(2*b*x + 2*a) - I*b^4))/b

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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