\(\int (c+d x)^4 \tan (a+b x) \, dx\) [209]

Optimal result

Integrand size = 14, antiderivative size = 158 \[ \int (c+d x)^4 \tan (a+b x) \, dx=\frac {i (c+d x)^5}{5 d}-\frac {(c+d x)^4 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^3}-\frac {3 i d^3 (c+d x) \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{b^4}+\frac {3 d^4 \operatorname {PolyLog}\left (5,-e^{2 i (a+b x)}\right )}{2 b^5} \]

[Out]

1/5*I*(d*x+c)^5/d-(d*x+c)^4*ln(1+exp(2*I*(b*x+a)))/b+2*I*d*(d*x+c)^3*polylog(2,-exp(2*I*(b*x+a)))/b^2-3*d^2*(d
*x+c)^2*polylog(3,-exp(2*I*(b*x+a)))/b^3-3*I*d^3*(d*x+c)*polylog(4,-exp(2*I*(b*x+a)))/b^4+3/2*d^4*polylog(5,-e
xp(2*I*(b*x+a)))/b^5

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3800, 2221, 2611, 6744, 2320, 6724} \[ \int (c+d x)^4 \tan (a+b x) \, dx=\frac {3 d^4 \operatorname {PolyLog}\left (5,-e^{2 i (a+b x)}\right )}{2 b^5}-\frac {3 i d^3 (c+d x) \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{b^4}-\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^3}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac {(c+d x)^4 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {i (c+d x)^5}{5 d} \]

[In]

Int[(c + d*x)^4*Tan[a + b*x],x]

[Out]

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

Rule 2221

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

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 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x
] - Dist[2*I, Int[(c + d*x)^m*(E^(2*I*(e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] &&
IGtQ[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 {i (c+d x)^5}{5 d}-2 i \int \frac {e^{2 i (a+b x)} (c+d x)^4}{1+e^{2 i (a+b x)}} \, dx \\ & = \frac {i (c+d x)^5}{5 d}-\frac {(c+d x)^4 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {(4 d) \int (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right ) \, dx}{b} \\ & = \frac {i (c+d x)^5}{5 d}-\frac {(c+d x)^4 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac {\left (6 i d^2\right ) \int (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right ) \, dx}{b^2} \\ & = \frac {i (c+d x)^5}{5 d}-\frac {(c+d x)^4 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^3}+\frac {\left (6 d^3\right ) \int (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right ) \, dx}{b^3} \\ & = \frac {i (c+d x)^5}{5 d}-\frac {(c+d x)^4 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^3}-\frac {3 i d^3 (c+d x) \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{b^4}+\frac {\left (3 i d^4\right ) \int \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right ) \, dx}{b^4} \\ & = \frac {i (c+d x)^5}{5 d}-\frac {(c+d x)^4 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^3}-\frac {3 i d^3 (c+d x) \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{b^4}+\frac {\left (3 d^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(4,-x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^5} \\ & = \frac {i (c+d x)^5}{5 d}-\frac {(c+d x)^4 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^3}-\frac {3 i d^3 (c+d x) \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{b^4}+\frac {3 d^4 \operatorname {PolyLog}\left (5,-e^{2 i (a+b x)}\right )}{2 b^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.99 \[ \int (c+d x)^4 \tan (a+b x) \, dx=\frac {i (c+d x)^5}{5 d}-\frac {(c+d x)^4 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac {3 d^2 \left (2 b^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )+d \left (2 i b (c+d x) \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )-d \operatorname {PolyLog}\left (5,-e^{2 i (a+b x)}\right )\right )\right )}{2 b^5} \]

[In]

Integrate[(c + d*x)^4*Tan[a + b*x],x]

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (141 ) = 282\).

Time = 1.24 (sec) , antiderivative size = 625, normalized size of antiderivative = 3.96

method	result	size
risch	\(\frac {4 i d \,c^{3} a^{2}}{b^{2}}+\frac {2 i d \,c^{3} \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (x b +a \right )}\right )}{b^{2}}-\frac {8 i d^{2} c^{2} a^{3}}{b^{3}}+2 i d \,c^{3} x^{2}-\frac {6 d^{3} c \operatorname {polylog}\left (3, -{\mathrm e}^{2 i \left (x b +a \right )}\right ) x}{b^{3}}-\frac {8 c \,d^{3} a^{3} \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}+i d^{3} c \,x^{4}-\frac {d^{4} \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) x^{4}}{b}+\frac {2 d^{4} a^{4} \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{5}}-\frac {2 i d^{4} a^{4} x}{b^{4}}-\frac {3 i d^{4} \operatorname {polylog}\left (4, -{\mathrm e}^{2 i \left (x b +a \right )}\right ) x}{b^{4}}+\frac {12 c^{2} d^{2} a^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {8 c^{3} d a \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}-\frac {6 d^{2} c^{2} \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) x^{2}}{b}-\frac {4 d^{3} c \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) x^{3}}{b}-\frac {4 d \,c^{3} \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) x}{b}+\frac {2 i d^{4} \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (x b +a \right )}\right ) x^{3}}{b^{2}}+\frac {6 i d^{3} c \,a^{4}}{b^{4}}-\frac {3 i d^{3} c \operatorname {polylog}\left (4, -{\mathrm e}^{2 i \left (x b +a \right )}\right )}{b^{4}}-\frac {3 d^{2} c^{2} \operatorname {polylog}\left (3, -{\mathrm e}^{2 i \left (x b +a \right )}\right )}{b^{3}}-\frac {3 d^{4} \operatorname {polylog}\left (3, -{\mathrm e}^{2 i \left (x b +a \right )}\right ) x^{2}}{b^{3}}-\frac {8 i d^{4} a^{5}}{5 b^{5}}+2 i d^{2} c^{2} x^{3}+\frac {6 i d^{2} c^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (x b +a \right )}\right ) x}{b^{2}}+\frac {8 i d \,c^{3} x a}{b}+\frac {8 i d^{3} c \,a^{3} x}{b^{3}}+\frac {6 i d^{3} c \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (x b +a \right )}\right ) x^{2}}{b^{2}}-\frac {12 i d^{2} c^{2} a^{2} x}{b^{2}}-i c^{4} x -\frac {i c^{5}}{5 d}-\frac {c^{4} \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )}{b}+\frac {2 c^{4} \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b}+\frac {i d^{4} x^{5}}{5}+\frac {3 d^{4} \operatorname {polylog}\left (5, -{\mathrm e}^{2 i \left (x b +a \right )}\right )}{2 b^{5}}\)	\(625\)

[In]

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

[Out]

6*I/b^2*d^2*c^2*polylog(2,-exp(2*I*(b*x+a)))*x+8*I/b*d*c^3*x*a+8*I/b^3*d^3*c*a^3*x+6*I/b^2*d^3*c*polylog(2,-ex
p(2*I*(b*x+a)))*x^2-12*I/b^2*d^2*c^2*a^2*x-I*c^4*x-1/5*I/d*c^5+I*d^3*c*x^4-1/b*c^4*ln(exp(2*I*(b*x+a))+1)+2/b*
c^4*ln(exp(I*(b*x+a)))+1/5*I*d^4*x^5+3/2*d^4*polylog(5,-exp(2*I*(b*x+a)))/b^5-1/b*d^4*ln(exp(2*I*(b*x+a))+1)*x
^4+2/b^5*d^4*a^4*ln(exp(I*(b*x+a)))+4*I/b^2*d*c^3*a^2+2*I/b^2*d*c^3*polylog(2,-exp(2*I*(b*x+a)))-8*I/b^3*d^2*c
^2*a^3-2*I/b^4*d^4*a^4*x-3*I/b^4*d^4*polylog(4,-exp(2*I*(b*x+a)))*x+12/b^3*c^2*d^2*a^2*ln(exp(I*(b*x+a)))-8/b^
2*c^3*d*a*ln(exp(I*(b*x+a)))-6/b*d^2*c^2*ln(exp(2*I*(b*x+a))+1)*x^2-4/b*d^3*c*ln(exp(2*I*(b*x+a))+1)*x^3-4/b*d
*c^3*ln(exp(2*I*(b*x+a))+1)*x+2*I/b^2*d^4*polylog(2,-exp(2*I*(b*x+a)))*x^3+6*I/b^4*d^3*c*a^4-3*I/b^4*d^3*c*pol
ylog(4,-exp(2*I*(b*x+a)))-3/b^3*d^2*c^2*polylog(3,-exp(2*I*(b*x+a)))-3/b^3*d^4*polylog(3,-exp(2*I*(b*x+a)))*x^
2-8/5*I/b^5*d^4*a^5+2*I*d^2*c^2*x^3+2*I*d*c^3*x^2-6/b^3*d^3*c*polylog(3,-exp(2*I*(b*x+a)))*x-8/b^4*c*d^3*a^3*l
n(exp(I*(b*x+a)))

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. 1410 vs. \(2 (137) = 274\).

Time = 0.31 (sec) , antiderivative size = 1410, normalized size of antiderivative = 8.92 \[ \int (c+d x)^4 \tan (a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*(24*d^4*polylog(5, I*cos(b*x + a) + sin(b*x + a)) + 24*d^4*polylog(5, I*cos(b*x + a) - sin(b*x + a)) + 24*
d^4*polylog(5, -I*cos(b*x + a) + sin(b*x + a)) + 24*d^4*polylog(5, -I*cos(b*x + a) - sin(b*x + a)) - 4*(I*b^3*
d^4*x^3 + 3*I*b^3*c*d^3*x^2 + 3*I*b^3*c^2*d^2*x + I*b^3*c^3*d)*dilog(I*cos(b*x + a) + sin(b*x + a)) - 4*(-I*b^
3*d^4*x^3 - 3*I*b^3*c*d^3*x^2 - 3*I*b^3*c^2*d^2*x - I*b^3*c^3*d)*dilog(I*cos(b*x + a) - sin(b*x + a)) - 4*(-I*
b^3*d^4*x^3 - 3*I*b^3*c*d^3*x^2 - 3*I*b^3*c^2*d^2*x - I*b^3*c^3*d)*dilog(-I*cos(b*x + a) + sin(b*x + a)) - 4*(
I*b^3*d^4*x^3 + 3*I*b^3*c*d^3*x^2 + 3*I*b^3*c^2*d^2*x + I*b^3*c^3*d)*dilog(-I*cos(b*x + a) - sin(b*x + a)) - (
b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*log(cos(b*x + a) + I*sin(b*x + a) + I)
- (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*log(cos(b*x + a) - I*sin(b*x + a) +
I) - (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 + 4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 +
4*a^3*b*c*d^3 - a^4*d^4)*log(I*cos(b*x + a) + sin(b*x + a) + 1) - (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 + 4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - a^4*d^4)*log(I*cos(b*x + a) - sin
(b*x + a) + 1) - (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 + 4*a*b^3*c^3*d - 6*a^2*b^
2*c^2*d^2 + 4*a^3*b*c*d^3 - a^4*d^4)*log(-I*cos(b*x + a) + sin(b*x + a) + 1) - (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 + 4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 - a^4*d^4)*log(-I*cos(
b*x + a) - sin(b*x + a) + 1) - (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*log(-co
s(b*x + a) + I*sin(b*x + a) + I) - (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*log
(-cos(b*x + a) - I*sin(b*x + a) + I) - 24*(-I*b*d^4*x - I*b*c*d^3)*polylog(4, I*cos(b*x + a) + sin(b*x + a)) -
24*(I*b*d^4*x + I*b*c*d^3)*polylog(4, I*cos(b*x + a) - sin(b*x + a)) - 24*(I*b*d^4*x + I*b*c*d^3)*polylog(4,
-I*cos(b*x + a) + sin(b*x + a)) - 24*(-I*b*d^4*x - I*b*c*d^3)*polylog(4, -I*cos(b*x + a) - sin(b*x + a)) - 12*
(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*polylog(3, I*cos(b*x + a) + sin(b*x + a)) - 12*(b^2*d^4*x^2 + 2*b^
2*c*d^3*x + b^2*c^2*d^2)*polylog(3, I*cos(b*x + a) - sin(b*x + a)) - 12*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2
*d^2)*polylog(3, -I*cos(b*x + a) + sin(b*x + a)) - 12*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2)*polylog(3, -
I*cos(b*x + a) - sin(b*x + a)))/b^5

Sympy [F]

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

[In]

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

[Out]

Integral((c + d*x)**4*sin(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. 803 vs. \(2 (137) = 274\).

Time = 0.41 (sec) , antiderivative size = 803, normalized size of antiderivative = 5.08 \[ \int (c+d x)^4 \tan (a+b x) \, dx=-\frac {15 \, c^{4} \log \left (-\sin \left (b x + a\right )^{2} + 1\right ) - \frac {60 \, a c^{3} d \log \left (-\sin \left (b x + a\right )^{2} + 1\right )}{b} + \frac {90 \, a^{2} c^{2} d^{2} \log \left (-\sin \left (b x + a\right )^{2} + 1\right )}{b^{2}} - \frac {60 \, a^{3} c d^{3} \log \left (-\sin \left (b x + a\right )^{2} + 1\right )}{b^{3}} + \frac {15 \, a^{4} d^{4} \log \left (-\sin \left (b x + a\right )^{2} + 1\right )}{b^{4}} + \frac {2 \, {\left (-3 i \, {\left (b x + a\right )}^{5} d^{4} - 15 \, {\left (i \, b c d^{3} - i \, a d^{4}\right )} {\left (b x + a\right )}^{4} - 45 \, d^{4} {\rm Li}_{5}(-e^{\left (2 i \, b x + 2 i \, a\right )}) - 30 \, {\left (i \, b^{2} c^{2} d^{2} - 2 i \, a b c d^{3} + i \, a^{2} d^{4}\right )} {\left (b x + a\right )}^{3} - 30 \, {\left (i \, b^{3} c^{3} d - 3 i \, a b^{2} c^{2} d^{2} + 3 i \, a^{2} b c d^{3} - i \, a^{3} d^{4}\right )} {\left (b x + a\right )}^{2} - 10 \, {\left (-3 i \, {\left (b x + a\right )}^{4} d^{4} + 8 \, {\left (-i \, b c d^{3} + i \, a d^{4}\right )} {\left (b x + a\right )}^{3} + 9 \, {\left (-i \, b^{2} c^{2} d^{2} + 2 i \, a b c d^{3} - i \, a^{2} d^{4}\right )} {\left (b x + a\right )}^{2} + 6 \, {\left (-i \, b^{3} c^{3} d + 3 i \, a b^{2} c^{2} d^{2} - 3 i \, a^{2} b c d^{3} + i \, a^{3} d^{4}\right )} {\left (b x + a\right )}\right )} \arctan \left (\sin \left (2 \, b x + 2 \, a\right ), \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) - 30 \, {\left (i \, b^{3} c^{3} d - 3 i \, a b^{2} c^{2} d^{2} + 3 i \, a^{2} b c d^{3} + 2 i \, {\left (b x + a\right )}^{3} d^{4} - i \, a^{3} d^{4} + 4 \, {\left (i \, b c d^{3} - i \, a d^{4}\right )} {\left (b x + a\right )}^{2} + 3 \, {\left (i \, b^{2} c^{2} d^{2} - 2 i \, a b c d^{3} + i \, a^{2} d^{4}\right )} {\left (b x + a\right )}\right )} {\rm Li}_2\left (-e^{\left (2 i \, b x + 2 i \, a\right )}\right ) + 5 \, {\left (3 \, {\left (b x + a\right )}^{4} d^{4} + 8 \, {\left (b c d^{3} - a d^{4}\right )} {\left (b x + a\right )}^{3} + 9 \, {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} {\left (b x + a\right )}^{2} + 6 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} {\left (b x + a\right )}\right )} \log \left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) - 30 \, {\left (-2 i \, b c d^{3} - 3 i \, {\left (b x + a\right )} d^{4} + 2 i \, a d^{4}\right )} {\rm Li}_{4}(-e^{\left (2 i \, b x + 2 i \, a\right )}) + 15 \, {\left (3 \, b^{2} c^{2} d^{2} - 6 \, a b c d^{3} + 6 \, {\left (b x + a\right )}^{2} d^{4} + 3 \, a^{2} d^{4} + 8 \, {\left (b c d^{3} - a d^{4}\right )} {\left (b x + a\right )}\right )} {\rm Li}_{3}(-e^{\left (2 i \, b x + 2 i \, a\right )})\right )}}{b^{4}}}{30 \, b} \]

[In]

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

[Out]

-1/30*(15*c^4*log(-sin(b*x + a)^2 + 1) - 60*a*c^3*d*log(-sin(b*x + a)^2 + 1)/b + 90*a^2*c^2*d^2*log(-sin(b*x +
a)^2 + 1)/b^2 - 60*a^3*c*d^3*log(-sin(b*x + a)^2 + 1)/b^3 + 15*a^4*d^4*log(-sin(b*x + a)^2 + 1)/b^4 + 2*(-3*I
*(b*x + a)^5*d^4 - 15*(I*b*c*d^3 - I*a*d^4)*(b*x + a)^4 - 45*d^4*polylog(5, -e^(2*I*b*x + 2*I*a)) - 30*(I*b^2*
c^2*d^2 - 2*I*a*b*c*d^3 + I*a^2*d^4)*(b*x + a)^3 - 30*(I*b^3*c^3*d - 3*I*a*b^2*c^2*d^2 + 3*I*a^2*b*c*d^3 - I*a
^3*d^4)*(b*x + a)^2 - 10*(-3*I*(b*x + a)^4*d^4 + 8*(-I*b*c*d^3 + I*a*d^4)*(b*x + a)^3 + 9*(-I*b^2*c^2*d^2 + 2*
I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a)^2 + 6*(-I*b^3*c^3*d + 3*I*a*b^2*c^2*d^2 - 3*I*a^2*b*c*d^3 + I*a^3*d^4)*(b*x
+ a))*arctan2(sin(2*b*x + 2*a), cos(2*b*x + 2*a) + 1) - 30*(I*b^3*c^3*d - 3*I*a*b^2*c^2*d^2 + 3*I*a^2*b*c*d^3
+ 2*I*(b*x + a)^3*d^4 - I*a^3*d^4 + 4*(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))*dilog(-e^(2*I*b*x + 2*I*a)) + 5*(3*(b*x + a)^4*d^4 + 8*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 9
*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 6*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)
*(b*x + a))*log(cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1) - 30*(-2*I*b*c*d^3 - 3*I*(b*
x + a)*d^4 + 2*I*a*d^4)*polylog(4, -e^(2*I*b*x + 2*I*a)) + 15*(3*b^2*c^2*d^2 - 6*a*b*c*d^3 + 6*(b*x + a)^2*d^4
+ 3*a^2*d^4 + 8*(b*c*d^3 - a*d^4)*(b*x + a))*polylog(3, -e^(2*I*b*x + 2*I*a)))/b^4)/b

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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