\(\int (c+d x)^4 \sec ^2(a+b x) \tan (a+b x) \, dx\) [291]

Optimal result

Integrand size = 22, antiderivative size = 139 \[ \int (c+d x)^4 \sec ^2(a+b x) \tan (a+b x) \, dx=\frac {2 i d (c+d x)^3}{b^2}-\frac {6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {6 i d^3 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^4}-\frac {3 d^4 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^5}+\frac {(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2} \]

[Out]

2*I*d*(d*x+c)^3/b^2-6*d^2*(d*x+c)^2*ln(1+exp(2*I*(b*x+a)))/b^3+6*I*d^3*(d*x+c)*polylog(2,-exp(2*I*(b*x+a)))/b^
4-3*d^4*polylog(3,-exp(2*I*(b*x+a)))/b^5+1/2*(d*x+c)^4*sec(b*x+a)^2/b-2*d*(d*x+c)^3*tan(b*x+a)/b^2

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {4494, 4269, 3800, 2221, 2611, 2320, 6724} \[ \int (c+d x)^4 \sec ^2(a+b x) \tan (a+b x) \, dx=-\frac {3 d^4 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^5}+\frac {6 i d^3 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^4}-\frac {6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac {(c+d x)^4 \sec ^2(a+b x)}{2 b}+\frac {2 i d (c+d x)^3}{b^2} \]

[In]

Int[(c + d*x)^4*Sec[a + b*x]^2*Tan[a + b*x],x]

[Out]

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

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 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[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]

Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac {(2 d) \int (c+d x)^3 \sec ^2(a+b x) \, dx}{b} \\ & = \frac {(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac {\left (6 d^2\right ) \int (c+d x)^2 \tan (a+b x) \, dx}{b^2} \\ & = \frac {2 i d (c+d x)^3}{b^2}+\frac {(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}-\frac {\left (12 i d^2\right ) \int \frac {e^{2 i (a+b x)} (c+d x)^2}{1+e^{2 i (a+b x)}} \, dx}{b^2} \\ & = \frac {2 i d (c+d x)^3}{b^2}-\frac {6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac {\left (12 d^3\right ) \int (c+d x) \log \left (1+e^{2 i (a+b x)}\right ) \, dx}{b^3} \\ & = \frac {2 i d (c+d x)^3}{b^2}-\frac {6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {6 i d^3 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^4}+\frac {(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}-\frac {\left (6 i d^4\right ) \int \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right ) \, dx}{b^4} \\ & = \frac {2 i d (c+d x)^3}{b^2}-\frac {6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {6 i d^3 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^4}+\frac {(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}-\frac {\left (3 d^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{b^5} \\ & = \frac {2 i d (c+d x)^3}{b^2}-\frac {6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {6 i d^3 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^4}-\frac {3 d^4 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^5}+\frac {(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(418\) vs. \(2(139)=278\).

Time = 6.60 (sec) , antiderivative size = 418, normalized size of antiderivative = 3.01 \[ \int (c+d x)^4 \sec ^2(a+b x) \tan (a+b x) \, dx=-\frac {i d^4 e^{-i a} \left (2 b^2 x^2 \left (2 b x-3 i \left (1+e^{2 i a}\right ) \log \left (1+e^{-2 i (a+b x)}\right )\right )+6 b \left (1+e^{2 i a}\right ) x \operatorname {PolyLog}\left (2,-e^{-2 i (a+b x)}\right )-3 i \left (1+e^{2 i a}\right ) \operatorname {PolyLog}\left (3,-e^{-2 i (a+b x)}\right )\right ) \sec (a)}{2 b^5}+\frac {(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac {6 c^2 d^2 \sec (a) (\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x))+b x \sin (a))}{b^3 \left (\cos ^2(a)+\sin ^2(a)\right )}-\frac {6 c d^3 \csc (a) \left (b^2 e^{-i \arctan (\cot (a))} x^2-\frac {\cot (a) \left (i b x (-\pi -2 \arctan (\cot (a)))-\pi \log \left (1+e^{-2 i b x}\right )-2 (b x-\arctan (\cot (a))) \log \left (1-e^{2 i (b x-\arctan (\cot (a)))}\right )+\pi \log (\cos (b x))-2 \arctan (\cot (a)) \log (\sin (b x-\arctan (\cot (a))))+i \operatorname {PolyLog}\left (2,e^{2 i (b x-\arctan (\cot (a)))}\right )\right )}{\sqrt {1+\cot ^2(a)}}\right ) \sec (a)}{b^4 \sqrt {\csc ^2(a) \left (\cos ^2(a)+\sin ^2(a)\right )}}-\frac {2 \sec (a) \sec (a+b x) \left (c^3 d \sin (b x)+3 c^2 d^2 x \sin (b x)+3 c d^3 x^2 \sin (b x)+d^4 x^3 \sin (b x)\right )}{b^2} \]

[In]

Integrate[(c + d*x)^4*Sec[a + b*x]^2*Tan[a + b*x],x]

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 488 vs. \(2 (129 ) = 258\).

Time = 4.89 (sec) , antiderivative size = 489, normalized size of antiderivative = 3.52

method	result	size
risch	\(\frac {2 b \,d^{4} x^{4} {\mathrm e}^{2 i \left (x b +a \right )}+8 b c \,d^{3} x^{3} {\mathrm e}^{2 i \left (x b +a \right )}+12 b \,c^{2} d^{2} x^{2} {\mathrm e}^{2 i \left (x b +a \right )}+8 b \,c^{3} d x \,{\mathrm e}^{2 i \left (x b +a \right )}-4 i d^{4} x^{3} {\mathrm e}^{2 i \left (x b +a \right )}+2 b \,c^{4} {\mathrm e}^{2 i \left (x b +a \right )}-12 i c \,d^{3} x^{2} {\mathrm e}^{2 i \left (x b +a \right )}-12 i c^{2} d^{2} x \,{\mathrm e}^{2 i \left (x b +a \right )}-4 i d^{4} x^{3}-4 i c^{3} d \,{\mathrm e}^{2 i \left (x b +a \right )}-12 i c \,d^{3} x^{2}-12 i c^{2} d^{2} x -4 i c^{3} d}{b^{2} \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )^{2}}-\frac {12 i d^{4} a^{2} x}{b^{4}}+\frac {12 d^{4} a^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{5}}-\frac {6 d^{4} \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) x^{2}}{b^{3}}-\frac {3 d^{4} \operatorname {polylog}\left (3, -{\mathrm e}^{2 i \left (x b +a \right )}\right )}{b^{5}}+\frac {12 i d^{3} c \,a^{2}}{b^{4}}+\frac {12 i d^{3} c \,x^{2}}{b^{2}}+\frac {24 i d^{3} c x a}{b^{3}}+\frac {4 i d^{4} x^{3}}{b^{2}}-\frac {6 d^{2} c^{2} \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )}{b^{3}}+\frac {12 d^{2} c^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {12 d^{3} c \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) x}{b^{3}}-\frac {24 d^{3} c a \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}+\frac {6 i d^{4} \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (x b +a \right )}\right ) x}{b^{4}}+\frac {6 i d^{3} c \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (x b +a \right )}\right )}{b^{4}}-\frac {8 i d^{4} a^{3}}{b^{5}}\)	\(489\)

[In]

int((d*x+c)^4*sec(b*x+a)^2*tan(b*x+a),x,method=_RETURNVERBOSE)

[Out]

2*(b*d^4*x^4*exp(2*I*(b*x+a))+4*b*c*d^3*x^3*exp(2*I*(b*x+a))+6*b*c^2*d^2*x^2*exp(2*I*(b*x+a))+4*b*c^3*d*x*exp(
2*I*(b*x+a))-2*I*d^4*x^3*exp(2*I*(b*x+a))+b*c^4*exp(2*I*(b*x+a))-6*I*c*d^3*x^2*exp(2*I*(b*x+a))-6*I*c^2*d^2*x*
exp(2*I*(b*x+a))-2*I*d^4*x^3-2*I*c^3*d*exp(2*I*(b*x+a))-6*I*c*d^3*x^2-6*I*c^2*d^2*x-2*I*c^3*d)/b^2/(exp(2*I*(b
*x+a))+1)^2-12*I/b^4*d^4*a^2*x+12/b^5*d^4*a^2*ln(exp(I*(b*x+a)))-6/b^3*d^4*ln(exp(2*I*(b*x+a))+1)*x^2-3*d^4*po
lylog(3,-exp(2*I*(b*x+a)))/b^5+12*I/b^4*d^3*c*a^2+12*I/b^2*d^3*c*x^2+24*I/b^3*d^3*c*x*a+4*I/b^2*d^4*x^3-6/b^3*
d^2*c^2*ln(exp(2*I*(b*x+a))+1)+12/b^3*d^2*c^2*ln(exp(I*(b*x+a)))-12/b^3*d^3*c*ln(exp(2*I*(b*x+a))+1)*x-24/b^4*
d^3*c*a*ln(exp(I*(b*x+a)))+6*I/b^4*d^4*polylog(2,-exp(2*I*(b*x+a)))*x+6*I/b^4*d^3*c*polylog(2,-exp(2*I*(b*x+a)
))-8*I/b^5*d^4*a^3

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. 892 vs. \(2 (126) = 252\).

Time = 0.31 (sec) , antiderivative size = 892, normalized size of antiderivative = 6.42 \[ \int (c+d x)^4 \sec ^2(a+b x) \tan (a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^4*sec(b*x+a)^2*tan(b*x+a),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int (c+d x)^4 \sec ^2(a+b x) \tan (a+b x) \, dx=\int \left (c + d x\right )^{4} \tan {\left (a + b x \right )} \sec ^{2}{\left (a + b x \right )}\, dx \]

[In]

integrate((d*x+c)**4*sec(b*x+a)**2*tan(b*x+a),x)

[Out]

Integral((c + d*x)**4*tan(a + b*x)*sec(a + b*x)**2, 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. 3446 vs. \(2 (126) = 252\).

Time = 0.49 (sec) , antiderivative size = 3446, normalized size of antiderivative = 24.79 \[ \int (c+d x)^4 \sec ^2(a+b x) \tan (a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^4*sec(b*x+a)^2*tan(b*x+a),x, algorithm="maxima")

[Out]

1/2*(c^4*tan(b*x + a)^2 - 4*a*c^3*d*tan(b*x + a)^2/b + 6*a^2*c^2*d^2*tan(b*x + a)^2/b^2 - 4*a^3*c*d^3*tan(b*x
+ a)^2/b^3 + a^4*d^4*tan(b*x + a)^2/b^4 + 8*(4*(b*x + a)*cos(2*b*x + 2*a)^2 + 4*(b*x + a)*sin(2*b*x + 2*a)^2 +
(2*(b*x + a)*cos(2*b*x + 2*a) + sin(2*b*x + 2*a))*cos(4*b*x + 4*a) + 2*(b*x + a)*cos(2*b*x + 2*a) + (2*(b*x +
a)*sin(2*b*x + 2*a) - cos(2*b*x + 2*a) - 1)*sin(4*b*x + 4*a) - sin(2*b*x + 2*a))*c^3*d/((2*(2*cos(2*b*x + 2*a
) + 1)*cos(4*b*x + 4*a) + cos(4*b*x + 4*a)^2 + 4*cos(2*b*x + 2*a)^2 + sin(4*b*x + 4*a)^2 + 4*sin(4*b*x + 4*a)*
sin(2*b*x + 2*a) + 4*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x + 2*a) + 1)*b) - 24*(4*(b*x + a)*cos(2*b*x + 2*a)^2 + 4*
(b*x + a)*sin(2*b*x + 2*a)^2 + (2*(b*x + a)*cos(2*b*x + 2*a) + sin(2*b*x + 2*a))*cos(4*b*x + 4*a) + 2*(b*x + a
)*cos(2*b*x + 2*a) + (2*(b*x + a)*sin(2*b*x + 2*a) - cos(2*b*x + 2*a) - 1)*sin(4*b*x + 4*a) - sin(2*b*x + 2*a)
)*a*c^2*d^2/((2*(2*cos(2*b*x + 2*a) + 1)*cos(4*b*x + 4*a) + cos(4*b*x + 4*a)^2 + 4*cos(2*b*x + 2*a)^2 + sin(4*
b*x + 4*a)^2 + 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x + 2*a) + 1)*b^2) + 24*
(4*(b*x + a)*cos(2*b*x + 2*a)^2 + 4*(b*x + a)*sin(2*b*x + 2*a)^2 + (2*(b*x + a)*cos(2*b*x + 2*a) + sin(2*b*x +
2*a))*cos(4*b*x + 4*a) + 2*(b*x + a)*cos(2*b*x + 2*a) + (2*(b*x + a)*sin(2*b*x + 2*a) - cos(2*b*x + 2*a) - 1)
*sin(4*b*x + 4*a) - sin(2*b*x + 2*a))*a^2*c*d^3/((2*(2*cos(2*b*x + 2*a) + 1)*cos(4*b*x + 4*a) + cos(4*b*x + 4*
a)^2 + 4*cos(2*b*x + 2*a)^2 + sin(4*b*x + 4*a)^2 + 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*sin(2*b*x + 2*a)^2
+ 4*cos(2*b*x + 2*a) + 1)*b^3) - 8*(4*(b*x + a)*cos(2*b*x + 2*a)^2 + 4*(b*x + a)*sin(2*b*x + 2*a)^2 + (2*(b*x
+ a)*cos(2*b*x + 2*a) + sin(2*b*x + 2*a))*cos(4*b*x + 4*a) + 2*(b*x + a)*cos(2*b*x + 2*a) + (2*(b*x + a)*sin(2
*b*x + 2*a) - cos(2*b*x + 2*a) - 1)*sin(4*b*x + 4*a) - sin(2*b*x + 2*a))*a^3*d^4/((2*(2*cos(2*b*x + 2*a) + 1)*
cos(4*b*x + 4*a) + cos(4*b*x + 4*a)^2 + 4*cos(2*b*x + 2*a)^2 + sin(4*b*x + 4*a)^2 + 4*sin(4*b*x + 4*a)*sin(2*b
*x + 2*a) + 4*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x + 2*a) + 1)*b^4) + 6*(8*(b*x + a)^2*cos(2*b*x + 2*a)^2 + 8*(b*x
+ a)^2*sin(2*b*x + 2*a)^2 + 4*(b*x + a)^2*cos(2*b*x + 2*a) + 4*((b*x + a)^2*cos(2*b*x + 2*a) + (b*x + a)*sin(
2*b*x + 2*a))*cos(4*b*x + 4*a) - (2*(2*cos(2*b*x + 2*a) + 1)*cos(4*b*x + 4*a) + cos(4*b*x + 4*a)^2 + 4*cos(2*b
*x + 2*a)^2 + sin(4*b*x + 4*a)^2 + 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x +
2*a) + 1)*log(cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1) + 4*((b*x + a)^2*sin(2*b*x + 2
*a) - b*x - (b*x + a)*cos(2*b*x + 2*a) - a)*sin(4*b*x + 4*a) - 4*(b*x + a)*sin(2*b*x + 2*a))*c^2*d^2/((2*(2*co
s(2*b*x + 2*a) + 1)*cos(4*b*x + 4*a) + cos(4*b*x + 4*a)^2 + 4*cos(2*b*x + 2*a)^2 + sin(4*b*x + 4*a)^2 + 4*sin(
4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x + 2*a) + 1)*b^2) - 12*(8*(b*x + a)^2*cos(2*
b*x + 2*a)^2 + 8*(b*x + a)^2*sin(2*b*x + 2*a)^2 + 4*(b*x + a)^2*cos(2*b*x + 2*a) + 4*((b*x + a)^2*cos(2*b*x +
2*a) + (b*x + a)*sin(2*b*x + 2*a))*cos(4*b*x + 4*a) - (2*(2*cos(2*b*x + 2*a) + 1)*cos(4*b*x + 4*a) + cos(4*b*x
+ 4*a)^2 + 4*cos(2*b*x + 2*a)^2 + sin(4*b*x + 4*a)^2 + 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*sin(2*b*x + 2*
a)^2 + 4*cos(2*b*x + 2*a) + 1)*log(cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1) + 4*((b*x
+ a)^2*sin(2*b*x + 2*a) - b*x - (b*x + a)*cos(2*b*x + 2*a) - a)*sin(4*b*x + 4*a) - 4*(b*x + a)*sin(2*b*x + 2*
a))*a*c*d^3/((2*(2*cos(2*b*x + 2*a) + 1)*cos(4*b*x + 4*a) + cos(4*b*x + 4*a)^2 + 4*cos(2*b*x + 2*a)^2 + sin(4*
b*x + 4*a)^2 + 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x + 2*a) + 1)*b^3) + 6*(
8*(b*x + a)^2*cos(2*b*x + 2*a)^2 + 8*(b*x + a)^2*sin(2*b*x + 2*a)^2 + 4*(b*x + a)^2*cos(2*b*x + 2*a) + 4*((b*x
+ a)^2*cos(2*b*x + 2*a) + (b*x + a)*sin(2*b*x + 2*a))*cos(4*b*x + 4*a) - (2*(2*cos(2*b*x + 2*a) + 1)*cos(4*b*
x + 4*a) + cos(4*b*x + 4*a)^2 + 4*cos(2*b*x + 2*a)^2 + sin(4*b*x + 4*a)^2 + 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a
) + 4*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x + 2*a) + 1)*log(cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x +
2*a) + 1) + 4*((b*x + a)^2*sin(2*b*x + 2*a) - b*x - (b*x + a)*cos(2*b*x + 2*a) - a)*sin(4*b*x + 4*a) - 4*(b*x
+ a)*sin(2*b*x + 2*a))*a^2*d^4/((2*(2*cos(2*b*x + 2*a) + 1)*cos(4*b*x + 4*a) + cos(4*b*x + 4*a)^2 + 4*cos(2*b
*x + 2*a)^2 + sin(4*b*x + 4*a)^2 + 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x +
2*a) + 1)*b^4) - 2*(6*((b*x + a)^2*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a) + ((b*x + a)^2*d^4 + 2*(b*c*d^3 - a*d^4
)*(b*x + a))*cos(4*b*x + 4*a) + 2*((b*x + a)^2*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a))*cos(2*b*x + 2*a) - (-I*(b*
x + a)^2*d^4 + 2*(-I*b*c*d^3 + I*a*d^4)*(b*x + a))*sin(4*b*x + 4*a) - 2*(-I*(b*x + a)^2*d^4 + 2*(-I*b*c*d^3 +
I*a*d^4)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(sin(2*b*x + 2*a), cos(2*b*x + 2*a) + 1) - 4*((b*x + a)^3*d^4 + 3
*(b*c*d^3 - a*d^4)*(b*x + a)^2)*cos(4*b*x + 4*a) - 2*(-I*(b*x + a)^4*d^4 + 2*(-2*I*b*c*d^3 + (2*I*a + 1)*d^4)*
(b*x + a)^3 + 6*(b*c*d^3 - a*d^4)*(b*x + a)^2)*cos(2*b*x + 2*a) - 6*(b*c*d^3 + (b*x + a)*d^4 - a*d^4 + (b*c*d^
3 + (b*x + a)*d^4 - a*d^4)*cos(4*b*x + 4*a) + 2*(b*c*d^3 + (b*x + a)*d^4 - a*d^4)*cos(2*b*x + 2*a) + (I*b*c*d^
3 + I*(b*x + a)*d^4 - I*a*d^4)*sin(4*b*x + 4*a) + 2*(I*b*c*d^3 + I*(b*x + a)*d^4 - I*a*d^4)*sin(2*b*x + 2*a))*
dilog(-e^(2*I*b*x + 2*I*a)) - 3*(I*(b*x + a)^2*d^4 + 2*(I*b*c*d^3 - I*a*d^4)*(b*x + a) + (I*(b*x + a)^2*d^4 +
2*(I*b*c*d^3 - I*a*d^4)*(b*x + a))*cos(4*b*x + 4*a) + 2*(I*(b*x + a)^2*d^4 + 2*(I*b*c*d^3 - I*a*d^4)*(b*x + a)
)*cos(2*b*x + 2*a) - ((b*x + a)^2*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a))*sin(4*b*x + 4*a) - 2*((b*x + a)^2*d^4 +
2*(b*c*d^3 - a*d^4)*(b*x + a))*sin(2*b*x + 2*a))*log(cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x +
2*a) + 1) - 3*(I*d^4*cos(4*b*x + 4*a) + 2*I*d^4*cos(2*b*x + 2*a) - d^4*sin(4*b*x + 4*a) - 2*d^4*sin(2*b*x + 2*
a) + I*d^4)*polylog(3, -e^(2*I*b*x + 2*I*a)) - 4*(I*(b*x + a)^3*d^4 + 3*(I*b*c*d^3 - I*a*d^4)*(b*x + a)^2)*sin
(4*b*x + 4*a) - 2*((b*x + a)^4*d^4 + 2*(2*b*c*d^3 - (2*a - I)*d^4)*(b*x + a)^3 + 6*(I*b*c*d^3 - I*a*d^4)*(b*x
+ a)^2)*sin(2*b*x + 2*a))/(-I*b^4*cos(4*b*x + 4*a) - 2*I*b^4*cos(2*b*x + 2*a) + b^4*sin(4*b*x + 4*a) + 2*b^4*s
in(2*b*x + 2*a) - I*b^4))/b

Giac [F]

\[ \int (c+d x)^4 \sec ^2(a+b x) \tan (a+b x) \, dx=\int { {\left (d x + c\right )}^{4} \sec \left (b x + a\right )^{2} \tan \left (b x + a\right ) \,d x } \]

[In]

integrate((d*x+c)^4*sec(b*x+a)^2*tan(b*x+a),x, algorithm="giac")

[Out]

integrate((d*x + c)^4*sec(b*x + a)^2*tan(b*x + a), x)

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^4 \sec ^2(a+b x) \tan (a+b x) \, dx=\int \frac {\mathrm {tan}\left (a+b\,x\right )\,{\left (c+d\,x\right )}^4}{{\cos \left (a+b\,x\right )}^2} \,d x \]

[In]

int((tan(a + b*x)*(c + d*x)^4)/cos(a + b*x)^2,x)

[Out]

int((tan(a + b*x)*(c + d*x)^4)/cos(a + b*x)^2, x)