Integrand size = 22, antiderivative size = 139 \[ \int (c+d x)^4 \sec ^2(a+b x) \tan (a+b x) \, dx=\frac {2 i d (c+d x)^3}{b^2}-\frac {6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {6 i d^3 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^4}-\frac {3 d^4 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^5}+\frac {(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2} \]
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
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 \[ \int (c+d x)^4 \sec ^2(a+b x) \tan (a+b x) \, dx=-\frac {i d^4 e^{-i a} \left (2 b^2 x^2 \left (2 b x-3 i \left (1+e^{2 i a}\right ) \log \left (1+e^{-2 i (a+b x)}\right )\right )+6 b \left (1+e^{2 i a}\right ) x \operatorname {PolyLog}\left (2,-e^{-2 i (a+b x)}\right )-3 i \left (1+e^{2 i a}\right ) \operatorname {PolyLog}\left (3,-e^{-2 i (a+b x)}\right )\right ) \sec (a)}{2 b^5}+\frac {(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac {6 c^2 d^2 \sec (a) (\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x))+b x \sin (a))}{b^3 \left (\cos ^2(a)+\sin ^2(a)\right )}-\frac {6 c d^3 \csc (a) \left (b^2 e^{-i \arctan (\cot (a))} x^2-\frac {\cot (a) \left (i b x (-\pi -2 \arctan (\cot (a)))-\pi \log \left (1+e^{-2 i b x}\right )-2 (b x-\arctan (\cot (a))) \log \left (1-e^{2 i (b x-\arctan (\cot (a)))}\right )+\pi \log (\cos (b x))-2 \arctan (\cot (a)) \log (\sin (b x-\arctan (\cot (a))))+i \operatorname {PolyLog}\left (2,e^{2 i (b x-\arctan (\cot (a)))}\right )\right )}{\sqrt {1+\cot ^2(a)}}\right ) \sec (a)}{b^4 \sqrt {\csc ^2(a) \left (\cos ^2(a)+\sin ^2(a)\right )}}-\frac {2 \sec (a) \sec (a+b x) \left (c^3 d \sin (b x)+3 c^2 d^2 x \sin (b x)+3 c d^3 x^2 \sin (b x)+d^4 x^3 \sin (b x)\right )}{b^2} \]
((-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 + Si n[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
Time = 0.78 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.19, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {4909, 3042, 4672, 25, 3042, 4202, 2620, 3011, 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.
| \(\displaystyle \int (c+d x)^4 \tan (a+b x) \sec ^2(a+b x) \, dx\) |
| \(\Big \downarrow \) 4909 |
| \(\displaystyle \frac {(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac {2 d \int (c+d x)^3 \sec ^2(a+b x)dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac {2 d \int (c+d x)^3 \csc \left (a+b x+\frac {\pi }{2}\right )^2dx}{b}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle \frac {(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac {2 d \left (\frac {3 d \int -(c+d x)^2 \tan (a+b x)dx}{b}+\frac {(c+d x)^3 \tan (a+b x)}{b}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac {2 d \left (\frac {(c+d x)^3 \tan (a+b x)}{b}-\frac {3 d \int (c+d x)^2 \tan (a+b x)dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac {2 d \left (\frac {(c+d x)^3 \tan (a+b x)}{b}-\frac {3 d \int (c+d x)^2 \tan (a+b x)dx}{b}\right )}{b}\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle \frac {(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac {2 d \left (\frac {(c+d x)^3 \tan (a+b x)}{b}-\frac {3 d \left (\frac {i (c+d x)^3}{3 d}-2 i \int \frac {e^{2 i (a+b x)} (c+d x)^2}{1+e^{2 i (a+b x)}}dx\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac {2 d \left (\frac {(c+d x)^3 \tan (a+b x)}{b}-\frac {3 d \left (\frac {i (c+d x)^3}{3 d}-2 i \left (\frac {i d \int (c+d x) \log \left (1+e^{2 i (a+b x)}\right )dx}{b}-\frac {i (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac {2 d \left (\frac {(c+d x)^3 \tan (a+b x)}{b}-\frac {3 d \left (\frac {i (c+d x)^3}{3 d}-2 i \left (\frac {i d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \int \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )dx}{2 b}\right )}{b}-\frac {i (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac {2 d \left (\frac {(c+d x)^3 \tan (a+b x)}{b}-\frac {3 d \left (\frac {i (c+d x)^3}{3 d}-2 i \left (\frac {i d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {d \int e^{-2 i (a+b x)} \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )de^{2 i (a+b x)}}{4 b^2}\right )}{b}-\frac {i (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {(c+d x)^4 \sec ^2(a+b x)}{2 b}-\frac {2 d \left (\frac {(c+d x)^3 \tan (a+b x)}{b}-\frac {3 d \left (\frac {i (c+d x)^3}{3 d}-2 i \left (\frac {i d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {d \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{4 b^2}\right )}{b}-\frac {i (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )\right )}{b}\right )}{b}\) |
((c + d*x)^4*Sec[a + b*x]^2)/(2*b) - (2*d*((-3*d*(((I/3)*(c + d*x)^3)/d - (2*I)*(((-1/2*I)*(c + d*x)^2*Log[1 + E^((2*I)*(a + b*x))])/b + (I*d*(((I/2 )*(c + d*x)*PolyLog[2, -E^((2*I)*(a + b*x))])/b - (d*PolyLog[3, -E^((2*I)* (a + b*x))])/(4*b^2)))/b)))/b + ((c + d*x)^3*Tan[a + b*x])/b))/b
3.3.91.3.1 Defintions of rubi rules used
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] - Si mp[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]
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]]]
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]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I *((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[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] && IGt Q[m, 0]
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1) *Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
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]
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]
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 | \(\frac {2 b \,d^{4} x^{4} {\mathrm e}^{2 i \left (x b +a \right )}+8 b c \,d^{3} x^{3} {\mathrm e}^{2 i \left (x b +a \right )}+12 b \,c^{2} d^{2} x^{2} {\mathrm e}^{2 i \left (x b +a \right )}+8 b \,c^{3} d x \,{\mathrm e}^{2 i \left (x b +a \right )}-4 i d^{4} x^{3} {\mathrm e}^{2 i \left (x b +a \right )}+2 b \,c^{4} {\mathrm e}^{2 i \left (x b +a \right )}-12 i c \,d^{3} x^{2} {\mathrm e}^{2 i \left (x b +a \right )}-12 i c^{2} d^{2} x \,{\mathrm e}^{2 i \left (x b +a \right )}-4 i d^{4} x^{3}-4 i c^{3} d \,{\mathrm e}^{2 i \left (x b +a \right )}-12 i c \,d^{3} x^{2}-12 i c^{2} d^{2} x -4 i c^{3} d}{b^{2} \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )^{2}}-\frac {12 i d^{4} a^{2} x}{b^{4}}+\frac {12 d^{4} a^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{5}}-\frac {6 d^{4} \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) x^{2}}{b^{3}}-\frac {3 d^{4} \operatorname {polylog}\left (3, -{\mathrm e}^{2 i \left (x b +a \right )}\right )}{b^{5}}+\frac {12 i d^{3} c \,a^{2}}{b^{4}}+\frac {12 i d^{3} c \,x^{2}}{b^{2}}+\frac {24 i d^{3} c x a}{b^{3}}+\frac {4 i d^{4} x^{3}}{b^{2}}-\frac {6 d^{2} c^{2} \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )}{b^{3}}+\frac {12 d^{2} c^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {12 d^{3} c \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) x}{b^{3}}-\frac {24 d^{3} c a \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}+\frac {6 i d^{4} \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (x b +a \right )}\right ) x}{b^{4}}+\frac {6 i d^{3} c \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (x b +a \right )}\right )}{b^{4}}-\frac {8 i d^{4} a^{3}}{b^{5}}\) | \(489\) |
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*ex p(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*polylo g(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
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 \[ \int (c+d x)^4 \sec ^2(a+b x) \tan (a+b x) \, dx=\text {Too large to display} \]
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*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*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)*c os(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) + s in(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(-c...
\[ \int (c+d x)^4 \sec ^2(a+b x) \tan (a+b x) \, dx=\int \left (c + d x\right )^{4} \tan {\left (a + b x \right )} \sec ^{2}{\left (a + b x \right )}\, dx \]
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 \[ \int (c+d x)^4 \sec ^2(a+b x) \tan (a+b x) \, dx=\text {Too large to display} \]
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)^...
\[ \int (c+d x)^4 \sec ^2(a+b x) \tan (a+b x) \, dx=\int { {\left (d x + c\right )}^{4} \sec \left (b x + a\right )^{2} \tan \left (b x + a\right ) \,d x } \]
Timed out. \[ \int (c+d x)^4 \sec ^2(a+b x) \tan (a+b x) \, dx=\int \frac {\mathrm {tan}\left (a+b\,x\right )\,{\left (c+d\,x\right )}^4}{{\cos \left (a+b\,x\right )}^2} \,d x \]