Integrand size = 20, antiderivative size = 117 \[ \int (c+d x) \sec (a+b x) \tan ^2(a+b x) \, dx=\frac {i (c+d x) \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {i d \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{2 b^2}+\frac {i d \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{2 b^2}-\frac {d \sec (a+b x)}{2 b^2}+\frac {(c+d x) \sec (a+b x) \tan (a+b x)}{2 b} \]
I*(d*x+c)*arctan(exp(I*(b*x+a)))/b-1/2*I*d*polylog(2,-I*exp(I*(b*x+a)))/b^ 2+1/2*I*d*polylog(2,I*exp(I*(b*x+a)))/b^2-1/2*d*sec(b*x+a)/b^2+1/2*(d*x+c) *sec(b*x+a)*tan(b*x+a)/b
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(555\) vs. \(2(117)=234\).
Time = 7.03 (sec) , antiderivative size = 555, normalized size of antiderivative = 4.74 \[ \int (c+d x) \sec (a+b x) \tan ^2(a+b x) \, dx=-\frac {c \text {arctanh}(\sin (a+b x))}{2 b}+\frac {d x \left (a \log \left (1-\tan \left (\frac {1}{2} (a+b x)\right )\right )+i \log \left (1+i \tan \left (\frac {1}{2} (a+b x)\right )\right ) \log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (-1+\tan \left (\frac {1}{2} (a+b x)\right )\right )\right )-i \log \left (1-i \tan \left (\frac {1}{2} (a+b x)\right )\right ) \log \left (\left (-\frac {1}{2}+\frac {i}{2}\right ) \left (-1+\tan \left (\frac {1}{2} (a+b x)\right )\right )\right )-i \log \left (1+i \tan \left (\frac {1}{2} (a+b x)\right )\right ) \log \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \left (1+\tan \left (\frac {1}{2} (a+b x)\right )\right )\right )+i \log \left (1-i \tan \left (\frac {1}{2} (a+b x)\right )\right ) \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+\tan \left (\frac {1}{2} (a+b x)\right )\right )\right )-a \log \left (1+\tan \left (\frac {1}{2} (a+b x)\right )\right )-i \operatorname {PolyLog}\left (2,\frac {1}{2} \left ((1+i)-(1-i) \tan \left (\frac {1}{2} (a+b x)\right )\right )\right )+i \operatorname {PolyLog}\left (2,\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (i+\tan \left (\frac {1}{2} (a+b x)\right )\right )\right )-i \operatorname {PolyLog}\left (2,\frac {1}{2} \left ((1+i)+(1-i) \tan \left (\frac {1}{2} (a+b x)\right )\right )\right )+i \operatorname {PolyLog}\left (2,\frac {1}{2} \left ((1-i)+(1+i) \tan \left (\frac {1}{2} (a+b x)\right )\right )\right )\right )}{2 b \left (a-i \log \left (1-i \tan \left (\frac {1}{2} (a+b x)\right )\right )+i \log \left (1+i \tan \left (\frac {1}{2} (a+b x)\right )\right )\right )}+\frac {d x}{4 b \left (\cos \left (\frac {1}{2} (a+b x)\right )-\sin \left (\frac {1}{2} (a+b x)\right )\right )^2}-\frac {d \sin \left (\frac {1}{2} (a+b x)\right )}{2 b^2 \left (\cos \left (\frac {1}{2} (a+b x)\right )-\sin \left (\frac {1}{2} (a+b x)\right )\right )}-\frac {d x}{4 b \left (\cos \left (\frac {1}{2} (a+b x)\right )+\sin \left (\frac {1}{2} (a+b x)\right )\right )^2}+\frac {d \sin \left (\frac {1}{2} (a+b x)\right )}{2 b^2 \left (\cos \left (\frac {1}{2} (a+b x)\right )+\sin \left (\frac {1}{2} (a+b x)\right )\right )}+\frac {c \sec (a+b x) \tan (a+b x)}{2 b} \]
-1/2*(c*ArcTanh[Sin[a + b*x]])/b + (d*x*(a*Log[1 - Tan[(a + b*x)/2]] + I*L og[1 + I*Tan[(a + b*x)/2]]*Log[(-1/2 - I/2)*(-1 + Tan[(a + b*x)/2])] - I*L og[1 - I*Tan[(a + b*x)/2]]*Log[(-1/2 + I/2)*(-1 + Tan[(a + b*x)/2])] - I*L og[1 + I*Tan[(a + b*x)/2]]*Log[(1/2 - I/2)*(1 + Tan[(a + b*x)/2])] + I*Log [1 - I*Tan[(a + b*x)/2]]*Log[(1/2 + I/2)*(1 + Tan[(a + b*x)/2])] - a*Log[1 + Tan[(a + b*x)/2]] - I*PolyLog[2, ((1 + I) - (1 - I)*Tan[(a + b*x)/2])/2 ] + I*PolyLog[2, (-1/2 - I/2)*(I + Tan[(a + b*x)/2])] - I*PolyLog[2, ((1 + I) + (1 - I)*Tan[(a + b*x)/2])/2] + I*PolyLog[2, ((1 - I) + (1 + I)*Tan[( a + b*x)/2])/2]))/(2*b*(a - I*Log[1 - I*Tan[(a + b*x)/2]] + I*Log[1 + I*Ta n[(a + b*x)/2]])) + (d*x)/(4*b*(Cos[(a + b*x)/2] - Sin[(a + b*x)/2])^2) - (d*Sin[(a + b*x)/2])/(2*b^2*(Cos[(a + b*x)/2] - Sin[(a + b*x)/2])) - (d*x) /(4*b*(Cos[(a + b*x)/2] + Sin[(a + b*x)/2])^2) + (d*Sin[(a + b*x)/2])/(2*b ^2*(Cos[(a + b*x)/2] + Sin[(a + b*x)/2])) + (c*Sec[a + b*x]*Tan[a + b*x])/ (2*b)
Time = 0.69 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.64, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4913, 3042, 4669, 2715, 2838, 4673, 3042, 4669, 2715, 2838}
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.
\(\displaystyle \int (c+d x) \tan ^2(a+b x) \sec (a+b x) \, dx\) |
\(\Big \downarrow \) 4913 |
\(\displaystyle \int (c+d x) \sec ^3(a+b x)dx-\int (c+d x) \sec (a+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (c+d x) \csc \left (a+b x+\frac {\pi }{2}\right )^3dx-\int (c+d x) \csc \left (a+b x+\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 4669 |
\(\displaystyle \int (c+d x) \csc \left (a+b x+\frac {\pi }{2}\right )^3dx+\frac {d \int \log \left (1-i e^{i (a+b x)}\right )dx}{b}-\frac {d \int \log \left (1+i e^{i (a+b x)}\right )dx}{b}+\frac {2 i (c+d x) \arctan \left (e^{i (a+b x)}\right )}{b}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {i d \int e^{-i (a+b x)} \log \left (1-i e^{i (a+b x)}\right )de^{i (a+b x)}}{b^2}+\frac {i d \int e^{-i (a+b x)} \log \left (1+i e^{i (a+b x)}\right )de^{i (a+b x)}}{b^2}+\int (c+d x) \csc \left (a+b x+\frac {\pi }{2}\right )^3dx+\frac {2 i (c+d x) \arctan \left (e^{i (a+b x)}\right )}{b}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \int (c+d x) \csc \left (a+b x+\frac {\pi }{2}\right )^3dx+\frac {2 i (c+d x) \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {i d \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}+\frac {i d \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}\) |
\(\Big \downarrow \) 4673 |
\(\displaystyle \frac {1}{2} \int (c+d x) \sec (a+b x)dx+\frac {2 i (c+d x) \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {i d \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}+\frac {i d \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {d \sec (a+b x)}{2 b^2}+\frac {(c+d x) \tan (a+b x) \sec (a+b x)}{2 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \int (c+d x) \csc \left (a+b x+\frac {\pi }{2}\right )dx+\frac {2 i (c+d x) \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {i d \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}+\frac {i d \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {d \sec (a+b x)}{2 b^2}+\frac {(c+d x) \tan (a+b x) \sec (a+b x)}{2 b}\) |
\(\Big \downarrow \) 4669 |
\(\displaystyle \frac {1}{2} \left (-\frac {d \int \log \left (1-i e^{i (a+b x)}\right )dx}{b}+\frac {d \int \log \left (1+i e^{i (a+b x)}\right )dx}{b}-\frac {2 i (c+d x) \arctan \left (e^{i (a+b x)}\right )}{b}\right )+\frac {2 i (c+d x) \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {i d \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}+\frac {i d \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {d \sec (a+b x)}{2 b^2}+\frac {(c+d x) \tan (a+b x) \sec (a+b x)}{2 b}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {1}{2} \left (\frac {i d \int e^{-i (a+b x)} \log \left (1-i e^{i (a+b x)}\right )de^{i (a+b x)}}{b^2}-\frac {i d \int e^{-i (a+b x)} \log \left (1+i e^{i (a+b x)}\right )de^{i (a+b x)}}{b^2}-\frac {2 i (c+d x) \arctan \left (e^{i (a+b x)}\right )}{b}\right )+\frac {2 i (c+d x) \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {i d \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}+\frac {i d \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {d \sec (a+b x)}{2 b^2}+\frac {(c+d x) \tan (a+b x) \sec (a+b x)}{2 b}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {1}{2} \left (-\frac {2 i (c+d x) \arctan \left (e^{i (a+b x)}\right )}{b}+\frac {i d \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac {i d \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}\right )+\frac {2 i (c+d x) \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {i d \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}+\frac {i d \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {d \sec (a+b x)}{2 b^2}+\frac {(c+d x) \tan (a+b x) \sec (a+b x)}{2 b}\) |
((2*I)*(c + d*x)*ArcTan[E^(I*(a + b*x))])/b - (I*d*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^2 + (I*d*PolyLog[2, I*E^(I*(a + b*x))])/b^2 + (((-2*I)*(c + d *x)*ArcTan[E^(I*(a + b*x))])/b + (I*d*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^ 2 - (I*d*PolyLog[2, I*E^(I*(a + b*x))])/b^2)/2 - (d*Sec[a + b*x])/(2*b^2) + ((c + d*x)*Sec[a + b*x]*Tan[a + b*x])/(2*b)
3.3.100.3.1 Defintions of rubi rules used
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
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]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(-b^2)*(c + d*x)*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (-Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x] + S imp[b^2*((n - 2)/(n - 1)) Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
Int[((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]*Tan[(a_.) + (b_.)*(x _)]^(p_), x_Symbol] :> -Int[(c + d*x)^m*Sec[a + b*x]*Tan[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Sec[a + b*x]^3*Tan[a + b*x]^(p - 2), x] /; FreeQ[{a, b , c, d, m}, x] && IGtQ[p/2, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (98 ) = 196\).
Time = 0.70 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.28
method | result | size |
risch | \(-\frac {i \left (d x b \,{\mathrm e}^{3 i \left (x b +a \right )}-i d \,{\mathrm e}^{3 i \left (x b +a \right )}+c b \,{\mathrm e}^{3 i \left (x b +a \right )}-d x b \,{\mathrm e}^{i \left (x b +a \right )}-i d \,{\mathrm e}^{i \left (x b +a \right )}-c b \,{\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2} \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )^{2}}+\frac {i c \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b}+\frac {d \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{2 b}+\frac {d \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) a}{2 b^{2}}-\frac {d \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{2 b}-\frac {d \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) a}{2 b^{2}}-\frac {i d \operatorname {dilog}\left (1+i {\mathrm e}^{i \left (x b +a \right )}\right )}{2 b^{2}}+\frac {i d \operatorname {dilog}\left (1-i {\mathrm e}^{i \left (x b +a \right )}\right )}{2 b^{2}}-\frac {i d a \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}\) | \(267\) |
-I/b^2/(exp(2*I*(b*x+a))+1)^2*(d*x*b*exp(3*I*(b*x+a))-I*d*exp(3*I*(b*x+a)) +c*b*exp(3*I*(b*x+a))-d*x*b*exp(I*(b*x+a))-I*d*exp(I*(b*x+a))-c*b*exp(I*(b *x+a)))+I/b*c*arctan(exp(I*(b*x+a)))+1/2/b*d*ln(1+I*exp(I*(b*x+a)))*x+1/2/ b^2*d*ln(1+I*exp(I*(b*x+a)))*a-1/2/b*d*ln(1-I*exp(I*(b*x+a)))*x-1/2/b^2*d* ln(1-I*exp(I*(b*x+a)))*a-1/2*I/b^2*d*dilog(1+I*exp(I*(b*x+a)))+1/2*I/b^2*d *dilog(1-I*exp(I*(b*x+a)))-I/b^2*d*a*arctan(exp(I*(b*x+a)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (93) = 186\).
Time = 0.30 (sec) , antiderivative size = 435, normalized size of antiderivative = 3.72 \[ \int (c+d x) \sec (a+b x) \tan ^2(a+b x) \, dx=\frac {i \, d \cos \left (b x + a\right )^{2} {\rm Li}_2\left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) + i \, d \cos \left (b x + a\right )^{2} {\rm Li}_2\left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) - i \, d \cos \left (b x + a\right )^{2} {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) - i \, d \cos \left (b x + a\right )^{2} {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) - {\left (b c - a d\right )} \cos \left (b x + a\right )^{2} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) + {\left (b c - a d\right )} \cos \left (b x + a\right )^{2} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right ) - {\left (b d x + a d\right )} \cos \left (b x + a\right )^{2} \log \left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) + {\left (b d x + a d\right )} \cos \left (b x + a\right )^{2} \log \left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) - {\left (b d x + a d\right )} \cos \left (b x + a\right )^{2} \log \left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) + {\left (b d x + a d\right )} \cos \left (b x + a\right )^{2} \log \left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) - {\left (b c - a d\right )} \cos \left (b x + a\right )^{2} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) + {\left (b c - a d\right )} \cos \left (b x + a\right )^{2} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right ) - 2 \, d \cos \left (b x + a\right ) + 2 \, {\left (b d x + b c\right )} \sin \left (b x + a\right )}{4 \, b^{2} \cos \left (b x + a\right )^{2}} \]
1/4*(I*d*cos(b*x + a)^2*dilog(I*cos(b*x + a) + sin(b*x + a)) + I*d*cos(b*x + a)^2*dilog(I*cos(b*x + a) - sin(b*x + a)) - I*d*cos(b*x + a)^2*dilog(-I *cos(b*x + a) + sin(b*x + a)) - I*d*cos(b*x + a)^2*dilog(-I*cos(b*x + a) - sin(b*x + a)) - (b*c - a*d)*cos(b*x + a)^2*log(cos(b*x + a) + I*sin(b*x + a) + I) + (b*c - a*d)*cos(b*x + a)^2*log(cos(b*x + a) - I*sin(b*x + a) + I) - (b*d*x + a*d)*cos(b*x + a)^2*log(I*cos(b*x + a) + sin(b*x + a) + 1) + (b*d*x + a*d)*cos(b*x + a)^2*log(I*cos(b*x + a) - sin(b*x + a) + 1) - (b* d*x + a*d)*cos(b*x + a)^2*log(-I*cos(b*x + a) + sin(b*x + a) + 1) + (b*d*x + a*d)*cos(b*x + a)^2*log(-I*cos(b*x + a) - sin(b*x + a) + 1) - (b*c - a* d)*cos(b*x + a)^2*log(-cos(b*x + a) + I*sin(b*x + a) + I) + (b*c - a*d)*co s(b*x + a)^2*log(-cos(b*x + a) - I*sin(b*x + a) + I) - 2*d*cos(b*x + a) + 2*(b*d*x + b*c)*sin(b*x + a))/(b^2*cos(b*x + a)^2)
\[ \int (c+d x) \sec (a+b x) \tan ^2(a+b x) \, dx=\int \left (c + d x\right ) \tan ^{2}{\left (a + b x \right )} \sec {\left (a + b x \right )}\, dx \]
\[ \int (c+d x) \sec (a+b x) \tan ^2(a+b x) \, dx=\int { {\left (d x + c\right )} \sec \left (b x + a\right ) \tan \left (b x + a\right )^{2} \,d x } \]
-1/4*(4*(d*cos(3*b*x + 3*a) + d*cos(b*x + a) - (b*d*x + b*c)*sin(3*b*x + 3 *a) + (b*d*x + b*c)*sin(b*x + a))*cos(4*b*x + 4*a) + 4*(2*d*cos(2*b*x + 2* a) + 2*(b*d*x + b*c)*sin(2*b*x + 2*a) + d)*cos(3*b*x + 3*a) + 8*(d*cos(b*x + a) + (b*d*x + b*c)*sin(b*x + a))*cos(2*b*x + 2*a) + 4*d*cos(b*x + a) + 4*(b^2*d*cos(4*b*x + 4*a)^2 + 4*b^2*d*cos(2*b*x + 2*a)^2 + b^2*d*sin(4*b*x + 4*a)^2 + 4*b^2*d*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*b^2*d*sin(2*b*x + 2*a)^2 + 4*b^2*d*cos(2*b*x + 2*a) + b^2*d + 2*(2*b^2*d*cos(2*b*x + 2*a) + b^2*d)*cos(4*b*x + 4*a))*integrate((x*cos(2*b*x + 2*a)*cos(b*x + a) + x* sin(2*b*x + 2*a)*sin(b*x + a) + x*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), x) + (b*c*cos(4*b*x + 4*a)^2 + 4 *b*c*cos(2*b*x + 2*a)^2 + b*c*sin(4*b*x + 4*a)^2 + 4*b*c*sin(4*b*x + 4*a)* sin(2*b*x + 2*a) + 4*b*c*sin(2*b*x + 2*a)^2 + 4*b*c*cos(2*b*x + 2*a) + b*c + 2*(2*b*c*cos(2*b*x + 2*a) + b*c)*cos(4*b*x + 4*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*sin(b*x + a) + 1) - (b*c*cos(4*b*x + 4*a)^2 + 4*b*c*co s(2*b*x + 2*a)^2 + b*c*sin(4*b*x + 4*a)^2 + 4*b*c*sin(4*b*x + 4*a)*sin(2*b *x + 2*a) + 4*b*c*sin(2*b*x + 2*a)^2 + 4*b*c*cos(2*b*x + 2*a) + b*c + 2*(2 *b*c*cos(2*b*x + 2*a) + b*c)*cos(4*b*x + 4*a))*log(cos(b*x + a)^2 + sin(b* x + a)^2 - 2*sin(b*x + a) + 1) + 4*((b*d*x + b*c)*cos(3*b*x + 3*a) - (b*d* x + b*c)*cos(b*x + a) + d*sin(3*b*x + 3*a) + d*sin(b*x + a))*sin(4*b*x + 4 *a) - 4*(b*d*x + b*c + 2*(b*d*x + b*c)*cos(2*b*x + 2*a) - 2*d*sin(2*b*x...
\[ \int (c+d x) \sec (a+b x) \tan ^2(a+b x) \, dx=\int { {\left (d x + c\right )} \sec \left (b x + a\right ) \tan \left (b x + a\right )^{2} \,d x } \]
Timed out. \[ \int (c+d x) \sec (a+b x) \tan ^2(a+b x) \, dx=\text {Hanged} \]