Integrand size = 20, antiderivative size = 186 \[ \int (c+d x)^2 \sin (a+b x) \tan (a+b x) \, dx=-\frac {2 i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {2 d (c+d x) \cos (a+b x)}{b^2}+\frac {2 i d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac {2 i d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {2 d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^3}+\frac {2 d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^3}+\frac {2 d^2 \sin (a+b x)}{b^3}-\frac {(c+d x)^2 \sin (a+b x)}{b} \]
-2*I*(d*x+c)^2*arctan(exp(I*(b*x+a)))/b-2*d*(d*x+c)*cos(b*x+a)/b^2+2*I*d*( d*x+c)*polylog(2,-I*exp(I*(b*x+a)))/b^2-2*I*d*(d*x+c)*polylog(2,I*exp(I*(b *x+a)))/b^2-2*d^2*polylog(3,-I*exp(I*(b*x+a)))/b^3+2*d^2*polylog(3,I*exp(I *(b*x+a)))/b^3+2*d^2*sin(b*x+a)/b^3-(d*x+c)^2*sin(b*x+a)/b
Time = 0.93 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.69 \[ \int (c+d x)^2 \sin (a+b x) \tan (a+b x) \, dx=-\frac {2 i b^2 c^2 \arctan \left (e^{i (a+b x)}\right )+2 b c d \cos (a+b x)+2 b d^2 x \cos (a+b x)-2 b^2 c d x \log \left (1-i e^{i (a+b x)}\right )-b^2 d^2 x^2 \log \left (1-i e^{i (a+b x)}\right )+2 b^2 c d x \log \left (1+i e^{i (a+b x)}\right )+b^2 d^2 x^2 \log \left (1+i e^{i (a+b x)}\right )-2 i b d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )+2 i b d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )+2 d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )-2 d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )+b^2 c^2 \sin (a+b x)-2 d^2 \sin (a+b x)+2 b^2 c d x \sin (a+b x)+b^2 d^2 x^2 \sin (a+b x)}{b^3} \]
-(((2*I)*b^2*c^2*ArcTan[E^(I*(a + b*x))] + 2*b*c*d*Cos[a + b*x] + 2*b*d^2* x*Cos[a + b*x] - 2*b^2*c*d*x*Log[1 - I*E^(I*(a + b*x))] - b^2*d^2*x^2*Log[ 1 - I*E^(I*(a + b*x))] + 2*b^2*c*d*x*Log[1 + I*E^(I*(a + b*x))] + b^2*d^2* x^2*Log[1 + I*E^(I*(a + b*x))] - (2*I)*b*d*(c + d*x)*PolyLog[2, (-I)*E^(I* (a + b*x))] + (2*I)*b*d*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))] + 2*d^2*Po lyLog[3, (-I)*E^(I*(a + b*x))] - 2*d^2*PolyLog[3, I*E^(I*(a + b*x))] + b^2 *c^2*Sin[a + b*x] - 2*d^2*Sin[a + b*x] + 2*b^2*c*d*x*Sin[a + b*x] + b^2*d^ 2*x^2*Sin[a + b*x])/b^3)
Time = 0.81 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.06, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4907, 3042, 3777, 25, 3042, 3777, 3042, 3117, 4669, 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.
| \(\displaystyle \int (c+d x)^2 \sin (a+b x) \tan (a+b x) \, dx\) |
| \(\Big \downarrow \) 4907 |
| \(\displaystyle \int (c+d x)^2 \sec (a+b x)dx-\int (c+d x)^2 \cos (a+b x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x)^2 \csc \left (a+b x+\frac {\pi }{2}\right )dx-\int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )dx\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {2 d \int -((c+d x) \sin (a+b x))dx}{b}+\int (c+d x)^2 \csc \left (a+b x+\frac {\pi }{2}\right )dx-\frac {(c+d x)^2 \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 d \int (c+d x) \sin (a+b x)dx}{b}+\int (c+d x)^2 \csc \left (a+b x+\frac {\pi }{2}\right )dx-\frac {(c+d x)^2 \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 d \int (c+d x) \sin (a+b x)dx}{b}+\int (c+d x)^2 \csc \left (a+b x+\frac {\pi }{2}\right )dx-\frac {(c+d x)^2 \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {2 d \left (\frac {d \int \cos (a+b x)dx}{b}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}+\int (c+d x)^2 \csc \left (a+b x+\frac {\pi }{2}\right )dx-\frac {(c+d x)^2 \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x)^2 \csc \left (a+b x+\frac {\pi }{2}\right )dx+\frac {2 d \left (\frac {d \int \sin \left (a+b x+\frac {\pi }{2}\right )dx}{b}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \int (c+d x)^2 \csc \left (a+b x+\frac {\pi }{2}\right )dx+\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle -\frac {2 d \int (c+d x) \log \left (1-i e^{i (a+b x)}\right )dx}{b}+\frac {2 d \int (c+d x) \log \left (1+i e^{i (a+b x)}\right )dx}{b}-\frac {2 i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}+\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b}-\frac {i d \int \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )dx}{b}\right )}{b}-\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b}-\frac {i d \int \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )dx}{b}\right )}{b}-\frac {2 i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}+\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b}-\frac {d \int e^{-i (a+b x)} \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )de^{i (a+b x)}}{b^2}\right )}{b}-\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b}-\frac {d \int e^{-i (a+b x)} \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )de^{i (a+b x)}}{b^2}\right )}{b}-\frac {2 i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}+\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {2 i (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b}+\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b}-\frac {d \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^2}\right )}{b}-\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b}-\frac {d \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^2}\right )}{b}+\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \sin (a+b x)}{b}\) |
((-2*I)*(c + d*x)^2*ArcTan[E^(I*(a + b*x))])/b + (2*d*((I*(c + d*x)*PolyLo g[2, (-I)*E^(I*(a + b*x))])/b - (d*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^2)) /b - (2*d*((I*(c + d*x)*PolyLog[2, I*E^(I*(a + b*x))])/b - (d*PolyLog[3, I *E^(I*(a + b*x))])/b^2))/b - ((c + d*x)^2*Sin[a + b*x])/b + (2*d*(-(((c + d*x)*Cos[a + b*x])/b) + (d*Sin[a + b*x])/b^2))/b
3.3.17.3.1 Defintions of rubi rules used
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[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
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[((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.)*Tan[(a_.) + (b _.)*(x_)]^(p_.), x_Symbol] :> -Int[(c + d*x)^m*Sin[a + b*x]^n*Tan[a + b*x]^ (p - 2), x] + Int[(c + d*x)^m*Sin[a + b*x]^(n - 2)*Tan[a + b*x]^p, x] /; Fr eeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 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. 511 vs. \(2 (169 ) = 338\).
Time = 2.19 (sec) , antiderivative size = 512, normalized size of antiderivative = 2.75
| method | result | size |
| risch | \(-\frac {2 i c^{2} \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b}-\frac {i \left (x^{2} d^{2} b^{2}+2 b^{2} c d x -2 i b \,d^{2} x +b^{2} c^{2}-2 i b c d -2 d^{2}\right ) {\mathrm e}^{-i \left (x b +a \right )}}{2 b^{3}}+\frac {2 c d \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{2}}-\frac {a^{2} d^{2} \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}+\frac {a^{2} d^{2} \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}+\frac {2 d^{2} \operatorname {polylog}\left (3, i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {2 c d \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{2}}+\frac {d^{2} \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) x^{2}}{b}+\frac {i \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +2 i b \,d^{2} x +b^{2} c^{2}+2 i b c d -2 d^{2}\right ) {\mathrm e}^{i \left (x b +a \right )}}{2 b^{3}}+\frac {2 i d^{2} \operatorname {polylog}\left (2, -i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}-\frac {2 i d^{2} a^{2} \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {d^{2} \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) x^{2}}{b}-\frac {2 d^{2} \operatorname {polylog}\left (3, -i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}+\frac {2 i c d \operatorname {polylog}\left (2, -i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}+\frac {4 i c d a \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}-\frac {2 i c d \operatorname {polylog}\left (2, i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}-\frac {2 i d^{2} \operatorname {polylog}\left (2, i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}-\frac {2 c d \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b}+\frac {2 c d \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b}\) | \(512\) |
-2*I/b*c^2*arctan(exp(I*(b*x+a)))-1/2*I*(x^2*d^2*b^2+2*b^2*c*d*x+b^2*c^2-2 *I*b*d^2*x-2*d^2-2*I*b*c*d)/b^3*exp(-I*(b*x+a))+2/b^2*c*d*ln(1-I*exp(I*(b* x+a)))*a-1/b^3*a^2*d^2*ln(1-I*exp(I*(b*x+a)))+1/b^3*a^2*d^2*ln(1+I*exp(I*( b*x+a)))+2*d^2*polylog(3,I*exp(I*(b*x+a)))/b^3-2/b^2*c*d*ln(1+I*exp(I*(b*x +a)))*a+1/b*d^2*ln(1-I*exp(I*(b*x+a)))*x^2+1/2*I*(x^2*d^2*b^2+2*b^2*c*d*x+ b^2*c^2+2*I*b*d^2*x-2*d^2+2*I*b*c*d)/b^3*exp(I*(b*x+a))+2*I/b^2*d^2*polylo g(2,-I*exp(I*(b*x+a)))*x-2*I/b^3*d^2*a^2*arctan(exp(I*(b*x+a)))-1/b*d^2*ln (1+I*exp(I*(b*x+a)))*x^2-2*d^2*polylog(3,-I*exp(I*(b*x+a)))/b^3+2*I/b^2*c* d*polylog(2,-I*exp(I*(b*x+a)))+4*I/b^2*c*d*a*arctan(exp(I*(b*x+a)))-2*I/b^ 2*c*d*polylog(2,I*exp(I*(b*x+a)))-2*I/b^2*d^2*polylog(2,I*exp(I*(b*x+a)))* x-2/b*c*d*ln(1+I*exp(I*(b*x+a)))*x+2/b*c*d*ln(1-I*exp(I*(b*x+a)))*x
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 656 vs. \(2 (160) = 320\).
Time = 0.29 (sec) , antiderivative size = 656, normalized size of antiderivative = 3.53 \[ \int (c+d x)^2 \sin (a+b x) \tan (a+b x) \, dx=-\frac {2 \, d^{2} {\rm polylog}\left (3, i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) - 2 \, d^{2} {\rm polylog}\left (3, i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) + 2 \, d^{2} {\rm polylog}\left (3, -i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) - 2 \, d^{2} {\rm polylog}\left (3, -i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) + 4 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right ) + 2 \, {\left (i \, b d^{2} x + i \, b c d\right )} {\rm Li}_2\left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) + 2 \, {\left (i \, b d^{2} x + i \, b c d\right )} {\rm Li}_2\left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) + 2 \, {\left (-i \, b d^{2} x - i \, b c d\right )} {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) + 2 \, {\left (-i \, b d^{2} x - i \, b c d\right )} {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) - {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\right )} \log \left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\right )} \log \left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\right )} \log \left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\right )} \log \left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) - {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right ) + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, d^{2}\right )} \sin \left (b x + a\right )}{2 \, b^{3}} \]
-1/2*(2*d^2*polylog(3, I*cos(b*x + a) + sin(b*x + a)) - 2*d^2*polylog(3, I *cos(b*x + a) - sin(b*x + a)) + 2*d^2*polylog(3, -I*cos(b*x + a) + sin(b*x + a)) - 2*d^2*polylog(3, -I*cos(b*x + a) - sin(b*x + a)) + 4*(b*d^2*x + b *c*d)*cos(b*x + a) + 2*(I*b*d^2*x + I*b*c*d)*dilog(I*cos(b*x + a) + sin(b* x + a)) + 2*(I*b*d^2*x + I*b*c*d)*dilog(I*cos(b*x + a) - sin(b*x + a)) + 2 *(-I*b*d^2*x - I*b*c*d)*dilog(-I*cos(b*x + a) + sin(b*x + a)) + 2*(-I*b*d^ 2*x - I*b*c*d)*dilog(-I*cos(b*x + a) - sin(b*x + a)) - (b^2*c^2 - 2*a*b*c* d + a^2*d^2)*log(cos(b*x + a) + I*sin(b*x + a) + I) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log(cos(b*x + a) - I*sin(b*x + a) + I) - (b^2*d^2*x^2 + 2*b^2* c*d*x + 2*a*b*c*d - a^2*d^2)*log(I*cos(b*x + a) + sin(b*x + a) + 1) + (b^2 *d^2*x^2 + 2*b^2*c*d*x + 2*a*b*c*d - a^2*d^2)*log(I*cos(b*x + a) - sin(b*x + a) + 1) - (b^2*d^2*x^2 + 2*b^2*c*d*x + 2*a*b*c*d - a^2*d^2)*log(-I*cos( b*x + a) + sin(b*x + a) + 1) + (b^2*d^2*x^2 + 2*b^2*c*d*x + 2*a*b*c*d - a^ 2*d^2)*log(-I*cos(b*x + a) - sin(b*x + a) + 1) - (b^2*c^2 - 2*a*b*c*d + a^ 2*d^2)*log(-cos(b*x + a) + I*sin(b*x + a) + I) + (b^2*c^2 - 2*a*b*c*d + a^ 2*d^2)*log(-cos(b*x + a) - I*sin(b*x + a) + I) + 2*(b^2*d^2*x^2 + 2*b^2*c* d*x + b^2*c^2 - 2*d^2)*sin(b*x + a))/b^3
\[ \int (c+d x)^2 \sin (a+b x) \tan (a+b x) \, dx=\int \left (c + d x\right )^{2} \sin ^{2}{\left (a + b x \right )} \sec {\left (a + b x \right )}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 516 vs. \(2 (160) = 320\).
Time = 0.39 (sec) , antiderivative size = 516, normalized size of antiderivative = 2.77 \[ \int (c+d x)^2 \sin (a+b x) \tan (a+b x) \, dx=\frac {c^{2} {\left (\log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (\sin \left (b x + a\right ) - 1\right ) - 2 \, \sin \left (b x + a\right )\right )} - \frac {2 \, a c d {\left (\log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (\sin \left (b x + a\right ) - 1\right ) - 2 \, \sin \left (b x + a\right )\right )}}{b} + \frac {a^{2} d^{2} {\left (\log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (\sin \left (b x + a\right ) - 1\right ) - 2 \, \sin \left (b x + a\right )\right )}}{b^{2}} + \frac {4 \, d^{2} {\rm Li}_{3}(i \, e^{\left (i \, b x + i \, a\right )}) - 4 \, d^{2} {\rm Li}_{3}(-i \, e^{\left (i \, b x + i \, a\right )}) - 2 \, {\left (i \, {\left (b x + a\right )}^{2} d^{2} + 2 \, {\left (i \, b c d - i \, a d^{2}\right )} {\left (b x + a\right )}\right )} \arctan \left (\cos \left (b x + a\right ), \sin \left (b x + a\right ) + 1\right ) - 2 \, {\left (i \, {\left (b x + a\right )}^{2} d^{2} + 2 \, {\left (i \, b c d - i \, a d^{2}\right )} {\left (b x + a\right )}\right )} \arctan \left (\cos \left (b x + a\right ), -\sin \left (b x + a\right ) + 1\right ) - 4 \, {\left (b c d + {\left (b x + a\right )} d^{2} - a d^{2}\right )} \cos \left (b x + a\right ) - 4 \, {\left (i \, b c d + i \, {\left (b x + a\right )} d^{2} - i \, a d^{2}\right )} {\rm Li}_2\left (i \, e^{\left (i \, b x + i \, a\right )}\right ) - 4 \, {\left (-i \, b c d - i \, {\left (b x + a\right )} d^{2} + i \, a d^{2}\right )} {\rm Li}_2\left (-i \, e^{\left (i \, b x + i \, a\right )}\right ) + {\left ({\left (b x + a\right )}^{2} d^{2} + 2 \, {\left (b c d - a d^{2}\right )} {\left (b x + a\right )}\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \sin \left (b x + a\right ) + 1\right ) - {\left ({\left (b x + a\right )}^{2} d^{2} + 2 \, {\left (b c d - a d^{2}\right )} {\left (b x + a\right )}\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \sin \left (b x + a\right ) + 1\right ) - 2 \, {\left ({\left (b x + a\right )}^{2} d^{2} + 2 \, {\left (b c d - a d^{2}\right )} {\left (b x + a\right )} - 2 \, d^{2}\right )} \sin \left (b x + a\right )}{b^{2}}}{2 \, b} \]
1/2*(c^2*(log(sin(b*x + a) + 1) - log(sin(b*x + a) - 1) - 2*sin(b*x + a)) - 2*a*c*d*(log(sin(b*x + a) + 1) - log(sin(b*x + a) - 1) - 2*sin(b*x + a)) /b + a^2*d^2*(log(sin(b*x + a) + 1) - log(sin(b*x + a) - 1) - 2*sin(b*x + a))/b^2 + (4*d^2*polylog(3, I*e^(I*b*x + I*a)) - 4*d^2*polylog(3, -I*e^(I* b*x + I*a)) - 2*(I*(b*x + a)^2*d^2 + 2*(I*b*c*d - I*a*d^2)*(b*x + a))*arct an2(cos(b*x + a), sin(b*x + a) + 1) - 2*(I*(b*x + a)^2*d^2 + 2*(I*b*c*d - I*a*d^2)*(b*x + a))*arctan2(cos(b*x + a), -sin(b*x + a) + 1) - 4*(b*c*d + (b*x + a)*d^2 - a*d^2)*cos(b*x + a) - 4*(I*b*c*d + I*(b*x + a)*d^2 - I*a*d ^2)*dilog(I*e^(I*b*x + I*a)) - 4*(-I*b*c*d - I*(b*x + a)*d^2 + I*a*d^2)*di log(-I*e^(I*b*x + I*a)) + ((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a))* log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*sin(b*x + a) + 1) - ((b*x + a)^2*d ^2 + 2*(b*c*d - a*d^2)*(b*x + a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2* sin(b*x + a) + 1) - 2*((b*x + a)^2*d^2 + 2*(b*c*d - a*d^2)*(b*x + a) - 2*d ^2)*sin(b*x + a))/b^2)/b
\[ \int (c+d x)^2 \sin (a+b x) \tan (a+b x) \, dx=\int { {\left (d x + c\right )}^{2} \sec \left (b x + a\right ) \sin \left (b x + a\right )^{2} \,d x } \]
Timed out. \[ \int (c+d x)^2 \sin (a+b x) \tan (a+b x) \, dx=\int \frac {{\sin \left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}{\cos \left (a+b\,x\right )} \,d x \]