Integrand size = 16, antiderivative size = 162 \[ \int \frac {\sin ^2(a+b x)}{(c+d x)^4} \, dx=-\frac {b^2}{3 d^3 (c+d x)}-\frac {2 b^3 \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{3 d^4}-\frac {b \cos (a+b x) \sin (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sin ^2(a+b x)}{3 d (c+d x)^3}+\frac {2 b^2 \sin ^2(a+b x)}{3 d^3 (c+d x)}-\frac {2 b^3 \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{3 d^4} \]
-1/3*b^2/d^3/(d*x+c)-2/3*b^3*cos(2*a-2*b*c/d)*Si(2*b*c/d+2*b*x)/d^4-2/3*b^ 3*Ci(2*b*c/d+2*b*x)*sin(2*a-2*b*c/d)/d^4-1/3*b*cos(b*x+a)*sin(b*x+a)/d^2/( d*x+c)^2-1/3*sin(b*x+a)^2/d/(d*x+c)^3+2/3*b^2*sin(b*x+a)^2/d^3/(d*x+c)
Time = 0.89 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.75 \[ \int \frac {\sin ^2(a+b x)}{(c+d x)^4} \, dx=-\frac {4 b^3 \operatorname {CosIntegral}\left (\frac {2 b (c+d x)}{d}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )+\frac {d \left (\left (-d^2+2 b^2 (c+d x)^2\right ) \cos (2 (a+b x))+d (d+b (c+d x) \sin (2 (a+b x)))\right )}{(c+d x)^3}+4 b^3 \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b (c+d x)}{d}\right )}{6 d^4} \]
-1/6*(4*b^3*CosIntegral[(2*b*(c + d*x))/d]*Sin[2*a - (2*b*c)/d] + (d*((-d^ 2 + 2*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] + d*(d + b*(c + d*x)*Sin[2*(a + b* x)])))/(c + d*x)^3 + 4*b^3*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x) )/d])/d^4
Time = 0.72 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {3042, 3795, 17, 3042, 3794, 27, 3042, 3784, 3042, 3780, 3783}
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.
| \(\displaystyle \int \frac {\sin ^2(a+b x)}{(c+d x)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (a+b x)^2}{(c+d x)^4}dx\) |
| \(\Big \downarrow \) 3795 |
| \(\displaystyle -\frac {2 b^2 \int \frac {\sin ^2(a+b x)}{(c+d x)^2}dx}{3 d^2}+\frac {b^2 \int \frac {1}{(c+d x)^2}dx}{3 d^2}-\frac {b \sin (a+b x) \cos (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sin ^2(a+b x)}{3 d (c+d x)^3}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {2 b^2 \int \frac {\sin ^2(a+b x)}{(c+d x)^2}dx}{3 d^2}-\frac {b \sin (a+b x) \cos (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sin ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 b^2 \int \frac {\sin (a+b x)^2}{(c+d x)^2}dx}{3 d^2}-\frac {b \sin (a+b x) \cos (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sin ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)}\) |
| \(\Big \downarrow \) 3794 |
| \(\displaystyle -\frac {2 b^2 \left (\frac {2 b \int \frac {\sin (2 a+2 b x)}{2 (c+d x)}dx}{d}-\frac {\sin ^2(a+b x)}{d (c+d x)}\right )}{3 d^2}-\frac {b \sin (a+b x) \cos (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sin ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b^2 \left (\frac {b \int \frac {\sin (2 a+2 b x)}{c+d x}dx}{d}-\frac {\sin ^2(a+b x)}{d (c+d x)}\right )}{3 d^2}-\frac {b \sin (a+b x) \cos (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sin ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 b^2 \left (\frac {b \int \frac {\sin (2 a+2 b x)}{c+d x}dx}{d}-\frac {\sin ^2(a+b x)}{d (c+d x)}\right )}{3 d^2}-\frac {b \sin (a+b x) \cos (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sin ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle -\frac {2 b^2 \left (\frac {b \left (\sin \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{c+d x}dx+\cos \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{c+d x}dx\right )}{d}-\frac {\sin ^2(a+b x)}{d (c+d x)}\right )}{3 d^2}-\frac {b \sin (a+b x) \cos (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sin ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 b^2 \left (\frac {b \left (\sin \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x+\frac {\pi }{2}\right )}{c+d x}dx+\cos \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{c+d x}dx\right )}{d}-\frac {\sin ^2(a+b x)}{d (c+d x)}\right )}{3 d^2}-\frac {b \sin (a+b x) \cos (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sin ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)}\) |
| \(\Big \downarrow \) 3780 |
| \(\displaystyle -\frac {2 b^2 \left (\frac {b \left (\sin \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x+\frac {\pi }{2}\right )}{c+d x}dx+\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d}\right )}{d}-\frac {\sin ^2(a+b x)}{d (c+d x)}\right )}{3 d^2}-\frac {b \sin (a+b x) \cos (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sin ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)}\) |
| \(\Big \downarrow \) 3783 |
| \(\displaystyle -\frac {2 b^2 \left (\frac {b \left (\frac {\sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{d}+\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d}\right )}{d}-\frac {\sin ^2(a+b x)}{d (c+d x)}\right )}{3 d^2}-\frac {b \sin (a+b x) \cos (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sin ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)}\) |
-1/3*b^2/(d^3*(c + d*x)) - (b*Cos[a + b*x]*Sin[a + b*x])/(3*d^2*(c + d*x)^ 2) - Sin[a + b*x]^2/(3*d*(c + d*x)^3) - (2*b^2*(-(Sin[a + b*x]^2/(d*(c + d *x))) + (b*((CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/d + (Cos [2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/d))/d))/(3*d^2)
3.1.15.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* f)/d] Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Si mp[(c + d*x)^(m + 1)*(Sin[e + f*x]^n/(d*(m + 1))), x] - Simp[f*(n/(d*(m + 1 ))) Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] & & LtQ[m, -1]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[(c + d*x)^(m + 1)*((b*Sin[e + f*x])^n/(d*(m + 1))), x] + (-Simp[ b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(d^2*(m + 1) *(m + 2))), x] + Simp[b^2*f^2*n*((n - 1)/(d^2*(m + 1)*(m + 2))) Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[f^2*(n^2/(d^2*(m + 1)* (m + 2))) Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]
Time = 0.50 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.41
| method | result | size |
| derivativedivides | \(\frac {-\frac {b^{4}}{6 \left (-d a +c b +d \left (b x +a \right )\right )^{3} d}-\frac {b^{4} \left (-\frac {2 \cos \left (2 b x +2 a \right )}{3 \left (-d a +c b +d \left (b x +a \right )\right )^{3} d}-\frac {2 \left (-\frac {\sin \left (2 b x +2 a \right )}{\left (-d a +c b +d \left (b x +a \right )\right )^{2} d}+\frac {-\frac {2 \cos \left (2 b x +2 a \right )}{\left (-d a +c b +d \left (b x +a \right )\right ) d}-\frac {2 \left (-\frac {2 \,\operatorname {Si}\left (-2 b x -2 a -\frac {2 \left (-d a +c b \right )}{d}\right ) \cos \left (\frac {-2 d a +2 c b}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 b x +2 a +\frac {-2 d a +2 c b}{d}\right ) \sin \left (\frac {-2 d a +2 c b}{d}\right )}{d}\right )}{d}}{d}\right )}{3 d}\right )}{4}}{b}\) | \(229\) |
| default | \(\frac {-\frac {b^{4}}{6 \left (-d a +c b +d \left (b x +a \right )\right )^{3} d}-\frac {b^{4} \left (-\frac {2 \cos \left (2 b x +2 a \right )}{3 \left (-d a +c b +d \left (b x +a \right )\right )^{3} d}-\frac {2 \left (-\frac {\sin \left (2 b x +2 a \right )}{\left (-d a +c b +d \left (b x +a \right )\right )^{2} d}+\frac {-\frac {2 \cos \left (2 b x +2 a \right )}{\left (-d a +c b +d \left (b x +a \right )\right ) d}-\frac {2 \left (-\frac {2 \,\operatorname {Si}\left (-2 b x -2 a -\frac {2 \left (-d a +c b \right )}{d}\right ) \cos \left (\frac {-2 d a +2 c b}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 b x +2 a +\frac {-2 d a +2 c b}{d}\right ) \sin \left (\frac {-2 d a +2 c b}{d}\right )}{d}\right )}{d}}{d}\right )}{3 d}\right )}{4}}{b}\) | \(229\) |
| risch | \(\frac {i b^{3} {\mathrm e}^{-\frac {2 i \left (d a -c b \right )}{d}} \operatorname {Ei}_{1}\left (2 i b x +2 i a -\frac {2 i \left (d a -c b \right )}{d}\right )}{3 d^{4}}-\frac {i b^{3} {\mathrm e}^{\frac {2 i \left (d a -c b \right )}{d}} \operatorname {Ei}_{1}\left (-2 i b x -2 i a -\frac {2 \left (-i a d +i c b \right )}{d}\right )}{3 d^{4}}-\frac {1}{6 d \left (d x +c \right )^{3}}+\frac {\left (-4 b^{5} d^{5} x^{5}-20 b^{5} c \,d^{4} x^{4}-40 b^{5} c^{2} d^{3} x^{3}-40 b^{5} c^{3} d^{2} x^{2}-20 b^{5} c^{4} d x +2 b^{3} d^{5} x^{3}-4 b^{5} c^{5}+6 b^{3} c \,d^{4} x^{2}+6 b^{3} c^{2} d^{3} x +2 b^{3} c^{3} d^{2}\right ) \cos \left (2 b x +2 a \right )}{12 d^{3} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right ) \left (d x +c \right )^{3}}+\frac {i \left (2 i b^{4} d^{5} x^{4}+8 i b^{4} c \,d^{4} x^{3}+12 i b^{4} c^{2} d^{3} x^{2}+8 i b^{4} c^{3} d^{2} x +2 i b^{4} c^{4} d \right ) \sin \left (2 b x +2 a \right )}{12 d^{3} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right ) \left (d x +c \right )^{3}}\) | \(423\) |
1/b*(-1/6*b^4/(-d*a+c*b+d*(b*x+a))^3/d-1/4*b^4*(-2/3*cos(2*b*x+2*a)/(-d*a+ c*b+d*(b*x+a))^3/d-2/3*(-sin(2*b*x+2*a)/(-d*a+c*b+d*(b*x+a))^2/d+(-2*cos(2 *b*x+2*a)/(-d*a+c*b+d*(b*x+a))/d-2*(-2*Si(-2*b*x-2*a-2*(-a*d+b*c)/d)*cos(2 *(-a*d+b*c)/d)/d-2*Ci(2*b*x+2*a+2*(-a*d+b*c)/d)*sin(2*(-a*d+b*c)/d)/d)/d)/ d)/d))
Time = 0.32 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.75 \[ \int \frac {\sin ^2(a+b x)}{(c+d x)^4} \, dx=\frac {b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d - d^{3} - {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{2} - {\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 2 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \operatorname {Ci}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - 2 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right )}{3 \, {\left (d^{7} x^{3} + 3 \, c d^{6} x^{2} + 3 \, c^{2} d^{5} x + c^{3} d^{4}\right )}} \]
1/3*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d - d^3 - (2*b^2*d^3*x^2 + 4*b^ 2*c*d^2*x + 2*b^2*c^2*d - d^3)*cos(b*x + a)^2 - (b*d^3*x + b*c*d^2)*cos(b* x + a)*sin(b*x + a) - 2*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b ^3*c^3)*cos_integral(2*(b*d*x + b*c)/d)*sin(-2*(b*c - a*d)/d) - 2*(b^3*d^3 *x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*cos(-2*(b*c - a*d)/d)*si n_integral(2*(b*d*x + b*c)/d))/(d^7*x^3 + 3*c*d^6*x^2 + 3*c^2*d^5*x + c^3* d^4)
\[ \int \frac {\sin ^2(a+b x)}{(c+d x)^4} \, dx=\int \frac {\sin ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{4}}\, dx \]
Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.59 \[ \int \frac {\sin ^2(a+b x)}{(c+d x)^4} \, dx=\frac {3 \, b^{4} {\left (E_{4}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{4}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 3 \, b^{4} {\left (i \, E_{4}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) - i \, E_{4}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - 2 \, b^{4}}{12 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} + {\left (b x + a\right )}^{3} d^{4} - a^{3} d^{4} + 3 \, {\left (b c d^{3} - a d^{4}\right )} {\left (b x + a\right )}^{2} + 3 \, {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} {\left (b x + a\right )}\right )} b} \]
1/12*(3*b^4*(exp_integral_e(4, 2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + exp _integral_e(4, -2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*cos(-2*(b*c - a*d)/ d) + 3*b^4*(I*exp_integral_e(4, 2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) - I* exp_integral_e(4, -2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*sin(-2*(b*c - a* d)/d) - 2*b^4)/((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 + (b*x + a)^3 *d^4 - a^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b* c*d^3 + a^2*d^4)*(b*x + a))*b)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.54 (sec) , antiderivative size = 7832, normalized size of antiderivative = 48.35 \[ \int \frac {\sin ^2(a+b x)}{(c+d x)^4} \, dx=\text {Too large to display} \]
-1/3*(b^3*d^3*x^3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan( a)^2*tan(b*c/d)^2 - b^3*d^3*x^3*imag_part(cos_integral(-2*b*x - 2*b*c/d))* tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 + 2*b^3*d^3*x^3*sin_integral(2*(b*d*x + b *c)/d)*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 + 2*b^3*d^3*x^3*real_part(cos_inte gral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d) + 2*b^3*d^3*x^3*real _part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d) - 2*b ^3*d^3*x^3*real_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)*tan( b*c/d)^2 - 2*b^3*d^3*x^3*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x )^2*tan(a)*tan(b*c/d)^2 + 3*b^3*c*d^2*x^2*imag_part(cos_integral(2*b*x + 2 *b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 - 3*b^3*c*d^2*x^2*imag_part(cos_ integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 + 6*b^3*c*d^2 *x^2*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(a)^2*tan(b*c/d)^2 - b^ 3*d^3*x^3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)^2 + b ^3*d^3*x^3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2*tan(a)^2 - 2*b^3*d^3*x^3*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x)^2*tan(a)^2 + 4*b^3 *d^3*x^3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(b*x)^2*tan(a)*tan(b* c/d) - 4*b^3*d^3*x^3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(b*x)^2* tan(a)*tan(b*c/d) + 8*b^3*d^3*x^3*sin_integral(2*(b*d*x + b*c)/d)*tan(b*x) ^2*tan(a)*tan(b*c/d) + 6*b^3*c*d^2*x^2*real_part(cos_integral(2*b*x + 2*b* c/d))*tan(b*x)^2*tan(a)^2*tan(b*c/d) + 6*b^3*c*d^2*x^2*real_part(cos_in...
Timed out. \[ \int \frac {\sin ^2(a+b x)}{(c+d x)^4} \, dx=\int \frac {{\sin \left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^4} \,d x \]