Integrand size = 21, antiderivative size = 152 \[ \int \frac {\sin ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\left (a^2-6 b^2\right ) x}{2 a^4}-\frac {2 b \left (2 a^2-3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 \sqrt {a-b} \sqrt {a+b} d}+\frac {3 b \sin (c+d x)}{a^3 d}-\frac {3 \cos (c+d x) \sin (c+d x)}{2 a^2 d}+\frac {\cos ^2(c+d x) \sin (c+d x)}{a d (b+a \cos (c+d x))} \]
1/2*(a^2-6*b^2)*x/a^4+3*b*sin(d*x+c)/a^3/d-3/2*cos(d*x+c)*sin(d*x+c)/a^2/d +cos(d*x+c)^2*sin(d*x+c)/a/d/(b+a*cos(d*x+c))-2*b*(2*a^2-3*b^2)*arctanh((a -b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^4/d/(a-b)^(1/2)/(a+b)^(1/2)
Time = 0.73 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.17 \[ \int \frac {\sin ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\frac {16 b \left (2 a^2-3 b^2\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {4 a^2 b c-24 b^3 c+4 a^2 b d x-24 b^3 d x+4 a \left (a^2-6 b^2\right ) (c+d x) \cos (c+d x)-a \left (a^2-24 b^2\right ) \sin (c+d x)+6 a^2 b \sin (2 (c+d x))-a^3 \sin (3 (c+d x))}{b+a \cos (c+d x)}}{8 a^4 d} \]
((16*b*(2*a^2 - 3*b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2] ])/Sqrt[a^2 - b^2] + (4*a^2*b*c - 24*b^3*c + 4*a^2*b*d*x - 24*b^3*d*x + 4* a*(a^2 - 6*b^2)*(c + d*x)*Cos[c + d*x] - a*(a^2 - 24*b^2)*Sin[c + d*x] + 6 *a^2*b*Sin[2*(c + d*x)] - a^3*Sin[3*(c + d*x)])/(b + a*Cos[c + d*x]))/(8*a ^4*d)
Time = 1.31 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.41, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.810, Rules used = {3042, 4360, 3042, 3368, 3042, 3527, 25, 3042, 3529, 25, 3042, 3502, 3042, 3214, 3042, 3138, 221}
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.
| \(\displaystyle \int \frac {\sin ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos \left (c+d x-\frac {\pi }{2}\right )^2}{\left (a-b \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \frac {\sin ^2(c+d x) \cos ^2(c+d x)}{(-a \cos (c+d x)-b)^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \cos \left (c+d x+\frac {\pi }{2}\right )^2}{\left (-a \sin \left (c+d x+\frac {\pi }{2}\right )-b\right )^2}dx\) |
| \(\Big \downarrow \) 3368 |
| \(\displaystyle \int \frac {\cos ^2(c+d x) \left (1-\cos ^2(c+d x)\right )}{(-a \cos (c+d x)-b)^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (1-\sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (-a \sin \left (c+d x+\frac {\pi }{2}\right )-b\right )^2}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \frac {\sin (c+d x) \cos ^2(c+d x)}{a d (a \cos (c+d x)+b)}-\frac {\int -\frac {\cos (c+d x) \left (2 \left (a^2-b^2\right )-3 \left (a^2-b^2\right ) \cos ^2(c+d x)\right )}{b+a \cos (c+d x)}dx}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\cos (c+d x) \left (2 \left (a^2-b^2\right )-3 \left (a^2-b^2\right ) \cos ^2(c+d x)\right )}{b+a \cos (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^2(c+d x)}{a d (a \cos (c+d x)+b)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (2 \left (a^2-b^2\right )-3 \left (a^2-b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^2(c+d x)}{a d (a \cos (c+d x)+b)}\) |
| \(\Big \downarrow \) 3529 |
| \(\displaystyle \frac {\frac {\int -\frac {-6 b \left (a^2-b^2\right ) \cos ^2(c+d x)-a \left (a^2-b^2\right ) \cos (c+d x)+3 b \left (a^2-b^2\right )}{b+a \cos (c+d x)}dx}{2 a}-\frac {3 \left (a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^2(c+d x)}{a d (a \cos (c+d x)+b)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \frac {-6 b \left (a^2-b^2\right ) \cos ^2(c+d x)-a \left (a^2-b^2\right ) \cos (c+d x)+3 b \left (a^2-b^2\right )}{b+a \cos (c+d x)}dx}{2 a}-\frac {3 \left (a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^2(c+d x)}{a d (a \cos (c+d x)+b)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {-6 b \left (a^2-b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-a \left (a^2-b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 b \left (a^2-b^2\right )}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{2 a}-\frac {3 \left (a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^2(c+d x)}{a d (a \cos (c+d x)+b)}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {3 a b \left (a^2-b^2\right )-\left (a^2-6 b^2\right ) \left (a^2-b^2\right ) \cos (c+d x)}{b+a \cos (c+d x)}dx}{a}-\frac {6 b \left (a^2-b^2\right ) \sin (c+d x)}{a d}}{2 a}-\frac {3 \left (a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^2(c+d x)}{a d (a \cos (c+d x)+b)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {3 a b \left (a^2-b^2\right )-\left (a^2-6 b^2\right ) \left (a^2-b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}-\frac {6 b \left (a^2-b^2\right ) \sin (c+d x)}{a d}}{2 a}-\frac {3 \left (a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^2(c+d x)}{a d (a \cos (c+d x)+b)}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {-\frac {\frac {\frac {2 b \left (2 a^2-3 b^2\right ) \left (a^2-b^2\right ) \int \frac {1}{b+a \cos (c+d x)}dx}{a}-\frac {x \left (a^2-6 b^2\right ) \left (a^2-b^2\right )}{a}}{a}-\frac {6 b \left (a^2-b^2\right ) \sin (c+d x)}{a d}}{2 a}-\frac {3 \left (a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^2(c+d x)}{a d (a \cos (c+d x)+b)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\frac {2 b \left (2 a^2-3 b^2\right ) \left (a^2-b^2\right ) \int \frac {1}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}-\frac {x \left (a^2-6 b^2\right ) \left (a^2-b^2\right )}{a}}{a}-\frac {6 b \left (a^2-b^2\right ) \sin (c+d x)}{a d}}{2 a}-\frac {3 \left (a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^2(c+d x)}{a d (a \cos (c+d x)+b)}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {-\frac {\frac {\frac {4 b \left (2 a^2-3 b^2\right ) \left (a^2-b^2\right ) \int \frac {1}{-\left ((a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}-\frac {x \left (a^2-6 b^2\right ) \left (a^2-b^2\right )}{a}}{a}-\frac {6 b \left (a^2-b^2\right ) \sin (c+d x)}{a d}}{2 a}-\frac {3 \left (a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^2(c+d x)}{a d (a \cos (c+d x)+b)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {-\frac {\frac {\frac {4 b \left (2 a^2-3 b^2\right ) \left (a^2-b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}-\frac {x \left (a^2-6 b^2\right ) \left (a^2-b^2\right )}{a}}{a}-\frac {6 b \left (a^2-b^2\right ) \sin (c+d x)}{a d}}{2 a}-\frac {3 \left (a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{2 a d}}{a \left (a^2-b^2\right )}+\frac {\sin (c+d x) \cos ^2(c+d x)}{a d (a \cos (c+d x)+b)}\) |
(Cos[c + d*x]^2*Sin[c + d*x])/(a*d*(b + a*Cos[c + d*x])) + ((-3*(a^2 - b^2 )*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((-(((a^2 - 6*b^2)*(a^2 - b^2)*x)/a ) + (4*b*(2*a^2 - 3*b^2)*(a^2 - b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2] )/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d))/a - (6*b*(a^2 - b^2)*Sin[c + d*x])/(a*d))/(2*a))/(a*(a^2 - b^2))
3.3.18.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, n }, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^ 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A* d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(m + n + 2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] : > Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x ])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*( n + 1)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + C* (a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f , A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 1.30 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.31
| method | result | size |
| derivativedivides | \(\frac {\frac {2 b \left (-\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b}-\frac {\left (2 a^{2}-3 b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{4}}+\frac {\frac {2 \left (\left (\frac {1}{2} a^{2}+2 a b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (2 a b -\frac {1}{2} a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\left (a^{2}-6 b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}}{d}\) | \(199\) |
| default | \(\frac {\frac {2 b \left (-\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b}-\frac {\left (2 a^{2}-3 b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{4}}+\frac {\frac {2 \left (\left (\frac {1}{2} a^{2}+2 a b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (2 a b -\frac {1}{2} a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\left (a^{2}-6 b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}}{d}\) | \(199\) |
| risch | \(\frac {x}{2 a^{2}}-\frac {3 x \,b^{2}}{a^{4}}+\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{2} d}-\frac {i b \,{\mathrm e}^{i \left (d x +c \right )}}{a^{3} d}+\frac {i b \,{\mathrm e}^{-i \left (d x +c \right )}}{a^{3} d}-\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{2} d}+\frac {2 i b^{2} \left (b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}{a^{4} d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}-\frac {2 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, d \,a^{2}}+\frac {3 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, d \,a^{4}}+\frac {2 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, d \,a^{2}}-\frac {3 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, d \,a^{4}}\) | \(439\) |
1/d*(2*b/a^4*(-a*b*tan(1/2*d*x+1/2*c)/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+ 1/2*c)^2*b-a-b)-(2*a^2-3*b^2)/((a-b)*(a+b))^(1/2)*arctanh((a-b)*tan(1/2*d* x+1/2*c)/((a-b)*(a+b))^(1/2)))+2/a^4*(((1/2*a^2+2*a*b)*tan(1/2*d*x+1/2*c)^ 3+(2*a*b-1/2*a^2)*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^2)^2+1/2*(a^2- 6*b^2)*arctan(tan(1/2*d*x+1/2*c))))
Time = 0.31 (sec) , antiderivative size = 551, normalized size of antiderivative = 3.62 \[ \int \frac {\sin ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\left [\frac {{\left (a^{5} - 7 \, a^{3} b^{2} + 6 \, a b^{4}\right )} d x \cos \left (d x + c\right ) + {\left (a^{4} b - 7 \, a^{2} b^{3} + 6 \, b^{5}\right )} d x - {\left (2 \, a^{2} b^{2} - 3 \, b^{4} + {\left (2 \, a^{3} b - 3 \, a b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) + {\left (6 \, a^{3} b^{2} - 6 \, a b^{4} - {\left (a^{5} - a^{3} b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (a^{4} b - a^{2} b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{7} - a^{5} b^{2}\right )} d \cos \left (d x + c\right ) + {\left (a^{6} b - a^{4} b^{3}\right )} d\right )}}, \frac {{\left (a^{5} - 7 \, a^{3} b^{2} + 6 \, a b^{4}\right )} d x \cos \left (d x + c\right ) + {\left (a^{4} b - 7 \, a^{2} b^{3} + 6 \, b^{5}\right )} d x - 2 \, {\left (2 \, a^{2} b^{2} - 3 \, b^{4} + {\left (2 \, a^{3} b - 3 \, a b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) + {\left (6 \, a^{3} b^{2} - 6 \, a b^{4} - {\left (a^{5} - a^{3} b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (a^{4} b - a^{2} b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{7} - a^{5} b^{2}\right )} d \cos \left (d x + c\right ) + {\left (a^{6} b - a^{4} b^{3}\right )} d\right )}}\right ] \]
[1/2*((a^5 - 7*a^3*b^2 + 6*a*b^4)*d*x*cos(d*x + c) + (a^4*b - 7*a^2*b^3 + 6*b^5)*d*x - (2*a^2*b^2 - 3*b^4 + (2*a^3*b - 3*a*b^3)*cos(d*x + c))*sqrt(a ^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 + 2*sqrt( a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) + (6*a^3*b^2 - 6*a*b^4 - (a^5 - a^3*b^2 )*cos(d*x + c)^2 + 3*(a^4*b - a^2*b^3)*cos(d*x + c))*sin(d*x + c))/((a^7 - a^5*b^2)*d*cos(d*x + c) + (a^6*b - a^4*b^3)*d), 1/2*((a^5 - 7*a^3*b^2 + 6 *a*b^4)*d*x*cos(d*x + c) + (a^4*b - 7*a^2*b^3 + 6*b^5)*d*x - 2*(2*a^2*b^2 - 3*b^4 + (2*a^3*b - 3*a*b^3)*cos(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt( -a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) + (6*a^3*b^2 - 6*a*b^4 - (a^5 - a^3*b^2)*cos(d*x + c)^2 + 3*(a^4*b - a^2*b^3)*cos(d*x + c))*sin(d*x + c))/((a^7 - a^5*b^2)*d*cos(d*x + c) + (a^6*b - a^4*b^3)*d)]
\[ \int \frac {\sin ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\int \frac {\sin ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {\sin ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f or more de
Time = 0.32 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.58 \[ \int \frac {\sin ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx=-\frac {\frac {4 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )} a^{3}} - \frac {{\left (a^{2} - 6 \, b^{2}\right )} {\left (d x + c\right )}}{a^{4}} + \frac {4 \, {\left (2 \, a^{2} b - 3 \, b^{3}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{\sqrt {-a^{2} + b^{2}} a^{4}} - \frac {2 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{3}}}{2 \, d} \]
-1/2*(4*b^2*tan(1/2*d*x + 1/2*c)/((a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d* x + 1/2*c)^2 - a - b)*a^3) - (a^2 - 6*b^2)*(d*x + c)/a^4 + 4*(2*a^2*b - 3* b^3)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/ 2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(-a^2 + b^2)))/(sqrt(-a^2 + b ^2)*a^4) - 2*(a*tan(1/2*d*x + 1/2*c)^3 + 4*b*tan(1/2*d*x + 1/2*c)^3 - a*ta n(1/2*d*x + 1/2*c) + 4*b*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^2*a^3))/d
Time = 16.44 (sec) , antiderivative size = 1655, normalized size of antiderivative = 10.89 \[ \int \frac {\sin ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \]
((2*tan(c/2 + (d*x)/2)^3*(a^2 + 6*b^2))/a^3 + (tan(c/2 + (d*x)/2)*(3*a*b - a^2 + 6*b^2))/a^3 - (tan(c/2 + (d*x)/2)^5*(3*a*b + a^2 - 6*b^2))/a^3)/(d* (a + b + tan(c/2 + (d*x)/2)^2*(a + 3*b) - tan(c/2 + (d*x)/2)^4*(a - 3*b) - tan(c/2 + (d*x)/2)^6*(a - b))) + (atan((8*tan(c/2 + (d*x)/2))/((8*b)/a + (24*b^2)/a^2 - (24*b^3)/a^3 + (144*b^4)/a^4 - (144*b^5)/a^5 - 8) + (8*b*ta n(c/2 + (d*x)/2))/(8*a - 8*b - (24*b^2)/a + (24*b^3)/a^2 - (144*b^4)/a^3 + (144*b^5)/a^4) - (24*b^2*tan(c/2 + (d*x)/2))/(8*a*b - 8*a^2 + 24*b^2 - (2 4*b^3)/a + (144*b^4)/a^2 - (144*b^5)/a^3) + (24*b^3*tan(c/2 + (d*x)/2))/(2 4*a*b^2 + 8*a^2*b - 8*a^3 - 24*b^3 + (144*b^4)/a - (144*b^5)/a^2) + (144*b ^4*tan(c/2 + (d*x)/2))/(24*a*b^3 - 8*a^3*b + 8*a^4 - 144*b^4 - 24*a^2*b^2 + (144*b^5)/a) + (144*b^5*tan(c/2 + (d*x)/2))/(144*a*b^4 + 8*a^4*b - 8*a^5 - 144*b^5 - 24*a^2*b^3 + 24*a^3*b^2))*(a^2*1i - b^2*6i)*1i)/(a^4*d) - (b* atan(((b*((a + b)*(a - b))^(1/2)*(2*a^2 - 3*b^2)*((8*tan(c/2 + (d*x)/2)*(1 44*a*b^6 - 3*a^6*b + a^7 - 72*b^7 - 48*a^2*b^5 - 48*a^3*b^4 + 19*a^4*b^3 + 7*a^5*b^2))/a^6 + (b*((a + b)*(a - b))^(1/2)*((8*(2*a^12 - 10*a^11*b - 12 *a^8*b^4 + 18*a^9*b^3 + 2*a^10*b^2))/a^9 - (8*b*tan(c/2 + (d*x)/2)*((a + b )*(a - b))^(1/2)*(2*a^2 - 3*b^2)*(8*a^10*b + 8*a^8*b^3 - 16*a^9*b^2))/(a^6 *(a^6 - a^4*b^2)))*(2*a^2 - 3*b^2))/(a^6 - a^4*b^2))*1i)/(a^6 - a^4*b^2) + (b*((a + b)*(a - b))^(1/2)*(2*a^2 - 3*b^2)*((8*tan(c/2 + (d*x)/2)*(144*a* b^6 - 3*a^6*b + a^7 - 72*b^7 - 48*a^2*b^5 - 48*a^3*b^4 + 19*a^4*b^3 + 7...