Integrand size = 21, antiderivative size = 161 \[ \int \frac {\sin ^4(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\left (3 a^4-12 a^2 b^2+8 b^4\right ) x}{8 a^5}-\frac {2 (a-b)^{3/2} b (a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d}+\frac {\left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cos (c+d x)\right ) \sin (c+d x)}{8 a^4 d}+\frac {(4 b-3 a \cos (c+d x)) \sin ^3(c+d x)}{12 a^2 d} \]
1/8*(3*a^4-12*a^2*b^2+8*b^4)*x/a^5-2*(a-b)^(3/2)*b*(a+b)^(3/2)*arctanh((a- b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^5/d+1/8*(8*b*(a^2-b^2)-a*(3*a^2 -4*b^2)*cos(d*x+c))*sin(d*x+c)/a^4/d+1/12*(4*b-3*a*cos(d*x+c))*sin(d*x+c)^ 3/a^2/d
Time = 0.75 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.07 \[ \int \frac {\sin ^4(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {36 a^4 c-144 a^2 b^2 c+96 b^4 c+36 a^4 d x-144 a^2 b^2 d x+96 b^4 d x+192 b \left (a^2-b^2\right )^{3/2} \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+24 a b \left (5 a^2-4 b^2\right ) \sin (c+d x)-24 \left (a^4-a^2 b^2\right ) \sin (2 (c+d x))-8 a^3 b \sin (3 (c+d x))+3 a^4 \sin (4 (c+d x))}{96 a^5 d} \]
(36*a^4*c - 144*a^2*b^2*c + 96*b^4*c + 36*a^4*d*x - 144*a^2*b^2*d*x + 96*b ^4*d*x + 192*b*(a^2 - b^2)^(3/2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[ a^2 - b^2]] + 24*a*b*(5*a^2 - 4*b^2)*Sin[c + d*x] - 24*(a^4 - a^2*b^2)*Sin [2*(c + d*x)] - 8*a^3*b*Sin[3*(c + d*x)] + 3*a^4*Sin[4*(c + d*x)])/(96*a^5 *d)
Time = 0.90 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.16, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 4360, 25, 25, 3042, 3344, 25, 3042, 3344, 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.
| \(\displaystyle \int \frac {\sin ^4(c+d x)}{a+b \sec (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos \left (c+d x-\frac {\pi }{2}\right )^4}{a-b \csc \left (c+d x-\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\frac {\sin ^4(c+d x) \cos (c+d x)}{-a \cos (c+d x)-b}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\frac {\cos (c+d x) \sin ^4(c+d x)}{b+a \cos (c+d x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\sin ^4(c+d x) \cos (c+d x)}{a \cos (c+d x)+b}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \cos \left (c+d x+\frac {\pi }{2}\right )^4}{a \sin \left (c+d x+\frac {\pi }{2}\right )+b}dx\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle \frac {\int -\frac {\left (a b-\left (3 a^2-4 b^2\right ) \cos (c+d x)\right ) \sin ^2(c+d x)}{b+a \cos (c+d x)}dx}{4 a^2}+\frac {\sin ^3(c+d x) (4 b-3 a \cos (c+d x))}{12 a^2 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sin ^3(c+d x) (4 b-3 a \cos (c+d x))}{12 a^2 d}-\frac {\int \frac {\left (a b-\left (3 a^2-4 b^2\right ) \cos (c+d x)\right ) \sin ^2(c+d x)}{b+a \cos (c+d x)}dx}{4 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^3(c+d x) (4 b-3 a \cos (c+d x))}{12 a^2 d}-\frac {\int \frac {\cos \left (c+d x+\frac {\pi }{2}\right )^2 \left (a b+\left (4 b^2-3 a^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{4 a^2}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle \frac {\sin ^3(c+d x) (4 b-3 a \cos (c+d x))}{12 a^2 d}-\frac {\frac {\int \frac {a b \left (5 a^2-4 b^2\right )-\left (3 a^4-12 b^2 a^2+8 b^4\right ) \cos (c+d x)}{b+a \cos (c+d x)}dx}{2 a^2}-\frac {\sin (c+d x) \left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cos (c+d x)\right )}{2 a^2 d}}{4 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^3(c+d x) (4 b-3 a \cos (c+d x))}{12 a^2 d}-\frac {\frac {\int \frac {a b \left (5 a^2-4 b^2\right )+\left (-3 a^4+12 b^2 a^2-8 b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{2 a^2}-\frac {\sin (c+d x) \left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cos (c+d x)\right )}{2 a^2 d}}{4 a^2}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\sin ^3(c+d x) (4 b-3 a \cos (c+d x))}{12 a^2 d}-\frac {\frac {\frac {8 b \left (a^2-b^2\right )^2 \int \frac {1}{b+a \cos (c+d x)}dx}{a}-\frac {x \left (3 a^4-12 a^2 b^2+8 b^4\right )}{a}}{2 a^2}-\frac {\sin (c+d x) \left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cos (c+d x)\right )}{2 a^2 d}}{4 a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin ^3(c+d x) (4 b-3 a \cos (c+d x))}{12 a^2 d}-\frac {\frac {\frac {8 b \left (a^2-b^2\right )^2 \int \frac {1}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}-\frac {x \left (3 a^4-12 a^2 b^2+8 b^4\right )}{a}}{2 a^2}-\frac {\sin (c+d x) \left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cos (c+d x)\right )}{2 a^2 d}}{4 a^2}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\sin ^3(c+d x) (4 b-3 a \cos (c+d x))}{12 a^2 d}-\frac {\frac {\frac {16 b \left (a^2-b^2\right )^2 \int \frac {1}{-\left ((a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}-\frac {x \left (3 a^4-12 a^2 b^2+8 b^4\right )}{a}}{2 a^2}-\frac {\sin (c+d x) \left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cos (c+d x)\right )}{2 a^2 d}}{4 a^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\sin ^3(c+d x) (4 b-3 a \cos (c+d x))}{12 a^2 d}-\frac {\frac {\frac {16 b \left (a^2-b^2\right )^2 \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}-\frac {x \left (3 a^4-12 a^2 b^2+8 b^4\right )}{a}}{2 a^2}-\frac {\sin (c+d x) \left (8 b \left (a^2-b^2\right )-a \left (3 a^2-4 b^2\right ) \cos (c+d x)\right )}{2 a^2 d}}{4 a^2}\) |
((4*b - 3*a*Cos[c + d*x])*Sin[c + d*x]^3)/(12*a^2*d) - ((-(((3*a^4 - 12*a^ 2*b^2 + 8*b^4)*x)/a) + (16*b*(a^2 - b^2)^2*ArcTanh[(Sqrt[a - b]*Tan[(c + d *x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d))/(2*a^2) - ((8*b*(a^2 - b^2) - a*(3*a^2 - 4*b^2)*Cos[c + d*x])*Sin[c + d*x])/(2*a^2*d))/(4*a^2)
3.3.4.3.1 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_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g*(g* Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) - a*d* p + b*d*(m + p)*Sin[e + f*x])/(b^2*f*(m + p)*(m + p + 1))), x] + Simp[g^2*( (p - 1)/(b^2*(m + p)*(m + p + 1))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Si n[e + f*x])^m*Simp[b*(a*d*m + b*c*(m + p + 1)) + (a*b*c*(m + p + 1) - d*(a^ 2*p - b^2*(m + p)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && NeQ[m + p + 1 , 0] && IntegerQ[2*m]
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 = 0.97 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.64
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (a -b \right )^{2} \left (a +b \right )^{2} b \,\operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {\frac {2 \left (\left (\frac {3}{8} a^{4}+a^{3} b -\frac {1}{2} a^{2} b^{2}-a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (\frac {13}{3} a^{3} b -\frac {1}{2} a^{2} b^{2}-3 a \,b^{3}+\frac {11}{8} a^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {11}{8} a^{4}+\frac {1}{2} a^{2} b^{2}+\frac {13}{3} a^{3} b -3 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (a^{3} b -a \,b^{3}-\frac {3}{8} a^{4}+\frac {1}{2} a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}+\frac {\left (3 a^{4}-12 a^{2} b^{2}+8 b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{a^{5}}}{d}\) | \(264\) |
| default | \(\frac {-\frac {2 \left (a -b \right )^{2} \left (a +b \right )^{2} b \,\operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {\frac {2 \left (\left (\frac {3}{8} a^{4}+a^{3} b -\frac {1}{2} a^{2} b^{2}-a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (\frac {13}{3} a^{3} b -\frac {1}{2} a^{2} b^{2}-3 a \,b^{3}+\frac {11}{8} a^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {11}{8} a^{4}+\frac {1}{2} a^{2} b^{2}+\frac {13}{3} a^{3} b -3 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (a^{3} b -a \,b^{3}-\frac {3}{8} a^{4}+\frac {1}{2} a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}+\frac {\left (3 a^{4}-12 a^{2} b^{2}+8 b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{a^{5}}}{d}\) | \(264\) |
| risch | \(\frac {3 x}{8 a}-\frac {3 x \,b^{2}}{2 a^{3}}+\frac {x \,b^{4}}{a^{5}}-\frac {5 i b \,{\mathrm e}^{i \left (d x +c \right )}}{8 d \,a^{2}}+\frac {i b^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d \,a^{4}}+\frac {5 i b \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d \,a^{2}}-\frac {i b^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d \,a^{4}}+\frac {\sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right )}{d \,a^{3}}-\frac {\sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+b \sqrt {a^{2}-b^{2}}}{a \sqrt {a^{2}-b^{2}}}\right )}{d \,a^{5}}-\frac {\sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{d \,a^{3}}+\frac {\sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{d \,a^{5}}+\frac {\sin \left (4 d x +4 c \right )}{32 a d}-\frac {b \sin \left (3 d x +3 c \right )}{12 a^{2} d}-\frac {\sin \left (2 d x +2 c \right )}{4 d a}+\frac {\sin \left (2 d x +2 c \right ) b^{2}}{4 d \,a^{3}}\) | \(467\) |
1/d*(-2*(a-b)^2*(a+b)^2*b/a^5/((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^5*(((3/8*a^4+a^3*b-1/2*a^2*b^2-a*b^3)*ta n(1/2*d*x+1/2*c)^7+(13/3*a^3*b-1/2*a^2*b^2-3*a*b^3+11/8*a^4)*tan(1/2*d*x+1 /2*c)^5+(-11/8*a^4+1/2*a^2*b^2+13/3*a^3*b-3*a*b^3)*tan(1/2*d*x+1/2*c)^3+(a ^3*b-a*b^3-3/8*a^4+1/2*a^2*b^2)*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^ 2)^4+1/8*(3*a^4-12*a^2*b^2+8*b^4)*arctan(tan(1/2*d*x+1/2*c))))
Time = 0.30 (sec) , antiderivative size = 393, normalized size of antiderivative = 2.44 \[ \int \frac {\sin ^4(c+d x)}{a+b \sec (c+d x)} \, dx=\left [\frac {3 \, {\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\right )} d x - 12 \, {\left (a^{2} b - b^{3}\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) + {\left (6 \, a^{4} \cos \left (d x + c\right )^{3} - 8 \, a^{3} b \cos \left (d x + c\right )^{2} + 32 \, a^{3} b - 24 \, a b^{3} - 3 \, {\left (5 \, a^{4} - 4 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, a^{5} d}, \frac {3 \, {\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\right )} d x - 24 \, {\left (a^{2} b - b^{3}\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) + {\left (6 \, a^{4} \cos \left (d x + c\right )^{3} - 8 \, a^{3} b \cos \left (d x + c\right )^{2} + 32 \, a^{3} b - 24 \, a b^{3} - 3 \, {\left (5 \, a^{4} - 4 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, a^{5} d}\right ] \]
[1/24*(3*(3*a^4 - 12*a^2*b^2 + 8*b^4)*d*x - 12*(a^2*b - b^3)*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^4*cos(d*x + c)^3 - 8*a^3*b*cos(d*x + c) ^2 + 32*a^3*b - 24*a*b^3 - 3*(5*a^4 - 4*a^2*b^2)*cos(d*x + c))*sin(d*x + c ))/(a^5*d), 1/24*(3*(3*a^4 - 12*a^2*b^2 + 8*b^4)*d*x - 24*(a^2*b - b^3)*sq rt(-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^4*cos(d*x + c)^3 - 8*a^3*b*cos(d*x + c)^2 + 32*a^3*b - 24*a*b^3 - 3*(5*a^4 - 4*a^2*b^2)*cos(d*x + c))*sin(d*x + c))/(a^5*d)]
\[ \int \frac {\sin ^4(c+d x)}{a+b \sec (c+d x)} \, dx=\int \frac {\sin ^{4}{\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \]
Exception generated. \[ \int \frac {\sin ^4(c+d x)}{a+b \sec (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (145) = 290\).
Time = 0.31 (sec) , antiderivative size = 407, normalized size of antiderivative = 2.53 \[ \int \frac {\sin ^4(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\frac {3 \, {\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\right )} {\left (d x + c\right )}}{a^{5}} - \frac {48 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{\sqrt {-a^{2} + b^{2}} a^{5}} + \frac {2 \, {\left (9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 12 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 33 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 104 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 72 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 33 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 104 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a^{4}}}{24 \, d} \]
1/24*(3*(3*a^4 - 12*a^2*b^2 + 8*b^4)*(d*x + c)/a^5 - 48*(a^4*b - 2*a^2*b^3 + b^5)*(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^5) + 2*(9*a^3*tan(1/2*d*x + 1/2*c)^7 + 24*a^2*b*tan(1/2*d*x + 1/2 *c)^7 - 12*a*b^2*tan(1/2*d*x + 1/2*c)^7 - 24*b^3*tan(1/2*d*x + 1/2*c)^7 + 33*a^3*tan(1/2*d*x + 1/2*c)^5 + 104*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 12*a*b^ 2*tan(1/2*d*x + 1/2*c)^5 - 72*b^3*tan(1/2*d*x + 1/2*c)^5 - 33*a^3*tan(1/2* d*x + 1/2*c)^3 + 104*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 12*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 72*b^3*tan(1/2*d*x + 1/2*c)^3 - 9*a^3*tan(1/2*d*x + 1/2*c) + 2 4*a^2*b*tan(1/2*d*x + 1/2*c) + 12*a*b^2*tan(1/2*d*x + 1/2*c) - 24*b^3*tan( 1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^4*a^4))/d
Time = 14.78 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.97 \[ \int \frac {\sin ^4(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\frac {5\,b\,\sin \left (c+d\,x\right )}{4}-\frac {b\,\sin \left (3\,c+3\,d\,x\right )}{12}}{a^2\,d}-\frac {3\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )-\frac {b^2\,\sin \left (2\,c+2\,d\,x\right )}{4}}{a^3\,d}+\frac {\frac {3\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4}-\frac {\sin \left (2\,c+2\,d\,x\right )}{4}+\frac {\sin \left (4\,c+4\,d\,x\right )}{32}}{a\,d}-\frac {b^3\,\sin \left (c+d\,x\right )}{a^4\,d}+\frac {2\,b^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{a^5\,d}-\frac {2\,b\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3+\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b-\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^2-\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^3}\right )\,\sqrt {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}}{a^5\,d} \]
((5*b*sin(c + d*x))/4 - (b*sin(3*c + 3*d*x))/12)/(a^2*d) - (3*b^2*atan(sin (c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)) - (b^2*sin(2*c + 2*d*x))/4)/(a^3*d) + ((3*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/4 - sin(2*c + 2*d*x)/4 + sin(4*c + 4*d*x)/32)/(a*d) - (b^3*sin(c + d*x))/(a^4*d) + (2*b^4*atan(sin( c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/(a^5*d) - (2*b*atanh((sin(c/2 + (d*x)/ 2)*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)^(1/2))/(a^3*cos(c/2 + (d*x)/2) - b^ 3*cos(c/2 + (d*x)/2) - a*b^2*cos(c/2 + (d*x)/2) + a^2*b*cos(c/2 + (d*x)/2) ))*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)^(1/2))/(a^5*d)