Integrand size = 21, antiderivative size = 108 \[ \int \cos ^2(c+d x) (a+b \sec (c+d x))^4 \, dx=\frac {1}{2} a^2 \left (a^2+12 b^2\right ) x+\frac {4 a b^3 \text {arctanh}(\sin (c+d x))}{d}+\frac {3 a^3 b \sin (c+d x)}{d}+\frac {a^2 \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{2 d}-\frac {b^2 \left (a^2-2 b^2\right ) \tan (c+d x)}{2 d} \]
1/2*a^2*(a^2+12*b^2)*x+4*a*b^3*arctanh(sin(d*x+c))/d+3*a^3*b*sin(d*x+c)/d+ 1/2*a^2*cos(d*x+c)*(a+b*sec(d*x+c))^2*sin(d*x+c)/d-1/2*b^2*(a^2-2*b^2)*tan (d*x+c)/d
Time = 0.98 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.10 \[ \int \cos ^2(c+d x) (a+b \sec (c+d x))^4 \, dx=\frac {2 a \left (a \left (a^2+12 b^2\right ) (c+d x)-8 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+8 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+16 a^3 b \sin (c+d x)+a^4 \sin (2 (c+d x))+4 b^4 \tan (c+d x)}{4 d} \]
(2*a*(a*(a^2 + 12*b^2)*(c + d*x) - 8*b^3*Log[Cos[(c + d*x)/2] - Sin[(c + d *x)/2]] + 8*b^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 16*a^3*b*Sin[c + d*x] + a^4*Sin[2*(c + d*x)] + 4*b^4*Tan[c + d*x])/(4*d)
Time = 0.81 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3042, 4328, 3042, 4564, 3042, 4535, 24, 3042, 4533, 3042, 4257}
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.
| \(\displaystyle \int \cos ^2(c+d x) (a+b \sec (c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx\) |
| \(\Big \downarrow \) 4328 |
| \(\displaystyle \frac {1}{2} \int \cos (c+d x) (a+b \sec (c+d x)) \left (6 b a^2+\left (a^2+6 b^2\right ) \sec (c+d x) a-b \left (a^2-2 b^2\right ) \sec ^2(c+d x)\right )dx+\frac {a^2 \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (6 b a^2+\left (a^2+6 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a-b \left (a^2-2 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a^2 \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\) |
| \(\Big \downarrow \) 4564 |
| \(\displaystyle \frac {1}{2} \left (\int \cos (c+d x) \left (6 b a^3+\left (a^2+12 b^2\right ) \sec (c+d x) a^2+8 b^3 \sec ^2(c+d x) a\right )dx-\frac {b^2 \left (a^2-2 b^2\right ) \tan (c+d x)}{d}\right )+\frac {a^2 \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\int \frac {6 b a^3+\left (a^2+12 b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2+8 b^3 \csc \left (c+d x+\frac {\pi }{2}\right )^2 a}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {b^2 \left (a^2-2 b^2\right ) \tan (c+d x)}{d}\right )+\frac {a^2 \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{2} \left (\int \cos (c+d x) \left (6 b a^3+8 b^3 \sec ^2(c+d x) a\right )dx+a^2 \left (a^2+12 b^2\right ) \int 1dx-\frac {b^2 \left (a^2-2 b^2\right ) \tan (c+d x)}{d}\right )+\frac {a^2 \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{2} \left (\int \cos (c+d x) \left (6 b a^3+8 b^3 \sec ^2(c+d x) a\right )dx-\frac {b^2 \left (a^2-2 b^2\right ) \tan (c+d x)}{d}+a^2 x \left (a^2+12 b^2\right )\right )+\frac {a^2 \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (\int \frac {6 b a^3+8 b^3 \csc \left (c+d x+\frac {\pi }{2}\right )^2 a}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {b^2 \left (a^2-2 b^2\right ) \tan (c+d x)}{d}+a^2 x \left (a^2+12 b^2\right )\right )+\frac {a^2 \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\) |
| \(\Big \downarrow \) 4533 |
| \(\displaystyle \frac {1}{2} \left (8 a b^3 \int \sec (c+d x)dx+\frac {6 a^3 b \sin (c+d x)}{d}-\frac {b^2 \left (a^2-2 b^2\right ) \tan (c+d x)}{d}+a^2 x \left (a^2+12 b^2\right )\right )+\frac {a^2 \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (8 a b^3 \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {6 a^3 b \sin (c+d x)}{d}-\frac {b^2 \left (a^2-2 b^2\right ) \tan (c+d x)}{d}+a^2 x \left (a^2+12 b^2\right )\right )+\frac {a^2 \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {a^2 \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}+\frac {1}{2} \left (\frac {6 a^3 b \sin (c+d x)}{d}-\frac {b^2 \left (a^2-2 b^2\right ) \tan (c+d x)}{d}+a^2 x \left (a^2+12 b^2\right )+\frac {8 a b^3 \text {arctanh}(\sin (c+d x))}{d}\right )\) |
(a^2*Cos[c + d*x]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + (a^2*(a^2 + 12*b^2)*x + (8*a*b^3*ArcTanh[Sin[c + d*x]])/d + (6*a^3*b*Sin[c + d*x])/d - (b^2*(a^2 - 2*b^2)*Tan[c + d*x])/d)/2
3.5.81.3.1 Defintions of rubi rules used
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_), x_Symbol] :> Simp[a^2*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)* ((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(d*n) Int[(a + b*Csc[e + f*x])^(m - 3)*(d*Csc[e + f*x])^(n + 1)*Simp[a^2*b*(m - 2*n - 2) - a*(3*b^2*n + a^2* (n + 1))*Csc[e + f*x] - b*(b^2*n + a^2*(m + n - 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 2] && ((Int egerQ[m] && LtQ[n, -1]) || (IntegersQ[m + 1/2, 2*n] && LeQ[n, -1]))
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[A*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*m)), x] + Simp[(C*m + A*(m + 1))/(b^2*m) Int[(b*Csc[e + f*x])^(m + 2), x], x] /; Fr eeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]* (B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.)), x_Symbol] :> Simp[B/b Int[(b*Cs c[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2) , x] /; FreeQ[{b, e, f, A, B, C, m}, x]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[(-b)*C*Csc[e + f*x]*Cot[e + f*x]*((d*Csc[e + f*x])^ n/(f*(n + 2))), x] + Simp[1/(n + 2) Int[(d*Csc[e + f*x])^n*Simp[A*a*(n + 2) + (B*a*(n + 2) + b*(C*(n + 1) + A*(n + 2)))*Csc[e + f*x] + (a*C + B*b)*( n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && !LtQ[n, -1]
Time = 0.90 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.81
| method | result | size |
| derivativedivides | \(\frac {a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{3} b \sin \left (d x +c \right )+6 a^{2} b^{2} \left (d x +c \right )+4 a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+\tan \left (d x +c \right ) b^{4}}{d}\) | \(87\) |
| default | \(\frac {a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{3} b \sin \left (d x +c \right )+6 a^{2} b^{2} \left (d x +c \right )+4 a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+\tan \left (d x +c \right ) b^{4}}{d}\) | \(87\) |
| parallelrisch | \(\frac {-32 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a \,b^{3} \cos \left (d x +c \right )+32 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a \,b^{3} \cos \left (d x +c \right )+16 a^{3} b \sin \left (2 d x +2 c \right )+a^{4} \sin \left (3 d x +3 c \right )+4 a^{2} d x \left (a^{2}+12 b^{2}\right ) \cos \left (d x +c \right )+\sin \left (d x +c \right ) \left (a^{4}+8 b^{4}\right )}{8 d \cos \left (d x +c \right )}\) | \(129\) |
| risch | \(\frac {a^{4} x}{2}+6 a^{2} b^{2} x -\frac {i a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {2 i a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}}{d}+\frac {2 i a^{3} b \,{\mathrm e}^{-i \left (d x +c \right )}}{d}+\frac {i a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {2 i b^{4}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {4 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {4 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}\) | \(157\) |
| norman | \(\frac {\left (-\frac {1}{2} a^{4}-6 a^{2} b^{2}\right ) x +\left (-\frac {1}{2} a^{4}-6 a^{2} b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (\frac {1}{2} a^{4}+6 a^{2} b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {1}{2} a^{4}+6 a^{2} b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (-a^{4}-12 a^{2} b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (a^{4}+12 a^{2} b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\frac {2 \left (3 a^{4}-2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}-\frac {\left (a^{4}-8 a^{3} b +2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}-\frac {\left (a^{4}+8 a^{3} b +2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {4 a^{3} \left (a -4 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}+\frac {4 a^{3} \left (4 b +a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-\frac {4 a \,b^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}+\frac {4 a \,b^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(360\) |
1/d*(a^4*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+4*a^3*b*sin(d*x+c)+6*a^ 2*b^2*(d*x+c)+4*a*b^3*ln(sec(d*x+c)+tan(d*x+c))+tan(d*x+c)*b^4)
Time = 0.30 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.07 \[ \int \cos ^2(c+d x) (a+b \sec (c+d x))^4 \, dx=\frac {4 \, a b^{3} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 4 \, a b^{3} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (a^{4} + 12 \, a^{2} b^{2}\right )} d x \cos \left (d x + c\right ) + {\left (a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{3} b \cos \left (d x + c\right ) + 2 \, b^{4}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
1/2*(4*a*b^3*cos(d*x + c)*log(sin(d*x + c) + 1) - 4*a*b^3*cos(d*x + c)*log (-sin(d*x + c) + 1) + (a^4 + 12*a^2*b^2)*d*x*cos(d*x + c) + (a^4*cos(d*x + c)^2 + 8*a^3*b*cos(d*x + c) + 2*b^4)*sin(d*x + c))/(d*cos(d*x + c))
\[ \int \cos ^2(c+d x) (a+b \sec (c+d x))^4 \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{4} \cos ^{2}{\left (c + d x \right )}\, dx \]
Time = 0.22 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int \cos ^2(c+d x) (a+b \sec (c+d x))^4 \, dx=\frac {{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} + 24 \, {\left (d x + c\right )} a^{2} b^{2} + 8 \, a b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 16 \, a^{3} b \sin \left (d x + c\right ) + 4 \, b^{4} \tan \left (d x + c\right )}{4 \, d} \]
1/4*((2*d*x + 2*c + sin(2*d*x + 2*c))*a^4 + 24*(d*x + c)*a^2*b^2 + 8*a*b^3 *(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 16*a^3*b*sin(d*x + c) + 4*b^4*tan(d*x + c))/d
Time = 0.34 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.57 \[ \int \cos ^2(c+d x) (a+b \sec (c+d x))^4 \, dx=\frac {8 \, a b^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 8 \, a b^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {4 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + {\left (a^{4} + 12 \, a^{2} b^{2}\right )} {\left (d x + c\right )} - \frac {2 \, {\left (a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
1/2*(8*a*b^3*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 8*a*b^3*log(abs(tan(1/2* d*x + 1/2*c) - 1)) - 4*b^4*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 - 1) + (a^4 + 12*a^2*b^2)*(d*x + c) - 2*(a^4*tan(1/2*d*x + 1/2*c)^3 - 8*a^3* b*tan(1/2*d*x + 1/2*c)^3 - a^4*tan(1/2*d*x + 1/2*c) - 8*a^3*b*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^2)/d
Time = 14.20 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.39 \[ \int \cos ^2(c+d x) (a+b \sec (c+d x))^4 \, dx=\frac {a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {b^4\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {12\,a^2\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d}+\frac {4\,a^3\,b\,\sin \left (c+d\,x\right )}{d}+\frac {8\,a\,b^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]