Integrand size = 21, antiderivative size = 73 \[ \int (a+a \cos (c+d x))^4 \sec ^4(c+d x) \, dx=a^4 x+\frac {6 a^4 \text {arctanh}(\sin (c+d x))}{d}+\frac {7 a^4 \tan (c+d x)}{d}+\frac {2 a^4 \sec (c+d x) \tan (c+d x)}{d}+\frac {a^4 \tan ^3(c+d x)}{3 d} \]
a^4*x+6*a^4*arctanh(sin(d*x+c))/d+7*a^4*tan(d*x+c)/d+2*a^4*sec(d*x+c)*tan( d*x+c)/d+1/3*a^4*tan(d*x+c)^3/d
Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84 \[ \int (a+a \cos (c+d x))^4 \sec ^4(c+d x) \, dx=a^4 \left (x+\frac {6 \text {arctanh}(\sin (c+d x))}{d}+\frac {7 \tan (c+d x)}{d}+\frac {2 \sec (c+d x) \tan (c+d x)}{d}+\frac {\tan ^3(c+d x)}{3 d}\right ) \]
a^4*(x + (6*ArcTanh[Sin[c + d*x]])/d + (7*Tan[c + d*x])/d + (2*Sec[c + d*x ]*Tan[c + d*x])/d + Tan[c + d*x]^3/(3*d))
Time = 0.30 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3236, 2009}
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.
| \(\displaystyle \int \sec ^4(c+d x) (a \cos (c+d x)+a)^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 3236 |
| \(\displaystyle \int \left (a^4 \sec ^4(c+d x)+4 a^4 \sec ^3(c+d x)+6 a^4 \sec ^2(c+d x)+4 a^4 \sec (c+d x)+a^4\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6 a^4 \text {arctanh}(\sin (c+d x))}{d}+\frac {a^4 \tan ^3(c+d x)}{3 d}+\frac {7 a^4 \tan (c+d x)}{d}+\frac {2 a^4 \tan (c+d x) \sec (c+d x)}{d}+a^4 x\) |
a^4*x + (6*a^4*ArcTanh[Sin[c + d*x]])/d + (7*a^4*Tan[c + d*x])/d + (2*a^4* Sec[c + d*x]*Tan[c + d*x])/d + (a^4*Tan[c + d*x]^3)/(3*d)
3.1.39.3.1 Defintions of rubi rules used
Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)])^(m_.), x_Symbol] :> Int[ExpandTrig[(a + b*sin[e + f*x])^m*(d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IGt Q[m, 0] && RationalQ[n]
Time = 3.30 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.32
| method | result | size |
| parts | \(-\frac {a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {a^{4} \left (d x +c \right )}{d}+\frac {2 a^{4} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{d}+\frac {6 a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {6 a^{4} \tan \left (d x +c \right )}{d}\) | \(96\) |
| derivativedivides | \(\frac {a^{4} \left (d x +c \right )+4 a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 a^{4} \tan \left (d x +c \right )+4 a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(104\) |
| default | \(\frac {a^{4} \left (d x +c \right )+4 a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 a^{4} \tan \left (d x +c \right )+4 a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(104\) |
| risch | \(a^{4} x -\frac {4 i a^{4} \left (3 \,{\mathrm e}^{5 i \left (d x +c \right )}-9 \,{\mathrm e}^{4 i \left (d x +c \right )}-21 \,{\mathrm e}^{2 i \left (d x +c \right )}-3 \,{\mathrm e}^{i \left (d x +c \right )}-10\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {6 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {6 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}\) | \(117\) |
| parallelrisch | \(-\frac {18 \left (\left (\cos \left (d x +c \right )+\frac {\cos \left (3 d x +3 c \right )}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (-\cos \left (d x +c \right )-\frac {\cos \left (3 d x +3 c \right )}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {d x \cos \left (d x +c \right )}{6}-\frac {d x \cos \left (3 d x +3 c \right )}{18}-\frac {4 \sin \left (d x +c \right )}{9}-\frac {2 \sin \left (2 d x +2 c \right )}{9}-\frac {10 \sin \left (3 d x +3 c \right )}{27}\right ) a^{4}}{d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(147\) |
| norman | \(\frac {a^{4} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a^{4} x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a^{4} x -\frac {18 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {140 a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {50 a^{4} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {40 a^{4} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {70 a^{4} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {44 a^{4} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {10 a^{4} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-a^{4} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a^{4} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a^{4} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 a^{4} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 a^{4} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {6 a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}+\frac {6 a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(328\) |
-a^4/d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+a^4/d*(d*x+c)+2*a^4*sec(d*x+c)*t an(d*x+c)/d+6*a^4/d*ln(sec(d*x+c)+tan(d*x+c))+6*a^4*tan(d*x+c)/d
Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.51 \[ \int (a+a \cos (c+d x))^4 \sec ^4(c+d x) \, dx=\frac {3 \, a^{4} d x \cos \left (d x + c\right )^{3} + 9 \, a^{4} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 9 \, a^{4} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (20 \, a^{4} \cos \left (d x + c\right )^{2} + 6 \, a^{4} \cos \left (d x + c\right ) + a^{4}\right )} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )^{3}} \]
1/3*(3*a^4*d*x*cos(d*x + c)^3 + 9*a^4*cos(d*x + c)^3*log(sin(d*x + c) + 1) - 9*a^4*cos(d*x + c)^3*log(-sin(d*x + c) + 1) + (20*a^4*cos(d*x + c)^2 + 6*a^4*cos(d*x + c) + a^4)*sin(d*x + c))/(d*cos(d*x + c)^3)
\[ \int (a+a \cos (c+d x))^4 \sec ^4(c+d x) \, dx=a^{4} \left (\int 4 \cos {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int 6 \cos ^{2}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int 4 \cos ^{3}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \cos ^{4}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \sec ^{4}{\left (c + d x \right )}\, dx\right ) \]
a**4*(Integral(4*cos(c + d*x)*sec(c + d*x)**4, x) + Integral(6*cos(c + d*x )**2*sec(c + d*x)**4, x) + Integral(4*cos(c + d*x)**3*sec(c + d*x)**4, x) + Integral(cos(c + d*x)**4*sec(c + d*x)**4, x) + Integral(sec(c + d*x)**4, x))
Time = 0.30 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.64 \[ \int (a+a \cos (c+d x))^4 \sec ^4(c+d x) \, dx=\frac {{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{4} + 3 \, {\left (d x + c\right )} a^{4} - 3 \, a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 18 \, a^{4} \tan \left (d x + c\right )}{3 \, d} \]
1/3*((tan(d*x + c)^3 + 3*tan(d*x + c))*a^4 + 3*(d*x + c)*a^4 - 3*a^4*(2*si n(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 6*a^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 18*a^4*ta n(d*x + c))/d
Time = 0.35 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.59 \[ \int (a+a \cos (c+d x))^4 \sec ^4(c+d x) \, dx=\frac {3 \, {\left (d x + c\right )} a^{4} + 18 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 18 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (15 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 38 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 27 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{3 \, d} \]
1/3*(3*(d*x + c)*a^4 + 18*a^4*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 18*a^4* log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(15*a^4*tan(1/2*d*x + 1/2*c)^5 - 38 *a^4*tan(1/2*d*x + 1/2*c)^3 + 27*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^3)/d
Time = 14.64 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.60 \[ \int (a+a \cos (c+d x))^4 \sec ^4(c+d x) \, dx=a^4\,x+\frac {12\,a^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {10\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-\frac {76\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+18\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]