Integrand size = 21, antiderivative size = 48 \[ \int (a+a \cos (c+d x))^3 \sec ^2(c+d x) \, dx=3 a^3 x+\frac {3 a^3 \text {arctanh}(\sin (c+d x))}{d}+\frac {a^3 \sin (c+d x)}{d}+\frac {a^3 \tan (c+d x)}{d} \]
Leaf count is larger than twice the leaf count of optimal. \(211\) vs. \(2(48)=96\).
Time = 1.08 (sec) , antiderivative size = 211, normalized size of antiderivative = 4.40 \[ \int (a+a \cos (c+d x))^3 \sec ^2(c+d x) \, dx=\frac {1}{8} a^3 (1+\cos (c+d x))^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (3 x-\frac {3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {\cos (d x) \sin (c)}{d}+\frac {\cos (c) \sin (d x)}{d}+\frac {\sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {\sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}\right ) \]
(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(3*x - (3*Log[Cos[(c + d*x)/2 ] - Sin[(c + d*x)/2]])/d + (3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (Cos[d*x]*Sin[c])/d + (Cos[c]*Sin[d*x])/d + Sin[(d*x)/2]/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + Sin[(d*x)/2]/(d*(Cos[c/ 2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/8
Time = 0.26 (sec) , antiderivative size = 48, 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.
\(\displaystyle \int \sec ^2(c+d x) (a \cos (c+d x)+a)^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx\) |
\(\Big \downarrow \) 3236 |
\(\displaystyle \int \left (a^3 \cos (c+d x)+a^3 \sec ^2(c+d x)+3 a^3 \sec (c+d x)+3 a^3\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 a^3 \text {arctanh}(\sin (c+d x))}{d}+\frac {a^3 \sin (c+d x)}{d}+\frac {a^3 \tan (c+d x)}{d}+3 a^3 x\) |
3.1.28.3.1 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 = 2.26 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.15
method | result | size |
derivativedivides | \(\frac {a^{3} \sin \left (d x +c \right )+3 a^{3} \left (d x +c \right )+3 a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{3} \tan \left (d x +c \right )}{d}\) | \(55\) |
default | \(\frac {a^{3} \sin \left (d x +c \right )+3 a^{3} \left (d x +c \right )+3 a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{3} \tan \left (d x +c \right )}{d}\) | \(55\) |
parts | \(\frac {a^{3} \tan \left (d x +c \right )}{d}+\frac {a^{3} \sin \left (d x +c \right )}{d}+\frac {3 a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {3 a^{3} \left (d x +c \right )}{d}\) | \(63\) |
parallelrisch | \(\frac {a^{3} \left (6 d x \cos \left (d x +c \right )-6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )+6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )+2 \sin \left (d x +c \right )+\sin \left (2 d x +2 c \right )\right )}{2 d \cos \left (d x +c \right )}\) | \(85\) |
risch | \(3 a^{3} x -\frac {i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {2 i a^{3}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}\) | \(108\) |
norman | \(\frac {-3 a^{3} x -\frac {4 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {8 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-6 a^{3} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a^{3} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a^{3} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {3 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}+\frac {3 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(186\) |
Time = 0.30 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.90 \[ \int (a+a \cos (c+d x))^3 \sec ^2(c+d x) \, dx=\frac {6 \, a^{3} d x \cos \left (d x + c\right ) + 3 \, a^{3} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, a^{3} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (a^{3} \cos \left (d x + c\right ) + a^{3}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \]
1/2*(6*a^3*d*x*cos(d*x + c) + 3*a^3*cos(d*x + c)*log(sin(d*x + c) + 1) - 3 *a^3*cos(d*x + c)*log(-sin(d*x + c) + 1) + 2*(a^3*cos(d*x + c) + a^3)*sin( d*x + c))/(d*cos(d*x + c))
\[ \int (a+a \cos (c+d x))^3 \sec ^2(c+d x) \, dx=a^{3} \left (\int 3 \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 3 \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \cos ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
a**3*(Integral(3*cos(c + d*x)*sec(c + d*x)**2, x) + Integral(3*cos(c + d*x )**2*sec(c + d*x)**2, x) + Integral(cos(c + d*x)**3*sec(c + d*x)**2, x) + Integral(sec(c + d*x)**2, x))
Time = 0.22 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.33 \[ \int (a+a \cos (c+d x))^3 \sec ^2(c+d x) \, dx=\frac {6 \, {\left (d x + c\right )} a^{3} + 3 \, a^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, a^{3} \sin \left (d x + c\right ) + 2 \, a^{3} \tan \left (d x + c\right )}{2 \, d} \]
1/2*(6*(d*x + c)*a^3 + 3*a^3*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1 )) + 2*a^3*sin(d*x + c) + 2*a^3*tan(d*x + c))/d
Time = 0.45 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.67 \[ \int (a+a \cos (c+d x))^3 \sec ^2(c+d x) \, dx=\frac {3 \, {\left (d x + c\right )} a^{3} + 3 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {4 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1}}{d} \]
(3*(d*x + c)*a^3 + 3*a^3*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 3*a^3*log(ab s(tan(1/2*d*x + 1/2*c) - 1)) - 4*a^3*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1 /2*c)^4 - 1))/d
Time = 14.89 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.19 \[ \int (a+a \cos (c+d x))^3 \sec ^2(c+d x) \, dx=3\,a^3\,x+\frac {6\,a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {4\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-1\right )} \]