Integrand size = 19, antiderivative size = 62 \[ \int (a+a \sec (c+d x))^3 \sin (c+d x) \, dx=-\frac {a^3 \cos (c+d x)}{d}-\frac {3 a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \]
Time = 0.18 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.05 \[ \int (a+a \sec (c+d x))^3 \sin (c+d x) \, dx=-\frac {a^3 (-4-9 \cos (c+d x)+\cos (3 (c+d x))+6 \log (\cos (c+d x))+\cos (2 (c+d x)) (-2+6 \log (\cos (c+d x)))) \sec ^2(c+d x)}{4 d} \]
-1/4*(a^3*(-4 - 9*Cos[c + d*x] + Cos[3*(c + d*x)] + 6*Log[Cos[c + d*x]] + Cos[2*(c + d*x)]*(-2 + 6*Log[Cos[c + d*x]]))*Sec[c + d*x]^2)/d
Time = 0.35 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.82, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {3042, 4360, 25, 25, 3042, 25, 3312, 27, 49, 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.
\(\displaystyle \int \sin (c+d x) (a \sec (c+d x)+a)^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \cos \left (c+d x-\frac {\pi }{2}\right ) \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3dx\) |
\(\Big \downarrow \) 4360 |
\(\displaystyle \int \tan (c+d x) \sec ^2(c+d x) \left (-(a (-\cos (c+d x))-a)^3\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int -(\cos (c+d x) a+a)^3 \sec ^2(c+d x) \tan (c+d x)dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\cos \left (c+d x+\frac {\pi }{2}\right ) \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right ) \left (\sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^3}{\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3}dx\) |
\(\Big \downarrow \) 3312 |
\(\displaystyle -\frac {\int (\cos (c+d x) a+a)^3 \sec ^3(c+d x)d(a \cos (c+d x))}{a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {a^2 \int \frac {(\cos (c+d x) a+a)^3 \sec ^3(c+d x)}{a^3}d(a \cos (c+d x))}{d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle -\frac {a^2 \int \left (\sec ^3(c+d x)+3 \sec ^2(c+d x)+3 \sec (c+d x)+1\right )d(a \cos (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^2 \left (a \cos (c+d x)-\frac {1}{2} a \sec ^2(c+d x)-3 a \sec (c+d x)+3 a \log (a \cos (c+d x))\right )}{d}\) |
-((a^2*(a*Cos[c + d*x] + 3*a*Log[a*Cos[c + d*x]] - 3*a*Sec[c + d*x] - (a*S ec[c + d*x]^2)/2))/d)
3.1.42.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*(( c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b*f) Su bst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
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 = 0.98 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.74
method | result | size |
derivativedivides | \(\frac {a^{3} \left (\frac {\sec \left (d x +c \right )^{2}}{2}+3 \sec \left (d x +c \right )+3 \ln \left (\sec \left (d x +c \right )\right )-\frac {1}{\sec \left (d x +c \right )}\right )}{d}\) | \(46\) |
default | \(\frac {a^{3} \left (\frac {\sec \left (d x +c \right )^{2}}{2}+3 \sec \left (d x +c \right )+3 \ln \left (\sec \left (d x +c \right )\right )-\frac {1}{\sec \left (d x +c \right )}\right )}{d}\) | \(46\) |
parts | \(-\frac {a^{3} \cos \left (d x +c \right )}{d}+\frac {a^{3} \sec \left (d x +c \right )^{2}}{2 d}+\frac {3 a^{3} \ln \left (\sec \left (d x +c \right )\right )}{d}+\frac {3 a^{3} \sec \left (d x +c \right )}{d}\) | \(61\) |
risch | \(3 i a^{3} x -\frac {a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {6 i a^{3} c}{d}+\frac {2 a^{3} \left (3 \,{\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {3 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(126\) |
norman | \(\frac {\frac {4 a^{3}}{d}+\frac {6 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}-\frac {6 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}-\frac {3 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {3 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}+\frac {3 a^{3} \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d}\) | \(142\) |
parallelrisch | \(\frac {a^{3} \left (-6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (2 d x +2 c \right )-6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (2 d x +2 c \right )+6 \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \cos \left (2 d x +2 c \right )+9 \cos \left (d x +c \right )-\cos \left (3 d x +3 c \right )+3 \cos \left (2 d x +2 c \right )-6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+6 \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+5\right )}{2 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(165\) |
Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.05 \[ \int (a+a \sec (c+d x))^3 \sin (c+d x) \, dx=-\frac {2 \, a^{3} \cos \left (d x + c\right )^{3} + 6 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) - 6 \, a^{3} \cos \left (d x + c\right ) - a^{3}}{2 \, d \cos \left (d x + c\right )^{2}} \]
-1/2*(2*a^3*cos(d*x + c)^3 + 6*a^3*cos(d*x + c)^2*log(-cos(d*x + c)) - 6*a ^3*cos(d*x + c) - a^3)/(d*cos(d*x + c)^2)
\[ \int (a+a \sec (c+d x))^3 \sin (c+d x) \, dx=a^{3} \left (\int 3 \sin {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 \sin {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \sin {\left (c + d x \right )}\, dx\right ) \]
a**3*(Integral(3*sin(c + d*x)*sec(c + d*x), x) + Integral(3*sin(c + d*x)*s ec(c + d*x)**2, x) + Integral(sin(c + d*x)*sec(c + d*x)**3, x) + Integral( sin(c + d*x), x))
Time = 0.21 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.89 \[ \int (a+a \sec (c+d x))^3 \sin (c+d x) \, dx=-\frac {2 \, a^{3} \cos \left (d x + c\right ) + 6 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) - \frac {6 \, a^{3}}{\cos \left (d x + c\right )} - \frac {a^{3}}{\cos \left (d x + c\right )^{2}}}{2 \, d} \]
-1/2*(2*a^3*cos(d*x + c) + 6*a^3*log(cos(d*x + c)) - 6*a^3/cos(d*x + c) - a^3/cos(d*x + c)^2)/d
Time = 0.33 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.03 \[ \int (a+a \sec (c+d x))^3 \sin (c+d x) \, dx=-\frac {a^{3} \cos \left (d x + c\right )}{d} - \frac {3 \, a^{3} \log \left (\frac {{\left | \cos \left (d x + c\right ) \right |}}{{\left | d \right |}}\right )}{d} + \frac {6 \, a^{3} \cos \left (d x + c\right ) + a^{3}}{2 \, d \cos \left (d x + c\right )^{2}} \]
-a^3*cos(d*x + c)/d - 3*a^3*log(abs(cos(d*x + c))/abs(d))/d + 1/2*(6*a^3*c os(d*x + c) + a^3)/(d*cos(d*x + c)^2)
Time = 0.06 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.84 \[ \int (a+a \sec (c+d x))^3 \sin (c+d x) \, dx=\frac {a^3\,\left (3\,\cos \left (c+d\,x\right )-{\cos \left (c+d\,x\right )}^3-3\,{\cos \left (c+d\,x\right )}^2\,\ln \left (\cos \left (c+d\,x\right )\right )+\frac {1}{2}\right )}{d\,{\cos \left (c+d\,x\right )}^2} \]