Integrand size = 21, antiderivative size = 98 \[ \int (a+a \sec (c+d x))^3 \sin ^3(c+d x) \, dx=\frac {2 a^3 \cos (c+d x)}{d}+\frac {3 a^3 \cos ^2(c+d x)}{2 d}+\frac {a^3 \cos ^3(c+d x)}{3 d}-\frac {2 a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \]
2*a^3*cos(d*x+c)/d+3/2*a^3*cos(d*x+c)^2/d+1/3*a^3*cos(d*x+c)^3/d-2*a^3*ln( cos(d*x+c))/d+3*a^3*sec(d*x+c)/d+1/2*a^3*sec(d*x+c)^2/d
Time = 0.16 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.88 \[ \int (a+a \sec (c+d x))^3 \sin ^3(c+d x) \, dx=\frac {a^3 (-41+226 \cos (c+d x)+29 \cos (3 (c+d x))+9 \cos (4 (c+d x))+\cos (5 (c+d x))-48 \log (\cos (c+d x))-8 \cos (2 (c+d x)) (7+6 \log (\cos (c+d x)))) \sec ^2(c+d x)}{48 d} \]
(a^3*(-41 + 226*Cos[c + d*x] + 29*Cos[3*(c + d*x)] + 9*Cos[4*(c + d*x)] + Cos[5*(c + d*x)] - 48*Log[Cos[c + d*x]] - 8*Cos[2*(c + d*x)]*(7 + 6*Log[Co s[c + d*x]]))*Sec[c + d*x]^2)/(48*d)
Time = 0.36 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.89, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 4359, 25, 25, 3042, 25, 3186, 84, 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.
| \(\displaystyle \int \sin ^3(c+d x) (a \sec (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos \left (c+d x-\frac {\pi }{2}\right )^3 \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3dx\) |
| \(\Big \downarrow \) 4359 |
| \(\displaystyle \int \tan ^3(c+d x) \left (-(a (-\cos (c+d x))-a)^3\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -(\cos (c+d x) a+a)^3 \tan ^3(c+d x)dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \tan ^3(c+d x) (a \cos (c+d x)+a)^3dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\left (a-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3}{\tan \left (c+d x-\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\left (a-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )\right )^3}{\tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^3}dx\) |
| \(\Big \downarrow \) 3186 |
| \(\displaystyle -\frac {\int \frac {(a-a \cos (c+d x)) (\cos (c+d x) a+a)^4 \sec ^3(c+d x)}{a^3}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 84 |
| \(\displaystyle -\frac {\int \left (a^2 \sec ^3(c+d x)+3 a^2 \sec ^2(c+d x)+2 a^2 \sec (c+d x)-2 a^2-a^2 \cos ^2(c+d x)-3 a^2 \cos (c+d x)\right )d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {1}{3} a^3 \cos ^3(c+d x)-\frac {3}{2} a^3 \cos ^2(c+d x)-2 a^3 \cos (c+d x)-\frac {1}{2} a^3 \sec ^2(c+d x)-3 a^3 \sec (c+d x)+2 a^3 \log (a \cos (c+d x))}{d}\) |
-((-2*a^3*Cos[c + d*x] - (3*a^3*Cos[c + d*x]^2)/2 - (a^3*Cos[c + d*x]^3)/3 + 2*a^3*Log[a*Cos[c + d*x]] - 3*a^3*Sec[c + d*x] - (a^3*Sec[c + d*x]^2)/2 )/d)
3.1.41.3.1 Defintions of rubi rules used
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] && !(ILtQ[n + p + 2, 0 ] && GtQ[n + 2*p, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p _.), x_Symbol] :> Simp[1/f Subst[Int[x^p*((a + x)^(m - (p + 1)/2)/(a - x) ^((p + 1)/2)), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && E qQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m _.), x_Symbol] :> Int[Cot[e + f*x]^p*(b + a*Sin[e + f*x])^m, x] /; FreeQ[{a , b, e, f, p}, x] && IntegerQ[m] && EqQ[m, p]
Time = 2.17 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.16
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (\frac {\tan \left (d x +c \right )^{2}}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{3} \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}}{d}\) | \(114\) |
| default | \(\frac {a^{3} \left (\frac {\tan \left (d x +c \right )^{2}}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{3} \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}}{d}\) | \(114\) |
| parts | \(-\frac {a^{3} \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3 d}+\frac {a^{3} \left (\frac {\tan \left (d x +c \right )^{2}}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )}{d}+\frac {3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}+\frac {3 a^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )}{d}\) | \(122\) |
| parallelrisch | \(\frac {a^{3} \left (48 \left (1+\cos \left (2 d x +2 c \right )\right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-48 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (2 d x +2 c \right )-48 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (2 d x +2 c \right )-48 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-48 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+226 \cos \left (d x +c \right )+116 \cos \left (2 d x +2 c \right )+29 \cos \left (3 d x +3 c \right )+9 \cos \left (4 d x +4 c \right )+\cos \left (5 d x +5 c \right )+131\right )}{24 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(173\) |
| norman | \(\frac {\frac {32 a^{3}}{3 d}-\frac {4 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}-\frac {4 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d}+\frac {20 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3 d}+\frac {20 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-\frac {2 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {2 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}+\frac {2 a^{3} \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d}\) | \(180\) |
| risch | \(2 i a^{3} x +\frac {a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{24 d}+\frac {3 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {9 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {9 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {3 a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}+\frac {4 i a^{3} c}{d}+\frac {2 a^{3} \left (3 \,{\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {2 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(194\) |
1/d*(a^3*(1/2*tan(d*x+c)^2+ln(cos(d*x+c)))+3*a^3*(sin(d*x+c)^4/cos(d*x+c)+ (2+sin(d*x+c)^2)*cos(d*x+c))+3*a^3*(-1/2*sin(d*x+c)^2-ln(cos(d*x+c)))-1/3* a^3*(2+sin(d*x+c)^2)*cos(d*x+c))
Time = 0.26 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.06 \[ \int (a+a \sec (c+d x))^3 \sin ^3(c+d x) \, dx=\frac {4 \, a^{3} \cos \left (d x + c\right )^{5} + 18 \, a^{3} \cos \left (d x + c\right )^{4} + 24 \, a^{3} \cos \left (d x + c\right )^{3} - 24 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) - 9 \, a^{3} \cos \left (d x + c\right )^{2} + 36 \, a^{3} \cos \left (d x + c\right ) + 6 \, a^{3}}{12 \, d \cos \left (d x + c\right )^{2}} \]
1/12*(4*a^3*cos(d*x + c)^5 + 18*a^3*cos(d*x + c)^4 + 24*a^3*cos(d*x + c)^3 - 24*a^3*cos(d*x + c)^2*log(-cos(d*x + c)) - 9*a^3*cos(d*x + c)^2 + 36*a^ 3*cos(d*x + c) + 6*a^3)/(d*cos(d*x + c)^2)
\[ \int (a+a \sec (c+d x))^3 \sin ^3(c+d x) \, dx=a^{3} \left (\int 3 \sin ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 \sin ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )}\, dx\right ) \]
a**3*(Integral(3*sin(c + d*x)**3*sec(c + d*x), x) + Integral(3*sin(c + d*x )**3*sec(c + d*x)**2, x) + Integral(sin(c + d*x)**3*sec(c + d*x)**3, x) + Integral(sin(c + d*x)**3, x))
Time = 0.21 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82 \[ \int (a+a \sec (c+d x))^3 \sin ^3(c+d x) \, dx=\frac {2 \, a^{3} \cos \left (d x + c\right )^{3} + 9 \, a^{3} \cos \left (d x + c\right )^{2} + 12 \, a^{3} \cos \left (d x + c\right ) - 12 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) + \frac {3 \, {\left (6 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )}}{\cos \left (d x + c\right )^{2}}}{6 \, d} \]
1/6*(2*a^3*cos(d*x + c)^3 + 9*a^3*cos(d*x + c)^2 + 12*a^3*cos(d*x + c) - 1 2*a^3*log(cos(d*x + c)) + 3*(6*a^3*cos(d*x + c) + a^3)/cos(d*x + c)^2)/d
Time = 0.38 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.04 \[ \int (a+a \sec (c+d x))^3 \sin ^3(c+d x) \, dx=-\frac {2 \, a^{3} \log \left (\frac {{\left | \cos \left (d x + c\right ) \right |}}{{\left | d \right |}}\right )}{d} + \frac {6 \, a^{3} \cos \left (d x + c\right ) + a^{3}}{2 \, d \cos \left (d x + c\right )^{2}} + \frac {2 \, a^{3} d^{8} \cos \left (d x + c\right )^{3} + 9 \, a^{3} d^{8} \cos \left (d x + c\right )^{2} + 12 \, a^{3} d^{8} \cos \left (d x + c\right )}{6 \, d^{9}} \]
-2*a^3*log(abs(cos(d*x + c))/abs(d))/d + 1/2*(6*a^3*cos(d*x + c) + a^3)/(d *cos(d*x + c)^2) + 1/6*(2*a^3*d^8*cos(d*x + c)^3 + 9*a^3*d^8*cos(d*x + c)^ 2 + 12*a^3*d^8*cos(d*x + c))/d^9
Time = 13.28 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82 \[ \int (a+a \sec (c+d x))^3 \sin ^3(c+d x) \, dx=\frac {\frac {3\,a^3\,\cos \left (c+d\,x\right )+\frac {a^3}{2}}{{\cos \left (c+d\,x\right )}^2}+2\,a^3\,\cos \left (c+d\,x\right )+\frac {3\,a^3\,{\cos \left (c+d\,x\right )}^2}{2}+\frac {a^3\,{\cos \left (c+d\,x\right )}^3}{3}-2\,a^3\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]