Integrand size = 21, antiderivative size = 62 \[ \int (a+a \sec (c+d x))^2 \sin ^3(c+d x) \, dx=\frac {a^2 \cos ^2(c+d x)}{d}+\frac {a^2 \cos ^3(c+d x)}{3 d}-\frac {2 a^2 \log (\cos (c+d x))}{d}+\frac {a^2 \sec (c+d x)}{d} \]
Time = 0.21 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.05 \[ \int (a+a \sec (c+d x))^2 \sin ^3(c+d x) \, dx=\frac {a^2 (27+4 \cos (2 (c+d x))+6 \cos (3 (c+d x))+\cos (4 (c+d x))-6 \cos (c+d x) (1+8 \log (\cos (c+d x)))) \sec (c+d x)}{24 d} \]
(a^2*(27 + 4*Cos[2*(c + d*x)] + 6*Cos[3*(c + d*x)] + Cos[4*(c + d*x)] - 6* Cos[c + d*x]*(1 + 8*Log[Cos[c + d*x]]))*Sec[c + d*x])/(24*d)
Time = 0.38 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 4360, 3042, 25, 3315, 27, 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.
\(\displaystyle \int \sin ^3(c+d x) (a \sec (c+d x)+a)^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \cos \left (c+d x-\frac {\pi }{2}\right )^3 \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2dx\) |
\(\Big \downarrow \) 4360 |
\(\displaystyle \int \sin (c+d x) \tan ^2(c+d x) (a (-\cos (c+d x))-a)^2dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\cos \left (c+d x+\frac {\pi }{2}\right )^3 \left (a \left (-\sin \left (c+d x+\frac {\pi }{2}\right )\right )-a\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )^3 \left (\sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^2}{\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^2}dx\) |
\(\Big \downarrow \) 3315 |
\(\displaystyle -\frac {\int (a-a \cos (c+d x)) (\cos (c+d x) a+a)^3 \sec ^2(c+d x)d(a \cos (c+d x))}{a^3 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {(a-a \cos (c+d x)) (\cos (c+d x) a+a)^3 \sec ^2(c+d x)}{a^2}d(a \cos (c+d x))}{a d}\) |
\(\Big \downarrow \) 84 |
\(\displaystyle -\frac {\int \left (-\cos ^2(c+d x) a^2+\sec ^2(c+d x) a^2-2 \cos (c+d x) a^2+2 \sec (c+d x) a^2\right )d(a \cos (c+d x))}{a d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\frac {1}{3} a^3 \cos ^3(c+d x)-a^3 \cos ^2(c+d x)-a^3 \sec (c+d x)+2 a^3 \log (a \cos (c+d x))}{a d}\) |
-((-(a^3*Cos[c + d*x]^2) - (a^3*Cos[c + d*x]^3)/3 + 2*a^3*Log[a*Cos[c + d* x]] - a^3*Sec[c + d*x])/(a*d))
3.1.22.3.1 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[((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[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege rQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
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 = 2.02 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.47
method | result | size |
derivativedivides | \(\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )+2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{2} \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}}{d}\) | \(91\) |
default | \(\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )+2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{2} \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}}{d}\) | \(91\) |
parts | \(-\frac {a^{2} \left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3 d}+\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )}{d}-\frac {a^{2} \sin \left (d x +c \right )^{2}}{d}-\frac {2 a^{2} \ln \left (\cos \left (d x +c \right )\right )}{d}\) | \(99\) |
parallelrisch | \(\frac {a^{2} \left (-48 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )-48 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )+48 \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \cos \left (d x +c \right )+26 \cos \left (d x +c \right )+6 \cos \left (3 d x +3 c \right )+\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+27\right )}{24 d \cos \left (d x +c \right )}\) | \(118\) |
risch | \(2 i a^{2} x +\frac {a^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{4 d}+\frac {a^{2} {\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {a^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{4 d}+\frac {4 i a^{2} c}{d}+\frac {2 a^{2} {\mathrm e}^{i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {2 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}+\frac {a^{2} \cos \left (3 d x +3 c \right )}{12 d}\) | \(154\) |
norman | \(\frac {-\frac {8 a^{2}}{3 d}-\frac {8 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}-\frac {4 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}-\frac {4 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}+\frac {2 a^{2} \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d}\) | \(161\) |
1/d*(a^2*(sin(d*x+c)^4/cos(d*x+c)+(2+sin(d*x+c)^2)*cos(d*x+c))+2*a^2*(-1/2 *sin(d*x+c)^2-ln(cos(d*x+c)))-1/3*a^2*(2+sin(d*x+c)^2)*cos(d*x+c))
Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.23 \[ \int (a+a \sec (c+d x))^2 \sin ^3(c+d x) \, dx=\frac {2 \, a^{2} \cos \left (d x + c\right )^{4} + 6 \, a^{2} \cos \left (d x + c\right )^{3} - 12 \, a^{2} \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) - 3 \, a^{2} \cos \left (d x + c\right ) + 6 \, a^{2}}{6 \, d \cos \left (d x + c\right )} \]
1/6*(2*a^2*cos(d*x + c)^4 + 6*a^2*cos(d*x + c)^3 - 12*a^2*cos(d*x + c)*log (-cos(d*x + c)) - 3*a^2*cos(d*x + c) + 6*a^2)/(d*cos(d*x + c))
\[ \int (a+a \sec (c+d x))^2 \sin ^3(c+d x) \, dx=a^{2} \left (\int 2 \sin ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )}\, dx\right ) \]
a**2*(Integral(2*sin(c + d*x)**3*sec(c + d*x), x) + Integral(sin(c + d*x)* *3*sec(c + d*x)**2, x) + Integral(sin(c + d*x)**3, x))
Time = 0.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.90 \[ \int (a+a \sec (c+d x))^2 \sin ^3(c+d x) \, dx=\frac {a^{2} \cos \left (d x + c\right )^{3} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - 6 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) + \frac {3 \, a^{2}}{\cos \left (d x + c\right )}}{3 \, d} \]
1/3*(a^2*cos(d*x + c)^3 + 3*a^2*cos(d*x + c)^2 - 6*a^2*log(cos(d*x + c)) + 3*a^2/cos(d*x + c))/d
Time = 0.35 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.19 \[ \int (a+a \sec (c+d x))^2 \sin ^3(c+d x) \, dx=-\frac {2 \, a^{2} \log \left (\frac {{\left | \cos \left (d x + c\right ) \right |}}{{\left | d \right |}}\right )}{d} + \frac {a^{2}}{d \cos \left (d x + c\right )} + \frac {a^{2} d^{5} \cos \left (d x + c\right )^{3} + 3 \, a^{2} d^{5} \cos \left (d x + c\right )^{2}}{3 \, d^{6}} \]
-2*a^2*log(abs(cos(d*x + c))/abs(d))/d + a^2/(d*cos(d*x + c)) + 1/3*(a^2*d ^5*cos(d*x + c)^3 + 3*a^2*d^5*cos(d*x + c)^2)/d^6
Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.87 \[ \int (a+a \sec (c+d x))^2 \sin ^3(c+d x) \, dx=\frac {\frac {a^2}{\cos \left (c+d\,x\right )}+a^2\,{\cos \left (c+d\,x\right )}^2+\frac {a^2\,{\cos \left (c+d\,x\right )}^3}{3}-2\,a^2\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]