Integrand size = 21, antiderivative size = 39 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \log (1-\sin (c+d x))}{d}+\frac {2 a^3}{d (1-\sin (c+d x))} \] Output:
a^3*ln(1-sin(d*x+c))/d+2*a^3/d/(1-sin(d*x+c))
Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.51 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \sec ^2(c+d x) \left (\log (1-\sin (c+d x))+\frac {2}{1-\sin (c+d x)}\right ) (1-\sin (c+d x)) (1+\sin (c+d x))}{d} \] Input:
Integrate[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]
Output:
(a^3*Sec[c + d*x]^2*(Log[1 - Sin[c + d*x]] + 2/(1 - Sin[c + d*x]))*(1 - Si n[c + d*x])*(1 + Sin[c + d*x]))/d
Time = 0.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3146, 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.
\(\displaystyle \int \sec ^3(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a \sin (c+d x)+a)^3}{\cos (c+d x)^3}dx\) |
\(\Big \downarrow \) 3146 |
\(\displaystyle \frac {a^3 \int \frac {\sin (c+d x) a+a}{(a-a \sin (c+d x))^2}d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {a^3 \int \left (\frac {2 a}{(a-a \sin (c+d x))^2}+\frac {1}{a \sin (c+d x)-a}\right )d(a \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^3 \left (\frac {2 a}{a-a \sin (c+d x)}+\log (a-a \sin (c+d x))\right )}{d}\) |
Input:
Int[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]
Output:
(a^3*(Log[a - a*Sin[c + d*x]] + (2*a)/(a - a*Sin[c + d*x])))/d
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_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(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] && I ntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] || !IntegerQ[m + 1/ 2])
Result contains complex when optimal does not.
Time = 0.21 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.85
method | result | size |
risch | \(-i a^{3} x -\frac {2 i a^{3} c}{d}-\frac {4 i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{\left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{2} d}+\frac {2 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}\) | \(72\) |
parallelrisch | \(-\frac {a^{3} \left (\left (\sin \left (d x +c \right )-1\right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+\left (2-2 \sin \left (d x +c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+2 \sin \left (d x +c \right )\right )}{d \left (\sin \left (d x +c \right )-1\right )}\) | \(72\) |
derivativedivides | \(\frac {a^{3} \left (\frac {\tan \left (d x +c \right )^{2}}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{3}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {3 a^{3}}{2 \cos \left (d x +c \right )^{2}}+a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(124\) |
default | \(\frac {a^{3} \left (\frac {\tan \left (d x +c \right )^{2}}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{3}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+\frac {3 a^{3}}{2 \cos \left (d x +c \right )^{2}}+a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(124\) |
norman | \(\frac {-\frac {8 a^{3}}{d}+\frac {4 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {16 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}+\frac {24 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}+\frac {16 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}+\frac {4 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}-\frac {8 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}+\frac {40 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}+\frac {40 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}+\frac {2 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {a^{3} \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d}\) | \(234\) |
Input:
int(sec(d*x+c)^3*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-I*a^3*x-2*I*a^3/d*c-4*I*a^3*exp(I*(d*x+c))/(exp(I*(d*x+c))-I)^2/d+2*a^3/d *ln(exp(I*(d*x+c))-I)
Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.31 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {2 \, a^{3} - {\left (a^{3} \sin \left (d x + c\right ) - a^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{d \sin \left (d x + c\right ) - d} \] Input:
integrate(sec(d*x+c)^3*(a+a*sin(d*x+c))^3,x, algorithm="fricas")
Output:
-(2*a^3 - (a^3*sin(d*x + c) - a^3)*log(-sin(d*x + c) + 1))/(d*sin(d*x + c) - d)
\[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=a^{3} \left (\int 3 \sin {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \sec ^{3}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate(sec(d*x+c)**3*(a+a*sin(d*x+c))**3,x)
Output:
a**3*(Integral(3*sin(c + d*x)*sec(c + d*x)**3, x) + Integral(3*sin(c + d*x )**2*sec(c + d*x)**3, x) + Integral(sin(c + d*x)**3*sec(c + d*x)**3, x) + Integral(sec(c + d*x)**3, x))
Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, a^{3}}{\sin \left (d x + c\right ) - 1}}{d} \] Input:
integrate(sec(d*x+c)^3*(a+a*sin(d*x+c))^3,x, algorithm="maxima")
Output:
(a^3*log(sin(d*x + c) - 1) - 2*a^3/(sin(d*x + c) - 1))/d
Time = 0.13 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.92 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{d} - \frac {2 \, a^{3}}{d {\left (\sin \left (d x + c\right ) - 1\right )}} \] Input:
integrate(sec(d*x+c)^3*(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
a^3*log(abs(sin(d*x + c) - 1))/d - 2*a^3/(d*(sin(d*x + c) - 1))
Time = 25.93 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.90 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{d}-\frac {2\,a^3}{d\,\left (\sin \left (c+d\,x\right )-1\right )} \] Input:
int((a + a*sin(c + d*x))^3/cos(c + d*x)^3,x)
Output:
(a^3*log(sin(c + d*x) - 1))/d - (2*a^3)/(d*(sin(c + d*x) - 1))
Time = 0.16 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.28 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (-\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )+\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )+2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )-2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-2\right )}{d \left (\sin \left (d x +c \right )-1\right )} \] Input:
int(sec(d*x+c)^3*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*( - log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x) + log(tan((c + d*x)/2) **2 + 1) + 2*log(tan((c + d*x)/2) - 1)*sin(c + d*x) - 2*log(tan((c + d*x)/ 2) - 1) - 2))/(d*(sin(c + d*x) - 1))