Integrand size = 25, antiderivative size = 60 \[ \int \sec (c+d x) (a+a \sin (c+d x)) \tan ^3(c+d x) \, dx=a x-\frac {a \sec (c+d x)}{d}+\frac {a \sec ^3(c+d x)}{3 d}-\frac {a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{3 d} \] Output:
a*x-a*sec(d*x+c)/d+1/3*a*sec(d*x+c)^3/d-a*tan(d*x+c)/d+1/3*a*tan(d*x+c)^3/ d
Time = 0.18 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.15 \[ \int \sec (c+d x) (a+a \sin (c+d x)) \tan ^3(c+d x) \, dx=\frac {a \arctan (\tan (c+d x))}{d}-\frac {a \sec (c+d x)}{d}+\frac {a \sec ^3(c+d x)}{3 d}-\frac {a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{3 d} \] Input:
Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])*Tan[c + d*x]^3,x]
Output:
(a*ArcTan[Tan[c + d*x]])/d - (a*Sec[c + d*x])/d + (a*Sec[c + d*x]^3)/(3*d) - (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)
Time = 0.44 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.95, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 3317, 3042, 3086, 2009, 3954, 3042, 3954, 24}
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.
\(\displaystyle \int \tan ^3(c+d x) \sec (c+d x) (a \sin (c+d x)+a) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (c+d x)^3 (a \sin (c+d x)+a)}{\cos (c+d x)^4}dx\) |
\(\Big \downarrow \) 3317 |
\(\displaystyle a \int \tan ^4(c+d x)dx+a \int \sec (c+d x) \tan ^3(c+d x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int \tan (c+d x)^4dx+a \int \sec (c+d x) \tan (c+d x)^3dx\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle a \int \tan (c+d x)^4dx+\frac {a \int \left (\sec ^2(c+d x)-1\right )d\sec (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a \int \tan (c+d x)^4dx+\frac {a \left (\frac {1}{3} \sec ^3(c+d x)-\sec (c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3954 |
\(\displaystyle a \left (\frac {\tan ^3(c+d x)}{3 d}-\int \tan ^2(c+d x)dx\right )+\frac {a \left (\frac {1}{3} \sec ^3(c+d x)-\sec (c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \left (\frac {\tan ^3(c+d x)}{3 d}-\int \tan (c+d x)^2dx\right )+\frac {a \left (\frac {1}{3} \sec ^3(c+d x)-\sec (c+d x)\right )}{d}\) |
\(\Big \downarrow \) 3954 |
\(\displaystyle a \left (\int 1dx+\frac {\tan ^3(c+d x)}{3 d}-\frac {\tan (c+d x)}{d}\right )+\frac {a \left (\frac {1}{3} \sec ^3(c+d x)-\sec (c+d x)\right )}{d}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle a \left (\frac {\tan ^3(c+d x)}{3 d}-\frac {\tan (c+d x)}{d}+x\right )+\frac {a \left (\frac {1}{3} \sec ^3(c+d x)-\sec (c+d x)\right )}{d}\) |
Input:
Int[Sec[c + d*x]*(a + a*Sin[c + d*x])*Tan[c + d*x]^3,x]
Output:
(a*(-Sec[c + d*x] + Sec[c + d*x]^3/3))/d + a*(x - Tan[c + d*x]/d + Tan[c + d*x]^3/(3*d))
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[(g*Co s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.07
method | result | size |
risch | \(a x -\frac {2 a \left (3 \,{\mathrm e}^{3 i \left (d x +c \right )}-4 i+5 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d}\) | \(64\) |
derivativedivides | \(\frac {a \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+d x +c \right )+a \left (\frac {\sin \left (d x +c \right )^{4}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}\right )}{d}\) | \(88\) |
default | \(\frac {a \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+d x +c \right )+a \left (\frac {\sin \left (d x +c \right )^{4}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}\right )}{d}\) | \(88\) |
Input:
int(sec(d*x+c)*(a+a*sin(d*x+c))*tan(d*x+c)^3,x,method=_RETURNVERBOSE)
Output:
a*x-2/3*a*(3*exp(3*I*(d*x+c))-4*I+5*exp(I*(d*x+c)))/(exp(I*(d*x+c))+I)/(ex p(I*(d*x+c))-I)^3/d
Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.25 \[ \int \sec (c+d x) (a+a \sin (c+d x)) \tan ^3(c+d x) \, dx=-\frac {3 \, a d x \cos \left (d x + c\right ) - 4 \, a \cos \left (d x + c\right )^{2} - {\left (3 \, a d x \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a}{3 \, {\left (d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - d \cos \left (d x + c\right )\right )}} \] Input:
integrate(sec(d*x+c)*(a+a*sin(d*x+c))*tan(d*x+c)^3,x, algorithm="fricas")
Output:
-1/3*(3*a*d*x*cos(d*x + c) - 4*a*cos(d*x + c)^2 - (3*a*d*x*cos(d*x + c) + a)*sin(d*x + c) + 2*a)/(d*cos(d*x + c)*sin(d*x + c) - d*cos(d*x + c))
\[ \int \sec (c+d x) (a+a \sin (c+d x)) \tan ^3(c+d x) \, dx=a \left (\int \tan ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \sin {\left (c + d x \right )} \tan ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx\right ) \] Input:
integrate(sec(d*x+c)*(a+a*sin(d*x+c))*tan(d*x+c)**3,x)
Output:
a*(Integral(tan(c + d*x)**3*sec(c + d*x), x) + Integral(sin(c + d*x)*tan(c + d*x)**3*sec(c + d*x), x))
Time = 0.11 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.92 \[ \int \sec (c+d x) (a+a \sin (c+d x)) \tan ^3(c+d x) \, dx=\frac {{\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a - \frac {{\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} a}{\cos \left (d x + c\right )^{3}}}{3 \, d} \] Input:
integrate(sec(d*x+c)*(a+a*sin(d*x+c))*tan(d*x+c)^3,x, algorithm="maxima")
Output:
1/3*((tan(d*x + c)^3 + 3*d*x + 3*c - 3*tan(d*x + c))*a - (3*cos(d*x + c)^2 - 1)*a/cos(d*x + c)^3)/d
Time = 0.34 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.23 \[ \int \sec (c+d x) (a+a \sin (c+d x)) \tan ^3(c+d x) \, dx=\frac {6 \, {\left (d x + c\right )} a + \frac {3 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1} + \frac {9 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11 \, a}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}}}{6 \, d} \] Input:
integrate(sec(d*x+c)*(a+a*sin(d*x+c))*tan(d*x+c)^3,x, algorithm="giac")
Output:
1/6*(6*(d*x + c)*a + 3*a/(tan(1/2*d*x + 1/2*c) + 1) + (9*a*tan(1/2*d*x + 1 /2*c)^2 - 24*a*tan(1/2*d*x + 1/2*c) + 11*a)/(tan(1/2*d*x + 1/2*c) - 1)^3)/ d
Time = 32.55 (sec) , antiderivative size = 135, normalized size of antiderivative = 2.25 \[ \int \sec (c+d x) (a+a \sin (c+d x)) \tan ^3(c+d x) \, dx=a\,x-\frac {\left (\frac {a\,\left (6\,c+6\,d\,x-6\right )}{3}-2\,a\,\left (c+d\,x\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\left (2\,a\,\left (c+d\,x\right )-\frac {a\,\left (6\,c+6\,d\,x-2\right )}{3}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a\,\left (3\,c+3\,d\,x-4\right )}{3}-a\,\left (c+d\,x\right )}{d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )} \] Input:
int((tan(c + d*x)^3*(a + a*sin(c + d*x)))/cos(c + d*x),x)
Output:
a*x - ((a*(3*c + 3*d*x - 4))/3 - tan(c/2 + (d*x)/2)*((a*(6*c + 6*d*x - 2)) /3 - 2*a*(c + d*x)) - a*(c + d*x) + tan(c/2 + (d*x)/2)^3*((a*(6*c + 6*d*x - 6))/3 - 2*a*(c + d*x)) + 4*a*tan(c/2 + (d*x)/2)^2)/(d*(tan(c/2 + (d*x)/2 ) - 1)^3*(tan(c/2 + (d*x)/2) + 1))
Time = 0.19 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.30 \[ \int \sec (c+d x) (a+a \sin (c+d x)) \tan ^3(c+d x) \, dx=\frac {a \left (-\cos \left (d x +c \right )^{2} \sec \left (d x +c \right ) \tan \left (d x +c \right )+2 \cos \left (d x +c \right ) \sec \left (d x +c \right ) \sin \left (d x +c \right ) \tan \left (d x +c \right )^{2}+\cos \left (d x +c \right ) \sec \left (d x +c \right ) \sin \left (d x +c \right )+2 \cos \left (d x +c \right ) \sec \left (d x +c \right ) \tan \left (d x +c \right )^{2}-4 \cos \left (d x +c \right ) \sec \left (d x +c \right )+6 \cos \left (d x +c \right ) d x -6 \sin \left (d x +c \right )\right )}{6 \cos \left (d x +c \right ) d} \] Input:
int(sec(d*x+c)*(a+a*sin(d*x+c))*tan(d*x+c)^3,x)
Output:
(a*( - cos(c + d*x)**2*sec(c + d*x)*tan(c + d*x) + 2*cos(c + d*x)*sec(c + d*x)*sin(c + d*x)*tan(c + d*x)**2 + cos(c + d*x)*sec(c + d*x)*sin(c + d*x) + 2*cos(c + d*x)*sec(c + d*x)*tan(c + d*x)**2 - 4*cos(c + d*x)*sec(c + d* x) + 6*cos(c + d*x)*d*x - 6*sin(c + d*x)))/(6*cos(c + d*x)*d)