Integrand size = 25, antiderivative size = 67 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx=-\frac {9 a^3 x}{2}+\frac {6 a^3 \cos (c+d x)}{d}+\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {\sec (c+d x) (a+a \sin (c+d x))^3}{d} \] Output:
-9/2*a^3*x+6*a^3*cos(d*x+c)/d+3/2*a^3*cos(d*x+c)*sin(d*x+c)/d+sec(d*x+c)*( a+a*sin(d*x+c))^3/d
Leaf count is larger than twice the leaf count of optimal. \(145\) vs. \(2(67)=134\).
Time = 6.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 2.16 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx=-\frac {a^3 (1+\sin (c+d x))^3 \left (\cos \left (\frac {1}{2} (c+d x)\right ) (18 (c+d x)-12 \cos (c+d x)-\sin (2 (c+d x)))+\sin \left (\frac {1}{2} (c+d x)\right ) (-2 (16+9 c+9 d x)+12 \cos (c+d x)+\sin (2 (c+d x)))\right )}{4 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \] Input:
Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])^3*Tan[c + d*x],x]
Output:
-1/4*(a^3*(1 + Sin[c + d*x])^3*(Cos[(c + d*x)/2]*(18*(c + d*x) - 12*Cos[c + d*x] - Sin[2*(c + d*x)]) + Sin[(c + d*x)/2]*(-2*(16 + 9*c + 9*d*x) + 12* Cos[c + d*x] + Sin[2*(c + d*x)])))/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2] )*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)
Time = 0.31 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3042, 3334, 3042, 3123}
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.
\(\displaystyle \int \tan (c+d x) \sec (c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (c+d x) (a \sin (c+d x)+a)^3}{\cos (c+d x)^2}dx\) |
\(\Big \downarrow \) 3334 |
\(\displaystyle \frac {\sec (c+d x) (a \sin (c+d x)+a)^3}{d}-3 a \int (\sin (c+d x) a+a)^2dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sec (c+d x) (a \sin (c+d x)+a)^3}{d}-3 a \int (\sin (c+d x) a+a)^2dx\) |
\(\Big \downarrow \) 3123 |
\(\displaystyle \frac {\sec (c+d x) (a \sin (c+d x)+a)^3}{d}-3 a \left (-\frac {2 a^2 \cos (c+d x)}{d}-\frac {a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {3 a^2 x}{2}\right )\) |
Input:
Int[Sec[c + d*x]*(a + a*Sin[c + d*x])^3*Tan[c + d*x],x]
Output:
(Sec[c + d*x]*(a + a*Sin[c + d*x])^3)/d - 3*a*((3*a^2*x)/2 - (2*a^2*Cos[c + d*x])/d - (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d))
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^2, x_Symbol] :> Simp[(2*a^2 + b^ 2)*(x/2), x] + (-Simp[2*a*b*(Cos[c + d*x]/d), x] - Simp[b^2*Cos[c + d*x]*(S in[c + d*x]/(2*d)), x]) /; FreeQ[{a, b, c, d}, x]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b* c + a*d))*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(p + 1))) , x] + Simp[b*((a*d*m + b*c*(m + p + 1))/(a*g^2*(p + 1))) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, -1] && LtQ[p, -1]
Result contains complex when optimal does not.
Time = 2.65 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.21
method | result | size |
risch | \(-\frac {9 a^{3} x}{2}+\frac {3 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {3 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {8 a^{3}}{d \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}+\frac {a^{3} \sin \left (2 d x +2 c \right )}{4 d}\) | \(81\) |
derivativedivides | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{5}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (\tan \left (d x +c \right )-d x -c \right )+\frac {a^{3}}{\cos \left (d x +c \right )}}{d}\) | \(130\) |
default | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{5}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{\cos \left (d x +c \right )}+\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (\tan \left (d x +c \right )-d x -c \right )+\frac {a^{3}}{\cos \left (d x +c \right )}}{d}\) | \(130\) |
Input:
int(sec(d*x+c)*(a+a*sin(d*x+c))^3*tan(d*x+c),x,method=_RETURNVERBOSE)
Output:
-9/2*a^3*x+3/2*a^3/d*exp(I*(d*x+c))+3/2*a^3/d*exp(-I*(d*x+c))+8*a^3/d/(exp (I*(d*x+c))-I)+1/4*a^3/d*sin(2*d*x+2*c)
Time = 0.08 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.87 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx=\frac {a^{3} \cos \left (d x + c\right )^{3} - 9 \, a^{3} d x + 6 \, a^{3} \cos \left (d x + c\right )^{2} + 8 \, a^{3} - {\left (9 \, a^{3} d x - 13 \, a^{3}\right )} \cos \left (d x + c\right ) + {\left (9 \, a^{3} d x + a^{3} \cos \left (d x + c\right )^{2} - 5 \, a^{3} \cos \left (d x + c\right ) + 8 \, a^{3}\right )} \sin \left (d x + c\right )}{2 \, {\left (d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d\right )}} \] Input:
integrate(sec(d*x+c)*(a+a*sin(d*x+c))^3*tan(d*x+c),x, algorithm="fricas")
Output:
1/2*(a^3*cos(d*x + c)^3 - 9*a^3*d*x + 6*a^3*cos(d*x + c)^2 + 8*a^3 - (9*a^ 3*d*x - 13*a^3)*cos(d*x + c) + (9*a^3*d*x + a^3*cos(d*x + c)^2 - 5*a^3*cos (d*x + c) + 8*a^3)*sin(d*x + c))/(d*cos(d*x + c) - d*sin(d*x + c) + d)
\[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx=a^{3} \left (\int \tan {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 \sin {\left (c + d x \right )} \tan {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \tan {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \tan {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx\right ) \] Input:
integrate(sec(d*x+c)*(a+a*sin(d*x+c))**3*tan(d*x+c),x)
Output:
a**3*(Integral(tan(c + d*x)*sec(c + d*x), x) + Integral(3*sin(c + d*x)*tan (c + d*x)*sec(c + d*x), x) + Integral(3*sin(c + d*x)**2*tan(c + d*x)*sec(c + d*x), x) + Integral(sin(c + d*x)**3*tan(c + d*x)*sec(c + d*x), x))
Time = 0.11 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.45 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx=-\frac {{\left (3 \, d x + 3 \, c - \frac {\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} a^{3} + 6 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{3} - 6 \, a^{3} {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} - \frac {2 \, a^{3}}{\cos \left (d x + c\right )}}{2 \, d} \] Input:
integrate(sec(d*x+c)*(a+a*sin(d*x+c))^3*tan(d*x+c),x, algorithm="maxima")
Output:
-1/2*((3*d*x + 3*c - tan(d*x + c)/(tan(d*x + c)^2 + 1) - 2*tan(d*x + c))*a ^3 + 6*(d*x + c - tan(d*x + c))*a^3 - 6*a^3*(1/cos(d*x + c) + cos(d*x + c) ) - 2*a^3/cos(d*x + c))/d
Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.52 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx=-\frac {9 \, {\left (d x + c\right )} a^{3} + \frac {16 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1} + \frac {2 \, {\left (a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \] Input:
integrate(sec(d*x+c)*(a+a*sin(d*x+c))^3*tan(d*x+c),x, algorithm="giac")
Output:
-1/2*(9*(d*x + c)*a^3 + 16*a^3/(tan(1/2*d*x + 1/2*c) - 1) + 2*(a^3*tan(1/2 *d*x + 1/2*c)^3 - 6*a^3*tan(1/2*d*x + 1/2*c)^2 - a^3*tan(1/2*d*x + 1/2*c) - 6*a^3)/(tan(1/2*d*x + 1/2*c)^2 + 1)^2)/d
Time = 33.58 (sec) , antiderivative size = 213, normalized size of antiderivative = 3.18 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx=-\frac {9\,a^3\,x}{2}-\frac {\frac {9\,a^3\,\left (c+d\,x\right )}{2}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {9\,a^3\,\left (c+d\,x\right )}{2}-\frac {a^3\,\left (9\,c+9\,d\,x-10\right )}{2}\right )-\frac {a^3\,\left (9\,c+9\,d\,x-28\right )}{2}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {9\,a^3\,\left (c+d\,x\right )}{2}-\frac {a^3\,\left (9\,c+9\,d\,x-18\right )}{2}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (9\,a^3\,\left (c+d\,x\right )-\frac {a^3\,\left (18\,c+18\,d\,x-14\right )}{2}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (9\,a^3\,\left (c+d\,x\right )-\frac {a^3\,\left (18\,c+18\,d\,x-42\right )}{2}\right )}{d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2} \] Input:
int((tan(c + d*x)*(a + a*sin(c + d*x))^3)/cos(c + d*x),x)
Output:
- (9*a^3*x)/2 - ((9*a^3*(c + d*x))/2 - tan(c/2 + (d*x)/2)*((9*a^3*(c + d*x ))/2 - (a^3*(9*c + 9*d*x - 10))/2) - (a^3*(9*c + 9*d*x - 28))/2 + tan(c/2 + (d*x)/2)^4*((9*a^3*(c + d*x))/2 - (a^3*(9*c + 9*d*x - 18))/2) - tan(c/2 + (d*x)/2)^3*(9*a^3*(c + d*x) - (a^3*(18*c + 18*d*x - 14))/2) + tan(c/2 + (d*x)/2)^2*(9*a^3*(c + d*x) - (a^3*(18*c + 18*d*x - 42))/2))/(d*(tan(c/2 + (d*x)/2) - 1)*(tan(c/2 + (d*x)/2)^2 + 1)^2)
Time = 0.16 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.72 \[ \int \sec (c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx=\frac {a^{3} \left (\cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+5 \cos \left (d x +c \right ) \sin \left (d x +c \right )-9 \cos \left (d x +c \right ) d x -18 \cos \left (d x +c \right )-\sin \left (d x +c \right )^{3}-6 \sin \left (d x +c \right )^{2}-9 \sin \left (d x +c \right ) d x +5 \sin \left (d x +c \right )+9 d x +18\right )}{2 d \left (\cos \left (d x +c \right )+\sin \left (d x +c \right )-1\right )} \] Input:
int(sec(d*x+c)*(a+a*sin(d*x+c))^3*tan(d*x+c),x)
Output:
(a**3*(cos(c + d*x)*sin(c + d*x)**2 + 5*cos(c + d*x)*sin(c + d*x) - 9*cos( c + d*x)*d*x - 18*cos(c + d*x) - sin(c + d*x)**3 - 6*sin(c + d*x)**2 - 9*s in(c + d*x)*d*x + 5*sin(c + d*x) + 9*d*x + 18))/(2*d*(cos(c + d*x) + sin(c + d*x) - 1))