Integrand size = 21, antiderivative size = 31 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^6 \cos ^3(c+d x)}{3 d (a-a \sin (c+d x))^3} \] Output:
1/3*a^6*cos(d*x+c)^3/d/(a-a*sin(d*x+c))^3
Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \sec ^3(c+d x) (1+\sin (c+d x))^3}{3 d} \] Input:
Integrate[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]
Output:
(a^3*Sec[c + d*x]^3*(1 + Sin[c + d*x])^3)/(3*d)
Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3149, 3042, 3150}
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.
\(\displaystyle \int \sec ^4(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a \sin (c+d x)+a)^3}{\cos (c+d x)^4}dx\) |
\(\Big \downarrow \) 3149 |
\(\displaystyle a^6 \int \frac {\cos ^2(c+d x)}{(a-a \sin (c+d x))^3}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^6 \int \frac {\cos (c+d x)^2}{(a-a \sin (c+d x))^3}dx\) |
\(\Big \downarrow \) 3150 |
\(\displaystyle \frac {a^6 \cos ^3(c+d x)}{3 d (a-a \sin (c+d x))^3}\) |
Input:
Int[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]
Output:
(a^6*Cos[c + d*x]^3)/(3*d*(a - a*Sin[c + d*x])^3)
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(a/g)^(2*m) Int[(g*Cos[e + f*x])^(2*m + p)/( a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2 , 0] && IntegerQ[m] && LtQ[p, -1] && GeQ[2*m + p, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^m/(a*f*g*m)), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && EqQ[Simplify[m + p + 1], 0] && !ILtQ[p, 0]
Time = 0.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19
method | result | size |
parallelrisch | \(\frac {2 a^{3} \left (-3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}{3 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\) | \(37\) |
risch | \(-\frac {2 \left (3 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-a^{3}\right )}{3 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d}\) | \(40\) |
derivativedivides | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}\right )+\frac {a^{3} \sin \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{3}}+\frac {a^{3}}{\cos \left (d x +c \right )^{3}}-a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(120\) |
default | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{3}\right )+\frac {a^{3} \sin \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{3}}+\frac {a^{3}}{\cos \left (d x +c \right )^{3}}-a^{3} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(120\) |
norman | \(\frac {-\frac {2 a^{3}}{3 d}-\frac {2 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {6 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}-\frac {38 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}-\frac {28 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}-\frac {92 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}-\frac {28 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d}-\frac {38 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3 d}-\frac {6 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}-\frac {2 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{d}-\frac {20 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}-\frac {22 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{3} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}\) | \(248\) |
Input:
int(sec(d*x+c)^4*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
2/3*a^3*(-3*tan(1/2*d*x+1/2*c)^2-1)/d/(tan(1/2*d*x+1/2*c)-1)^3
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (30) = 60\).
Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.19 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \cos \left (d x + c\right )^{2} - a^{3} \cos \left (d x + c\right ) - 2 \, a^{3} - {\left (a^{3} \cos \left (d x + c\right ) + 2 \, a^{3}\right )} \sin \left (d x + c\right )}{3 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \] Input:
integrate(sec(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="fricas")
Output:
1/3*(a^3*cos(d*x + c)^2 - a^3*cos(d*x + c) - 2*a^3 - (a^3*cos(d*x + c) + 2 *a^3)*sin(d*x + c))/(d*cos(d*x + c)^2 - d*cos(d*x + c) + (d*cos(d*x + c) + 2*d)*sin(d*x + c) - 2*d)
\[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=a^{3} \left (\int 3 \sin {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \sec ^{4}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate(sec(d*x+c)**4*(a+a*sin(d*x+c))**3,x)
Output:
a**3*(Integral(3*sin(c + d*x)*sec(c + d*x)**4, x) + Integral(3*sin(c + d*x )**2*sec(c + d*x)**4, x) + Integral(sin(c + d*x)**3*sec(c + d*x)**4, x) + Integral(sec(c + d*x)**4, x))
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (30) = 60\).
Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.52 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {3 \, a^{3} \tan \left (d x + c\right )^{3} + {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{3} - \frac {{\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} a^{3}}{\cos \left (d x + c\right )^{3}} + \frac {3 \, a^{3}}{\cos \left (d x + c\right )^{3}}}{3 \, d} \] Input:
integrate(sec(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="maxima")
Output:
1/3*(3*a^3*tan(d*x + c)^3 + (tan(d*x + c)^3 + 3*tan(d*x + c))*a^3 - (3*cos (d*x + c)^2 - 1)*a^3/cos(d*x + c)^3 + 3*a^3/cos(d*x + c)^3)/d
Time = 0.15 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {2 \, {\left (3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a^{3}\right )}}{3 \, d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} \] Input:
integrate(sec(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
-2/3*(3*a^3*tan(1/2*d*x + 1/2*c)^2 + a^3)/(d*(tan(1/2*d*x + 1/2*c) - 1)^3)
Time = 25.74 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.77 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {2\,a^3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-3\right )}{3\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^3} \] Input:
int((a + a*sin(c + d*x))^3/cos(c + d*x)^4,x)
Output:
-(2*a^3*cos(c/2 + (d*x)/2)*(2*cos(c/2 + (d*x)/2)^2 - 3))/(3*d*(cos(c/2 + ( d*x)/2) - sin(c/2 + (d*x)/2))^3)
Time = 0.18 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.29 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3} \left (-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-3\right )}{3 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )} \] Input:
int(sec(d*x+c)^4*(a+a*sin(d*x+c))^3,x)
Output:
(2*tan((c + d*x)/2)*a**3*( - tan((c + d*x)/2)**2 - 3))/(3*d*(tan((c + d*x) /2)**3 - 3*tan((c + d*x)/2)**2 + 3*tan((c + d*x)/2) - 1))