Integrand size = 27, antiderivative size = 64 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx=a^3 x+\frac {\sec ^3(c+d x) (a+a \sin (c+d x))^3}{3 d}-\frac {2 a^5 \cos (c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )} \]
Time = 5.74 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.67 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx=-\frac {a^3 \left (-9 (2+c+d x) \cos \left (\frac {1}{2} (c+d x)\right )+(14+3 c+3 d x) \cos \left (\frac {3}{2} (c+d x)\right )+6 (2 (2+c+d x)+(c+d x) \cos (c+d x)) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{6 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
-1/6*(a^3*(-9*(2 + c + d*x)*Cos[(c + d*x)/2] + (14 + 3*c + 3*d*x)*Cos[(3*( c + d*x))/2] + 6*(2*(2 + c + d*x) + (c + d*x)*Cos[c + d*x])*Sin[(c + d*x)/ 2]))/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3)
Time = 0.41 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3042, 3334, 3042, 3149, 3042, 3159, 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.
\(\displaystyle \int \tan (c+d x) \sec ^3(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (c+d x) (a \sin (c+d x)+a)^3}{\cos (c+d x)^4}dx\) |
\(\Big \downarrow \) 3334 |
\(\displaystyle \frac {\sec ^3(c+d x) (a \sin (c+d x)+a)^3}{3 d}-a \int \sec ^2(c+d x) (\sin (c+d x) a+a)^2dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sec ^3(c+d x) (a \sin (c+d x)+a)^3}{3 d}-a \int \frac {(\sin (c+d x) a+a)^2}{\cos (c+d x)^2}dx\) |
\(\Big \downarrow \) 3149 |
\(\displaystyle \frac {\sec ^3(c+d x) (a \sin (c+d x)+a)^3}{3 d}-a^5 \int \frac {\cos ^2(c+d x)}{(a-a \sin (c+d x))^2}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sec ^3(c+d x) (a \sin (c+d x)+a)^3}{3 d}-a^5 \int \frac {\cos (c+d x)^2}{(a-a \sin (c+d x))^2}dx\) |
\(\Big \downarrow \) 3159 |
\(\displaystyle \frac {\sec ^3(c+d x) (a \sin (c+d x)+a)^3}{3 d}-a^5 \left (\frac {2 \cos (c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {\int 1dx}{a^2}\right )\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\sec ^3(c+d x) (a \sin (c+d x)+a)^3}{3 d}-a^5 \left (\frac {2 \cos (c+d x)}{d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {x}{a^2}\right )\) |
(Sec[c + d*x]^3*(a + a*Sin[c + d*x])^3)/(3*d) - a^5*(-(x/a^2) + (2*Cos[c + d*x])/(d*(a^2 - a^2*Sin[c + d*x])))
3.9.14.3.1 Defintions of rubi rules used
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[2*g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f *x])^(m + 1)/(b*f*(2*m + p + 1))), x] + Simp[g^2*((p - 1)/(b^2*(2*m + p + 1 ))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] & & NeQ[2*m + p + 1, 0] && !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*p]
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 = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.84
method | result | size |
risch | \(a^{3} x -\frac {2 a^{3} \left (-12 i {\mathrm e}^{i \left (d x +c \right )}+9 \,{\mathrm e}^{2 i \left (d x +c \right )}-7\right )}{3 d \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3}}\) | \(54\) |
parallelrisch | \(\frac {a^{3} \left (3 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) d x -9 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) x d +9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) d x +6 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 d x -24 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+10\right )}{3 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\) | \(95\) |
derivativedivides | \(\frac {a^{3} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )+3 a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )+\frac {a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{3}}+\frac {a^{3}}{3 \cos \left (d x +c \right )^{3}}}{d}\) | \(126\) |
default | \(\frac {a^{3} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )+3 a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )+\frac {a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{3}}+\frac {a^{3}}{3 \cos \left (d x +c \right )^{3}}}{d}\) | \(126\) |
norman | \(\frac {a^{3} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a^{3} x +\frac {10 a^{3}}{3 d}+\frac {2 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {26 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {36 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {116 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {36 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {26 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {2 a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+3 a^{3} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 a^{3} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {18 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {28 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}\) | \(304\) |
Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (63) = 126\).
Time = 0.28 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.23 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx=-\frac {6 \, a^{3} d x + 2 \, a^{3} - {\left (3 \, a^{3} d x + 7 \, a^{3}\right )} \cos \left (d x + c\right )^{2} + {\left (3 \, a^{3} d x - 5 \, a^{3}\right )} \cos \left (d x + c\right ) - {\left (6 \, a^{3} d x - 2 \, a^{3} + {\left (3 \, a^{3} d x - 7 \, a^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \]
-1/3*(6*a^3*d*x + 2*a^3 - (3*a^3*d*x + 7*a^3)*cos(d*x + c)^2 + (3*a^3*d*x - 5*a^3)*cos(d*x + c) - (6*a^3*d*x - 2*a^3 + (3*a^3*d*x - 7*a^3)*cos(d*x + c))*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)
Timed out. \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.31 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx=\frac {3 \, a^{3} \tan \left (d x + c\right )^{3} + {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{3} - \frac {3 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} a^{3}}{\cos \left (d x + c\right )^{3}} + \frac {a^{3}}{\cos \left (d x + c\right )^{3}}}{3 \, d} \]
1/3*(3*a^3*tan(d*x + c)^3 + (tan(d*x + c)^3 + 3*d*x + 3*c - 3*tan(d*x + c) )*a^3 - 3*(3*cos(d*x + c)^2 - 1)*a^3/cos(d*x + c)^3 + a^3/cos(d*x + c)^3)/ d
Time = 0.31 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.05 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx=\frac {3 \, {\left (d x + c\right )} a^{3} + \frac {2 \, {\left (3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}}}{3 \, d} \]
1/3*(3*(d*x + c)*a^3 + 2*(3*a^3*tan(1/2*d*x + 1/2*c)^2 - 12*a^3*tan(1/2*d* x + 1/2*c) + 5*a^3)/(tan(1/2*d*x + 1/2*c) - 1)^3)/d
Time = 10.70 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.59 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \tan (c+d x) \, dx=a^3\,x+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {a^3\,\left (9\,d\,x-24\right )}{3}-3\,a^3\,d\,x\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^3\,\left (9\,d\,x-6\right )}{3}-3\,a^3\,d\,x\right )-\frac {a^3\,\left (3\,d\,x-10\right )}{3}+a^3\,d\,x}{d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3} \]