Integrand size = 27, antiderivative size = 55 \[ \int \frac {\sec (c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\sec ^3(c+d x)}{3 a d}+\frac {\sec ^5(c+d x)}{5 a d}-\frac {\tan ^5(c+d x)}{5 a d} \]
Time = 0.70 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.93 \[ \int \frac {\sec (c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\sec ^3(c+d x) (40+66 \cos (c+d x)-192 \cos (2 (c+d x))+22 \cos (3 (c+d x))+24 \cos (4 (c+d x))+16 \sin (c+d x)+22 \sin (2 (c+d x))-48 \sin (3 (c+d x))+11 \sin (4 (c+d x)))}{960 a d (1+\sin (c+d x))} \]
(Sec[c + d*x]^3*(40 + 66*Cos[c + d*x] - 192*Cos[2*(c + d*x)] + 22*Cos[3*(c + d*x)] + 24*Cos[4*(c + d*x)] + 16*Sin[c + d*x] + 22*Sin[2*(c + d*x)] - 4 8*Sin[3*(c + d*x)] + 11*Sin[4*(c + d*x)]))/(960*a*d*(1 + Sin[c + d*x]))
Time = 0.41 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3318, 3042, 3086, 25, 244, 2009, 3087, 15}
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.
\(\displaystyle \int \frac {\tan ^3(c+d x) \sec (c+d x)}{a \sin (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (c+d x)^3}{\cos (c+d x)^4 (a \sin (c+d x)+a)}dx\) |
\(\Big \downarrow \) 3318 |
\(\displaystyle \frac {\int \sec ^3(c+d x) \tan ^3(c+d x)dx}{a}-\frac {\int \sec ^2(c+d x) \tan ^4(c+d x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \sec (c+d x)^3 \tan (c+d x)^3dx}{a}-\frac {\int \sec (c+d x)^2 \tan (c+d x)^4dx}{a}\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle \frac {\int -\sec ^2(c+d x) \left (1-\sec ^2(c+d x)\right )d\sec (c+d x)}{a d}-\frac {\int \sec (c+d x)^2 \tan (c+d x)^4dx}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \sec ^2(c+d x) \left (1-\sec ^2(c+d x)\right )d\sec (c+d x)}{a d}-\frac {\int \sec (c+d x)^2 \tan (c+d x)^4dx}{a}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle -\frac {\int \left (\sec ^2(c+d x)-\sec ^4(c+d x)\right )d\sec (c+d x)}{a d}-\frac {\int \sec (c+d x)^2 \tan (c+d x)^4dx}{a}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{5} \sec ^5(c+d x)-\frac {1}{3} \sec ^3(c+d x)}{a d}-\frac {\int \sec (c+d x)^2 \tan (c+d x)^4dx}{a}\) |
\(\Big \downarrow \) 3087 |
\(\displaystyle \frac {\frac {1}{5} \sec ^5(c+d x)-\frac {1}{3} \sec ^3(c+d x)}{a d}-\frac {\int \tan ^4(c+d x)d\tan (c+d x)}{a d}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {\frac {1}{5} \sec ^5(c+d x)-\frac {1}{3} \sec ^3(c+d x)}{a d}-\frac {\tan ^5(c+d x)}{5 a d}\) |
3.9.24.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
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[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> Simp[1/f Subst[Int[(b*x)^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] && !(IntegerQ[(n - 1) /2] && LtQ[0, n, m - 1])
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^( n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g^2/a Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[g^2/(b*d) Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0]
Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.58
method | result | size |
parallelrisch | \(\frac {\frac {4}{15}-4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15}+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{15}}{d a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\) | \(87\) |
risch | \(-\frac {2 i \left (8 i {\mathrm e}^{3 i \left (d x +c \right )}+9 \,{\mathrm e}^{2 i \left (d x +c \right )}+6 i {\mathrm e}^{i \left (d x +c \right )}-5 \,{\mathrm e}^{4 i \left (d x +c \right )}-3+10 i {\mathrm e}^{5 i \left (d x +c \right )}+15 \,{\mathrm e}^{6 i \left (d x +c \right )}\right )}{15 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{5} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d a}\) | \(109\) |
norman | \(\frac {-\frac {4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {4}{15 a d}+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{15 d a}-\frac {8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d a}-\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\) | \(111\) |
derivativedivides | \(\frac {-\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {16}{128 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-128}+\frac {2}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {2}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d a}\) | \(115\) |
default | \(\frac {-\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {16}{128 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-128}+\frac {2}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {2}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d a}\) | \(115\) |
4/15*(1-15*tan(1/2*d*x+1/2*c)^4-6*tan(1/2*d*x+1/2*c)^3-2*tan(1/2*d*x+1/2*c )^2+2*tan(1/2*d*x+1/2*c))/d/a/(tan(1/2*d*x+1/2*c)-1)^3/(tan(1/2*d*x+1/2*c) +1)^5
Time = 0.26 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.36 \[ \int \frac {\sec (c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 \, \cos \left (d x + c\right )^{4} - 9 \, \cos \left (d x + c\right )^{2} - {\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) + 4}{15 \, {\left (a d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{3}\right )}} \]
1/15*(3*cos(d*x + c)^4 - 9*cos(d*x + c)^2 - (3*cos(d*x + c)^2 - 1)*sin(d*x + c) + 4)/(a*d*cos(d*x + c)^3*sin(d*x + c) + a*d*cos(d*x + c)^3)
Timed out. \[ \int \frac {\sec (c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (49) = 98\).
Time = 0.22 (sec) , antiderivative size = 234, normalized size of antiderivative = 4.25 \[ \int \frac {\sec (c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {4 \, {\left (\frac {2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {6 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 1\right )}}{15 \, {\left (a + \frac {2 \, a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {6 \, a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {6 \, a \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {2 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {2 \, a \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d} \]
-4/15*(2*sin(d*x + c)/(cos(d*x + c) + 1) - 2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 6*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 15*sin(d*x + c)^4/(cos(d* x + c) + 1)^4 + 1)/((a + 2*a*sin(d*x + c)/(cos(d*x + c) + 1) - 2*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 6*a*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 6 *a*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 2*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 2*a*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - a*sin(d*x + c)^8/(cos(d *x + c) + 1)^8)*d)
Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (49) = 98\).
Time = 0.32 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.18 \[ \int \frac {\sec (c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {5 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5\right )}}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} - \frac {15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 60 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 20 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{120 \, d} \]
1/120*(5*(3*tan(1/2*d*x + 1/2*c)^2 - 12*tan(1/2*d*x + 1/2*c) + 5)/(a*(tan( 1/2*d*x + 1/2*c) - 1)^3) - (15*tan(1/2*d*x + 1/2*c)^4 + 60*tan(1/2*d*x + 1 /2*c)^3 + 10*tan(1/2*d*x + 1/2*c)^2 + 20*tan(1/2*d*x + 1/2*c) + 7)/(a*(tan (1/2*d*x + 1/2*c) + 1)^5))/d
Time = 10.54 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.56 \[ \int \frac {\sec (c+d x) \tan ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {4\,\left (15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}{15\,a\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^5} \]