Integrand size = 27, antiderivative size = 105 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {x}{a}+\frac {\sec (c+d x)}{a d}-\frac {2 \sec ^3(c+d x)}{3 a d}+\frac {\sec ^5(c+d x)}{5 a d}-\frac {\tan (c+d x)}{a d}+\frac {\tan ^3(c+d x)}{3 a d}-\frac {\tan ^5(c+d x)}{5 a d} \]
x/a+sec(d*x+c)/a/d-2/3*sec(d*x+c)^3/a/d+1/5*sec(d*x+c)^5/a/d-tan(d*x+c)/a/ d+1/3*tan(d*x+c)^3/a/d-1/5*tan(d*x+c)^5/a/d
Time = 0.90 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.82 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\sec ^3(c+d x) \left (-25+\left (\frac {267}{4}-90 c-90 d x\right ) \cos (c+d x)-16 \cos (2 (c+d x))+\frac {89}{4} \cos (3 (c+d x))-30 c \cos (3 (c+d x))-30 d x \cos (3 (c+d x))-23 \cos (4 (c+d x))+8 \sin (c+d x)+\frac {89}{4} \sin (2 (c+d x))-30 c \sin (2 (c+d x))-30 d x \sin (2 (c+d x))+16 \sin (3 (c+d x))+\frac {89}{8} \sin (4 (c+d x))-15 c \sin (4 (c+d x))-15 d x \sin (4 (c+d x))\right )}{120 a d (1+\sin (c+d x))} \]
-1/120*(Sec[c + d*x]^3*(-25 + (267/4 - 90*c - 90*d*x)*Cos[c + d*x] - 16*Co s[2*(c + d*x)] + (89*Cos[3*(c + d*x)])/4 - 30*c*Cos[3*(c + d*x)] - 30*d*x* Cos[3*(c + d*x)] - 23*Cos[4*(c + d*x)] + 8*Sin[c + d*x] + (89*Sin[2*(c + d *x)])/4 - 30*c*Sin[2*(c + d*x)] - 30*d*x*Sin[2*(c + d*x)] + 16*Sin[3*(c + d*x)] + (89*Sin[4*(c + d*x)])/8 - 15*c*Sin[4*(c + d*x)] - 15*d*x*Sin[4*(c + d*x)]))/(a*d*(1 + Sin[c + d*x]))
Time = 0.53 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.84, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 3318, 3042, 3086, 210, 2009, 3954, 3042, 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.
| \(\displaystyle \int \frac {\sin (c+d x) \tan ^4(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^5}{\cos (c+d x)^4 (a \sin (c+d x)+a)}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \frac {\int \sec (c+d x) \tan ^5(c+d x)dx}{a}-\frac {\int \tan ^6(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sec (c+d x) \tan (c+d x)^5dx}{a}-\frac {\int \tan (c+d x)^6dx}{a}\) |
| \(\Big \downarrow \) 3086 |
| \(\displaystyle \frac {\int \left (\sec ^2(c+d x)-1\right )^2d\sec (c+d x)}{a d}-\frac {\int \tan (c+d x)^6dx}{a}\) |
| \(\Big \downarrow \) 210 |
| \(\displaystyle \frac {\int \left (\sec ^4(c+d x)-2 \sec ^2(c+d x)+1\right )d\sec (c+d x)}{a d}-\frac {\int \tan (c+d x)^6dx}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{5} \sec ^5(c+d x)-\frac {2}{3} \sec ^3(c+d x)+\sec (c+d x)}{a d}-\frac {\int \tan (c+d x)^6dx}{a}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {\frac {1}{5} \sec ^5(c+d x)-\frac {2}{3} \sec ^3(c+d x)+\sec (c+d x)}{a d}-\frac {\frac {\tan ^5(c+d x)}{5 d}-\int \tan ^4(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \sec ^5(c+d x)-\frac {2}{3} \sec ^3(c+d x)+\sec (c+d x)}{a d}-\frac {\frac {\tan ^5(c+d x)}{5 d}-\int \tan (c+d x)^4dx}{a}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {\frac {1}{5} \sec ^5(c+d x)-\frac {2}{3} \sec ^3(c+d x)+\sec (c+d x)}{a d}-\frac {\int \tan ^2(c+d x)dx+\frac {\tan ^5(c+d x)}{5 d}-\frac {\tan ^3(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \sec ^5(c+d x)-\frac {2}{3} \sec ^3(c+d x)+\sec (c+d x)}{a d}-\frac {\int \tan (c+d x)^2dx+\frac {\tan ^5(c+d x)}{5 d}-\frac {\tan ^3(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {\frac {1}{5} \sec ^5(c+d x)-\frac {2}{3} \sec ^3(c+d x)+\sec (c+d x)}{a d}-\frac {-\int 1dx+\frac {\tan ^5(c+d x)}{5 d}-\frac {\tan ^3(c+d x)}{3 d}+\frac {\tan (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {1}{5} \sec ^5(c+d x)-\frac {2}{3} \sec ^3(c+d x)+\sec (c+d x)}{a d}-\frac {\frac {\tan ^5(c+d x)}{5 d}-\frac {\tan ^3(c+d x)}{3 d}+\frac {\tan (c+d x)}{d}-x}{a}\) |
(Sec[c + d*x] - (2*Sec[c + d*x]^3)/3 + Sec[c + d*x]^5/5)/(a*d) - (-x + Tan [c + d*x]/d - Tan[c + d*x]^3/(3*d) + Tan[c + d*x]^5/(5*d))/a
3.9.22.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 )^p, x], x] /; FreeQ[{a, b}, 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[((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]
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 = 0.43 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.20
| method | result | size |
| risch | \(\frac {x}{a}+\frac {-2 i {\mathrm e}^{6 i \left (d x +c \right )}+2 \,{\mathrm e}^{7 i \left (d x +c \right )}+\frac {10 i {\mathrm e}^{4 i \left (d x +c \right )}}{3}+\frac {26 \,{\mathrm e}^{5 i \left (d x +c \right )}}{3}+\frac {62 i {\mathrm e}^{2 i \left (d x +c \right )}}{15}+\frac {146 \,{\mathrm e}^{3 i \left (d x +c \right )}}{15}+\frac {46 i}{15}+\frac {62 \,{\mathrm e}^{i \left (d x +c \right )}}{15}}{\left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{5} d a}\) | \(126\) |
| derivativedivides | \(\frac {-\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {5}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {2}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {11}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d a}\) | \(127\) |
| default | \(\frac {-\frac {1}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {5}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {2}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {11}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d a}\) | \(127\) |
| parallelrisch | \(\frac {15 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) x d +\left (30 d x +30\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-30 d x +60\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-90 d x -70\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-200 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (90 d x +26\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 d x +92\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-30 d x -2\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-15 d x -16}{15 d a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\) | \(175\) |
| norman | \(\frac {\frac {x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {x}{a}-\frac {44 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d a}-\frac {2 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}+\frac {4 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {2 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {2 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {4 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {2 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {16}{15 a d}+\frac {4 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {2 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {8 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {28 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {36 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}+\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{15 d a}+\frac {76 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(376\) |
x/a+2/15*(-15*I*exp(6*I*(d*x+c))+15*exp(7*I*(d*x+c))+25*I*exp(4*I*(d*x+c)) +65*exp(5*I*(d*x+c))+31*I*exp(2*I*(d*x+c))+73*exp(3*I*(d*x+c))+23*I+31*exp (I*(d*x+c)))/(exp(I*(d*x+c))-I)^3/(exp(I*(d*x+c))+I)^5/d/a
Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.93 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {15 \, d x \cos \left (d x + c\right )^{3} + 23 \, \cos \left (d x + c\right )^{4} - 19 \, \cos \left (d x + c\right )^{2} + {\left (15 \, d x \cos \left (d x + c\right )^{3} - 8 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + 4}{15 \, {\left (a d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{3}\right )}} \]
1/15*(15*d*x*cos(d*x + c)^3 + 23*cos(d*x + c)^4 - 19*cos(d*x + c)^2 + (15* d*x*cos(d*x + c)^3 - 8*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. \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (97) = 194\).
Time = 0.31 (sec) , antiderivative size = 318, normalized size of antiderivative = 3.03 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2 \, {\left (\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {46 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {13 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {100 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {35 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {30 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + 8}{a + \frac {2 \, a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {6 \, a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {6 \, a \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {2 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {2 \, a \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} + \frac {15 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}\right )}}{15 \, d} \]
2/15*((sin(d*x + c)/(cos(d*x + c) + 1) - 46*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 13*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 100*sin(d*x + c)^4/(cos(d *x + c) + 1)^4 + 35*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 30*sin(d*x + c)^ 6/(cos(d*x + c) + 1)^6 - 15*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 8)/(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) + 15*arct an(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d
Time = 0.33 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.24 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {120 \, {\left (d x + c\right )}}{a} + \frac {5 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 36 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 17\right )}}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {3 \, {\left (55 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 260 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 450 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 300 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 71\right )}}{a {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{120 \, d} \]
1/120*(120*(d*x + c)/a + 5*(15*tan(1/2*d*x + 1/2*c)^2 - 36*tan(1/2*d*x + 1 /2*c) + 17)/(a*(tan(1/2*d*x + 1/2*c) - 1)^3) + 3*(55*tan(1/2*d*x + 1/2*c)^ 4 + 260*tan(1/2*d*x + 1/2*c)^3 + 450*tan(1/2*d*x + 1/2*c)^2 + 300*tan(1/2* d*x + 1/2*c) + 71)/(a*(tan(1/2*d*x + 1/2*c) + 1)^5))/d
Time = 12.95 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.25 \[ \int \frac {\sin (c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {x}{a}-\frac {-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}+\frac {40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}-\frac {26\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{15}-\frac {92\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}+\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{15}+\frac {16}{15}}{a\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^5} \]