Integrand size = 21, antiderivative size = 119 \[ \int (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx=\frac {17 a^3 x}{2}-\frac {6 a^3 \cos (c+d x)}{d}+\frac {a^3 \cos ^3(c+d x)}{3 d}+\frac {2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac {25 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}-\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{2 d} \]
17/2*a^3*x-6*a^3*cos(d*x+c)/d+1/3*a^3*cos(d*x+c)^3/d+2/3*a^3*cos(d*x+c)/d/ (1-sin(d*x+c))^2-25/3*a^3*cos(d*x+c)/d/(1-sin(d*x+c))-3/2*a^3*cos(d*x+c)*s in(d*x+c)/d
Time = 1.23 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.49 \[ \int (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx=\frac {(a+a \sin (c+d x))^3 \left (102 (c+d x)-69 \cos (c+d x)+\cos (3 (c+d x))+\frac {8}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {16 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}-\frac {200 \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}-9 \sin (2 (c+d x))\right )}{12 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
((a + a*Sin[c + d*x])^3*(102*(c + d*x) - 69*Cos[c + d*x] + Cos[3*(c + d*x) ] + 8/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (16*Sin[(c + d*x)/2])/(Cos [(c + d*x)/2] - Sin[(c + d*x)/2])^3 - (200*Sin[(c + d*x)/2])/(Cos[(c + d*x )/2] - Sin[(c + d*x)/2]) - 9*Sin[2*(c + d*x)]))/(12*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)
Time = 0.37 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3188, 2009}
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.
| \(\displaystyle \int \tan ^4(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (c+d x)^4 (a \sin (c+d x)+a)^3dx\) |
| \(\Big \downarrow \) 3188 |
| \(\displaystyle a^4 \int \left (\frac {\sin ^3(c+d x)}{a}+\frac {3 \sin ^2(c+d x)}{a}+\frac {5 \sin (c+d x)}{a}+\frac {7}{a}-\frac {9}{a (1-\sin (c+d x))}+\frac {2}{a (1-\sin (c+d x))^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^4 \left (\frac {\cos ^3(c+d x)}{3 a d}-\frac {6 \cos (c+d x)}{a d}-\frac {3 \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {25 \cos (c+d x)}{3 a d (1-\sin (c+d x))}+\frac {2 \cos (c+d x)}{3 a d (1-\sin (c+d x))^2}+\frac {17 x}{2 a}\right )\) |
a^4*((17*x)/(2*a) - (6*Cos[c + d*x])/(a*d) + Cos[c + d*x]^3/(3*a*d) + (2*C os[c + d*x])/(3*a*d*(1 - Sin[c + d*x])^2) - (25*Cos[c + d*x])/(3*a*d*(1 - Sin[c + d*x])) - (3*Cos[c + d*x]*Sin[c + d*x])/(2*a*d))
3.9.11.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_ ), x_Symbol] :> Simp[a^p Int[ExpandIntegrand[Sin[e + f*x]^p*((a + b*Sin[e + f*x])^(m - p/2)/(a - b*Sin[e + f*x])^(p/2)), x], x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p/2] && (LtQ[p, 0] || GtQ[m - p/2, 0])
Time = 0.30 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.13
| method | result | size |
| parallelrisch | \(\frac {a^{3} \left (612 d x \cos \left (d x +c \right )+204 d x \cos \left (3 d x +3 c \right )+\cos \left (6 d x +6 c \right )-9 \sin \left (5 d x +5 c \right )-66 \cos \left (4 d x +4 c \right )-705 \cos \left (2 d x +2 c \right )-227 \sin \left (3 d x +3 c \right )-960 \cos \left (d x +c \right )-90 \sin \left (d x +c \right )-320 \cos \left (3 d x +3 c \right )-510\right )}{24 d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(134\) |
| risch | \(\frac {17 a^{3} x}{2}+\frac {3 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {23 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {23 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}-\frac {3 i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {2 \left (-48 i a^{3} {\mathrm e}^{i \left (d x +c \right )}+27 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-25 a^{3}\right )}{3 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d}+\frac {a^{3} \cos \left (3 d x +3 c \right )}{12 d}\) | \(149\) |
| norman | \(\frac {-\frac {17 a^{3} x}{2}+\frac {80 a^{3}}{3 d}+\frac {17 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {17 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {54 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {32 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {54 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {17 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {17 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {51 a^{3} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {51 a^{3} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {17 a^{3} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {80 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}\) | \(248\) |
| derivativedivides | \(\frac {a^{3} \left (\frac {\sin ^{8}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {5 \left (\sin ^{8}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {5 \left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )}{3}\right )+3 a^{3} \left (\frac {\sin ^{7}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+3 a^{3} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+a^{3} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )}{d}\) | \(266\) |
| default | \(\frac {a^{3} \left (\frac {\sin ^{8}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {5 \left (\sin ^{8}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {5 \left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )}{3}\right )+3 a^{3} \left (\frac {\sin ^{7}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+3 a^{3} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+a^{3} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )}{d}\) | \(266\) |
1/24/d*a^3*(612*d*x*cos(d*x+c)+204*d*x*cos(3*d*x+3*c)+cos(6*d*x+6*c)-9*sin (5*d*x+5*c)-66*cos(4*d*x+4*c)-705*cos(2*d*x+2*c)-227*sin(3*d*x+3*c)-960*co s(d*x+c)-90*sin(d*x+c)-320*cos(3*d*x+3*c)-510)/(cos(3*d*x+3*c)+3*cos(d*x+c ))
Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (105) = 210\).
Time = 0.26 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.85 \[ \int (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx=\frac {2 \, a^{3} \cos \left (d x + c\right )^{5} + 7 \, a^{3} \cos \left (d x + c\right )^{4} - 22 \, a^{3} \cos \left (d x + c\right )^{3} - 102 \, a^{3} d x - 4 \, a^{3} + {\left (51 \, a^{3} d x + 77 \, a^{3}\right )} \cos \left (d x + c\right )^{2} - {\left (51 \, a^{3} d x - 100 \, a^{3}\right )} \cos \left (d x + c\right ) + {\left (2 \, a^{3} \cos \left (d x + c\right )^{4} - 5 \, a^{3} \cos \left (d x + c\right )^{3} + 102 \, a^{3} d x - 27 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3} + {\left (51 \, a^{3} d x - 104 \, a^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \]
1/6*(2*a^3*cos(d*x + c)^5 + 7*a^3*cos(d*x + c)^4 - 22*a^3*cos(d*x + c)^3 - 102*a^3*d*x - 4*a^3 + (51*a^3*d*x + 77*a^3)*cos(d*x + c)^2 - (51*a^3*d*x - 100*a^3)*cos(d*x + c) + (2*a^3*cos(d*x + c)^4 - 5*a^3*cos(d*x + c)^3 + 1 02*a^3*d*x - 27*a^3*cos(d*x + c)^2 - 4*a^3 + (51*a^3*d*x - 104*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. \[ \int (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.39 \[ \int (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx=\frac {2 \, {\left (\cos \left (d x + c\right )^{3} - \frac {9 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} - 9 \, \cos \left (d x + c\right )\right )} a^{3} + 3 \, {\left (2 \, \tan \left (d x + c\right )^{3} + 15 \, d x + 15 \, c - \frac {3 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 12 \, \tan \left (d x + c\right )\right )} a^{3} + 2 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{3} - 6 \, a^{3} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )}}{6 \, d} \]
1/6*(2*(cos(d*x + c)^3 - (9*cos(d*x + c)^2 - 1)/cos(d*x + c)^3 - 9*cos(d*x + c))*a^3 + 3*(2*tan(d*x + c)^3 + 15*d*x + 15*c - 3*tan(d*x + c)/(tan(d*x + c)^2 + 1) - 12*tan(d*x + c))*a^3 + 2*(tan(d*x + c)^3 + 3*d*x + 3*c - 3* tan(d*x + c))*a^3 - 6*a^3*((6*cos(d*x + c)^2 - 1)/cos(d*x + c)^3 + 3*cos(d *x + c)))/d
Time = 0.36 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.57 \[ \int (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx=\frac {51 \, {\left (d x + c\right )} a^{3} + \frac {2 \, {\left (51 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 153 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 289 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 459 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 501 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 511 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 327 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 189 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 80 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}}}{6 \, d} \]
1/6*(51*(d*x + c)*a^3 + 2*(51*a^3*tan(1/2*d*x + 1/2*c)^8 - 153*a^3*tan(1/2 *d*x + 1/2*c)^7 + 289*a^3*tan(1/2*d*x + 1/2*c)^6 - 459*a^3*tan(1/2*d*x + 1 /2*c)^5 + 501*a^3*tan(1/2*d*x + 1/2*c)^4 - 511*a^3*tan(1/2*d*x + 1/2*c)^3 + 327*a^3*tan(1/2*d*x + 1/2*c)^2 - 189*a^3*tan(1/2*d*x + 1/2*c) + 80*a^3)/ (tan(1/2*d*x + 1/2*c)^3 - tan(1/2*d*x + 1/2*c)^2 + tan(1/2*d*x + 1/2*c) - 1)^3)/d
Time = 17.33 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.66 \[ \int (a+a \sin (c+d x))^3 \tan ^4(c+d x) \, dx=\frac {17\,a^3\,x}{2}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {a^3\,\left (153\,d\,x-378\right )}{6}-\frac {51\,a^3\,d\,x}{2}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {a^3\,\left (153\,d\,x-102\right )}{6}-\frac {51\,a^3\,d\,x}{2}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {a^3\,\left (306\,d\,x-306\right )}{6}-51\,a^3\,d\,x\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^3\,\left (306\,d\,x-654\right )}{6}-51\,a^3\,d\,x\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a^3\,\left (510\,d\,x-578\right )}{6}-85\,a^3\,d\,x\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {a^3\,\left (510\,d\,x-1022\right )}{6}-85\,a^3\,d\,x\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {a^3\,\left (612\,d\,x-918\right )}{6}-102\,a^3\,d\,x\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^3\,\left (612\,d\,x-1002\right )}{6}-102\,a^3\,d\,x\right )-\frac {a^3\,\left (51\,d\,x-160\right )}{6}+\frac {17\,a^3\,d\,x}{2}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3} \]
(17*a^3*x)/2 + (tan(c/2 + (d*x)/2)*((a^3*(153*d*x - 378))/6 - (51*a^3*d*x) /2) - tan(c/2 + (d*x)/2)^8*((a^3*(153*d*x - 102))/6 - (51*a^3*d*x)/2) + ta n(c/2 + (d*x)/2)^7*((a^3*(306*d*x - 306))/6 - 51*a^3*d*x) - tan(c/2 + (d*x )/2)^2*((a^3*(306*d*x - 654))/6 - 51*a^3*d*x) - tan(c/2 + (d*x)/2)^6*((a^3 *(510*d*x - 578))/6 - 85*a^3*d*x) + tan(c/2 + (d*x)/2)^3*((a^3*(510*d*x - 1022))/6 - 85*a^3*d*x) + tan(c/2 + (d*x)/2)^5*((a^3*(612*d*x - 918))/6 - 1 02*a^3*d*x) - tan(c/2 + (d*x)/2)^4*((a^3*(612*d*x - 1002))/6 - 102*a^3*d*x ) - (a^3*(51*d*x - 160))/6 + (17*a^3*d*x)/2)/(d*(tan(c/2 + (d*x)/2) - tan( c/2 + (d*x)/2)^2 + tan(c/2 + (d*x)/2)^3 - 1)^3)