Integrand size = 21, antiderivative size = 143 \[ \int (a+a \sin (c+d x))^4 \tan ^4(c+d x) \, dx=\frac {163 a^4 x}{8}-\frac {16 a^4 \cos (c+d x)}{d}+\frac {4 a^4 \cos ^3(c+d x)}{3 d}+\frac {4 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac {56 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))}-\frac {35 a^4 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d} \]
163/8*a^4*x-16*a^4*cos(d*x+c)/d+4/3*a^4*cos(d*x+c)^3/d+4/3*a^4*cos(d*x+c)/ d/(1-sin(d*x+c))^2-56/3*a^4*cos(d*x+c)/d/(1-sin(d*x+c))-35/8*a^4*cos(d*x+c )*sin(d*x+c)/d-1/4*a^4*cos(d*x+c)*sin(d*x+c)^3/d
Time = 0.86 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.76 \[ \int (a+a \sin (c+d x))^4 \tan ^4(c+d x) \, dx=\frac {a^4 \left (24 (209+489 c+489 d x) \cos \left (\frac {1}{2} (c+d x)\right )-24 (453+163 c+163 d x) \cos \left (\frac {3}{2} (c+d x)\right )+885 \cos \left (\frac {5}{2} (c+d x)\right )-129 \cos \left (\frac {7}{2} (c+d x)\right )-23 \cos \left (\frac {9}{2} (c+d x)\right )+3 \cos \left (\frac {11}{2} (c+d x)\right )-16488 \sin \left (\frac {1}{2} (c+d x)\right )-11736 c \sin \left (\frac {1}{2} (c+d x)\right )-11736 d x \sin \left (\frac {1}{2} (c+d x)\right )+3704 \sin \left (\frac {3}{2} (c+d x)\right )-3912 c \sin \left (\frac {3}{2} (c+d x)\right )-3912 d x \sin \left (\frac {3}{2} (c+d x)\right )+885 \sin \left (\frac {5}{2} (c+d x)\right )+129 \sin \left (\frac {7}{2} (c+d x)\right )-23 \sin \left (\frac {9}{2} (c+d x)\right )-3 \sin \left (\frac {11}{2} (c+d x)\right )\right )}{384 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
(a^4*(24*(209 + 489*c + 489*d*x)*Cos[(c + d*x)/2] - 24*(453 + 163*c + 163* d*x)*Cos[(3*(c + d*x))/2] + 885*Cos[(5*(c + d*x))/2] - 129*Cos[(7*(c + d*x ))/2] - 23*Cos[(9*(c + d*x))/2] + 3*Cos[(11*(c + d*x))/2] - 16488*Sin[(c + d*x)/2] - 11736*c*Sin[(c + d*x)/2] - 11736*d*x*Sin[(c + d*x)/2] + 3704*Si n[(3*(c + d*x))/2] - 3912*c*Sin[(3*(c + d*x))/2] - 3912*d*x*Sin[(3*(c + d* x))/2] + 885*Sin[(5*(c + d*x))/2] + 129*Sin[(7*(c + d*x))/2] - 23*Sin[(9*( c + d*x))/2] - 3*Sin[(11*(c + d*x))/2]))/(384*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3)
Time = 0.38 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.88, 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.
| \(\displaystyle \int \tan ^4(c+d x) (a \sin (c+d x)+a)^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (c+d x)^4 (a \sin (c+d x)+a)^4dx\) |
| \(\Big \downarrow \) 3188 |
| \(\displaystyle a^4 \int \left (\sin ^4(c+d x)+4 \sin ^3(c+d x)+8 \sin ^2(c+d x)+12 \sin (c+d x)-\frac {20}{1-\sin (c+d x)}+\frac {4}{(1-\sin (c+d x))^2}+16\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^4 \left (\frac {4 \cos ^3(c+d x)}{3 d}-\frac {16 \cos (c+d x)}{d}-\frac {\sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {35 \sin (c+d x) \cos (c+d x)}{8 d}-\frac {56 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac {4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac {163 x}{8}\right )\) |
a^4*((163*x)/8 - (16*Cos[c + d*x])/d + (4*Cos[c + d*x]^3)/(3*d) + (4*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])^2) - (56*Cos[c + d*x])/(3*d*(1 - Sin[c + d*x])) - (35*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (Cos[c + d*x]*Sin[c + d*x] ^3)/(4*d))
3.9.19.3.1 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])
Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.16
| method | result | size |
| risch | \(\frac {163 a^{4} x}{8}+\frac {9 i a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {15 a^{4} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {15 a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {9 i a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {8 \left (-27 i a^{4} {\mathrm e}^{i \left (d x +c \right )}+15 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-14 a^{4}\right )}{3 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d}+\frac {a^{4} \sin \left (4 d x +4 c \right )}{32 d}+\frac {a^{4} \cos \left (3 d x +3 c \right )}{3 d}\) | \(166\) |
| parallelrisch | \(\frac {\left (11736 a^{4} x d -12696 a^{4}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+a^{4} \left (3912 d x \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-11736 d x \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+3912 d x \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-13432 \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-885 \sin \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )-129 \sin \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )+23 \sin \left (\frac {9 d x}{2}+\frac {9 c}{2}\right )-3 \cos \left (\frac {11 d x}{2}+\frac {11 c}{2}\right )+3 \sin \left (\frac {11 d x}{2}+\frac {11 c}{2}\right )+24168 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+1144 \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-885 \cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+129 \cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )+23 \cos \left (\frac {9 d x}{2}+\frac {9 c}{2}\right )\right )}{192 d \left (\sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )\right )}\) | \(238\) |
| derivativedivides | \(\frac {a^{4} \left (\frac {\sin ^{9}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {2 \left (\sin ^{9}\left (d x +c \right )\right )}{\cos \left (d x +c \right )}-2 \left (\sin ^{7}\left (d x +c \right )+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )+\frac {35 d x}{8}+\frac {35 c}{8}\right )+4 a^{4} \left (\frac {\sin ^{8}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {5 \left (\sin ^{8}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {5 \left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )}{3}\right )+6 a^{4} \left (\frac {\sin ^{7}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+4 a^{4} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+a^{4} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )}{d}\) | \(360\) |
| default | \(\frac {a^{4} \left (\frac {\sin ^{9}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {2 \left (\sin ^{9}\left (d x +c \right )\right )}{\cos \left (d x +c \right )}-2 \left (\sin ^{7}\left (d x +c \right )+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )+\frac {35 d x}{8}+\frac {35 c}{8}\right )+4 a^{4} \left (\frac {\sin ^{8}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {5 \left (\sin ^{8}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {5 \left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )}{3}\right )+6 a^{4} \left (\frac {\sin ^{7}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+4 a^{4} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+a^{4} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )}{d}\) | \(360\) |
| norman | \(\frac {\frac {64 a^{4}}{d}+\frac {64 a^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {163 a^{4} x}{8}+\frac {163 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {163 a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {1609 a^{4} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {209 a^{4} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {1609 a^{4} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {163 a^{4} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {163 a^{4} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {163 a^{4} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {489 a^{4} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {489 a^{4} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {489 a^{4} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {489 a^{4} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {163 a^{4} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {163 a^{4} x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {192 a^{4} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {128 a^{4} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {704 a^{4} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}\) | \(373\) |
163/8*a^4*x+9/8*I*a^4/d*exp(2*I*(d*x+c))-15/2*a^4/d*exp(I*(d*x+c))-15/2*a^ 4/d*exp(-I*(d*x+c))-9/8*I*a^4/d*exp(-2*I*(d*x+c))-8/3*(-27*I*a^4*exp(I*(d* x+c))+15*a^4*exp(2*I*(d*x+c))-14*a^4)/(exp(I*(d*x+c))-I)^3/d+1/32*a^4/d*si n(4*d*x+4*c)+1/3*a^4/d*cos(3*d*x+3*c)
Time = 0.26 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.73 \[ \int (a+a \sin (c+d x))^4 \tan ^4(c+d x) \, dx=-\frac {6 \, a^{4} \cos \left (d x + c\right )^{6} - 20 \, a^{4} \cos \left (d x + c\right )^{5} - 85 \, a^{4} \cos \left (d x + c\right )^{4} + 214 \, a^{4} \cos \left (d x + c\right )^{3} + 978 \, a^{4} d x + 32 \, a^{4} - {\left (489 \, a^{4} d x + 721 \, a^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (489 \, a^{4} d x - 962 \, a^{4}\right )} \cos \left (d x + c\right ) - {\left (6 \, a^{4} \cos \left (d x + c\right )^{5} + 26 \, a^{4} \cos \left (d x + c\right )^{4} - 59 \, a^{4} \cos \left (d x + c\right )^{3} + 978 \, a^{4} d x - 273 \, a^{4} \cos \left (d x + c\right )^{2} - 32 \, a^{4} + {\left (489 \, a^{4} d x - 994 \, a^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \]
-1/24*(6*a^4*cos(d*x + c)^6 - 20*a^4*cos(d*x + c)^5 - 85*a^4*cos(d*x + c)^ 4 + 214*a^4*cos(d*x + c)^3 + 978*a^4*d*x + 32*a^4 - (489*a^4*d*x + 721*a^4 )*cos(d*x + c)^2 + (489*a^4*d*x - 962*a^4)*cos(d*x + c) - (6*a^4*cos(d*x + c)^5 + 26*a^4*cos(d*x + c)^4 - 59*a^4*cos(d*x + c)^3 + 978*a^4*d*x - 273* a^4*cos(d*x + c)^2 - 32*a^4 + (489*a^4*d*x - 994*a^4)*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. \[ \int (a+a \sin (c+d x))^4 \tan ^4(c+d x) \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.66 \[ \int (a+a \sin (c+d x))^4 \tan ^4(c+d x) \, dx=\frac {32 \, {\left (\cos \left (d x + c\right )^{3} - \frac {9 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} - 9 \, \cos \left (d x + c\right )\right )} a^{4} + {\left (8 \, \tan \left (d x + c\right )^{3} + 105 \, d x + 105 \, c - \frac {3 \, {\left (13 \, \tan \left (d x + c\right )^{3} + 11 \, \tan \left (d x + c\right )\right )}}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1} - 72 \, \tan \left (d x + c\right )\right )} a^{4} + 24 \, {\left (2 \, \tan \left (d x + c\right )^{3} + 15 \, d x + 15 \, c - \frac {3 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 12 \, \tan \left (d x + c\right )\right )} a^{4} + 8 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{4} - 32 \, a^{4} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )}}{24 \, d} \]
1/24*(32*(cos(d*x + c)^3 - (9*cos(d*x + c)^2 - 1)/cos(d*x + c)^3 - 9*cos(d *x + c))*a^4 + (8*tan(d*x + c)^3 + 105*d*x + 105*c - 3*(13*tan(d*x + c)^3 + 11*tan(d*x + c))/(tan(d*x + c)^4 + 2*tan(d*x + c)^2 + 1) - 72*tan(d*x + c))*a^4 + 24*(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^4 + 8*(tan(d*x + c)^3 + 3*d*x + 3*c - 3*ta n(d*x + c))*a^4 - 32*a^4*((6*cos(d*x + c)^2 - 1)/cos(d*x + c)^3 + 3*cos(d* x + c)))/d
Time = 0.35 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.40 \[ \int (a+a \sin (c+d x))^4 \tan ^4(c+d x) \, dx=\frac {489 \, {\left (d x + c\right )} a^{4} + \frac {64 \, {\left (12 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 27 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 13 \, a^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {2 \, {\left (105 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 288 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 129 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1056 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 129 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1120 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 105 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 352 \, a^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
1/24*(489*(d*x + c)*a^4 + 64*(12*a^4*tan(1/2*d*x + 1/2*c)^2 - 27*a^4*tan(1 /2*d*x + 1/2*c) + 13*a^4)/(tan(1/2*d*x + 1/2*c) - 1)^3 + 2*(105*a^4*tan(1/ 2*d*x + 1/2*c)^7 - 288*a^4*tan(1/2*d*x + 1/2*c)^6 + 129*a^4*tan(1/2*d*x + 1/2*c)^5 - 1056*a^4*tan(1/2*d*x + 1/2*c)^4 - 129*a^4*tan(1/2*d*x + 1/2*c)^ 3 - 1120*a^4*tan(1/2*d*x + 1/2*c)^2 - 105*a^4*tan(1/2*d*x + 1/2*c) - 352*a ^4)/(tan(1/2*d*x + 1/2*c)^2 + 1)^4)/d
Time = 19.52 (sec) , antiderivative size = 437, normalized size of antiderivative = 3.06 \[ \int (a+a \sin (c+d x))^4 \tan ^4(c+d x) \, dx=\frac {163\,a^4\,x}{8}+\frac {\frac {163\,a^4\,\left (c+d\,x\right )}{8}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {489\,a^4\,\left (c+d\,x\right )}{8}-\frac {a^4\,\left (1467\,c+1467\,d\,x-3630\right )}{24}\right )-\frac {a^4\,\left (489\,c+489\,d\,x-1536\right )}{24}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {489\,a^4\,\left (c+d\,x\right )}{8}-\frac {a^4\,\left (1467\,c+1467\,d\,x-978\right )}{24}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {1141\,a^4\,\left (c+d\,x\right )}{8}-\frac {a^4\,\left (3423\,c+3423\,d\,x-2934\right )}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {1141\,a^4\,\left (c+d\,x\right )}{8}-\frac {a^4\,\left (3423\,c+3423\,d\,x-7818\right )}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {2119\,a^4\,\left (c+d\,x\right )}{8}-\frac {a^4\,\left (6357\,c+6357\,d\,x-6520\right )}{24}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {2119\,a^4\,\left (c+d\,x\right )}{8}-\frac {a^4\,\left (6357\,c+6357\,d\,x-13448\right )}{24}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {1467\,a^4\,\left (c+d\,x\right )}{4}-\frac {a^4\,\left (8802\,c+8802\,d\,x-11736\right )}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {1467\,a^4\,\left (c+d\,x\right )}{4}-\frac {a^4\,\left (8802\,c+8802\,d\,x-15912\right )}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {1793\,a^4\,\left (c+d\,x\right )}{4}-\frac {a^4\,\left (10758\,c+10758\,d\,x-15364\right )}{24}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {1793\,a^4\,\left (c+d\,x\right )}{4}-\frac {a^4\,\left (10758\,c+10758\,d\,x-18428\right )}{24}\right )}{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 )}^2+1\right )}^4} \]
(163*a^4*x)/8 + ((163*a^4*(c + d*x))/8 - tan(c/2 + (d*x)/2)*((489*a^4*(c + d*x))/8 - (a^4*(1467*c + 1467*d*x - 3630))/24) - (a^4*(489*c + 489*d*x - 1536))/24 + tan(c/2 + (d*x)/2)^10*((489*a^4*(c + d*x))/8 - (a^4*(1467*c + 1467*d*x - 978))/24) - tan(c/2 + (d*x)/2)^9*((1141*a^4*(c + d*x))/8 - (a^4 *(3423*c + 3423*d*x - 2934))/24) + tan(c/2 + (d*x)/2)^2*((1141*a^4*(c + d* x))/8 - (a^4*(3423*c + 3423*d*x - 7818))/24) + tan(c/2 + (d*x)/2)^8*((2119 *a^4*(c + d*x))/8 - (a^4*(6357*c + 6357*d*x - 6520))/24) - tan(c/2 + (d*x) /2)^3*((2119*a^4*(c + d*x))/8 - (a^4*(6357*c + 6357*d*x - 13448))/24) - ta n(c/2 + (d*x)/2)^7*((1467*a^4*(c + d*x))/4 - (a^4*(8802*c + 8802*d*x - 117 36))/24) + tan(c/2 + (d*x)/2)^4*((1467*a^4*(c + d*x))/4 - (a^4*(8802*c + 8 802*d*x - 15912))/24) + tan(c/2 + (d*x)/2)^6*((1793*a^4*(c + d*x))/4 - (a^ 4*(10758*c + 10758*d*x - 15364))/24) - tan(c/2 + (d*x)/2)^5*((1793*a^4*(c + d*x))/4 - (a^4*(10758*c + 10758*d*x - 18428))/24))/(d*(tan(c/2 + (d*x)/2 ) - 1)^3*(tan(c/2 + (d*x)/2)^2 + 1)^4)