Integrand size = 21, antiderivative size = 143 \[ \int (a+a \sin (c+d x))^4 \tan ^4(c+d x) \, dx=\frac {163 a^4 x}{8}-\frac {16 a^4 \cos (c+d x)}{d}+\frac {4 a^4 \cos ^3(c+d x)}{3 d}+\frac {4 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac {56 a^4 \cos (c+d x)}{3 d (1-\sin (c+d x))}-\frac {35 a^4 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^4 \cos (c+d x) \sin ^3(c+d x)}{4 d} \] Output:
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 = 7.71 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.76 \[ \int (a+a \sin (c+d x))^4 \tan ^4(c+d x) \, dx=\frac {a^4 \left (24 (209+489 c+489 d x) \cos \left (\frac {1}{2} (c+d x)\right )-24 (453+163 c+163 d x) \cos \left (\frac {3}{2} (c+d x)\right )+885 \cos \left (\frac {5}{2} (c+d x)\right )-129 \cos \left (\frac {7}{2} (c+d x)\right )-23 \cos \left (\frac {9}{2} (c+d x)\right )+3 \cos \left (\frac {11}{2} (c+d x)\right )-16488 \sin \left (\frac {1}{2} (c+d x)\right )-11736 c \sin \left (\frac {1}{2} (c+d x)\right )-11736 d x \sin \left (\frac {1}{2} (c+d x)\right )+3704 \sin \left (\frac {3}{2} (c+d x)\right )-3912 c \sin \left (\frac {3}{2} (c+d x)\right )-3912 d x \sin \left (\frac {3}{2} (c+d x)\right )+885 \sin \left (\frac {5}{2} (c+d x)\right )+129 \sin \left (\frac {7}{2} (c+d x)\right )-23 \sin \left (\frac {9}{2} (c+d x)\right )-3 \sin \left (\frac {11}{2} (c+d x)\right )\right )}{384 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3} \] Input:
Integrate[(a + a*Sin[c + d*x])^4*Tan[c + d*x]^4,x]
Output:
(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 = 0.41 (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 |
| \(\displaystyle a^4 \left (\frac {4 \cos ^3(c+d x)}{3 d}-\frac {16 \cos (c+d x)}{d}-\frac {\sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {35 \sin (c+d x) \cos (c+d x)}{8 d}-\frac {56 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac {4 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac {163 x}{8}\right )\) |
Input:
Int[(a + a*Sin[c + d*x])^4*Tan[c + d*x]^4,x]
Output:
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))
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 = 28.14 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.16
| method | result | size |
| risch | \(\frac {163 a^{4} x}{8}+\frac {9 i a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {15 a^{4} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {15 a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {9 i a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {8 \left (-27 i a^{4} {\mathrm e}^{i \left (d x +c \right )}+15 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-14 a^{4}\right )}{3 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d}+\frac {a^{4} \sin \left (4 d x +4 c \right )}{32 d}+\frac {a^{4} \cos \left (3 d x +3 c \right )}{3 d}\) | \(166\) |
| derivativedivides | \(\frac {a^{4} \left (\frac {\sin \left (d x +c \right )^{9}}{3 \cos \left (d x +c \right )^{3}}-\frac {2 \sin \left (d x +c \right )^{9}}{\cos \left (d x +c \right )}-2 \left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )+\frac {35 d x}{8}+\frac {35 c}{8}\right )+4 a^{4} \left (\frac {\sin \left (d x +c \right )^{8}}{3 \cos \left (d x +c \right )^{3}}-\frac {5 \sin \left (d x +c \right )^{8}}{3 \cos \left (d x +c \right )}-\frac {5 \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{3}\right )+6 a^{4} \left (\frac {\sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+4 a^{4} \left (\frac {\sin \left (d x +c \right )^{6}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{6}}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )\right )+a^{4} \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+d x +c \right )}{d}\) | \(360\) |
| default | \(\frac {a^{4} \left (\frac {\sin \left (d x +c \right )^{9}}{3 \cos \left (d x +c \right )^{3}}-\frac {2 \sin \left (d x +c \right )^{9}}{\cos \left (d x +c \right )}-2 \left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )+\frac {35 d x}{8}+\frac {35 c}{8}\right )+4 a^{4} \left (\frac {\sin \left (d x +c \right )^{8}}{3 \cos \left (d x +c \right )^{3}}-\frac {5 \sin \left (d x +c \right )^{8}}{3 \cos \left (d x +c \right )}-\frac {5 \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{3}\right )+6 a^{4} \left (\frac {\sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )+4 a^{4} \left (\frac {\sin \left (d x +c \right )^{6}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{6}}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )\right )+a^{4} \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+d x +c \right )}{d}\) | \(360\) |
| parts | \(\frac {a^{4} \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {a^{4} \left (\frac {\sin \left (d x +c \right )^{9}}{3 \cos \left (d x +c \right )^{3}}-\frac {2 \sin \left (d x +c \right )^{9}}{\cos \left (d x +c \right )}-2 \left (\sin \left (d x +c \right )^{7}+\frac {7 \sin \left (d x +c \right )^{5}}{6}+\frac {35 \sin \left (d x +c \right )^{3}}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )+\frac {35 d x}{8}+\frac {35 c}{8}\right )}{d}+\frac {4 a^{4} \left (\frac {\sin \left (d x +c \right )^{6}}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{6}}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )\right )}{d}+\frac {6 a^{4} \left (\frac {\sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )^{3}}-\frac {4 \sin \left (d x +c \right )^{7}}{3 \cos \left (d x +c \right )}-\frac {4 \left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{3}+\frac {5 d x}{2}+\frac {5 c}{2}\right )}{d}+\frac {4 a^{4} \left (\frac {\sin \left (d x +c \right )^{8}}{3 \cos \left (d x +c \right )^{3}}-\frac {5 \sin \left (d x +c \right )^{8}}{3 \cos \left (d x +c \right )}-\frac {5 \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{3}\right )}{d}\) | \(374\) |
Input:
int((a+a*sin(d*x+c))^4*tan(d*x+c)^4,x,method=_RETURNVERBOSE)
Output:
163/8*a^4*x+9/8*I/d*a^4*exp(2*I*(d*x+c))-15/2/d*a^4*exp(I*(d*x+c))-15/2*a^ 4/d*exp(-I*(d*x+c))-9/8*I/d*a^4*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/d*a^4*si n(4*d*x+4*c)+1/3*a^4/d*cos(3*d*x+3*c)
Time = 0.13 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.73 \[ \int (a+a \sin (c+d x))^4 \tan ^4(c+d x) \, dx=-\frac {6 \, a^{4} \cos \left (d x + c\right )^{6} - 20 \, a^{4} \cos \left (d x + c\right )^{5} - 85 \, a^{4} \cos \left (d x + c\right )^{4} + 214 \, a^{4} \cos \left (d x + c\right )^{3} + 978 \, a^{4} d x + 32 \, a^{4} - {\left (489 \, a^{4} d x + 721 \, a^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (489 \, a^{4} d x - 962 \, a^{4}\right )} \cos \left (d x + c\right ) - {\left (6 \, a^{4} \cos \left (d x + c\right )^{5} + 26 \, a^{4} \cos \left (d x + c\right )^{4} - 59 \, a^{4} \cos \left (d x + c\right )^{3} + 978 \, a^{4} d x - 273 \, a^{4} \cos \left (d x + c\right )^{2} - 32 \, a^{4} + {\left (489 \, a^{4} d x - 994 \, a^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \] Input:
integrate((a+a*sin(d*x+c))^4*tan(d*x+c)^4,x, algorithm="fricas")
Output:
-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)
\[ \int (a+a \sin (c+d x))^4 \tan ^4(c+d x) \, dx=a^{4} \left (\int 4 \sin {\left (c + d x \right )} \tan ^{4}{\left (c + d x \right )}\, dx + \int 6 \sin ^{2}{\left (c + d x \right )} \tan ^{4}{\left (c + d x \right )}\, dx + \int 4 \sin ^{3}{\left (c + d x \right )} \tan ^{4}{\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )} \tan ^{4}{\left (c + d x \right )}\, dx + \int \tan ^{4}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate((a+a*sin(d*x+c))**4*tan(d*x+c)**4,x)
Output:
a**4*(Integral(4*sin(c + d*x)*tan(c + d*x)**4, x) + Integral(6*sin(c + d*x )**2*tan(c + d*x)**4, x) + Integral(4*sin(c + d*x)**3*tan(c + d*x)**4, x) + Integral(sin(c + d*x)**4*tan(c + d*x)**4, x) + Integral(tan(c + d*x)**4, x))
Time = 0.11 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.66 \[ \int (a+a \sin (c+d x))^4 \tan ^4(c+d x) \, dx=\frac {32 \, {\left (\cos \left (d x + c\right )^{3} - \frac {9 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} - 9 \, \cos \left (d x + c\right )\right )} a^{4} + {\left (8 \, \tan \left (d x + c\right )^{3} + 105 \, d x + 105 \, c - \frac {3 \, {\left (13 \, \tan \left (d x + c\right )^{3} + 11 \, \tan \left (d x + c\right )\right )}}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1} - 72 \, \tan \left (d x + c\right )\right )} a^{4} + 24 \, {\left (2 \, \tan \left (d x + c\right )^{3} + 15 \, d x + 15 \, c - \frac {3 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 12 \, \tan \left (d x + c\right )\right )} a^{4} + 8 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{4} - 32 \, a^{4} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )}}{24 \, d} \] Input:
integrate((a+a*sin(d*x+c))^4*tan(d*x+c)^4,x, algorithm="maxima")
Output:
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
Timed out. \[ \int (a+a \sin (c+d x))^4 \tan ^4(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(d*x+c))^4*tan(d*x+c)^4,x, algorithm="giac")
Output:
Timed out
Time = 20.68 (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} \] Input:
int(tan(c + d*x)^4*(a + a*sin(c + d*x))^4,x)
Output:
(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)
Time = 0.18 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.76 \[ \int (a+a \sin (c+d x))^4 \tan ^4(c+d x) \, dx=\frac {a^{4} \left (8 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} \tan \left (d x +c \right )^{3}-24 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} \tan \left (d x +c \right )+465 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} c +489 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} d x +768 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-8 \cos \left (d x +c \right ) \tan \left (d x +c \right )^{3}+24 \cos \left (d x +c \right ) \tan \left (d x +c \right )-465 \cos \left (d x +c \right ) c -489 \cos \left (d x +c \right ) d x -768 \cos \left (d x +c \right )+6 \sin \left (d x +c \right )^{7}+32 \sin \left (d x +c \right )^{6}+93 \sin \left (d x +c \right )^{5}+288 \sin \left (d x +c \right )^{4}-620 \sin \left (d x +c \right )^{3}-1152 \sin \left (d x +c \right )^{2}+465 \sin \left (d x +c \right )+768\right )}{24 \cos \left (d x +c \right ) d \left (\sin \left (d x +c \right )^{2}-1\right )} \] Input:
int((a+a*sin(d*x+c))^4*tan(d*x+c)^4,x)
Output:
(a**4*(8*cos(c + d*x)*sin(c + d*x)**2*tan(c + d*x)**3 - 24*cos(c + d*x)*si n(c + d*x)**2*tan(c + d*x) + 465*cos(c + d*x)*sin(c + d*x)**2*c + 489*cos( c + d*x)*sin(c + d*x)**2*d*x + 768*cos(c + d*x)*sin(c + d*x)**2 - 8*cos(c + d*x)*tan(c + d*x)**3 + 24*cos(c + d*x)*tan(c + d*x) - 465*cos(c + d*x)*c - 489*cos(c + d*x)*d*x - 768*cos(c + d*x) + 6*sin(c + d*x)**7 + 32*sin(c + d*x)**6 + 93*sin(c + d*x)**5 + 288*sin(c + d*x)**4 - 620*sin(c + d*x)**3 - 1152*sin(c + d*x)**2 + 465*sin(c + d*x) + 768))/(24*cos(c + d*x)*d*(sin (c + d*x)**2 - 1))