Integrand size = 21, antiderivative size = 180 \[ \int (a+a \sin (c+d x))^3 \tan ^6(c+d x) \, dx=-\frac {23 a^3 x}{2}+\frac {136 a^3 \cos (c+d x)}{5 d}-\frac {136 a^3 \cos ^3(c+d x)}{15 d}+\frac {23 a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a^6 \cos (c+d x) \sin ^5(c+d x)}{5 d (a-a \sin (c+d x))^3}-\frac {13 a^5 \cos (c+d x) \sin ^4(c+d x)}{15 d (a-a \sin (c+d x))^2}+\frac {23 a^6 \cos (c+d x) \sin ^3(c+d x)}{3 d \left (a^3-a^3 \sin (c+d x)\right )} \]
-23/2*a^3*x+136/5*a^3*cos(d*x+c)/d-136/15*a^3*cos(d*x+c)^3/d+23/2*a^3*cos( d*x+c)*sin(d*x+c)/d+1/5*a^6*cos(d*x+c)*sin(d*x+c)^5/d/(a-a*sin(d*x+c))^3-1 3/15*a^5*cos(d*x+c)*sin(d*x+c)^4/d/(a-a*sin(d*x+c))^2+23/3*a^6*cos(d*x+c)* sin(d*x+c)^3/d/(a^3-a^3*sin(d*x+c))
Time = 8.81 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.35 \[ \int (a+a \sin (c+d x))^3 \tan ^6(c+d x) \, dx=\frac {(a+a \sin (c+d x))^3 \left (-690 (c+d x)+405 \cos (c+d x)-5 \cos (3 (c+d x))+\frac {12}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^4}-\frac {112}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {24 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^5}-\frac {224 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {1576 \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+45 \sin (2 (c+d x))\right )}{60 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
((a + a*Sin[c + d*x])^3*(-690*(c + d*x) + 405*Cos[c + d*x] - 5*Cos[3*(c + d*x)] + 12/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4 - 112/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (24*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5 - (224*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2] )^3 + (1576*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + 45*S in[2*(c + d*x)]))/(60*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)
Time = 1.03 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.04, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.810, Rules used = {3042, 3187, 3042, 3244, 25, 3042, 3456, 3042, 3456, 27, 3042, 3227, 3042, 3113, 2009, 3115, 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.
| \(\displaystyle \int \tan ^6(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (c+d x)^6 (a \sin (c+d x)+a)^3dx\) |
| \(\Big \downarrow \) 3187 |
| \(\displaystyle a^6 \int \frac {\sin ^6(c+d x)}{(a-a \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^6 \int \frac {\sin (c+d x)^6}{(a-a \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3244 |
| \(\displaystyle a^6 \left (\frac {\int -\frac {\sin ^4(c+d x) (8 \sin (c+d x) a+5 a)}{(a-a \sin (c+d x))^2}dx}{5 a^2}+\frac {\sin ^5(c+d x) \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle a^6 \left (\frac {\sin ^5(c+d x) \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}-\frac {\int \frac {\sin ^4(c+d x) (8 \sin (c+d x) a+5 a)}{(a-a \sin (c+d x))^2}dx}{5 a^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^6 \left (\frac {\sin ^5(c+d x) \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}-\frac {\int \frac {\sin (c+d x)^4 (8 \sin (c+d x) a+5 a)}{(a-a \sin (c+d x))^2}dx}{5 a^2}\right )\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle a^6 \left (\frac {\sin ^5(c+d x) \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}-\frac {\frac {13 a \sin ^4(c+d x) \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac {\int \frac {\sin ^3(c+d x) \left (63 \sin (c+d x) a^2+52 a^2\right )}{a-a \sin (c+d x)}dx}{3 a^2}}{5 a^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^6 \left (\frac {\sin ^5(c+d x) \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}-\frac {\frac {13 a \sin ^4(c+d x) \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac {\int \frac {\sin (c+d x)^3 \left (63 \sin (c+d x) a^2+52 a^2\right )}{a-a \sin (c+d x)}dx}{3 a^2}}{5 a^2}\right )\) |
| \(\Big \downarrow \) 3456 |
| \(\displaystyle a^6 \left (\frac {\sin ^5(c+d x) \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}-\frac {\frac {13 a \sin ^4(c+d x) \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac {\frac {115 a^2 \sin ^3(c+d x) \cos (c+d x)}{d (a-a \sin (c+d x))}-\frac {\int 3 \sin ^2(c+d x) \left (136 \sin (c+d x) a^3+115 a^3\right )dx}{a^2}}{3 a^2}}{5 a^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle a^6 \left (\frac {\sin ^5(c+d x) \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}-\frac {\frac {13 a \sin ^4(c+d x) \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac {\frac {115 a^2 \sin ^3(c+d x) \cos (c+d x)}{d (a-a \sin (c+d x))}-\frac {3 \int \sin ^2(c+d x) \left (136 \sin (c+d x) a^3+115 a^3\right )dx}{a^2}}{3 a^2}}{5 a^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^6 \left (\frac {\sin ^5(c+d x) \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}-\frac {\frac {13 a \sin ^4(c+d x) \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac {\frac {115 a^2 \sin ^3(c+d x) \cos (c+d x)}{d (a-a \sin (c+d x))}-\frac {3 \int \sin (c+d x)^2 \left (136 \sin (c+d x) a^3+115 a^3\right )dx}{a^2}}{3 a^2}}{5 a^2}\right )\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle a^6 \left (\frac {\sin ^5(c+d x) \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}-\frac {\frac {13 a \sin ^4(c+d x) \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac {\frac {115 a^2 \sin ^3(c+d x) \cos (c+d x)}{d (a-a \sin (c+d x))}-\frac {3 \left (136 a^3 \int \sin ^3(c+d x)dx+115 a^3 \int \sin ^2(c+d x)dx\right )}{a^2}}{3 a^2}}{5 a^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^6 \left (\frac {\sin ^5(c+d x) \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}-\frac {\frac {13 a \sin ^4(c+d x) \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac {\frac {115 a^2 \sin ^3(c+d x) \cos (c+d x)}{d (a-a \sin (c+d x))}-\frac {3 \left (115 a^3 \int \sin (c+d x)^2dx+136 a^3 \int \sin (c+d x)^3dx\right )}{a^2}}{3 a^2}}{5 a^2}\right )\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle a^6 \left (\frac {\sin ^5(c+d x) \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}-\frac {\frac {13 a \sin ^4(c+d x) \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac {\frac {115 a^2 \sin ^3(c+d x) \cos (c+d x)}{d (a-a \sin (c+d x))}-\frac {3 \left (115 a^3 \int \sin (c+d x)^2dx-\frac {136 a^3 \int \left (1-\cos ^2(c+d x)\right )d\cos (c+d x)}{d}\right )}{a^2}}{3 a^2}}{5 a^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^6 \left (\frac {\sin ^5(c+d x) \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}-\frac {\frac {13 a \sin ^4(c+d x) \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac {\frac {115 a^2 \sin ^3(c+d x) \cos (c+d x)}{d (a-a \sin (c+d x))}-\frac {3 \left (115 a^3 \int \sin (c+d x)^2dx-\frac {136 a^3 \left (\cos (c+d x)-\frac {1}{3} \cos ^3(c+d x)\right )}{d}\right )}{a^2}}{3 a^2}}{5 a^2}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a^6 \left (\frac {\sin ^5(c+d x) \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}-\frac {\frac {13 a \sin ^4(c+d x) \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac {\frac {115 a^2 \sin ^3(c+d x) \cos (c+d x)}{d (a-a \sin (c+d x))}-\frac {3 \left (115 a^3 \left (\frac {\int 1dx}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {136 a^3 \left (\cos (c+d x)-\frac {1}{3} \cos ^3(c+d x)\right )}{d}\right )}{a^2}}{3 a^2}}{5 a^2}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a^6 \left (\frac {\sin ^5(c+d x) \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}-\frac {\frac {13 a \sin ^4(c+d x) \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac {\frac {115 a^2 \sin ^3(c+d x) \cos (c+d x)}{d (a-a \sin (c+d x))}-\frac {3 \left (115 a^3 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {136 a^3 \left (\cos (c+d x)-\frac {1}{3} \cos ^3(c+d x)\right )}{d}\right )}{a^2}}{3 a^2}}{5 a^2}\right )\) |
a^6*((Cos[c + d*x]*Sin[c + d*x]^5)/(5*d*(a - a*Sin[c + d*x])^3) - ((13*a*C os[c + d*x]*Sin[c + d*x]^4)/(3*d*(a - a*Sin[c + d*x])^2) - ((115*a^2*Cos[c + d*x]*Sin[c + d*x]^3)/(d*(a - a*Sin[c + d*x])) - (3*((-136*a^3*(Cos[c + d*x] - Cos[c + d*x]^3/3))/d + 115*a^3*(x/2 - (Cos[c + d*x]*Sin[c + d*x])/( 2*d))))/a^2)/(3*a^2))/(5*a^2))
3.1.29.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p _), x_Symbol] :> Simp[a^p Int[Sin[e + f*x]^p/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p] && EqQ [p, 2*m]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n - 1)/(a*f*(2*m + 1))), x] + Simp[1/(a*b* (2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)* Simp[b*(c^2*(m + 1) + d^2*(n - 1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/( a*f*(2*m + 1))), x] - Simp[1/(a*b*(2*m + 1)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x], x] /; Fre eQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] & & NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (In tegerQ[2*n] || EqQ[c, 0])
Result contains complex when optimal does not.
Time = 15.58 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.08
| method | result | size |
| risch | \(-\frac {23 a^{3} x}{2}-\frac {a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{24 d}-\frac {3 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {27 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {27 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {3 i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}+\frac {-\frac {464 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{3}-108 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+\frac {304 i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{3}+30 a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+\frac {394 a^{3}}{15}}{\left (-i+{\mathrm e}^{i \left (d x +c \right )}\right )^{5} d}\) | \(195\) |
| derivativedivides | \(\frac {a^{3} \left (\frac {\sin ^{10}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}-\frac {\sin ^{10}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}+\frac {7 \left (\sin ^{10}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}+\frac {7 \left (\frac {128}{35}+\sin ^{8}\left (d x +c \right )+\frac {8 \left (\sin ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\sin ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\sin ^{2}\left (d x +c \right )\right )}{35}\right ) \cos \left (d x +c \right )}{3}\right )+3 a^{3} \left (\frac {\sin ^{9}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}-\frac {4 \left (\sin ^{9}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}+\frac {8 \left (\sin ^{9}\left (d x +c \right )\right )}{5 \cos \left (d x +c \right )}+\frac {8 \left (\sin ^{7}\left (d x +c \right )+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )}{5}-\frac {7 d x}{2}-\frac {7 c}{2}\right )+3 a^{3} \left (\frac {\sin ^{8}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}-\frac {\sin ^{8}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{3}}+\frac {\sin ^{8}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )\right )+a^{3} \left (\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}+\tan \left (d x +c \right )-d x -c \right )}{d}\) | \(359\) |
| default | \(\frac {a^{3} \left (\frac {\sin ^{10}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}-\frac {\sin ^{10}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}+\frac {7 \left (\sin ^{10}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}+\frac {7 \left (\frac {128}{35}+\sin ^{8}\left (d x +c \right )+\frac {8 \left (\sin ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\sin ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\sin ^{2}\left (d x +c \right )\right )}{35}\right ) \cos \left (d x +c \right )}{3}\right )+3 a^{3} \left (\frac {\sin ^{9}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}-\frac {4 \left (\sin ^{9}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}+\frac {8 \left (\sin ^{9}\left (d x +c \right )\right )}{5 \cos \left (d x +c \right )}+\frac {8 \left (\sin ^{7}\left (d x +c \right )+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )}{5}-\frac {7 d x}{2}-\frac {7 c}{2}\right )+3 a^{3} \left (\frac {\sin ^{8}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}-\frac {\sin ^{8}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{3}}+\frac {\sin ^{8}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )\right )+a^{3} \left (\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}+\tan \left (d x +c \right )-d x -c \right )}{d}\) | \(359\) |
| parts | \(\frac {a^{3} \left (\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}+\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {a^{3} \left (\frac {\sin ^{10}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}-\frac {\sin ^{10}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}+\frac {7 \left (\sin ^{10}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )}+\frac {7 \left (\frac {128}{35}+\sin ^{8}\left (d x +c \right )+\frac {8 \left (\sin ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\sin ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\sin ^{2}\left (d x +c \right )\right )}{35}\right ) \cos \left (d x +c \right )}{3}\right )}{d}+\frac {3 a^{3} \left (\frac {\sin ^{8}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}-\frac {\sin ^{8}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{3}}+\frac {\sin ^{8}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )\right )}{d}+\frac {3 a^{3} \left (\frac {\sin ^{9}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}-\frac {4 \left (\sin ^{9}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}+\frac {8 \left (\sin ^{9}\left (d x +c \right )\right )}{5 \cos \left (d x +c \right )}+\frac {8 \left (\sin ^{7}\left (d x +c \right )+\frac {7 \left (\sin ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \sin \left (d x +c \right )}{16}\right ) \cos \left (d x +c \right )}{5}-\frac {7 d x}{2}-\frac {7 c}{2}\right )}{d}\) | \(369\) |
-23/2*a^3*x-1/24*a^3/d*exp(3*I*(d*x+c))-3/8*I/d*a^3*exp(2*I*(d*x+c))+27/8* a^3/d*exp(I*(d*x+c))+27/8*a^3/d*exp(-I*(d*x+c))+3/8*I/d*a^3*exp(-2*I*(d*x+ c))-1/24*a^3/d*exp(-3*I*(d*x+c))+2/15*(-1160*a^3*exp(2*I*(d*x+c))-810*I*a^ 3*exp(3*I*(d*x+c))+760*I*a^3*exp(I*(d*x+c))+225*a^3*exp(4*I*(d*x+c))+197*a ^3)/(-I+exp(I*(d*x+c)))^5/d
Time = 0.29 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.61 \[ \int (a+a \sin (c+d x))^3 \tan ^6(c+d x) \, dx=-\frac {10 \, a^{3} \cos \left (d x + c\right )^{6} - 15 \, a^{3} \cos \left (d x + c\right )^{5} - 140 \, a^{3} \cos \left (d x + c\right )^{4} - 1380 \, a^{3} d x + {\left (345 \, a^{3} d x - 839 \, a^{3}\right )} \cos \left (d x + c\right )^{3} + 6 \, a^{3} + {\left (1035 \, a^{3} d x + 668 \, a^{3}\right )} \cos \left (d x + c\right )^{2} - 6 \, {\left (115 \, a^{3} d x - 233 \, a^{3}\right )} \cos \left (d x + c\right ) - {\left (10 \, a^{3} \cos \left (d x + c\right )^{5} + 25 \, a^{3} \cos \left (d x + c\right )^{4} - 115 \, a^{3} \cos \left (d x + c\right )^{3} - 1380 \, a^{3} d x - 6 \, a^{3} + {\left (345 \, a^{3} d x + 724 \, a^{3}\right )} \cos \left (d x + c\right )^{2} - 6 \, {\left (115 \, a^{3} d x - 232 \, a^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{30 \, {\left (d \cos \left (d x + c\right )^{3} + 3 \, d \cos \left (d x + c\right )^{2} - 2 \, d \cos \left (d x + c\right ) - {\left (d \cos \left (d x + c\right )^{2} - 2 \, d \cos \left (d x + c\right ) - 4 \, d\right )} \sin \left (d x + c\right ) - 4 \, d\right )}} \]
-1/30*(10*a^3*cos(d*x + c)^6 - 15*a^3*cos(d*x + c)^5 - 140*a^3*cos(d*x + c )^4 - 1380*a^3*d*x + (345*a^3*d*x - 839*a^3)*cos(d*x + c)^3 + 6*a^3 + (103 5*a^3*d*x + 668*a^3)*cos(d*x + c)^2 - 6*(115*a^3*d*x - 233*a^3)*cos(d*x + c) - (10*a^3*cos(d*x + c)^5 + 25*a^3*cos(d*x + c)^4 - 115*a^3*cos(d*x + c) ^3 - 1380*a^3*d*x - 6*a^3 + (345*a^3*d*x + 724*a^3)*cos(d*x + c)^2 - 6*(11 5*a^3*d*x - 232*a^3)*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^3 + 3*d*c os(d*x + c)^2 - 2*d*cos(d*x + c) - (d*cos(d*x + c)^2 - 2*d*cos(d*x + c) - 4*d)*sin(d*x + c) - 4*d)
\[ \int (a+a \sin (c+d x))^3 \tan ^6(c+d x) \, dx=a^{3} \left (\int 3 \sin {\left (c + d x \right )} \tan ^{6}{\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \tan ^{6}{\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \tan ^{6}{\left (c + d x \right )}\, dx + \int \tan ^{6}{\left (c + d x \right )}\, dx\right ) \]
a**3*(Integral(3*sin(c + d*x)*tan(c + d*x)**6, x) + Integral(3*sin(c + d*x )**2*tan(c + d*x)**6, x) + Integral(sin(c + d*x)**3*tan(c + d*x)**6, x) + Integral(tan(c + d*x)**6, x))
Time = 0.29 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.16 \[ \int (a+a \sin (c+d x))^3 \tan ^6(c+d x) \, dx=\frac {3 \, {\left (6 \, \tan \left (d x + c\right )^{5} - 20 \, \tan \left (d x + c\right )^{3} - 105 \, d x - 105 \, c + \frac {15 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} + 90 \, \tan \left (d x + c\right )\right )} a^{3} + 2 \, {\left (3 \, \tan \left (d x + c\right )^{5} - 5 \, \tan \left (d x + c\right )^{3} - 15 \, d x - 15 \, c + 15 \, \tan \left (d x + c\right )\right )} a^{3} - 2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - \frac {90 \, \cos \left (d x + c\right )^{4} - 20 \, \cos \left (d x + c\right )^{2} + 3}{\cos \left (d x + c\right )^{5}} - 60 \, \cos \left (d x + c\right )\right )} a^{3} + 18 \, a^{3} {\left (\frac {15 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{2} + 1}{\cos \left (d x + c\right )^{5}} + 5 \, \cos \left (d x + c\right )\right )}}{30 \, d} \]
1/30*(3*(6*tan(d*x + c)^5 - 20*tan(d*x + c)^3 - 105*d*x - 105*c + 15*tan(d *x + c)/(tan(d*x + c)^2 + 1) + 90*tan(d*x + c))*a^3 + 2*(3*tan(d*x + c)^5 - 5*tan(d*x + c)^3 - 15*d*x - 15*c + 15*tan(d*x + c))*a^3 - 2*(5*cos(d*x + c)^3 - (90*cos(d*x + c)^4 - 20*cos(d*x + c)^2 + 3)/cos(d*x + c)^5 - 60*co s(d*x + c))*a^3 + 18*a^3*((15*cos(d*x + c)^4 - 5*cos(d*x + c)^2 + 1)/cos(d *x + c)^5 + 5*cos(d*x + c)))/d
Timed out. \[ \int (a+a \sin (c+d x))^3 \tan ^6(c+d x) \, dx=\text {Timed out} \]
Time = 10.39 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.43 \[ \int (a+a \sin (c+d x))^3 \tan ^6(c+d x) \, dx=-\frac {23\,a^3\,x}{2}-\frac {\frac {23\,a^3\,\left (c+d\,x\right )}{2}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {115\,a^3\,\left (c+d\,x\right )}{2}-\frac {a^3\,\left (1725\,c+1725\,d\,x-4750\right )}{30}\right )-\frac {a^3\,\left (345\,c+345\,d\,x-1088\right )}{30}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {115\,a^3\,\left (c+d\,x\right )}{2}-\frac {a^3\,\left (1725\,c+1725\,d\,x-690\right )}{30}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {299\,a^3\,\left (c+d\,x\right )}{2}-\frac {a^3\,\left (4485\,c+4485\,d\,x-3450\right )}{30}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {299\,a^3\,\left (c+d\,x\right )}{2}-\frac {a^3\,\left (4485\,c+4485\,d\,x-10694\right )}{30}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {575\,a^3\,\left (c+d\,x\right )}{2}-\frac {a^3\,\left (8625\,c+8625\,d\,x-8740\right )}{30}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {575\,a^3\,\left (c+d\,x\right )}{2}-\frac {a^3\,\left (8625\,c+8625\,d\,x-18460\right )}{30}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (437\,a^3\,\left (c+d\,x\right )-\frac {a^3\,\left (13110\,c+13110\,d\,x-16100\right )}{30}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (437\,a^3\,\left (c+d\,x\right )-\frac {a^3\,\left (13110\,c+13110\,d\,x-25244\right )}{30}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (529\,a^3\,\left (c+d\,x\right )-\frac {a^3\,\left (15870\,c+15870\,d\,x-23368\right )}{30}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (529\,a^3\,\left (c+d\,x\right )-\frac {a^3\,\left (15870\,c+15870\,d\,x-26680\right )}{30}\right )}{d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^5\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3} \]
- (23*a^3*x)/2 - ((23*a^3*(c + d*x))/2 - tan(c/2 + (d*x)/2)*((115*a^3*(c + d*x))/2 - (a^3*(1725*c + 1725*d*x - 4750))/30) - (a^3*(345*c + 345*d*x - 1088))/30 + tan(c/2 + (d*x)/2)^10*((115*a^3*(c + d*x))/2 - (a^3*(1725*c + 1725*d*x - 690))/30) - tan(c/2 + (d*x)/2)^9*((299*a^3*(c + d*x))/2 - (a^3* (4485*c + 4485*d*x - 3450))/30) + tan(c/2 + (d*x)/2)^2*((299*a^3*(c + d*x) )/2 - (a^3*(4485*c + 4485*d*x - 10694))/30) + tan(c/2 + (d*x)/2)^8*((575*a ^3*(c + d*x))/2 - (a^3*(8625*c + 8625*d*x - 8740))/30) - tan(c/2 + (d*x)/2 )^3*((575*a^3*(c + d*x))/2 - (a^3*(8625*c + 8625*d*x - 18460))/30) - tan(c /2 + (d*x)/2)^7*(437*a^3*(c + d*x) - (a^3*(13110*c + 13110*d*x - 16100))/3 0) + tan(c/2 + (d*x)/2)^4*(437*a^3*(c + d*x) - (a^3*(13110*c + 13110*d*x - 25244))/30) + tan(c/2 + (d*x)/2)^6*(529*a^3*(c + d*x) - (a^3*(15870*c + 1 5870*d*x - 23368))/30) - tan(c/2 + (d*x)/2)^5*(529*a^3*(c + d*x) - (a^3*(1 5870*c + 15870*d*x - 26680))/30))/(d*(tan(c/2 + (d*x)/2) - 1)^5*(tan(c/2 + (d*x)/2)^2 + 1)^3)