Integrand size = 21, antiderivative size = 181 \[ \int \tan ^3(c+d x) (a+b \tan (c+d x))^4 \, dx=4 a b \left (a^2-b^2\right ) x+\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (\cos (c+d x))}{d}-\frac {a b \left (a^2-3 b^2\right ) \tan (c+d x)}{d}-\frac {\left (a^2-b^2\right ) (a+b \tan (c+d x))^2}{2 d}-\frac {a (a+b \tan (c+d x))^3}{3 d}-\frac {(a+b \tan (c+d x))^4}{4 d}-\frac {a (a+b \tan (c+d x))^5}{30 b^2 d}+\frac {\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d} \]
4*a*b*(a^2-b^2)*x+(a^4-6*a^2*b^2+b^4)*ln(cos(d*x+c))/d-a*b*(a^2-3*b^2)*tan (d*x+c)/d-1/2*(a^2-b^2)*(a+b*tan(d*x+c))^2/d-1/3*a*(a+b*tan(d*x+c))^3/d-1/ 4*(a+b*tan(d*x+c))^4/d-1/30*a*(a+b*tan(d*x+c))^5/b^2/d+1/6*tan(d*x+c)*(a+b *tan(d*x+c))^5/b/d
Result contains complex when optimal does not.
Time = 1.24 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.05 \[ \int \tan ^3(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {-2 \left (a^6+15 (a+i b)^4 b^2 \log (i-\tan (c+d x))+15 (a-i b)^4 b^2 \log (i+\tan (c+d x))\right )-240 a b^3 \left (a^2-b^2\right ) \tan (c+d x)+30 b^2 \left (a^4-6 a^2 b^2+b^4\right ) \tan ^2(c+d x)+80 a b^3 \left (a^2-b^2\right ) \tan ^3(c+d x)-15 b^4 \left (-6 a^2+b^2\right ) \tan ^4(c+d x)+48 a b^5 \tan ^5(c+d x)+10 b^6 \tan ^6(c+d x)}{60 b^2 d} \]
(-2*(a^6 + 15*(a + I*b)^4*b^2*Log[I - Tan[c + d*x]] + 15*(a - I*b)^4*b^2*L og[I + Tan[c + d*x]]) - 240*a*b^3*(a^2 - b^2)*Tan[c + d*x] + 30*b^2*(a^4 - 6*a^2*b^2 + b^4)*Tan[c + d*x]^2 + 80*a*b^3*(a^2 - b^2)*Tan[c + d*x]^3 - 1 5*b^4*(-6*a^2 + b^2)*Tan[c + d*x]^4 + 48*a*b^5*Tan[c + d*x]^5 + 10*b^6*Tan [c + d*x]^6)/(60*b^2*d)
Time = 1.10 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.07, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {3042, 4049, 25, 3042, 4113, 27, 3042, 4011, 3042, 4011, 3042, 4011, 3042, 4008, 3042, 3956}
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.
| \(\displaystyle \int \tan ^3(c+d x) (a+b \tan (c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (c+d x)^3 (a+b \tan (c+d x))^4dx\) |
| \(\Big \downarrow \) 4049 |
| \(\displaystyle \frac {\int -(a+b \tan (c+d x))^4 \left (a \tan ^2(c+d x)+6 b \tan (c+d x)+a\right )dx}{6 b}+\frac {\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\frac {\int (a+b \tan (c+d x))^4 \left (a \tan ^2(c+d x)+6 b \tan (c+d x)+a\right )dx}{6 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\frac {\int (a+b \tan (c+d x))^4 \left (a \tan (c+d x)^2+6 b \tan (c+d x)+a\right )dx}{6 b}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\frac {\int 6 b \tan (c+d x) (a+b \tan (c+d x))^4dx+\frac {a (a+b \tan (c+d x))^5}{5 b d}}{6 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\frac {6 b \int \tan (c+d x) (a+b \tan (c+d x))^4dx+\frac {a (a+b \tan (c+d x))^5}{5 b d}}{6 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\frac {6 b \int \tan (c+d x) (a+b \tan (c+d x))^4dx+\frac {a (a+b \tan (c+d x))^5}{5 b d}}{6 b}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\frac {6 b \left (\int (a \tan (c+d x)-b) (a+b \tan (c+d x))^3dx+\frac {(a+b \tan (c+d x))^4}{4 d}\right )+\frac {a (a+b \tan (c+d x))^5}{5 b d}}{6 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\frac {6 b \left (\int (a \tan (c+d x)-b) (a+b \tan (c+d x))^3dx+\frac {(a+b \tan (c+d x))^4}{4 d}\right )+\frac {a (a+b \tan (c+d x))^5}{5 b d}}{6 b}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\frac {6 b \left (\int (a+b \tan (c+d x))^2 \left (\left (a^2-b^2\right ) \tan (c+d x)-2 a b\right )dx+\frac {(a+b \tan (c+d x))^4}{4 d}+\frac {a (a+b \tan (c+d x))^3}{3 d}\right )+\frac {a (a+b \tan (c+d x))^5}{5 b d}}{6 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\frac {6 b \left (\int (a+b \tan (c+d x))^2 \left (\left (a^2-b^2\right ) \tan (c+d x)-2 a b\right )dx+\frac {(a+b \tan (c+d x))^4}{4 d}+\frac {a (a+b \tan (c+d x))^3}{3 d}\right )+\frac {a (a+b \tan (c+d x))^5}{5 b d}}{6 b}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\frac {6 b \left (\int (a+b \tan (c+d x)) \left (a \left (a^2-3 b^2\right ) \tan (c+d x)-b \left (3 a^2-b^2\right )\right )dx+\frac {\left (a^2-b^2\right ) (a+b \tan (c+d x))^2}{2 d}+\frac {(a+b \tan (c+d x))^4}{4 d}+\frac {a (a+b \tan (c+d x))^3}{3 d}\right )+\frac {a (a+b \tan (c+d x))^5}{5 b d}}{6 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\frac {6 b \left (\int (a+b \tan (c+d x)) \left (a \left (a^2-3 b^2\right ) \tan (c+d x)-b \left (3 a^2-b^2\right )\right )dx+\frac {\left (a^2-b^2\right ) (a+b \tan (c+d x))^2}{2 d}+\frac {(a+b \tan (c+d x))^4}{4 d}+\frac {a (a+b \tan (c+d x))^3}{3 d}\right )+\frac {a (a+b \tan (c+d x))^5}{5 b d}}{6 b}\) |
| \(\Big \downarrow \) 4008 |
| \(\displaystyle \frac {\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\frac {6 b \left (\left (a^4-6 a^2 b^2+b^4\right ) \int \tan (c+d x)dx+\frac {\left (a^2-b^2\right ) (a+b \tan (c+d x))^2}{2 d}+\frac {a b \left (a^2-3 b^2\right ) \tan (c+d x)}{d}-4 a b x \left (a^2-b^2\right )+\frac {(a+b \tan (c+d x))^4}{4 d}+\frac {a (a+b \tan (c+d x))^3}{3 d}\right )+\frac {a (a+b \tan (c+d x))^5}{5 b d}}{6 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\frac {6 b \left (\left (a^4-6 a^2 b^2+b^4\right ) \int \tan (c+d x)dx+\frac {\left (a^2-b^2\right ) (a+b \tan (c+d x))^2}{2 d}+\frac {a b \left (a^2-3 b^2\right ) \tan (c+d x)}{d}-4 a b x \left (a^2-b^2\right )+\frac {(a+b \tan (c+d x))^4}{4 d}+\frac {a (a+b \tan (c+d x))^3}{3 d}\right )+\frac {a (a+b \tan (c+d x))^5}{5 b d}}{6 b}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {\tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\frac {6 b \left (\frac {\left (a^2-b^2\right ) (a+b \tan (c+d x))^2}{2 d}+\frac {a b \left (a^2-3 b^2\right ) \tan (c+d x)}{d}-4 a b x \left (a^2-b^2\right )-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (\cos (c+d x))}{d}+\frac {(a+b \tan (c+d x))^4}{4 d}+\frac {a (a+b \tan (c+d x))^3}{3 d}\right )+\frac {a (a+b \tan (c+d x))^5}{5 b d}}{6 b}\) |
(Tan[c + d*x]*(a + b*Tan[c + d*x])^5)/(6*b*d) - ((a*(a + b*Tan[c + d*x])^5 )/(5*b*d) + 6*b*(-4*a*b*(a^2 - b^2)*x - ((a^4 - 6*a^2*b^2 + b^4)*Log[Cos[c + d*x]])/d + (a*b*(a^2 - 3*b^2)*Tan[c + d*x])/d + ((a^2 - b^2)*(a + b*Tan [c + d*x])^2)/(2*d) + (a*(a + b*Tan[c + d*x])^3)/(3*d) + (a + b*Tan[c + d* x])^4/(4*d)))/(6*b)
3.5.45.3.1 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[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.) *(x_)]), x_Symbol] :> Simp[(a*c - b*d)*x, x] + (Simp[b*d*(Tan[e + f*x]/f), x] + Simp[(b*c + a*d) Int[Tan[e + f*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int [(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] , x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Simp[1/(d*(m + n - 1)) Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[ e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2 , 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || I ntegerQ[m]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])) )
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && !LeQ[m, -1]
Time = 0.36 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00
| method | result | size |
| norman | \(\frac {b^{4} \left (\tan ^{6}\left (d x +c \right )\right )}{6 d}+\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+4 a b \left (a^{2}-b^{2}\right ) x +\frac {4 a \,b^{3} \left (\tan ^{5}\left (d x +c \right )\right )}{5 d}+\frac {b^{2} \left (6 a^{2}-b^{2}\right ) \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}-\frac {4 a b \left (a^{2}-b^{2}\right ) \tan \left (d x +c \right )}{d}+\frac {4 a b \left (a^{2}-b^{2}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(181\) |
| derivativedivides | \(\frac {\frac {b^{4} \left (\tan ^{6}\left (d x +c \right )\right )}{6}+\frac {4 a \,b^{3} \left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {3 a^{2} b^{2} \left (\tan ^{4}\left (d x +c \right )\right )}{2}-\frac {b^{4} \left (\tan ^{4}\left (d x +c \right )\right )}{4}+\frac {4 a^{3} b \left (\tan ^{3}\left (d x +c \right )\right )}{3}-\frac {4 a \,b^{3} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {a^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{2}-3 a^{2} b^{2} \left (\tan ^{2}\left (d x +c \right )\right )+\frac {b^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{2}-4 a^{3} b \tan \left (d x +c \right )+4 a \,b^{3} \tan \left (d x +c \right )+\frac {\left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (4 a^{3} b -4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(209\) |
| default | \(\frac {\frac {b^{4} \left (\tan ^{6}\left (d x +c \right )\right )}{6}+\frac {4 a \,b^{3} \left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {3 a^{2} b^{2} \left (\tan ^{4}\left (d x +c \right )\right )}{2}-\frac {b^{4} \left (\tan ^{4}\left (d x +c \right )\right )}{4}+\frac {4 a^{3} b \left (\tan ^{3}\left (d x +c \right )\right )}{3}-\frac {4 a \,b^{3} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {a^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{2}-3 a^{2} b^{2} \left (\tan ^{2}\left (d x +c \right )\right )+\frac {b^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{2}-4 a^{3} b \tan \left (d x +c \right )+4 a \,b^{3} \tan \left (d x +c \right )+\frac {\left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (4 a^{3} b -4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(209\) |
| parts | \(\frac {a^{4} \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}+\frac {b^{4} \left (\frac {\left (\tan ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}+\frac {4 a \,b^{3} \left (\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}+\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {6 a^{2} b^{2} \left (\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}+\frac {4 a^{3} b \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(209\) |
| parallelrisch | \(-\frac {-10 b^{4} \left (\tan ^{6}\left (d x +c \right )\right )-48 a \,b^{3} \left (\tan ^{5}\left (d x +c \right )\right )-90 a^{2} b^{2} \left (\tan ^{4}\left (d x +c \right )\right )+15 b^{4} \left (\tan ^{4}\left (d x +c \right )\right )-80 a^{3} b \left (\tan ^{3}\left (d x +c \right )\right )+80 a \,b^{3} \left (\tan ^{3}\left (d x +c \right )\right )-240 a^{3} b d x +240 a \,b^{3} d x -30 a^{4} \left (\tan ^{2}\left (d x +c \right )\right )+180 a^{2} b^{2} \left (\tan ^{2}\left (d x +c \right )\right )-30 b^{4} \left (\tan ^{2}\left (d x +c \right )\right )+30 a^{4} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )-180 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2} b^{2}+30 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) b^{4}+240 a^{3} b \tan \left (d x +c \right )-240 a \,b^{3} \tan \left (d x +c \right )}{60 d}\) | \(224\) |
| risch | \(4 a^{3} b x -4 a \,b^{3} x -i a^{4} x +6 i a^{2} b^{2} x -i b^{4} x -\frac {2 i a^{4} c}{d}+\frac {12 i a^{2} b^{2} c}{d}-\frac {2 i b^{4} c}{d}+\frac {-96 a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-72 a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-24 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-72 a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-24 a^{2} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+\frac {184 i a \,b^{3}}{15}-\frac {32 i a^{3} b}{3}+\frac {248 i a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{5}-96 i a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}+\frac {368 i a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}}{3}-16 i a^{3} b \,{\mathrm e}^{10 i \left (d x +c \right )}+24 i a \,b^{3} {\mathrm e}^{10 i \left (d x +c \right )}+112 i a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-48 i a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}+72 i a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-\frac {320 i a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}}{3}-64 i a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}+\frac {68 b^{4} {\mathrm e}^{6 i \left (d x +c \right )}}{3}+6 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+2 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+12 b^{4} {\mathrm e}^{8 i \left (d x +c \right )}+12 a^{4} {\mathrm e}^{6 i \left (d x +c \right )}+2 a^{4} {\mathrm e}^{10 i \left (d x +c \right )}+6 b^{4} {\mathrm e}^{10 i \left (d x +c \right )}+8 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}+8 a^{4} {\mathrm e}^{8 i \left (d x +c \right )}+12 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{4}}{d}-\frac {6 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{2} b^{2}}{d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) b^{4}}{d}\) | \(552\) |
1/6*b^4/d*tan(d*x+c)^6+1/2*(a^4-6*a^2*b^2+b^4)/d*tan(d*x+c)^2+4*a*b*(a^2-b ^2)*x+4/5*a*b^3/d*tan(d*x+c)^5+1/4*b^2*(6*a^2-b^2)/d*tan(d*x+c)^4-4*a*b*(a ^2-b^2)/d*tan(d*x+c)+4/3*a*b*(a^2-b^2)/d*tan(d*x+c)^3-1/2*(a^4-6*a^2*b^2+b ^4)/d*ln(1+tan(d*x+c)^2)
Time = 0.25 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.94 \[ \int \tan ^3(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {10 \, b^{4} \tan \left (d x + c\right )^{6} + 48 \, a b^{3} \tan \left (d x + c\right )^{5} + 15 \, {\left (6 \, a^{2} b^{2} - b^{4}\right )} \tan \left (d x + c\right )^{4} + 80 \, {\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{3} + 240 \, {\left (a^{3} b - a b^{3}\right )} d x + 30 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{2} + 30 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 240 \, {\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )}{60 \, d} \]
1/60*(10*b^4*tan(d*x + c)^6 + 48*a*b^3*tan(d*x + c)^5 + 15*(6*a^2*b^2 - b^ 4)*tan(d*x + c)^4 + 80*(a^3*b - a*b^3)*tan(d*x + c)^3 + 240*(a^3*b - a*b^3 )*d*x + 30*(a^4 - 6*a^2*b^2 + b^4)*tan(d*x + c)^2 + 30*(a^4 - 6*a^2*b^2 + b^4)*log(1/(tan(d*x + c)^2 + 1)) - 240*(a^3*b - a*b^3)*tan(d*x + c))/d
Time = 0.20 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.53 \[ \int \tan ^3(c+d x) (a+b \tan (c+d x))^4 \, dx=\begin {cases} - \frac {a^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a^{4} \tan ^{2}{\left (c + d x \right )}}{2 d} + 4 a^{3} b x + \frac {4 a^{3} b \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac {4 a^{3} b \tan {\left (c + d x \right )}}{d} + \frac {3 a^{2} b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {3 a^{2} b^{2} \tan ^{4}{\left (c + d x \right )}}{2 d} - \frac {3 a^{2} b^{2} \tan ^{2}{\left (c + d x \right )}}{d} - 4 a b^{3} x + \frac {4 a b^{3} \tan ^{5}{\left (c + d x \right )}}{5 d} - \frac {4 a b^{3} \tan ^{3}{\left (c + d x \right )}}{3 d} + \frac {4 a b^{3} \tan {\left (c + d x \right )}}{d} - \frac {b^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {b^{4} \tan ^{6}{\left (c + d x \right )}}{6 d} - \frac {b^{4} \tan ^{4}{\left (c + d x \right )}}{4 d} + \frac {b^{4} \tan ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\left (c \right )}\right )^{4} \tan ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((-a**4*log(tan(c + d*x)**2 + 1)/(2*d) + a**4*tan(c + d*x)**2/(2* d) + 4*a**3*b*x + 4*a**3*b*tan(c + d*x)**3/(3*d) - 4*a**3*b*tan(c + d*x)/d + 3*a**2*b**2*log(tan(c + d*x)**2 + 1)/d + 3*a**2*b**2*tan(c + d*x)**4/(2 *d) - 3*a**2*b**2*tan(c + d*x)**2/d - 4*a*b**3*x + 4*a*b**3*tan(c + d*x)** 5/(5*d) - 4*a*b**3*tan(c + d*x)**3/(3*d) + 4*a*b**3*tan(c + d*x)/d - b**4* log(tan(c + d*x)**2 + 1)/(2*d) + b**4*tan(c + d*x)**6/(6*d) - b**4*tan(c + d*x)**4/(4*d) + b**4*tan(c + d*x)**2/(2*d), Ne(d, 0)), (x*(a + b*tan(c))* *4*tan(c)**3, True))
Time = 0.29 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.94 \[ \int \tan ^3(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {10 \, b^{4} \tan \left (d x + c\right )^{6} + 48 \, a b^{3} \tan \left (d x + c\right )^{5} + 15 \, {\left (6 \, a^{2} b^{2} - b^{4}\right )} \tan \left (d x + c\right )^{4} + 80 \, {\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{3} + 30 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{2} + 240 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )} - 30 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 240 \, {\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )}{60 \, d} \]
1/60*(10*b^4*tan(d*x + c)^6 + 48*a*b^3*tan(d*x + c)^5 + 15*(6*a^2*b^2 - b^ 4)*tan(d*x + c)^4 + 80*(a^3*b - a*b^3)*tan(d*x + c)^3 + 30*(a^4 - 6*a^2*b^ 2 + b^4)*tan(d*x + c)^2 + 240*(a^3*b - a*b^3)*(d*x + c) - 30*(a^4 - 6*a^2* b^2 + b^4)*log(tan(d*x + c)^2 + 1) - 240*(a^3*b - a*b^3)*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 3175 vs. \(2 (171) = 342\).
Time = 5.02 (sec) , antiderivative size = 3175, normalized size of antiderivative = 17.54 \[ \int \tan ^3(c+d x) (a+b \tan (c+d x))^4 \, dx=\text {Too large to display} \]
1/60*(240*a^3*b*d*x*tan(d*x)^6*tan(c)^6 - 240*a*b^3*d*x*tan(d*x)^6*tan(c)^ 6 + 30*a^4*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2 *tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^6*tan(c)^6 - 180*a^2*b^2* log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^6*tan(c)^6 + 30*b^4*log(4*(tan(d*x)^ 2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + ta n(c)^2 + 1))*tan(d*x)^6*tan(c)^6 - 1440*a^3*b*d*x*tan(d*x)^5*tan(c)^5 + 14 40*a*b^3*d*x*tan(d*x)^5*tan(c)^5 + 30*a^4*tan(d*x)^6*tan(c)^6 - 270*a^2*b^ 2*tan(d*x)^6*tan(c)^6 + 55*b^4*tan(d*x)^6*tan(c)^6 - 180*a^4*log(4*(tan(d* x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^5*tan(c)^5 + 1080*a^2*b^2*log(4*(tan(d*x)^2*tan(c )^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^5*tan(c)^5 - 180*b^4*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x )*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^ 5*tan(c)^5 + 240*a^3*b*tan(d*x)^6*tan(c)^5 - 240*a*b^3*tan(d*x)^6*tan(c)^5 + 240*a^3*b*tan(d*x)^5*tan(c)^6 - 240*a*b^3*tan(d*x)^5*tan(c)^6 + 3600*a^ 3*b*d*x*tan(d*x)^4*tan(c)^4 - 3600*a*b^3*d*x*tan(d*x)^4*tan(c)^4 + 30*a^4* tan(d*x)^6*tan(c)^4 - 180*a^2*b^2*tan(d*x)^6*tan(c)^4 + 30*b^4*tan(d*x)^6* tan(c)^4 - 120*a^4*tan(d*x)^5*tan(c)^5 + 1260*a^2*b^2*tan(d*x)^5*tan(c)^5 - 270*b^4*tan(d*x)^5*tan(c)^5 + 30*a^4*tan(d*x)^4*tan(c)^6 - 180*a^2*b^...
Time = 4.96 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.24 \[ \int \tan ^3(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {a^4}{2}-3\,a^2\,b^2+\frac {b^4}{2}\right )}{d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (\frac {b^4}{4}-\frac {3\,a^2\,b^2}{2}\right )}{d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {4\,a\,b^3}{3}-\frac {4\,a^3\,b}{3}\right )}{d}-\frac {\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )\,\left (\frac {a^4}{2}-3\,a^2\,b^2+\frac {b^4}{2}\right )}{d}+\frac {b^4\,{\mathrm {tan}\left (c+d\,x\right )}^6}{6\,d}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a\,b^3-4\,a^3\,b\right )}{d}+\frac {4\,a\,b^3\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5\,d}-\frac {4\,a\,b\,\mathrm {atan}\left (\frac {4\,a\,b\,\mathrm {tan}\left (c+d\,x\right )\,\left (a+b\right )\,\left (a-b\right )}{4\,a\,b^3-4\,a^3\,b}\right )\,\left (a+b\right )\,\left (a-b\right )}{d} \]
(tan(c + d*x)^2*(a^4/2 + b^4/2 - 3*a^2*b^2))/d - (tan(c + d*x)^4*(b^4/4 - (3*a^2*b^2)/2))/d - (tan(c + d*x)^3*((4*a*b^3)/3 - (4*a^3*b)/3))/d - (log( tan(c + d*x)^2 + 1)*(a^4/2 + b^4/2 - 3*a^2*b^2))/d + (b^4*tan(c + d*x)^6)/ (6*d) + (tan(c + d*x)*(4*a*b^3 - 4*a^3*b))/d + (4*a*b^3*tan(c + d*x)^5)/(5 *d) - (4*a*b*atan((4*a*b*tan(c + d*x)*(a + b)*(a - b))/(4*a*b^3 - 4*a^3*b) )*(a + b)*(a - b))/d