Integrand size = 14, antiderivative size = 255 \[ \int \left (a+b \tan ^3(c+d x)\right )^4 \, dx=\left (a^4-6 a^2 b^2+b^4\right ) x+\frac {4 a b \left (a^2-b^2\right ) \log (\cos (c+d x))}{d}+\frac {b^2 \left (6 a^2-b^2\right ) \tan (c+d x)}{d}+\frac {2 a b \left (a^2-b^2\right ) \tan ^2(c+d x)}{d}-\frac {b^2 \left (6 a^2-b^2\right ) \tan ^3(c+d x)}{3 d}+\frac {a b^3 \tan ^4(c+d x)}{d}+\frac {b^2 \left (6 a^2-b^2\right ) \tan ^5(c+d x)}{5 d}-\frac {2 a b^3 \tan ^6(c+d x)}{3 d}+\frac {b^4 \tan ^7(c+d x)}{7 d}+\frac {a b^3 \tan ^8(c+d x)}{2 d}-\frac {b^4 \tan ^9(c+d x)}{9 d}+\frac {b^4 \tan ^{11}(c+d x)}{11 d} \]
(a^4-6*a^2*b^2+b^4)*x+4*a*b*(a^2-b^2)*ln(cos(d*x+c))/d+b^2*(6*a^2-b^2)*tan (d*x+c)/d+2*a*b*(a^2-b^2)*tan(d*x+c)^2/d-1/3*b^2*(6*a^2-b^2)*tan(d*x+c)^3/ d+a*b^3*tan(d*x+c)^4/d+1/5*b^2*(6*a^2-b^2)*tan(d*x+c)^5/d-2/3*a*b^3*tan(d* x+c)^6/d+1/7*b^4*tan(d*x+c)^7/d+1/2*a*b^3*tan(d*x+c)^8/d-1/9*b^4*tan(d*x+c )^9/d+1/11*b^4*tan(d*x+c)^11/d
Result contains complex when optimal does not.
Time = 1.08 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.88 \[ \int \left (a+b \tan ^3(c+d x)\right )^4 \, dx=\frac {-3465 i \left ((a-i b)^4 \log (i-\tan (c+d x))-(a+i b)^4 \log (i+\tan (c+d x))\right )-6930 b^2 \left (-6 a^2+b^2\right ) \tan (c+d x)+13860 a b \left (a^2-b^2\right ) \tan ^2(c+d x)+2310 b^2 \left (-6 a^2+b^2\right ) \tan ^3(c+d x)+6930 a b^3 \tan ^4(c+d x)-1386 b^2 \left (-6 a^2+b^2\right ) \tan ^5(c+d x)-4620 a b^3 \tan ^6(c+d x)+990 b^4 \tan ^7(c+d x)+3465 a b^3 \tan ^8(c+d x)-770 b^4 \tan ^9(c+d x)+630 b^4 \tan ^{11}(c+d x)}{6930 d} \]
((-3465*I)*((a - I*b)^4*Log[I - Tan[c + d*x]] - (a + I*b)^4*Log[I + Tan[c + d*x]]) - 6930*b^2*(-6*a^2 + b^2)*Tan[c + d*x] + 13860*a*b*(a^2 - b^2)*Ta n[c + d*x]^2 + 2310*b^2*(-6*a^2 + b^2)*Tan[c + d*x]^3 + 6930*a*b^3*Tan[c + d*x]^4 - 1386*b^2*(-6*a^2 + b^2)*Tan[c + d*x]^5 - 4620*a*b^3*Tan[c + d*x] ^6 + 990*b^4*Tan[c + d*x]^7 + 3465*a*b^3*Tan[c + d*x]^8 - 770*b^4*Tan[c + d*x]^9 + 630*b^4*Tan[c + d*x]^11)/(6930*d)
Time = 0.38 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 4144, 2341, 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.
| \(\displaystyle \int \left (a+b \tan ^3(c+d x)\right )^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a+b \tan (c+d x)^3\right )^4dx\) |
| \(\Big \downarrow \) 4144 |
| \(\displaystyle \frac {\int \frac {\left (b \tan ^3(c+d x)+a\right )^4}{\tan ^2(c+d x)+1}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 2341 |
| \(\displaystyle \frac {\int \left (b^4 \tan ^{10}(c+d x)-b^4 \tan ^8(c+d x)+4 a b^3 \tan ^7(c+d x)+b^4 \tan ^6(c+d x)-4 a b^3 \tan ^5(c+d x)+b^2 \left (6 a^2-b^2\right ) \tan ^4(c+d x)+4 a b^3 \tan ^3(c+d x)-b^2 \left (6 a^2-b^2\right ) \tan ^2(c+d x)+4 a b \left (a^2-b^2\right ) \tan (c+d x)-b^4+6 a^2 b^2+\frac {a^4-6 b^2 a^2-4 b \left (a^2-b^2\right ) \tan (c+d x) a+b^4}{\tan ^2(c+d x)+1}\right )d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{5} b^2 \left (6 a^2-b^2\right ) \tan ^5(c+d x)-\frac {1}{3} b^2 \left (6 a^2-b^2\right ) \tan ^3(c+d x)+2 a b \left (a^2-b^2\right ) \tan ^2(c+d x)+b^2 \left (6 a^2-b^2\right ) \tan (c+d x)-2 a b \left (a^2-b^2\right ) \log \left (\tan ^2(c+d x)+1\right )+\left (a^4-6 a^2 b^2+b^4\right ) \arctan (\tan (c+d x))+\frac {1}{2} a b^3 \tan ^8(c+d x)-\frac {2}{3} a b^3 \tan ^6(c+d x)+a b^3 \tan ^4(c+d x)+\frac {1}{11} b^4 \tan ^{11}(c+d x)-\frac {1}{9} b^4 \tan ^9(c+d x)+\frac {1}{7} b^4 \tan ^7(c+d x)}{d}\) |
((a^4 - 6*a^2*b^2 + b^4)*ArcTan[Tan[c + d*x]] - 2*a*b*(a^2 - b^2)*Log[1 + Tan[c + d*x]^2] + b^2*(6*a^2 - b^2)*Tan[c + d*x] + 2*a*b*(a^2 - b^2)*Tan[c + d*x]^2 - (b^2*(6*a^2 - b^2)*Tan[c + d*x]^3)/3 + a*b^3*Tan[c + d*x]^4 + (b^2*(6*a^2 - b^2)*Tan[c + d*x]^5)/5 - (2*a*b^3*Tan[c + d*x]^6)/3 + (b^4*T an[c + d*x]^7)/7 + (a*b^3*Tan[c + d*x]^8)/2 - (b^4*Tan[c + d*x]^9)/9 + (b^ 4*Tan[c + d*x]^11)/11)/d
3.4.74.3.1 Defintions of rubi rules used
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq* (a + b*x^2)^p, x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[c*(ff/f) Subst[Int[(a + b* (ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])
Time = 0.19 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.89
| method | result | size |
| parts | \(a^{4} x +\frac {b^{4} \left (\frac {\tan \left (d x +c \right )^{11}}{11}-\frac {\tan \left (d x +c \right )^{9}}{9}+\frac {\tan \left (d x +c \right )^{7}}{7}-\frac {\tan \left (d x +c \right )^{5}}{5}+\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {4 a \,b^{3} \left (\frac {\tan \left (d x +c \right )^{8}}{8}-\frac {\tan \left (d x +c \right )^{6}}{6}+\frac {\tan \left (d x +c \right )^{4}}{4}-\frac {\tan \left (d x +c \right )^{2}}{2}+\frac {\ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}\right )}{d}+\frac {6 a^{2} b^{2} \left (\frac {\tan \left (d x +c \right )^{5}}{5}-\frac {\tan \left (d x +c \right )^{3}}{3}+\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {2 a^{3} b \tan \left (d x +c \right )^{2}}{d}-\frac {2 a^{3} b \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{d}\) | \(227\) |
| derivativedivides | \(\frac {\frac {b^{4} \tan \left (d x +c \right )^{11}}{11}-\frac {b^{4} \tan \left (d x +c \right )^{9}}{9}+\frac {a \,b^{3} \tan \left (d x +c \right )^{8}}{2}+\frac {b^{4} \tan \left (d x +c \right )^{7}}{7}-\frac {2 a \,b^{3} \tan \left (d x +c \right )^{6}}{3}+\frac {6 a^{2} b^{2} \tan \left (d x +c \right )^{5}}{5}-\frac {b^{4} \tan \left (d x +c \right )^{5}}{5}+a \,b^{3} \tan \left (d x +c \right )^{4}-2 a^{2} b^{2} \tan \left (d x +c \right )^{3}+\frac {b^{4} \tan \left (d x +c \right )^{3}}{3}+2 a^{3} b \tan \left (d x +c \right )^{2}-2 a \,b^{3} \tan \left (d x +c \right )^{2}+6 a^{2} b^{2} \tan \left (d x +c \right )-b^{4} \tan \left (d x +c \right )+\frac {\left (-4 a^{3} b +4 a \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(246\) |
| default | \(\frac {\frac {b^{4} \tan \left (d x +c \right )^{11}}{11}-\frac {b^{4} \tan \left (d x +c \right )^{9}}{9}+\frac {a \,b^{3} \tan \left (d x +c \right )^{8}}{2}+\frac {b^{4} \tan \left (d x +c \right )^{7}}{7}-\frac {2 a \,b^{3} \tan \left (d x +c \right )^{6}}{3}+\frac {6 a^{2} b^{2} \tan \left (d x +c \right )^{5}}{5}-\frac {b^{4} \tan \left (d x +c \right )^{5}}{5}+a \,b^{3} \tan \left (d x +c \right )^{4}-2 a^{2} b^{2} \tan \left (d x +c \right )^{3}+\frac {b^{4} \tan \left (d x +c \right )^{3}}{3}+2 a^{3} b \tan \left (d x +c \right )^{2}-2 a \,b^{3} \tan \left (d x +c \right )^{2}+6 a^{2} b^{2} \tan \left (d x +c \right )-b^{4} \tan \left (d x +c \right )+\frac {\left (-4 a^{3} b +4 a \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(246\) |
| norman | \(\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) x +\frac {a \,b^{3} \tan \left (d x +c \right )^{4}}{d}+\frac {b^{2} \left (6 a^{2}-b^{2}\right ) \tan \left (d x +c \right )}{d}+\frac {b^{4} \tan \left (d x +c \right )^{7}}{7 d}-\frac {b^{4} \tan \left (d x +c \right )^{9}}{9 d}+\frac {b^{4} \tan \left (d x +c \right )^{11}}{11 d}-\frac {2 a \,b^{3} \tan \left (d x +c \right )^{6}}{3 d}+\frac {a \,b^{3} \tan \left (d x +c \right )^{8}}{2 d}-\frac {b^{2} \left (6 a^{2}-b^{2}\right ) \tan \left (d x +c \right )^{3}}{3 d}+\frac {b^{2} \left (6 a^{2}-b^{2}\right ) \tan \left (d x +c \right )^{5}}{5 d}+\frac {2 a b \left (a^{2}-b^{2}\right ) \tan \left (d x +c \right )^{2}}{d}-\frac {2 a b \left (a^{2}-b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{d}\) | \(246\) |
| parallelrisch | \(-\frac {-630 b^{4} \tan \left (d x +c \right )^{11}+770 b^{4} \tan \left (d x +c \right )^{9}-3465 a \,b^{3} \tan \left (d x +c \right )^{8}-990 b^{4} \tan \left (d x +c \right )^{7}+4620 a \,b^{3} \tan \left (d x +c \right )^{6}-8316 a^{2} b^{2} \tan \left (d x +c \right )^{5}+1386 b^{4} \tan \left (d x +c \right )^{5}-6930 a \,b^{3} \tan \left (d x +c \right )^{4}+13860 a^{2} b^{2} \tan \left (d x +c \right )^{3}-2310 b^{4} \tan \left (d x +c \right )^{3}-6930 a^{4} d x +41580 a^{2} b^{2} d x -6930 b^{4} d x -13860 a^{3} b \tan \left (d x +c \right )^{2}+13860 a \,b^{3} \tan \left (d x +c \right )^{2}+13860 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) a^{3} b -13860 \ln \left (1+\tan \left (d x +c \right )^{2}\right ) a \,b^{3}-41580 a^{2} b^{2} \tan \left (d x +c \right )+6930 b^{4} \tan \left (d x +c \right )}{6930 d}\) | \(257\) |
| risch | \(-4 i a^{3} b x +4 i a \,b^{3} x +a^{4} x -6 a^{2} b^{2} x +b^{4} x -\frac {8 i a^{3} b c}{d}+\frac {8 i a \,b^{3} c}{d}+\frac {4 b \left (627165 i a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+144144 i a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+2189880 i a^{2} b \,{\mathrm e}^{14 i \left (d x +c \right )}+3477474 i a^{2} b \,{\mathrm e}^{12 i \left (d x +c \right )}+3886344 i a^{2} b \,{\mathrm e}^{10 i \left (d x +c \right )}+873180 a^{3} {\mathrm e}^{10 i \left (d x +c \right )}-1210440 a \,b^{2} {\mathrm e}^{14 i \left (d x +c \right )}-10395 i b^{3} {\mathrm e}^{20 i \left (d x +c \right )}-27720 a \,b^{2} {\mathrm e}^{20 i \left (d x +c \right )}-166320 a \,b^{2} {\mathrm e}^{18 i \left (d x +c \right )}-1727880 a \,b^{2} {\mathrm e}^{12 i \left (d x +c \right )}-1727880 a \,b^{2} {\mathrm e}^{10 i \left (d x +c \right )}-1210440 a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+15939 i a^{2} b +31185 i a^{2} b \,{\mathrm e}^{20 i \left (d x +c \right )}+249480 i a^{2} b \,{\mathrm e}^{18 i \left (d x +c \right )}+939015 i a^{2} b \,{\mathrm e}^{16 i \left (d x +c \right )}+3069990 i a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}+1690920 i a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-317460 i b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-126995 i b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-25399 i b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-751674 i b^{3} {\mathrm e}^{12 i \left (d x +c \right )}-751674 i b^{3} {\mathrm e}^{10 i \left (d x +c \right )}-634920 i b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-51975 i b^{3} {\mathrm e}^{18 i \left (d x +c \right )}-219450 i b^{3} {\mathrm e}^{16 i \left (d x +c \right )}-438900 i b^{3} {\mathrm e}^{14 i \left (d x +c \right )}-563640 a \,b^{2} {\mathrm e}^{16 i \left (d x +c \right )}-27720 a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-166320 a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-563640 a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+6930 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+62370 a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+249480 a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-3254 i b^{3}+873180 a^{3} {\mathrm e}^{12 i \left (d x +c \right )}+582120 a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+582120 a^{3} {\mathrm e}^{14 i \left (d x +c \right )}+249480 a^{3} {\mathrm e}^{16 i \left (d x +c \right )}+6930 a^{3} {\mathrm e}^{20 i \left (d x +c \right )}+62370 a^{3} {\mathrm e}^{18 i \left (d x +c \right )}\right )}{3465 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{11}}+\frac {4 a^{3} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}-\frac {4 a \,b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(734\) |
a^4*x+b^4/d*(1/11*tan(d*x+c)^11-1/9*tan(d*x+c)^9+1/7*tan(d*x+c)^7-1/5*tan( d*x+c)^5+1/3*tan(d*x+c)^3-tan(d*x+c)+arctan(tan(d*x+c)))+4*a*b^3/d*(1/8*ta n(d*x+c)^8-1/6*tan(d*x+c)^6+1/4*tan(d*x+c)^4-1/2*tan(d*x+c)^2+1/2*ln(1+tan (d*x+c)^2))+6*a^2*b^2/d*(1/5*tan(d*x+c)^5-1/3*tan(d*x+c)^3+tan(d*x+c)-arct an(tan(d*x+c)))+2*a^3*b/d*tan(d*x+c)^2-2*a^3*b/d*ln(1+tan(d*x+c)^2)
Time = 0.29 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.88 \[ \int \left (a+b \tan ^3(c+d x)\right )^4 \, dx=\frac {630 \, b^{4} \tan \left (d x + c\right )^{11} - 770 \, b^{4} \tan \left (d x + c\right )^{9} + 3465 \, a b^{3} \tan \left (d x + c\right )^{8} + 990 \, b^{4} \tan \left (d x + c\right )^{7} - 4620 \, a b^{3} \tan \left (d x + c\right )^{6} + 6930 \, a b^{3} \tan \left (d x + c\right )^{4} + 1386 \, {\left (6 \, a^{2} b^{2} - b^{4}\right )} \tan \left (d x + c\right )^{5} - 2310 \, {\left (6 \, a^{2} b^{2} - b^{4}\right )} \tan \left (d x + c\right )^{3} + 6930 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} d x + 13860 \, {\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{2} + 13860 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 6930 \, {\left (6 \, a^{2} b^{2} - b^{4}\right )} \tan \left (d x + c\right )}{6930 \, d} \]
1/6930*(630*b^4*tan(d*x + c)^11 - 770*b^4*tan(d*x + c)^9 + 3465*a*b^3*tan( d*x + c)^8 + 990*b^4*tan(d*x + c)^7 - 4620*a*b^3*tan(d*x + c)^6 + 6930*a*b ^3*tan(d*x + c)^4 + 1386*(6*a^2*b^2 - b^4)*tan(d*x + c)^5 - 2310*(6*a^2*b^ 2 - b^4)*tan(d*x + c)^3 + 6930*(a^4 - 6*a^2*b^2 + b^4)*d*x + 13860*(a^3*b - a*b^3)*tan(d*x + c)^2 + 13860*(a^3*b - a*b^3)*log(1/(tan(d*x + c)^2 + 1) ) + 6930*(6*a^2*b^2 - b^4)*tan(d*x + c))/d
Time = 0.45 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.18 \[ \int \left (a+b \tan ^3(c+d x)\right )^4 \, dx=\begin {cases} a^{4} x - \frac {2 a^{3} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {2 a^{3} b \tan ^{2}{\left (c + d x \right )}}{d} - 6 a^{2} b^{2} x + \frac {6 a^{2} b^{2} \tan ^{5}{\left (c + d x \right )}}{5 d} - \frac {2 a^{2} b^{2} \tan ^{3}{\left (c + d x \right )}}{d} + \frac {6 a^{2} b^{2} \tan {\left (c + d x \right )}}{d} + \frac {2 a b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {a b^{3} \tan ^{8}{\left (c + d x \right )}}{2 d} - \frac {2 a b^{3} \tan ^{6}{\left (c + d x \right )}}{3 d} + \frac {a b^{3} \tan ^{4}{\left (c + d x \right )}}{d} - \frac {2 a b^{3} \tan ^{2}{\left (c + d x \right )}}{d} + b^{4} x + \frac {b^{4} \tan ^{11}{\left (c + d x \right )}}{11 d} - \frac {b^{4} \tan ^{9}{\left (c + d x \right )}}{9 d} + \frac {b^{4} \tan ^{7}{\left (c + d x \right )}}{7 d} - \frac {b^{4} \tan ^{5}{\left (c + d x \right )}}{5 d} + \frac {b^{4} \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac {b^{4} \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tan ^{3}{\left (c \right )}\right )^{4} & \text {otherwise} \end {cases} \]
Piecewise((a**4*x - 2*a**3*b*log(tan(c + d*x)**2 + 1)/d + 2*a**3*b*tan(c + d*x)**2/d - 6*a**2*b**2*x + 6*a**2*b**2*tan(c + d*x)**5/(5*d) - 2*a**2*b* *2*tan(c + d*x)**3/d + 6*a**2*b**2*tan(c + d*x)/d + 2*a*b**3*log(tan(c + d *x)**2 + 1)/d + a*b**3*tan(c + d*x)**8/(2*d) - 2*a*b**3*tan(c + d*x)**6/(3 *d) + a*b**3*tan(c + d*x)**4/d - 2*a*b**3*tan(c + d*x)**2/d + b**4*x + b** 4*tan(c + d*x)**11/(11*d) - b**4*tan(c + d*x)**9/(9*d) + b**4*tan(c + d*x) **7/(7*d) - b**4*tan(c + d*x)**5/(5*d) + b**4*tan(c + d*x)**3/(3*d) - b**4 *tan(c + d*x)/d, Ne(d, 0)), (x*(a + b*tan(c)**3)**4, True))
Time = 0.29 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.02 \[ \int \left (a+b \tan ^3(c+d x)\right )^4 \, dx=a^{4} x + \frac {2 \, {\left (3 \, \tan \left (d x + c\right )^{5} - 5 \, \tan \left (d x + c\right )^{3} - 15 \, d x - 15 \, c + 15 \, \tan \left (d x + c\right )\right )} a^{2} b^{2}}{5 \, d} + \frac {{\left (315 \, \tan \left (d x + c\right )^{11} - 385 \, \tan \left (d x + c\right )^{9} + 495 \, \tan \left (d x + c\right )^{7} - 693 \, \tan \left (d x + c\right )^{5} + 1155 \, \tan \left (d x + c\right )^{3} + 3465 \, d x + 3465 \, c - 3465 \, \tan \left (d x + c\right )\right )} b^{4}}{3465 \, d} + \frac {a b^{3} {\left (\frac {48 \, \sin \left (d x + c\right )^{6} - 108 \, \sin \left (d x + c\right )^{4} + 88 \, \sin \left (d x + c\right )^{2} - 25}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - 12 \, \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )}}{6 \, d} - \frac {2 \, a^{3} b {\left (\frac {1}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )}}{d} \]
a^4*x + 2/5*(3*tan(d*x + c)^5 - 5*tan(d*x + c)^3 - 15*d*x - 15*c + 15*tan( d*x + c))*a^2*b^2/d + 1/3465*(315*tan(d*x + c)^11 - 385*tan(d*x + c)^9 + 4 95*tan(d*x + c)^7 - 693*tan(d*x + c)^5 + 1155*tan(d*x + c)^3 + 3465*d*x + 3465*c - 3465*tan(d*x + c))*b^4/d + 1/6*a*b^3*((48*sin(d*x + c)^6 - 108*si n(d*x + c)^4 + 88*sin(d*x + c)^2 - 25)/(sin(d*x + c)^8 - 4*sin(d*x + c)^6 + 6*sin(d*x + c)^4 - 4*sin(d*x + c)^2 + 1) - 12*log(sin(d*x + c)^2 - 1))/d - 2*a^3*b*(1/(sin(d*x + c)^2 - 1) - log(sin(d*x + c)^2 - 1))/d
Leaf count of result is larger than twice the leaf count of optimal. 5709 vs. \(2 (241) = 482\).
Time = 36.01 (sec) , antiderivative size = 5709, normalized size of antiderivative = 22.39 \[ \int \left (a+b \tan ^3(c+d x)\right )^4 \, dx=\text {Too large to display} \]
1/6930*(6930*a^4*d*x*tan(d*x)^11*tan(c)^11 - 41580*a^2*b^2*d*x*tan(d*x)^11 *tan(c)^11 + 6930*b^4*d*x*tan(d*x)^11*tan(c)^11 + 13860*a^3*b*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)^11*tan(c)^11 - 13860*a*b^3*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)^11*tan(c)^11 - 76230*a^4*d*x*tan(d*x)^10*tan(c)^10 + 4573 80*a^2*b^2*d*x*tan(d*x)^10*tan(c)^10 - 76230*b^4*d*x*tan(d*x)^10*tan(c)^10 + 13860*a^3*b*tan(d*x)^11*tan(c)^11 - 28875*a*b^3*tan(d*x)^11*tan(c)^11 - 152460*a^3*b*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)^10*tan(c)^10 + 152460* a*b^3*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)^10*tan(c)^10 - 41580*a^2*b^2*t an(d*x)^11*tan(c)^10 + 6930*b^4*tan(d*x)^11*tan(c)^10 - 41580*a^2*b^2*tan( d*x)^10*tan(c)^11 + 6930*b^4*tan(d*x)^10*tan(c)^11 + 381150*a^4*d*x*tan(d* x)^9*tan(c)^9 - 2286900*a^2*b^2*d*x*tan(d*x)^9*tan(c)^9 + 381150*b^4*d*x*t an(d*x)^9*tan(c)^9 + 13860*a^3*b*tan(d*x)^11*tan(c)^9 - 13860*a*b^3*tan(d* x)^11*tan(c)^9 - 124740*a^3*b*tan(d*x)^10*tan(c)^10 + 289905*a*b^3*tan(d*x )^10*tan(c)^10 + 13860*a^3*b*tan(d*x)^9*tan(c)^11 - 13860*a*b^3*tan(d*x)^9 *tan(c)^11 + 13860*a^2*b^2*tan(d*x)^11*tan(c)^8 - 2310*b^4*tan(d*x)^11*tan (c)^8 + 762300*a^3*b*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1...
Time = 11.51 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.22 \[ \int \left (a+b \tan ^3(c+d x)\right )^4 \, dx=\frac {\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )\,\left (2\,a\,b^3-2\,a^3\,b\right )}{d}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {b^4}{3}-2\,a^2\,b^2\right )}{d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (\frac {b^4}{5}-\frac {6\,a^2\,b^2}{5}\right )}{d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (2\,a\,b^3-2\,a^3\,b\right )}{d}-\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (b^4-6\,a^2\,b^2\right )}{d}+\frac {b^4\,{\mathrm {tan}\left (c+d\,x\right )}^7}{7\,d}-\frac {b^4\,{\mathrm {tan}\left (c+d\,x\right )}^9}{9\,d}+\frac {b^4\,{\mathrm {tan}\left (c+d\,x\right )}^{11}}{11\,d}+\frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (-a^2+2\,a\,b+b^2\right )\,\left (a^2+2\,a\,b-b^2\right )}{a^4-6\,a^2\,b^2+b^4}\right )\,\left (-a^2+2\,a\,b+b^2\right )\,\left (a^2+2\,a\,b-b^2\right )}{d}+\frac {a\,b^3\,{\mathrm {tan}\left (c+d\,x\right )}^4}{d}-\frac {2\,a\,b^3\,{\mathrm {tan}\left (c+d\,x\right )}^6}{3\,d}+\frac {a\,b^3\,{\mathrm {tan}\left (c+d\,x\right )}^8}{2\,d} \]
(log(tan(c + d*x)^2 + 1)*(2*a*b^3 - 2*a^3*b))/d + (tan(c + d*x)^3*(b^4/3 - 2*a^2*b^2))/d - (tan(c + d*x)^5*(b^4/5 - (6*a^2*b^2)/5))/d - (tan(c + d*x )^2*(2*a*b^3 - 2*a^3*b))/d - (tan(c + d*x)*(b^4 - 6*a^2*b^2))/d + (b^4*tan (c + d*x)^7)/(7*d) - (b^4*tan(c + d*x)^9)/(9*d) + (b^4*tan(c + d*x)^11)/(1 1*d) + (atan((tan(c + d*x)*(2*a*b - a^2 + b^2)*(2*a*b + a^2 - b^2))/(a^4 + b^4 - 6*a^2*b^2))*(2*a*b - a^2 + b^2)*(2*a*b + a^2 - b^2))/d + (a*b^3*tan (c + d*x)^4)/d - (2*a*b^3*tan(c + d*x)^6)/(3*d) + (a*b^3*tan(c + d*x)^8)/( 2*d)