Integrand size = 19, antiderivative size = 130 \[ \int \tan (c+d x) (a+b \tan (c+d x))^4 \, dx=-4 a b \left (a^2-b^2\right ) x-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (\cos (c+d x))}{d}+\frac {a b \left (a^2-3 b^2\right ) \tan (c+d x)}{d}+\frac {\left (a^2-b^2\right ) (a+b \tan (c+d x))^2}{2 d}+\frac {a (a+b \tan (c+d x))^3}{3 d}+\frac {(a+b \tan (c+d x))^4}{4 d} \]
-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)*ta n(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
Result contains complex when optimal does not.
Time = 1.79 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.95 \[ \int \tan (c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {6 \left ((a+i b)^4 \log (i-\tan (c+d x))+(a-i b)^4 \log (i+\tan (c+d x))\right )+48 a b \left (a^2-b^2\right ) \tan (c+d x)-6 b^2 \left (-6 a^2+b^2\right ) \tan ^2(c+d x)+16 a b^3 \tan ^3(c+d x)+3 b^4 \tan ^4(c+d x)}{12 d} \]
(6*((a + I*b)^4*Log[I - Tan[c + d*x]] + (a - I*b)^4*Log[I + Tan[c + d*x]]) + 48*a*b*(a^2 - b^2)*Tan[c + d*x] - 6*b^2*(-6*a^2 + b^2)*Tan[c + d*x]^2 + 16*a*b^3*Tan[c + d*x]^3 + 3*b^4*Tan[c + d*x]^4)/(12*d)
Time = 0.67 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {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.
| \(\displaystyle \int \tan (c+d x) (a+b \tan (c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (c+d x) (a+b \tan (c+d x))^4dx\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int (a \tan (c+d x)-b) (a+b \tan (c+d x))^3dx+\frac {(a+b \tan (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \tan (c+d x)-b) (a+b \tan (c+d x))^3dx+\frac {(a+b \tan (c+d x))^4}{4 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int (a+b \tan (c+d x))^2 \left (\left (a^2-b^2\right ) \tan (c+d x)-2 a b\right )dx+\frac {(a+b \tan (c+d x))^4}{4 d}+\frac {a (a+b \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \tan (c+d x))^2 \left (\left (a^2-b^2\right ) \tan (c+d x)-2 a b\right )dx+\frac {(a+b \tan (c+d x))^4}{4 d}+\frac {a (a+b \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int (a+b \tan (c+d x)) \left (a \left (a^2-3 b^2\right ) \tan (c+d x)-b \left (3 a^2-b^2\right )\right )dx+\frac {\left (a^2-b^2\right ) (a+b \tan (c+d x))^2}{2 d}+\frac {(a+b \tan (c+d x))^4}{4 d}+\frac {a (a+b \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \tan (c+d x)) \left (a \left (a^2-3 b^2\right ) \tan (c+d x)-b \left (3 a^2-b^2\right )\right )dx+\frac {\left (a^2-b^2\right ) (a+b \tan (c+d x))^2}{2 d}+\frac {(a+b \tan (c+d x))^4}{4 d}+\frac {a (a+b \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 4008 |
| \(\displaystyle \left (a^4-6 a^2 b^2+b^4\right ) \int \tan (c+d x)dx+\frac {\left (a^2-b^2\right ) (a+b \tan (c+d x))^2}{2 d}+\frac {a b \left (a^2-3 b^2\right ) \tan (c+d x)}{d}-4 a b x \left (a^2-b^2\right )+\frac {(a+b \tan (c+d x))^4}{4 d}+\frac {a (a+b \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \left (a^4-6 a^2 b^2+b^4\right ) \int \tan (c+d x)dx+\frac {\left (a^2-b^2\right ) (a+b \tan (c+d x))^2}{2 d}+\frac {a b \left (a^2-3 b^2\right ) \tan (c+d x)}{d}-4 a b x \left (a^2-b^2\right )+\frac {(a+b \tan (c+d x))^4}{4 d}+\frac {a (a+b \tan (c+d x))^3}{3 d}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) (a+b \tan (c+d x))^2}{2 d}+\frac {a b \left (a^2-3 b^2\right ) \tan (c+d x)}{d}-4 a b x \left (a^2-b^2\right )-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (\cos (c+d x))}{d}+\frac {(a+b \tan (c+d x))^4}{4 d}+\frac {a (a+b \tan (c+d x))^3}{3 d}\) |
-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)
3.5.47.3.1 Defintions of rubi rules used
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]
Time = 0.31 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00
| method | result | size |
| norman | \(\left (-4 a^{3} b +4 a \,b^{3}\right ) x +\frac {b^{4} \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}+\frac {4 a \,b^{3} \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}+\frac {b^{2} \left (6 a^{2}-b^{2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {4 a b \left (a^{2}-b^{2}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(130\) |
| derivativedivides | \(\frac {\frac {b^{4} \left (\tan ^{4}\left (d x +c \right )\right )}{4}+\frac {4 a \,b^{3} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+3 a^{2} b^{2} \left (\tan ^{2}\left (d x +c \right )\right )-\frac {b^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{2}+4 a^{3} b \tan \left (d x +c \right )-4 a \,b^{3} \tan \left (d x +c \right )+\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-4 a^{3} b +4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(135\) |
| default | \(\frac {\frac {b^{4} \left (\tan ^{4}\left (d x +c \right )\right )}{4}+\frac {4 a \,b^{3} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+3 a^{2} b^{2} \left (\tan ^{2}\left (d x +c \right )\right )-\frac {b^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{2}+4 a^{3} b \tan \left (d x +c \right )-4 a \,b^{3} \tan \left (d x +c \right )+\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-4 a^{3} b +4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(135\) |
| parallelrisch | \(\frac {3 b^{4} \left (\tan ^{4}\left (d x +c \right )\right )+16 a \,b^{3} \left (\tan ^{3}\left (d x +c \right )\right )-48 a^{3} b d x +48 a \,b^{3} d x +36 a^{2} b^{2} \left (\tan ^{2}\left (d x +c \right )\right )-6 b^{4} \left (\tan ^{2}\left (d x +c \right )\right )+6 a^{4} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )-36 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2} b^{2}+6 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) b^{4}+48 a^{3} b \tan \left (d x +c \right )-48 a \,b^{3} \tan \left (d x +c \right )}{12 d}\) | \(154\) |
| parts | \(\frac {a^{4} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {b^{4} \left (\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}+\frac {4 a \,b^{3} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {3 a^{2} b^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{d}-\frac {3 a^{2} b^{2} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}+\frac {4 a^{3} b \left (\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(163\) |
| risch | \(-4 a^{3} b x +4 a \,b^{3} x +i a^{4} x -6 i a^{2} b^{2} x +i b^{4} x +\frac {2 i a^{4} c}{d}-\frac {12 i a^{2} b^{2} c}{d}+\frac {2 i b^{4} c}{d}+\frac {4 i b \left (-9 i a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+3 i {\mathrm e}^{6 i \left (d x +c \right )} b^{3}+6 a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-12 a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-18 i a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 i {\mathrm e}^{4 i \left (d x +c \right )} b^{3}+18 a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-24 a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-9 i a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+3 i {\mathrm e}^{2 i \left (d x +c \right )} b^{3}+18 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-20 a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+6 a^{3}-8 a \,b^{2}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{4}}{d}+\frac {6 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a^{2} b^{2}}{d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) b^{4}}{d}\) | \(348\) |
(-4*a^3*b+4*a*b^3)*x+1/4*b^4/d*tan(d*x+c)^4+4/3*a*b^3/d*tan(d*x+c)^3+1/2*b ^2*(6*a^2-b^2)/d*tan(d*x+c)^2+4*a*b*(a^2-b^2)/d*tan(d*x+c)+1/2*(a^4-6*a^2* b^2+b^4)/d*ln(1+tan(d*x+c)^2)
Time = 0.24 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.95 \[ \int \tan (c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {3 \, b^{4} \tan \left (d x + c\right )^{4} + 16 \, a b^{3} \tan \left (d x + c\right )^{3} - 48 \, {\left (a^{3} b - a b^{3}\right )} d x + 6 \, {\left (6 \, a^{2} b^{2} - b^{4}\right )} \tan \left (d x + c\right )^{2} - 6 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 48 \, {\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )}{12 \, d} \]
1/12*(3*b^4*tan(d*x + c)^4 + 16*a*b^3*tan(d*x + c)^3 - 48*(a^3*b - a*b^3)* d*x + 6*(6*a^2*b^2 - b^4)*tan(d*x + c)^2 - 6*(a^4 - 6*a^2*b^2 + b^4)*log(1 /(tan(d*x + c)^2 + 1)) + 48*(a^3*b - a*b^3)*tan(d*x + c))/d
Time = 0.14 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.44 \[ \int \tan (c+d x) (a+b \tan (c+d x))^4 \, dx=\begin {cases} \frac {a^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - 4 a^{3} b x + \frac {4 a^{3} b \tan {\left (c + d x \right )}}{d} - \frac {3 a^{2} b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {3 a^{2} b^{2} \tan ^{2}{\left (c + d x \right )}}{d} + 4 a b^{3} x + \frac {4 a b^{3} \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac {4 a b^{3} \tan {\left (c + d x \right )}}{d} + \frac {b^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {b^{4} \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac {b^{4} \tan ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\left (c \right )}\right )^{4} \tan {\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((a**4*log(tan(c + d*x)**2 + 1)/(2*d) - 4*a**3*b*x + 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)**2/d + 4*a*b**3*x + 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)**4/(4*d ) - b**4*tan(c + d*x)**2/(2*d), Ne(d, 0)), (x*(a + b*tan(c))**4*tan(c), Tr ue))
Time = 0.37 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.95 \[ \int \tan (c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {3 \, b^{4} \tan \left (d x + c\right )^{4} + 16 \, a b^{3} \tan \left (d x + c\right )^{3} + 6 \, {\left (6 \, a^{2} b^{2} - b^{4}\right )} \tan \left (d x + c\right )^{2} - 48 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )} + 6 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 48 \, {\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )}{12 \, d} \]
1/12*(3*b^4*tan(d*x + c)^4 + 16*a*b^3*tan(d*x + c)^3 + 6*(6*a^2*b^2 - b^4) *tan(d*x + c)^2 - 48*(a^3*b - a*b^3)*(d*x + c) + 6*(a^4 - 6*a^2*b^2 + b^4) *log(tan(d*x + c)^2 + 1) + 48*(a^3*b - a*b^3)*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 1736 vs. \(2 (124) = 248\).
Time = 2.21 (sec) , antiderivative size = 1736, normalized size of antiderivative = 13.35 \[ \int \tan (c+d x) (a+b \tan (c+d x))^4 \, dx=\text {Too large to display} \]
-1/12*(48*a^3*b*d*x*tan(d*x)^4*tan(c)^4 - 48*a*b^3*d*x*tan(d*x)^4*tan(c)^4 + 6*a^4*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*t an(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^4*tan(c)^4 - 36*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 + ta n(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^4*tan(c)^4 + 6*b^4*log(4*(tan(d*x)^2*ta n(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)^4*tan(c)^4 - 192*a^3*b*d*x*tan(d*x)^3*tan(c)^3 + 192*a*b ^3*d*x*tan(d*x)^3*tan(c)^3 - 36*a^2*b^2*tan(d*x)^4*tan(c)^4 + 9*b^4*tan(d* x)^4*tan(c)^4 - 24*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)^3*tan(c)^3 + 144*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)^3*tan(c)^3 - 24*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)^3*tan(c)^3 + 48*a^3*b*tan(d*x)^4*tan(c)^ 3 - 48*a*b^3*tan(d*x)^4*tan(c)^3 + 48*a^3*b*tan(d*x)^3*tan(c)^4 - 48*a*b^3 *tan(d*x)^3*tan(c)^4 + 288*a^3*b*d*x*tan(d*x)^2*tan(c)^2 - 288*a*b^3*d*x*t an(d*x)^2*tan(c)^2 - 36*a^2*b^2*tan(d*x)^4*tan(c)^2 + 6*b^4*tan(d*x)^4*tan (c)^2 + 72*a^2*b^2*tan(d*x)^3*tan(c)^3 - 24*b^4*tan(d*x)^3*tan(c)^3 - 36*a ^2*b^2*tan(d*x)^2*tan(c)^4 + 6*b^4*tan(d*x)^2*tan(c)^4 + 16*a*b^3*tan(d*x) ^4*tan(c) + 36*a^4*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/...
Time = 5.03 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.29 \[ \int \tan (c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )\,\left (\frac {a^4}{2}-3\,a^2\,b^2+\frac {b^4}{2}\right )}{d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {b^4}{2}-3\,a^2\,b^2\right )}{d}+\frac {b^4\,{\mathrm {tan}\left (c+d\,x\right )}^4}{4\,d}-\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a\,b^3-4\,a^3\,b\right )}{d}+\frac {4\,a\,b^3\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3\,d}+\frac {4\,a\,b\,\mathrm {atan}\left (\frac {4\,a\,b\,\mathrm {tan}\left (c+d\,x\right )\,\left (a+b\right )\,\left (a-b\right )}{4\,a\,b^3-4\,a^3\,b}\right )\,\left (a+b\right )\,\left (a-b\right )}{d} \]