Integrand size = 19, antiderivative size = 60 \[ \int \tan ^3(c+d x) (a+b \tan (c+d x)) \, dx=b x+\frac {a \log (\cos (c+d x))}{d}-\frac {b \tan (c+d x)}{d}+\frac {a \tan ^2(c+d x)}{2 d}+\frac {b \tan ^3(c+d x)}{3 d} \] Output:
b*x+a*ln(cos(d*x+c))/d-b*tan(d*x+c)/d+1/2*a*tan(d*x+c)^2/d+1/3*b*tan(d*x+c )^3/d
Time = 0.10 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.12 \[ \int \tan ^3(c+d x) (a+b \tan (c+d x)) \, dx=\frac {b \arctan (\tan (c+d x))}{d}+\frac {a \left (2 \log (\cos (c+d x))+\sec ^2(c+d x)\right )}{2 d}-\frac {b \tan (c+d x)}{d}+\frac {b \tan ^3(c+d x)}{3 d} \] Input:
Integrate[Tan[c + d*x]^3*(a + b*Tan[c + d*x]),x]
Output:
(b*ArcTan[Tan[c + d*x]])/d + (a*(2*Log[Cos[c + d*x]] + Sec[c + d*x]^2))/(2 *d) - (b*Tan[c + d*x])/d + (b*Tan[c + d*x]^3)/(3*d)
Time = 0.42 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {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.
\(\displaystyle \int \tan ^3(c+d x) (a+b \tan (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \tan (c+d x)^3 (a+b \tan (c+d x))dx\) |
\(\Big \downarrow \) 4011 |
\(\displaystyle \int \tan ^2(c+d x) (a \tan (c+d x)-b)dx+\frac {b \tan ^3(c+d x)}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \tan (c+d x)^2 (a \tan (c+d x)-b)dx+\frac {b \tan ^3(c+d x)}{3 d}\) |
\(\Big \downarrow \) 4011 |
\(\displaystyle \int \tan (c+d x) (-a-b \tan (c+d x))dx+\frac {a \tan ^2(c+d x)}{2 d}+\frac {b \tan ^3(c+d x)}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \tan (c+d x) (-a-b \tan (c+d x))dx+\frac {a \tan ^2(c+d x)}{2 d}+\frac {b \tan ^3(c+d x)}{3 d}\) |
\(\Big \downarrow \) 4008 |
\(\displaystyle -a \int \tan (c+d x)dx+\frac {a \tan ^2(c+d x)}{2 d}+\frac {b \tan ^3(c+d x)}{3 d}-\frac {b \tan (c+d x)}{d}+b x\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -a \int \tan (c+d x)dx+\frac {a \tan ^2(c+d x)}{2 d}+\frac {b \tan ^3(c+d x)}{3 d}-\frac {b \tan (c+d x)}{d}+b x\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle \frac {a \tan ^2(c+d x)}{2 d}+\frac {a \log (\cos (c+d x))}{d}+\frac {b \tan ^3(c+d x)}{3 d}-\frac {b \tan (c+d x)}{d}+b x\) |
Input:
Int[Tan[c + d*x]^3*(a + b*Tan[c + d*x]),x]
Output:
b*x + (a*Log[Cos[c + d*x]])/d - (b*Tan[c + d*x])/d + (a*Tan[c + d*x]^2)/(2 *d) + (b*Tan[c + d*x]^3)/(3*d)
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 = 0.44 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.95
method | result | size |
parallelrisch | \(-\frac {-2 b \tan \left (d x +c \right )^{3}-6 b d x -3 a \tan \left (d x +c \right )^{2}+3 a \ln \left (1+\tan \left (d x +c \right )^{2}\right )+6 b \tan \left (d x +c \right )}{6 d}\) | \(57\) |
derivativedivides | \(\frac {\frac {b \tan \left (d x +c \right )^{3}}{3}+\frac {a \tan \left (d x +c \right )^{2}}{2}-b \tan \left (d x +c \right )-\frac {a \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+b \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(60\) |
default | \(\frac {\frac {b \tan \left (d x +c \right )^{3}}{3}+\frac {a \tan \left (d x +c \right )^{2}}{2}-b \tan \left (d x +c \right )-\frac {a \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+b \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(60\) |
norman | \(b x +\frac {a \tan \left (d x +c \right )^{2}}{2 d}-\frac {b \tan \left (d x +c \right )}{d}+\frac {b \tan \left (d x +c \right )^{3}}{3 d}-\frac {a \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d}\) | \(62\) |
parts | \(\frac {a \left (\frac {\tan \left (d x +c \right )^{2}}{2}-\frac {\ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}\right )}{d}+\frac {b \left (\frac {\tan \left (d x +c \right )^{3}}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(62\) |
risch | \(b x -i a x -\frac {2 i a c}{d}-\frac {2 i \left (3 i a \,{\mathrm e}^{4 i \left (d x +c \right )}+6 b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 i a \,{\mathrm e}^{2 i \left (d x +c \right )}+6 b \,{\mathrm e}^{2 i \left (d x +c \right )}+4 b \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(108\) |
Input:
int(tan(d*x+c)^3*(a+b*tan(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-1/6*(-2*b*tan(d*x+c)^3-6*b*d*x-3*a*tan(d*x+c)^2+3*a*ln(1+tan(d*x+c)^2)+6* b*tan(d*x+c))/d
Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.97 \[ \int \tan ^3(c+d x) (a+b \tan (c+d x)) \, dx=\frac {2 \, b \tan \left (d x + c\right )^{3} + 6 \, b d x + 3 \, a \tan \left (d x + c\right )^{2} + 3 \, a \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 6 \, b \tan \left (d x + c\right )}{6 \, d} \] Input:
integrate(tan(d*x+c)^3*(a+b*tan(d*x+c)),x, algorithm="fricas")
Output:
1/6*(2*b*tan(d*x + c)^3 + 6*b*d*x + 3*a*tan(d*x + c)^2 + 3*a*log(1/(tan(d* x + c)^2 + 1)) - 6*b*tan(d*x + c))/d
Time = 0.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.17 \[ \int \tan ^3(c+d x) (a+b \tan (c+d x)) \, dx=\begin {cases} - \frac {a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a \tan ^{2}{\left (c + d x \right )}}{2 d} + b x + \frac {b \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac {b \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\left (c \right )}\right ) \tan ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(tan(d*x+c)**3*(a+b*tan(d*x+c)),x)
Output:
Piecewise((-a*log(tan(c + d*x)**2 + 1)/(2*d) + a*tan(c + d*x)**2/(2*d) + b *x + b*tan(c + d*x)**3/(3*d) - b*tan(c + d*x)/d, Ne(d, 0)), (x*(a + b*tan( c))*tan(c)**3, True))
Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.98 \[ \int \tan ^3(c+d x) (a+b \tan (c+d x)) \, dx=\frac {2 \, b \tan \left (d x + c\right )^{3} + 3 \, a \tan \left (d x + c\right )^{2} + 6 \, {\left (d x + c\right )} b - 3 \, a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 \, b \tan \left (d x + c\right )}{6 \, d} \] Input:
integrate(tan(d*x+c)^3*(a+b*tan(d*x+c)),x, algorithm="maxima")
Output:
1/6*(2*b*tan(d*x + c)^3 + 3*a*tan(d*x + c)^2 + 6*(d*x + c)*b - 3*a*log(tan (d*x + c)^2 + 1) - 6*b*tan(d*x + c))/d
Time = 0.18 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.23 \[ \int \tan ^3(c+d x) (a+b \tan (c+d x)) \, dx=\frac {{\left (d x + c\right )} b}{d} - \frac {a \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} + \frac {2 \, b d^{2} \tan \left (d x + c\right )^{3} + 3 \, a d^{2} \tan \left (d x + c\right )^{2} - 6 \, b d^{2} \tan \left (d x + c\right )}{6 \, d^{3}} \] Input:
integrate(tan(d*x+c)^3*(a+b*tan(d*x+c)),x, algorithm="giac")
Output:
(d*x + c)*b/d - 1/2*a*log(tan(d*x + c)^2 + 1)/d + 1/6*(2*b*d^2*tan(d*x + c )^3 + 3*a*d^2*tan(d*x + c)^2 - 6*b*d^2*tan(d*x + c))/d^3
Time = 1.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.90 \[ \int \tan ^3(c+d x) (a+b \tan (c+d x)) \, dx=\frac {\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}-\frac {a\,\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}{2}-b\,\mathrm {tan}\left (c+d\,x\right )+\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3}+b\,d\,x}{d} \] Input:
int(tan(c + d*x)^3*(a + b*tan(c + d*x)),x)
Output:
((a*tan(c + d*x)^2)/2 - (a*log(tan(c + d*x)^2 + 1))/2 - b*tan(c + d*x) + ( b*tan(c + d*x)^3)/3 + b*d*x)/d
Time = 0.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.93 \[ \int \tan ^3(c+d x) (a+b \tan (c+d x)) \, dx=\frac {-3 \,\mathrm {log}\left (\tan \left (d x +c \right )^{2}+1\right ) a +2 \tan \left (d x +c \right )^{3} b +3 \tan \left (d x +c \right )^{2} a -6 \tan \left (d x +c \right ) b +6 b d x}{6 d} \] Input:
int(tan(d*x+c)^3*(a+b*tan(d*x+c)),x)
Output:
( - 3*log(tan(c + d*x)**2 + 1)*a + 2*tan(c + d*x)**3*b + 3*tan(c + d*x)**2 *a - 6*tan(c + d*x)*b + 6*b*d*x)/(6*d)