\(\int \tan ^5(c+d x) \, dx\) [5]

Optimal result

Integrand size = 8, antiderivative size = 43 \[ \int \tan ^5(c+d x) \, dx=-\frac {\log (\cos (c+d x))}{d}-\frac {\tan ^2(c+d x)}{2 d}+\frac {\tan ^4(c+d x)}{4 d} \]

[Out]

-ln(cos(d*x+c))/d-1/2*tan(d*x+c)^2/d+1/4*tan(d*x+c)^4/d

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3554, 3556} \[ \int \tan ^5(c+d x) \, dx=\frac {\tan ^4(c+d x)}{4 d}-\frac {\tan ^2(c+d x)}{2 d}-\frac {\log (\cos (c+d x))}{d} \]

[In]

Int[Tan[c + d*x]^5,x]

[Out]

-(Log[Cos[c + d*x]]/d) - Tan[c + d*x]^2/(2*d) + Tan[c + d*x]^4/(4*d)

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\tan ^4(c+d x)}{4 d}-\int \tan ^3(c+d x) \, dx \\ & = -\frac {\tan ^2(c+d x)}{2 d}+\frac {\tan ^4(c+d x)}{4 d}+\int \tan (c+d x) \, dx \\ & = -\frac {\log (\cos (c+d x))}{d}-\frac {\tan ^2(c+d x)}{2 d}+\frac {\tan ^4(c+d x)}{4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \tan ^5(c+d x) \, dx=-\frac {4 \log (\cos (c+d x))+2 \tan ^2(c+d x)-\tan ^4(c+d x)}{4 d} \]

[In]

Integrate[Tan[c + d*x]^5,x]

[Out]

-1/4*(4*Log[Cos[c + d*x]] + 2*Tan[c + d*x]^2 - Tan[c + d*x]^4)/d

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.88

method	result	size
parallelrisch	\(\frac {\tan ^{4}\left (d x +c \right )-2 \left (\tan ^{2}\left (d x +c \right )\right )+2 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{4 d}\)	\(38\)
derivativedivides	\(\frac {\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}}{d}\)	\(39\)
default	\(\frac {\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}}{d}\)	\(39\)
norman	\(-\frac {\tan ^{2}\left (d x +c \right )}{2 d}+\frac {\tan ^{4}\left (d x +c \right )}{4 d}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(44\)
risch	\(i x +\frac {2 i c}{d}-\frac {4 \left ({\mathrm e}^{6 i \left (d x +c \right )}+{\mathrm e}^{4 i \left (d x +c \right )}+{\mathrm e}^{2 i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\)	\(76\)

[In]

int(tan(d*x+c)^5,x,method=_RETURNVERBOSE)

[Out]

1/4*(tan(d*x+c)^4-2*tan(d*x+c)^2+2*ln(1+tan(d*x+c)^2))/d

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91 \[ \int \tan ^5(c+d x) \, dx=\frac {\tan \left (d x + c\right )^{4} - 2 \, \tan \left (d x + c\right )^{2} - 2 \, \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right )}{4 \, d} \]

[In]

integrate(tan(d*x+c)^5,x, algorithm="fricas")

[Out]

1/4*(tan(d*x + c)^4 - 2*tan(d*x + c)^2 - 2*log(1/(tan(d*x + c)^2 + 1)))/d

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.02 \[ \int \tan ^5(c+d x) \, dx=\begin {cases} \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {\tan ^{4}{\left (c + d x \right )}}{4 d} - \frac {\tan ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \tan ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(tan(d*x+c)**5,x)

[Out]

Piecewise((log(tan(c + d*x)**2 + 1)/(2*d) + tan(c + d*x)**4/(4*d) - tan(c + d*x)**2/(2*d), Ne(d, 0)), (x*tan(c
)**5, True))

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.26 \[ \int \tan ^5(c+d x) \, dx=\frac {\frac {4 \, \sin \left (d x + c\right )^{2} - 3}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 2 \, \log \left (\sin \left (d x + c\right )^{2} - 1\right )}{4 \, d} \]

[In]

integrate(tan(d*x+c)^5,x, algorithm="maxima")

[Out]

1/4*((4*sin(d*x + c)^2 - 3)/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 2*log(sin(d*x + c)^2 - 1))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 462 vs. \(2 (39) = 78\).

Time = 1.37 (sec) , antiderivative size = 462, normalized size of antiderivative = 10.74 \[ \int \tan ^5(c+d x) \, dx=-\frac {2 \, \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right )^{4} \tan \left (c\right )^{4} + 3 \, \tan \left (d x\right )^{4} \tan \left (c\right )^{4} - 8 \, \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right )^{3} \tan \left (c\right )^{3} + 2 \, \tan \left (d x\right )^{4} \tan \left (c\right )^{2} - 8 \, \tan \left (d x\right )^{3} \tan \left (c\right )^{3} + 2 \, \tan \left (d x\right )^{2} \tan \left (c\right )^{4} + 12 \, \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - \tan \left (d x\right )^{4} - 8 \, \tan \left (d x\right )^{3} \tan \left (c\right ) + 4 \, \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 8 \, \tan \left (d x\right ) \tan \left (c\right )^{3} - \tan \left (c\right )^{4} - 8 \, \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right ) \tan \left (c\right ) + 2 \, \tan \left (d x\right )^{2} - 8 \, \tan \left (d x\right ) \tan \left (c\right ) + 2 \, \tan \left (c\right )^{2} + 2 \, \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) + 3}{4 \, {\left (d \tan \left (d x\right )^{4} \tan \left (c\right )^{4} - 4 \, d \tan \left (d x\right )^{3} \tan \left (c\right )^{3} + 6 \, d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 4 \, d \tan \left (d x\right ) \tan \left (c\right ) + d\right )}} \]

[In]

integrate(tan(d*x+c)^5,x, algorithm="giac")

[Out]

-1/4*(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)^4*tan(c)^4 + 3*tan(d*x)^4*tan(c)^4 - 8*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 + 2*tan(d*x)^4*tan(c)^2 - 8*tan(d*x)^3*tan(c)^3
+ 2*tan(d*x)^2*tan(c)^4 + 12*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)^2*tan(c)^2 - tan(d*x)^4 - 8*tan(d*x)^3*tan(c) + 4*tan(d*x)^2*tan(c)^2 - 8*tan(d
*x)*tan(c)^3 - tan(c)^4 - 8*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)*tan(c) + 2*tan(d*x)^2 - 8*tan(d*x)*tan(c) + 2*tan(c)^2 + 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)) + 3)/(d*tan(d*x)^4*tan(c)^4
- 4*d*tan(d*x)^3*tan(c)^3 + 6*d*tan(d*x)^2*tan(c)^2 - 4*d*tan(d*x)*tan(c) + d)

Mupad [B] (verification not implemented)

Time = 2.74 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.88 \[ \int \tan ^5(c+d x) \, dx=\frac {\frac {\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}{2}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4}{4}}{d} \]

[In]

int(tan(c + d*x)^5,x)

[Out]

(log(tan(c + d*x)^2 + 1)/2 - tan(c + d*x)^2/2 + tan(c + d*x)^4/4)/d