Integrand size = 21, antiderivative size = 66 \[ \int \frac {\tan ^5(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\log (\cos (c+d x))}{a d}-\frac {\sec (c+d x)}{a d}-\frac {\sec ^2(c+d x)}{2 a d}+\frac {\sec ^3(c+d x)}{3 a d} \] Output:
-ln(cos(d*x+c))/a/d-sec(d*x+c)/a/d-1/2*sec(d*x+c)^2/a/d+1/3*sec(d*x+c)^3/a /d
Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.98 \[ \int \frac {\tan ^5(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {(2+6 \cos (2 (c+d x))+3 \cos (3 (c+d x)) \log (\cos (c+d x))+\cos (c+d x) (6+9 \log (\cos (c+d x)))) \sec ^3(c+d x)}{12 a d} \] Input:
Integrate[Tan[c + d*x]^5/(a + a*Sec[c + d*x]),x]
Output:
-1/12*((2 + 6*Cos[2*(c + d*x)] + 3*Cos[3*(c + d*x)]*Log[Cos[c + d*x]] + Co s[c + d*x]*(6 + 9*Log[Cos[c + d*x]]))*Sec[c + d*x]^3)/(a*d)
Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 25, 4367, 27, 84, 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.
\(\displaystyle \int \frac {\tan ^5(c+d x)}{a \sec (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\cot \left (c+d x+\frac {\pi }{2}\right )^5}{a \csc \left (c+d x+\frac {\pi }{2}\right )+a}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^5}{\csc \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a}dx\) |
\(\Big \downarrow \) 4367 |
\(\displaystyle -\frac {\int a^3 (1-\cos (c+d x))^2 (\cos (c+d x)+1) \sec ^4(c+d x)d\cos (c+d x)}{a^4 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int (1-\cos (c+d x))^2 (\cos (c+d x)+1) \sec ^4(c+d x)d\cos (c+d x)}{a d}\) |
\(\Big \downarrow \) 84 |
\(\displaystyle -\frac {\int \left (\sec ^4(c+d x)-\sec ^3(c+d x)-\sec ^2(c+d x)+\sec (c+d x)\right )d\cos (c+d x)}{a d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\frac {1}{3} \sec ^3(c+d x)+\frac {1}{2} \sec ^2(c+d x)+\sec (c+d x)+\log (\cos (c+d x))}{a d}\) |
Input:
Int[Tan[c + d*x]^5/(a + a*Sec[c + d*x]),x]
Output:
-((Log[Cos[c + d*x]] + Sec[c + d*x] + Sec[c + d*x]^2/2 - Sec[c + d*x]^3/3) /(a*d))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] && !(ILtQ[n + p + 2, 0 ] && GtQ[n + 2*p, 0])
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _.), x_Symbol] :> Simp[1/(a^(m - n - 1)*b^n*d) Subst[Int[(a - b*x)^((m - 1)/2)*((a + b*x)^((m - 1)/2 + n)/x^(m + n)), x], x, Sin[c + d*x]], x] /; Fr eeQ[{a, b, c, d}, x] && IntegerQ[(m - 1)/2] && EqQ[a^2 - b^2, 0] && Integer Q[n]
Time = 0.22 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(\frac {-\ln \left (\cos \left (d x +c \right )\right )-\frac {1}{2 \cos \left (d x +c \right )^{2}}-\frac {1}{\cos \left (d x +c \right )}+\frac {1}{3 \cos \left (d x +c \right )^{3}}}{d a}\) | \(48\) |
default | \(\frac {-\ln \left (\cos \left (d x +c \right )\right )-\frac {1}{2 \cos \left (d x +c \right )^{2}}-\frac {1}{\cos \left (d x +c \right )}+\frac {1}{3 \cos \left (d x +c \right )^{3}}}{d a}\) | \(48\) |
risch | \(\frac {i x}{a}+\frac {2 i c}{d a}-\frac {2 \left (3 \,{\mathrm e}^{5 i \left (d x +c \right )}+3 \,{\mathrm e}^{4 i \left (d x +c \right )}+2 \,{\mathrm e}^{3 i \left (d x +c \right )}+3 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d a}\) | \(116\) |
Input:
int(tan(d*x+c)^5/(a+a*sec(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d/a*(-ln(cos(d*x+c))-1/2/cos(d*x+c)^2-1/cos(d*x+c)+1/3/cos(d*x+c)^3)
Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.83 \[ \int \frac {\tan ^5(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {6 \, \cos \left (d x + c\right )^{3} \log \left (-\cos \left (d x + c\right )\right ) + 6 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) - 2}{6 \, a d \cos \left (d x + c\right )^{3}} \] Input:
integrate(tan(d*x+c)^5/(a+a*sec(d*x+c)),x, algorithm="fricas")
Output:
-1/6*(6*cos(d*x + c)^3*log(-cos(d*x + c)) + 6*cos(d*x + c)^2 + 3*cos(d*x + c) - 2)/(a*d*cos(d*x + c)^3)
\[ \int \frac {\tan ^5(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\tan ^{5}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \] Input:
integrate(tan(d*x+c)**5/(a+a*sec(d*x+c)),x)
Output:
Integral(tan(c + d*x)**5/(sec(c + d*x) + 1), x)/a
Time = 0.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.76 \[ \int \frac {\tan ^5(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {6 \, \log \left (\cos \left (d x + c\right )\right )}{a} + \frac {6 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) - 2}{a \cos \left (d x + c\right )^{3}}}{6 \, d} \] Input:
integrate(tan(d*x+c)^5/(a+a*sec(d*x+c)),x, algorithm="maxima")
Output:
-1/6*(6*log(cos(d*x + c))/a + (6*cos(d*x + c)^2 + 3*cos(d*x + c) - 2)/(a*c os(d*x + c)^3))/d
Time = 0.18 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.73 \[ \int \frac {\tan ^5(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {6 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) - 2}{\cos \left (d x + c\right )^{3}} + 6 \, \log \left ({\left | \cos \left (d x + c\right ) \right |}\right )}{6 \, a d} \] Input:
integrate(tan(d*x+c)^5/(a+a*sec(d*x+c)),x, algorithm="giac")
Output:
-1/6*((6*cos(d*x + c)^2 + 3*cos(d*x + c) - 2)/cos(d*x + c)^3 + 6*log(abs(c os(d*x + c))))/(a*d)
Time = 13.36 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.50 \[ \int \frac {\tan ^5(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {2\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{a\,d}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {4}{3}}{d\,\left (-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )} \] Input:
int(tan(c + d*x)^5/(a + a/cos(c + d*x)),x)
Output:
(2*atanh(tan(c/2 + (d*x)/2)^2))/(a*d) + (2*tan(c/2 + (d*x)/2)^2 + 2*tan(c/ 2 + (d*x)/2)^4 - 4/3)/(d*(a - 3*a*tan(c/2 + (d*x)/2)^2 + 3*a*tan(c/2 + (d* x)/2)^4 - a*tan(c/2 + (d*x)/2)^6))
Time = 0.17 (sec) , antiderivative size = 212, normalized size of antiderivative = 3.21 \[ \int \frac {\tan ^5(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {6 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{2}-6 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )-6 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2}+6 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-6 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2}+6 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+7 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-4 \cos \left (d x +c \right )-6 \sin \left (d x +c \right )^{2}+4}{6 \cos \left (d x +c \right ) a d \left (\sin \left (d x +c \right )^{2}-1\right )} \] Input:
int(tan(d*x+c)^5/(a+a*sec(d*x+c)),x)
Output:
(6*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2 - 6*cos(c + d *x)*log(tan((c + d*x)/2)**2 + 1) - 6*cos(c + d*x)*log(tan((c + d*x)/2) - 1 )*sin(c + d*x)**2 + 6*cos(c + d*x)*log(tan((c + d*x)/2) - 1) - 6*cos(c + d *x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2 + 6*cos(c + d*x)*log(tan((c + d*x)/2) + 1) + 7*cos(c + d*x)*sin(c + d*x)**2 - 4*cos(c + d*x) - 6*sin(c + d*x)**2 + 4)/(6*cos(c + d*x)*a*d*(sin(c + d*x)**2 - 1))