Integrand size = 21, antiderivative size = 108 \[ \int \frac {\tan ^5(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {\log (\cos (c+d x))}{a d}-\frac {\left (a^2-b^2\right )^2 \log (a+b \sec (c+d x))}{a b^4 d}+\frac {\left (a^2-2 b^2\right ) \sec (c+d x)}{b^3 d}-\frac {a \sec ^2(c+d x)}{2 b^2 d}+\frac {\sec ^3(c+d x)}{3 b d} \]
-ln(cos(d*x+c))/a/d-(a^2-b^2)^2*ln(a+b*sec(d*x+c))/a/b^4/d+(a^2-2*b^2)*sec (d*x+c)/b^3/d-1/2*a*sec(d*x+c)^2/b^2/d+1/3*sec(d*x+c)^3/b/d
Time = 0.36 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.91 \[ \int \frac {\tan ^5(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {-\frac {b^4 \log (\cos (c+d x))}{a}-\frac {\left (a^2-b^2\right )^2 \log (a+b \sec (c+d x))}{a}+b \left (a^2-2 b^2\right ) \sec (c+d x)-\frac {1}{2} a b^2 \sec ^2(c+d x)+\frac {1}{3} b^3 \sec ^3(c+d x)}{b^4 d} \]
(-((b^4*Log[Cos[c + d*x]])/a) - ((a^2 - b^2)^2*Log[a + b*Sec[c + d*x]])/a + b*(a^2 - 2*b^2)*Sec[c + d*x] - (a*b^2*Sec[c + d*x]^2)/2 + (b^3*Sec[c + d *x]^3)/3)/(b^4*d)
Time = 0.31 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 25, 4373, 522, 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.
| \(\displaystyle \int \frac {\tan ^5(c+d x)}{a+b \sec (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\cot \left (c+d x+\frac {\pi }{2}\right )^5}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^5}{a+b \csc \left (\frac {1}{2} (2 c+\pi )+d x\right )}dx\) |
| \(\Big \downarrow \) 4373 |
| \(\displaystyle \frac {\int \frac {\cos (c+d x) \left (b^2-b^2 \sec ^2(c+d x)\right )^2}{b (a+b \sec (c+d x))}d(b \sec (c+d x))}{b^4 d}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \frac {\int \left (\frac {\cos (c+d x) b^3}{a}+\sec ^2(c+d x) b^2-a \sec (c+d x) b+a^2 \left (1-\frac {2 b^2}{a^2}\right )-\frac {\left (a^2-b^2\right )^2}{a (a+b \sec (c+d x))}\right )d(b \sec (c+d x))}{b^4 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \left (a^2-2 b^2\right ) \sec (c+d x)-\frac {\left (a^2-b^2\right )^2 \log (a+b \sec (c+d x))}{a}+\frac {b^4 \log (b \sec (c+d x))}{a}-\frac {1}{2} a b^2 \sec ^2(c+d x)+\frac {1}{3} b^3 \sec ^3(c+d x)}{b^4 d}\) |
((b^4*Log[b*Sec[c + d*x]])/a - ((a^2 - b^2)^2*Log[a + b*Sec[c + d*x]])/a + b*(a^2 - 2*b^2)*Sec[c + d*x] - (a*b^2*Sec[c + d*x]^2)/2 + (b^3*Sec[c + d* x]^3)/3)/(b^4*d)
3.3.88.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _), x_Symbol] :> Simp[-(-1)^((m - 1)/2)/(d*b^(m - 1)) Subst[Int[(b^2 - x^ 2)^((m - 1)/2)*((a + x)^n/x), x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]
Time = 1.24 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.06
| method | result | size |
| derivativedivides | \(\frac {\frac {\left (-a^{4}+2 a^{2} b^{2}-b^{4}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{b^{4} a}-\frac {a}{2 b^{2} \cos \left (d x +c \right )^{2}}-\frac {-a^{2}+2 b^{2}}{b^{3} \cos \left (d x +c \right )}+\frac {\left (a^{2}-2 b^{2}\right ) a \ln \left (\cos \left (d x +c \right )\right )}{b^{4}}+\frac {1}{3 b \cos \left (d x +c \right )^{3}}}{d}\) | \(115\) |
| default | \(\frac {\frac {\left (-a^{4}+2 a^{2} b^{2}-b^{4}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{b^{4} a}-\frac {a}{2 b^{2} \cos \left (d x +c \right )^{2}}-\frac {-a^{2}+2 b^{2}}{b^{3} \cos \left (d x +c \right )}+\frac {\left (a^{2}-2 b^{2}\right ) a \ln \left (\cos \left (d x +c \right )\right )}{b^{4}}+\frac {1}{3 b \cos \left (d x +c \right )^{3}}}{d}\) | \(115\) |
| risch | \(\frac {i x}{a}+\frac {2 i c}{a d}+\frac {2 a^{2} {\mathrm e}^{5 i \left (d x +c \right )}-4 b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-2 a b \,{\mathrm e}^{4 i \left (d x +c \right )}+4 a^{2} {\mathrm e}^{3 i \left (d x +c \right )}-\frac {16 b^{2} {\mathrm e}^{3 i \left (d x +c \right )}}{3}-2 a b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 a^{2} {\mathrm e}^{i \left (d x +c \right )}-4 b^{2} {\mathrm e}^{i \left (d x +c \right )}}{d \,b^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}-\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{b^{4} d}+\frac {2 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{b^{2} d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a d}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{4} d}-\frac {2 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{b^{2} d}\) | \(303\) |
1/d*((-a^4+2*a^2*b^2-b^4)/b^4/a*ln(b+a*cos(d*x+c))-1/2/b^2*a/cos(d*x+c)^2- (-a^2+2*b^2)/b^3/cos(d*x+c)+(a^2-2*b^2)/b^4*a*ln(cos(d*x+c))+1/3/b/cos(d*x +c)^3)
Time = 0.32 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.19 \[ \int \frac {\tan ^5(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {3 \, a^{2} b^{2} \cos \left (d x + c\right ) + 6 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{3} \log \left (a \cos \left (d x + c\right ) + b\right ) - 6 \, {\left (a^{4} - 2 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} \log \left (-\cos \left (d x + c\right )\right ) - 2 \, a b^{3} - 6 \, {\left (a^{3} b - 2 \, a b^{3}\right )} \cos \left (d x + c\right )^{2}}{6 \, a b^{4} d \cos \left (d x + c\right )^{3}} \]
-1/6*(3*a^2*b^2*cos(d*x + c) + 6*(a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^3*lo g(a*cos(d*x + c) + b) - 6*(a^4 - 2*a^2*b^2)*cos(d*x + c)^3*log(-cos(d*x + c)) - 2*a*b^3 - 6*(a^3*b - 2*a*b^3)*cos(d*x + c)^2)/(a*b^4*d*cos(d*x + c)^ 3)
\[ \int \frac {\tan ^5(c+d x)}{a+b \sec (c+d x)} \, dx=\int \frac {\tan ^{5}{\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \]
Time = 0.22 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.02 \[ \int \frac {\tan ^5(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\frac {6 \, {\left (a^{3} - 2 \, a b^{2}\right )} \log \left (\cos \left (d x + c\right )\right )}{b^{4}} - \frac {6 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a b^{4}} - \frac {3 \, a b \cos \left (d x + c\right ) - 6 \, {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, b^{2}}{b^{3} \cos \left (d x + c\right )^{3}}}{6 \, d} \]
1/6*(6*(a^3 - 2*a*b^2)*log(cos(d*x + c))/b^4 - 6*(a^4 - 2*a^2*b^2 + b^4)*l og(a*cos(d*x + c) + b)/(a*b^4) - (3*a*b*cos(d*x + c) - 6*(a^2 - 2*b^2)*cos (d*x + c)^2 - 2*b^2)/(b^3*cos(d*x + c)^3))/d
Leaf count of result is larger than twice the leaf count of optimal. 560 vs. \(2 (104) = 208\).
Time = 1.53 (sec) , antiderivative size = 560, normalized size of antiderivative = 5.19 \[ \int \frac {\tan ^5(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {\frac {3 \, {\left (a^{3} - 2 \, a b^{2}\right )} \log \left ({\left | a + b - \frac {2 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} \right |}\right )}{b^{4}} - \frac {6 \, {\left (a^{3} - 2 \, a b^{2}\right )} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{b^{4}} - \frac {3 \, {\left (a^{4} - 2 \, a^{2} b^{2} + 2 \, b^{4}\right )} \log \left (\frac {{\left | 2 \, b + \frac {2 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {2 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - 2 \, {\left | a \right |} \right |}}{{\left | 2 \, b + \frac {2 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {2 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + 2 \, {\left | a \right |} \right |}}\right )}{b^{4} {\left | a \right |}} + \frac {11 \, a^{3} - 12 \, a^{2} b - 22 \, a b^{2} + 20 \, b^{3} + \frac {33 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {24 \, a^{2} b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {78 \, a b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {48 \, b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {33 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {12 \, a^{2} b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {78 \, a b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {12 \, b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {11 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {22 \, a b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{b^{4} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{3}}}{6 \, d} \]
-1/6*(3*(a^3 - 2*a*b^2)*log(abs(a + b - 2*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2))/b^4 - 6*(a^3 - 2*a*b^2)*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1))/b^4 - 3*(a^4 - 2*a^2*b^2 + 2*b^4)*log(abs(2 *b + 2*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 2*b*(cos(d*x + c) - 1)/(c os(d*x + c) + 1) - 2*abs(a))/abs(2*b + 2*a*(cos(d*x + c) - 1)/(cos(d*x + c ) + 1) - 2*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 2*abs(a)))/(b^4*abs(a )) + (11*a^3 - 12*a^2*b - 22*a*b^2 + 20*b^3 + 33*a^3*(cos(d*x + c) - 1)/(c os(d*x + c) + 1) - 24*a^2*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 78*a*b ^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 48*b^3*(cos(d*x + c) - 1)/(cos( d*x + c) + 1) + 33*a^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 12*a^2* b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 78*a*b^2*(cos(d*x + c) - 1)^ 2/(cos(d*x + c) + 1)^2 + 12*b^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 11*a^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 22*a*b^2*(cos(d*x + c ) - 1)^3/(cos(d*x + c) + 1)^3)/(b^4*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)^3))/d
Time = 15.08 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.10 \[ \int \frac {\tan ^5(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d}-\frac {\frac {2\,\left (3\,a^2-5\,b^2\right )}{3\,b^3}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (2\,a^2+a\,b-4\,b^2\right )}{b^3}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (a^2+a\,b-b^2\right )}{b^3}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}+\frac {a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )\,\left (a^2-2\,b^2\right )}{b^4\,d}-\frac {\ln \left (a+b-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )\,{\left (a^2-b^2\right )}^2}{a\,b^4\,d} \]
log(tan(c/2 + (d*x)/2)^2 + 1)/(a*d) - ((2*(3*a^2 - 5*b^2))/(3*b^3) - (2*ta n(c/2 + (d*x)/2)^2*(a*b + 2*a^2 - 4*b^2))/b^3 + (2*tan(c/2 + (d*x)/2)^4*(a *b + a^2 - b^2))/b^3)/(d*(3*tan(c/2 + (d*x)/2)^2 - 3*tan(c/2 + (d*x)/2)^4 + tan(c/2 + (d*x)/2)^6 - 1)) + (a*log(tan(c/2 + (d*x)/2)^2 - 1)*(a^2 - 2*b ^2))/(b^4*d) - (log(a + b - a*tan(c/2 + (d*x)/2)^2 + b*tan(c/2 + (d*x)/2)^ 2)*(a^2 - b^2)^2)/(a*b^4*d)