Integrand size = 23, antiderivative size = 108 \[ \int \frac {\cot ^5(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {(2 a+3 b) \csc ^2(e+f x)}{2 (a+b)^2 f}-\frac {\csc ^4(e+f x)}{4 (a+b) f}+\frac {b^3 \log \left (b+a \cos ^2(e+f x)\right )}{2 a (a+b)^3 f}+\frac {\left (a^2+3 a b+3 b^2\right ) \log (\sin (e+f x))}{(a+b)^3 f} \]
1/2*(2*a+3*b)*csc(f*x+e)^2/(a+b)^2/f-1/4*csc(f*x+e)^4/(a+b)/f+1/2*b^3*ln(b +a*cos(f*x+e)^2)/a/(a+b)^3/f+(a^2+3*a*b+3*b^2)*ln(sin(f*x+e))/(a+b)^3/f
Time = 0.72 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.28 \[ \int \frac {\cot ^5(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {(a+2 b+a \cos (2 e+2 f x)) \left (\frac {2 (2 a+3 b) \csc ^2(e+f x)}{(a+b)^2}-\frac {\csc ^4(e+f x)}{a+b}+\frac {4 \left (a^2+3 a b+3 b^2\right ) \log (\sin (e+f x))}{(a+b)^3}+\frac {2 b^3 \log \left (a+b-a \sin ^2(e+f x)\right )}{a (a+b)^3}\right ) \sec ^2(e+f x)}{8 f \left (a+b \sec ^2(e+f x)\right )} \]
((a + 2*b + a*Cos[2*e + 2*f*x])*((2*(2*a + 3*b)*Csc[e + f*x]^2)/(a + b)^2 - Csc[e + f*x]^4/(a + b) + (4*(a^2 + 3*a*b + 3*b^2)*Log[Sin[e + f*x]])/(a + b)^3 + (2*b^3*Log[a + b - a*Sin[e + f*x]^2])/(a*(a + b)^3))*Sec[e + f*x] ^2)/(8*f*(a + b*Sec[e + f*x]^2))
Time = 0.34 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 4626, 354, 99, 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.
| \(\displaystyle \int \frac {\cot ^5(e+f x)}{a+b \sec ^2(e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (e+f x)^5 \left (a+b \sec (e+f x)^2\right )}dx\) |
| \(\Big \downarrow \) 4626 |
| \(\displaystyle -\frac {\int \frac {\cos ^7(e+f x)}{\left (1-\cos ^2(e+f x)\right )^3 \left (a \cos ^2(e+f x)+b\right )}d\cos (e+f x)}{f}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle -\frac {\int \frac {\cos ^6(e+f x)}{\left (1-\cos ^2(e+f x)\right )^3 \left (a \cos ^2(e+f x)+b\right )}d\cos ^2(e+f x)}{2 f}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -\frac {\int \left (-\frac {b^3}{(a+b)^3 \left (a \cos ^2(e+f x)+b\right )}+\frac {-a^2-3 b a-3 b^2}{(a+b)^3 \left (\cos ^2(e+f x)-1\right )}+\frac {-2 a-3 b}{(a+b)^2 \left (\cos ^2(e+f x)-1\right )^2}-\frac {1}{(a+b) \left (\cos ^2(e+f x)-1\right )^3}\right )d\cos ^2(e+f x)}{2 f}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {\left (a^2+3 a b+3 b^2\right ) \log \left (1-\cos ^2(e+f x)\right )}{(a+b)^3}-\frac {b^3 \log \left (a \cos ^2(e+f x)+b\right )}{a (a+b)^3}-\frac {2 a+3 b}{(a+b)^2 \left (1-\cos ^2(e+f x)\right )}+\frac {1}{2 (a+b) \left (1-\cos ^2(e+f x)\right )^2}}{2 f}\) |
-1/2*(1/(2*(a + b)*(1 - Cos[e + f*x]^2)^2) - (2*a + 3*b)/((a + b)^2*(1 - C os[e + f*x]^2)) - ((a^2 + 3*a*b + 3*b^2)*Log[1 - Cos[e + f*x]^2])/(a + b)^ 3 - (b^3*Log[b + a*Cos[e + f*x]^2])/(a*(a + b)^3))/f
3.4.42.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_ )]^(m_.), x_Symbol] :> Module[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-(f *ff^(m + n*p - 1))^(-1) Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*(ff* x)^n)^p/x^(m + n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, n} , x] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p]
Time = 2.61 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.67
| method | result | size |
| derivativedivides | \(\frac {\frac {b^{3} \ln \left (b +a \cos \left (f x +e \right )^{2}\right )}{2 \left (a +b \right )^{3} a}-\frac {1}{2 \left (8 a +8 b \right ) \left (-1+\cos \left (f x +e \right )\right )^{2}}-\frac {7 a +11 b}{16 \left (a +b \right )^{2} \left (-1+\cos \left (f x +e \right )\right )}+\frac {\left (a^{2}+3 a b +3 b^{2}\right ) \ln \left (-1+\cos \left (f x +e \right )\right )}{2 \left (a +b \right )^{3}}-\frac {1}{2 \left (8 a +8 b \right ) \left (1+\cos \left (f x +e \right )\right )^{2}}-\frac {-7 a -11 b}{16 \left (a +b \right )^{2} \left (1+\cos \left (f x +e \right )\right )}+\frac {\left (a^{2}+3 a b +3 b^{2}\right ) \ln \left (1+\cos \left (f x +e \right )\right )}{2 \left (a +b \right )^{3}}}{f}\) | \(180\) |
| default | \(\frac {\frac {b^{3} \ln \left (b +a \cos \left (f x +e \right )^{2}\right )}{2 \left (a +b \right )^{3} a}-\frac {1}{2 \left (8 a +8 b \right ) \left (-1+\cos \left (f x +e \right )\right )^{2}}-\frac {7 a +11 b}{16 \left (a +b \right )^{2} \left (-1+\cos \left (f x +e \right )\right )}+\frac {\left (a^{2}+3 a b +3 b^{2}\right ) \ln \left (-1+\cos \left (f x +e \right )\right )}{2 \left (a +b \right )^{3}}-\frac {1}{2 \left (8 a +8 b \right ) \left (1+\cos \left (f x +e \right )\right )^{2}}-\frac {-7 a -11 b}{16 \left (a +b \right )^{2} \left (1+\cos \left (f x +e \right )\right )}+\frac {\left (a^{2}+3 a b +3 b^{2}\right ) \ln \left (1+\cos \left (f x +e \right )\right )}{2 \left (a +b \right )^{3}}}{f}\) | \(180\) |
| risch | \(\frac {i x}{a}-\frac {2 i a^{2} x}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}-\frac {2 i a^{2} e}{f \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {6 i a b x}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}-\frac {6 i a b e}{f \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {6 i b^{2} x}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}-\frac {6 i b^{2} e}{f \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 i b^{3} x}{a \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 i b^{3} e}{a f \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {2 \left (2 a \,{\mathrm e}^{6 i \left (f x +e \right )}+3 b \,{\mathrm e}^{6 i \left (f x +e \right )}-2 a \,{\mathrm e}^{4 i \left (f x +e \right )}-4 b \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+3 b \,{\mathrm e}^{2 i \left (f x +e \right )}\right )}{f \left (a +b \right )^{2} \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right ) a^{2}}{f \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right ) a b}{f \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right ) b^{2}}{f \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {b^{3} \ln \left ({\mathrm e}^{4 i \left (f x +e \right )}+\frac {2 \left (a +2 b \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{a}+1\right )}{2 a f \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}\) | \(529\) |
1/f*(1/2*b^3/(a+b)^3/a*ln(b+a*cos(f*x+e)^2)-1/2/(8*a+8*b)/(-1+cos(f*x+e))^ 2-1/16*(7*a+11*b)/(a+b)^2/(-1+cos(f*x+e))+1/2*(a^2+3*a*b+3*b^2)/(a+b)^3*ln (-1+cos(f*x+e))-1/2/(8*a+8*b)/(1+cos(f*x+e))^2-1/16*(-7*a-11*b)/(a+b)^2/(1 +cos(f*x+e))+1/2*(a^2+3*a*b+3*b^2)/(a+b)^3*ln(1+cos(f*x+e)))
Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (102) = 204\).
Time = 0.46 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.45 \[ \int \frac {\cot ^5(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {3 \, a^{3} + 8 \, a^{2} b + 5 \, a b^{2} - 2 \, {\left (2 \, a^{3} + 5 \, a^{2} b + 3 \, a b^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (b^{3} \cos \left (f x + e\right )^{4} - 2 \, b^{3} \cos \left (f x + e\right )^{2} + b^{3}\right )} \log \left (a \cos \left (f x + e\right )^{2} + b\right ) + 4 \, {\left ({\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2}\right )} \cos \left (f x + e\right )^{4} + a^{3} + 3 \, a^{2} b + 3 \, a b^{2} - 2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (\frac {1}{2} \, \sin \left (f x + e\right )\right )}{4 \, {\left ({\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} f \cos \left (f x + e\right )^{4} - 2 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} f\right )}} \]
1/4*(3*a^3 + 8*a^2*b + 5*a*b^2 - 2*(2*a^3 + 5*a^2*b + 3*a*b^2)*cos(f*x + e )^2 + 2*(b^3*cos(f*x + e)^4 - 2*b^3*cos(f*x + e)^2 + b^3)*log(a*cos(f*x + e)^2 + b) + 4*((a^3 + 3*a^2*b + 3*a*b^2)*cos(f*x + e)^4 + a^3 + 3*a^2*b + 3*a*b^2 - 2*(a^3 + 3*a^2*b + 3*a*b^2)*cos(f*x + e)^2)*log(1/2*sin(f*x + e) ))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^4 - 2*(a^4 + 3*a^3* b + 3*a^2*b^2 + a*b^3)*f*cos(f*x + e)^2 + (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b ^3)*f)
\[ \int \frac {\cot ^5(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\int \frac {\cot ^{5}{\left (e + f x \right )}}{a + b \sec ^{2}{\left (e + f x \right )}}\, dx \]
Time = 0.20 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.34 \[ \int \frac {\cot ^5(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {\frac {2 \, b^{3} \log \left (a \sin \left (f x + e\right )^{2} - a - b\right )}{a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}} + \frac {2 \, {\left (a^{2} + 3 \, a b + 3 \, b^{2}\right )} \log \left (\sin \left (f x + e\right )^{2}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {2 \, {\left (2 \, a + 3 \, b\right )} \sin \left (f x + e\right )^{2} - a - b}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \sin \left (f x + e\right )^{4}}}{4 \, f} \]
1/4*(2*b^3*log(a*sin(f*x + e)^2 - a - b)/(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^ 3) + 2*(a^2 + 3*a*b + 3*b^2)*log(sin(f*x + e)^2)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + (2*(2*a + 3*b)*sin(f*x + e)^2 - a - b)/((a^2 + 2*a*b + b^2)*sin(f *x + e)^4))/f
Leaf count of result is larger than twice the leaf count of optimal. 513 vs. \(2 (102) = 204\).
Time = 0.40 (sec) , antiderivative size = 513, normalized size of antiderivative = 4.75 \[ \int \frac {\cot ^5(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {\frac {32 \, b^{3} \log \left (a + b + \frac {2 \, a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {2 \, b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {a {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}{a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}} + \frac {32 \, {\left (a^{2} + 3 \, a b + 3 \, b^{2}\right )} \log \left (\frac {{\left | -\cos \left (f x + e\right ) + 1 \right |}}{{\left | \cos \left (f x + e\right ) + 1 \right |}}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {\frac {12 \, a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {20 \, b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {a {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}}{a^{2} + 2 \, a b + b^{2}} - \frac {{\left (a^{2} + 2 \, a b + b^{2} + \frac {12 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {32 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {20 \, b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {48 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {144 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {144 \, b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{2}}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}} - \frac {64 \, \log \left ({\left | -\frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} + 1 \right |}\right )}{a}}{64 \, f} \]
1/64*(32*b^3*log(a + b + 2*a*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) - 2*b*( cos(f*x + e) - 1)/(cos(f*x + e) + 1) + a*(cos(f*x + e) - 1)^2/(cos(f*x + e ) + 1)^2 + b*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2)/(a^4 + 3*a^3*b + 3 *a^2*b^2 + a*b^3) + 32*(a^2 + 3*a*b + 3*b^2)*log(abs(-cos(f*x + e) + 1)/ab s(cos(f*x + e) + 1))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) - (12*a*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 20*b*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + a* (cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 + b*(cos(f*x + e) - 1)^2/(cos(f* x + e) + 1)^2)/(a^2 + 2*a*b + b^2) - (a^2 + 2*a*b + b^2 + 12*a^2*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 32*a*b*(cos(f*x + e) - 1)/(cos(f*x + e) + 1 ) + 20*b^2*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 48*a^2*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 + 144*a*b*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1 )^2 + 144*b^2*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2)*(cos(f*x + e) + 1 )^2/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*(cos(f*x + e) - 1)^2) - 64*log(abs(-( cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 1))/a)/f
Time = 20.66 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.48 \[ \int \frac {\cot ^5(e+f x)}{a+b \sec ^2(e+f x)} \, dx=\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )\right )\,\left (a^2+3\,a\,b+3\,b^2\right )}{f\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}-\frac {\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )}{2\,a\,f}-\frac {\ln \left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a+b\right )\,\left (\frac {b}{2\,{\left (a+b\right )}^2}+\frac {1}{2\,\left (a+b\right )}+\frac {b^2}{2\,{\left (a+b\right )}^3}-\frac {1}{2\,a}\right )}{f}-\frac {{\mathrm {cot}\left (e+f\,x\right )}^4\,\left (\frac {1}{4\,\left (a+b\right )}-\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (a+2\,b\right )}{2\,{\left (a+b\right )}^2}\right )}{f} \]
(log(tan(e + f*x))*(3*a*b + a^2 + 3*b^2))/(f*(3*a*b^2 + 3*a^2*b + a^3 + b^ 3)) - log(tan(e + f*x)^2 + 1)/(2*a*f) - (log(a + b + b*tan(e + f*x)^2)*(b/ (2*(a + b)^2) + 1/(2*(a + b)) + b^2/(2*(a + b)^3) - 1/(2*a)))/f - (cot(e + f*x)^4*(1/(4*(a + b)) - (tan(e + f*x)^2*(a + 2*b))/(2*(a + b)^2)))/f