Integrand size = 21, antiderivative size = 169 \[ \int \frac {\csc ^6(c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {\left (a^2+b^2\right )^2 \cot (c+d x)}{a^5 d}+\frac {b \left (2 a^2+b^2\right ) \cot ^2(c+d x)}{2 a^4 d}-\frac {\left (2 a^2+b^2\right ) \cot ^3(c+d x)}{3 a^3 d}+\frac {b \cot ^4(c+d x)}{4 a^2 d}-\frac {\cot ^5(c+d x)}{5 a d}-\frac {b \left (a^2+b^2\right )^2 \log (\tan (c+d x))}{a^6 d}+\frac {b \left (a^2+b^2\right )^2 \log (a+b \tan (c+d x))}{a^6 d} \]
-(a^2+b^2)^2*cot(d*x+c)/a^5/d+1/2*b*(2*a^2+b^2)*cot(d*x+c)^2/a^4/d-1/3*(2* a^2+b^2)*cot(d*x+c)^3/a^3/d+1/4*b*cot(d*x+c)^4/a^2/d-1/5*cot(d*x+c)^5/a/d- b*(a^2+b^2)^2*ln(tan(d*x+c))/a^6/d+b*(a^2+b^2)^2*ln(a+b*tan(d*x+c))/a^6/d
Time = 6.17 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.89 \[ \int \frac {\csc ^6(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {-4 \cot (c+d x) \left (8 a^5+25 a^3 b^2+15 a b^4+a^3 \left (4 a^2+5 b^2\right ) \csc ^2(c+d x)+3 a^5 \csc ^4(c+d x)\right )+15 b \left (2 a^2 \left (a^2+b^2\right ) \csc ^2(c+d x)+a^4 \csc ^4(c+d x)-4 \left (a^2+b^2\right )^2 (\log (\sin (c+d x))-\log (a \cos (c+d x)+b \sin (c+d x)))\right )}{60 a^6 d} \]
(-4*Cot[c + d*x]*(8*a^5 + 25*a^3*b^2 + 15*a*b^4 + a^3*(4*a^2 + 5*b^2)*Csc[ c + d*x]^2 + 3*a^5*Csc[c + d*x]^4) + 15*b*(2*a^2*(a^2 + b^2)*Csc[c + d*x]^ 2 + a^4*Csc[c + d*x]^4 - 4*(a^2 + b^2)^2*(Log[Sin[c + d*x]] - Log[a*Cos[c + d*x] + b*Sin[c + d*x]])))/(60*a^6*d)
Time = 0.37 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3999, 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.
| \(\displaystyle \int \frac {\csc ^6(c+d x)}{a+b \tan (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (c+d x)^6 (a+b \tan (c+d x))}dx\) |
| \(\Big \downarrow \) 3999 |
| \(\displaystyle \frac {b \int \frac {\cot ^6(c+d x) \left (\tan ^2(c+d x) b^2+b^2\right )^2}{b^6 (a+b \tan (c+d x))}d(b \tan (c+d x))}{d}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \frac {b \int \left (\frac {\cot ^6(c+d x)}{a b^2}-\frac {\cot ^5(c+d x)}{a^2 b}+\frac {\left (b^4+2 a^2 b^2\right ) \cot ^4(c+d x)}{a^3 b^4}+\frac {\left (-2 a^2-b^2\right ) \cot ^3(c+d x)}{a^4 b}+\frac {\left (a^2+b^2\right )^2 \cot ^2(c+d x)}{a^5 b^2}-\frac {\left (a^2+b^2\right )^2 \cot (c+d x)}{a^6 b}+\frac {\left (a^2+b^2\right )^2}{a^6 (a+b \tan (c+d x))}\right )d(b \tan (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \left (\frac {\cot ^4(c+d x)}{4 a^2}-\frac {\left (a^2+b^2\right )^2 \log (b \tan (c+d x))}{a^6}+\frac {\left (a^2+b^2\right )^2 \log (a+b \tan (c+d x))}{a^6}-\frac {\left (a^2+b^2\right )^2 \cot (c+d x)}{a^5 b}+\frac {\left (2 a^2+b^2\right ) \cot ^2(c+d x)}{2 a^4}-\frac {\left (2 a^2+b^2\right ) \cot ^3(c+d x)}{3 a^3 b}-\frac {\cot ^5(c+d x)}{5 a b}\right )}{d}\) |
(b*(-(((a^2 + b^2)^2*Cot[c + d*x])/(a^5*b)) + ((2*a^2 + b^2)*Cot[c + d*x]^ 2)/(2*a^4) - ((2*a^2 + b^2)*Cot[c + d*x]^3)/(3*a^3*b) + Cot[c + d*x]^4/(4* a^2) - Cot[c + d*x]^5/(5*a*b) - ((a^2 + b^2)^2*Log[b*Tan[c + d*x]])/a^6 + ((a^2 + b^2)^2*Log[a + b*Tan[c + d*x]])/a^6))/d
3.1.60.3.1 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[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ ), x_Symbol] :> Simp[b/f Subst[Int[x^m*((a + x)^n/(b^2 + x^2)^(m/2 + 1)), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[m/2]
Time = 4.08 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.98
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{5 a \tan \left (d x +c \right )^{5}}-\frac {2 a^{2}+b^{2}}{3 a^{3} \tan \left (d x +c \right )^{3}}-\frac {a^{4}+2 a^{2} b^{2}+b^{4}}{a^{5} \tan \left (d x +c \right )}+\frac {b}{4 a^{2} \tan \left (d x +c \right )^{4}}+\frac {\left (2 a^{2}+b^{2}\right ) b}{2 a^{4} \tan \left (d x +c \right )^{2}}-\frac {\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b \ln \left (\tan \left (d x +c \right )\right )}{a^{6}}+\frac {\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{6}}}{d}\) | \(165\) |
| default | \(\frac {-\frac {1}{5 a \tan \left (d x +c \right )^{5}}-\frac {2 a^{2}+b^{2}}{3 a^{3} \tan \left (d x +c \right )^{3}}-\frac {a^{4}+2 a^{2} b^{2}+b^{4}}{a^{5} \tan \left (d x +c \right )}+\frac {b}{4 a^{2} \tan \left (d x +c \right )^{4}}+\frac {\left (2 a^{2}+b^{2}\right ) b}{2 a^{4} \tan \left (d x +c \right )^{2}}-\frac {\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b \ln \left (\tan \left (d x +c \right )\right )}{a^{6}}+\frac {\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{6}}}{d}\) | \(165\) |
| risch | \(-\frac {2 i \left (15 i a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}+15 i a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+15 a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+15 b^{4} {\mathrm e}^{8 i \left (d x +c \right )}-45 i a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-15 i a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}-90 a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-60 b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-75 i a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}+75 i a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}+80 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}+160 a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+90 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-15 i a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+45 i a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-40 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-110 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-60 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+8 a^{4}+25 a^{2} b^{2}+15 b^{4}\right )}{15 d \,a^{5} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{2} d}+\frac {2 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{4} d}+\frac {b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{6} d}-\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{2} d}-\frac {2 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{4} d}-\frac {b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{6} d}\) | \(506\) |
1/d*(-1/5/a/tan(d*x+c)^5-1/3*(2*a^2+b^2)/a^3/tan(d*x+c)^3-(a^4+2*a^2*b^2+b ^4)/a^5/tan(d*x+c)+1/4/a^2*b/tan(d*x+c)^4+1/2*(2*a^2+b^2)/a^4*b/tan(d*x+c) ^2-(a^4+2*a^2*b^2+b^4)/a^6*b*ln(tan(d*x+c))+(a^4+2*a^2*b^2+b^4)/a^6*b*ln(a +b*tan(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (161) = 322\).
Time = 0.29 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.28 \[ \int \frac {\csc ^6(c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {4 \, {\left (8 \, a^{5} + 25 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} - 20 \, {\left (4 \, a^{5} + 11 \, a^{3} b^{2} + 6 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} - 30 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) \sin \left (d x + c\right ) + 30 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{4} \, \cos \left (d x + c\right )^{2} + \frac {1}{4}\right ) \sin \left (d x + c\right ) + 60 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right ) - 15 \, {\left (3 \, a^{4} b + 2 \, a^{2} b^{3} - 2 \, {\left (a^{4} b + a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{60 \, {\left (a^{6} d \cos \left (d x + c\right )^{4} - 2 \, a^{6} d \cos \left (d x + c\right )^{2} + a^{6} d\right )} \sin \left (d x + c\right )} \]
-1/60*(4*(8*a^5 + 25*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^5 - 20*(4*a^5 + 11*a ^3*b^2 + 6*a*b^4)*cos(d*x + c)^3 - 30*(a^4*b + 2*a^2*b^3 + b^5 + (a^4*b + 2*a^2*b^3 + b^5)*cos(d*x + c)^4 - 2*(a^4*b + 2*a^2*b^3 + b^5)*cos(d*x + c) ^2)*log(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2 )*sin(d*x + c) + 30*(a^4*b + 2*a^2*b^3 + b^5 + (a^4*b + 2*a^2*b^3 + b^5)*c os(d*x + c)^4 - 2*(a^4*b + 2*a^2*b^3 + b^5)*cos(d*x + c)^2)*log(-1/4*cos(d *x + c)^2 + 1/4)*sin(d*x + c) + 60*(a^5 + 2*a^3*b^2 + a*b^4)*cos(d*x + c) - 15*(3*a^4*b + 2*a^2*b^3 - 2*(a^4*b + a^2*b^3)*cos(d*x + c)^2)*sin(d*x + c))/((a^6*d*cos(d*x + c)^4 - 2*a^6*d*cos(d*x + c)^2 + a^6*d)*sin(d*x + c))
\[ \int \frac {\csc ^6(c+d x)}{a+b \tan (c+d x)} \, dx=\int \frac {\csc ^{6}{\left (c + d x \right )}}{a + b \tan {\left (c + d x \right )}}\, dx \]
Time = 0.21 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.99 \[ \int \frac {\csc ^6(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {\frac {60 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6}} - \frac {60 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{6}} + \frac {15 \, a^{3} b \tan \left (d x + c\right ) - 60 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{4} - 12 \, a^{4} + 30 \, {\left (2 \, a^{3} b + a b^{3}\right )} \tan \left (d x + c\right )^{3} - 20 \, {\left (2 \, a^{4} + a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{a^{5} \tan \left (d x + c\right )^{5}}}{60 \, d} \]
1/60*(60*(a^4*b + 2*a^2*b^3 + b^5)*log(b*tan(d*x + c) + a)/a^6 - 60*(a^4*b + 2*a^2*b^3 + b^5)*log(tan(d*x + c))/a^6 + (15*a^3*b*tan(d*x + c) - 60*(a ^4 + 2*a^2*b^2 + b^4)*tan(d*x + c)^4 - 12*a^4 + 30*(2*a^3*b + a*b^3)*tan(d *x + c)^3 - 20*(2*a^4 + a^2*b^2)*tan(d*x + c)^2)/(a^5*tan(d*x + c)^5))/d
Time = 0.44 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.49 \[ \int \frac {\csc ^6(c+d x)}{a+b \tan (c+d x)} \, dx=-\frac {\frac {60 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{6}} - \frac {60 \, {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b} - \frac {137 \, a^{4} b \tan \left (d x + c\right )^{5} + 274 \, a^{2} b^{3} \tan \left (d x + c\right )^{5} + 137 \, b^{5} \tan \left (d x + c\right )^{5} - 60 \, a^{5} \tan \left (d x + c\right )^{4} - 120 \, a^{3} b^{2} \tan \left (d x + c\right )^{4} - 60 \, a b^{4} \tan \left (d x + c\right )^{4} + 60 \, a^{4} b \tan \left (d x + c\right )^{3} + 30 \, a^{2} b^{3} \tan \left (d x + c\right )^{3} - 40 \, a^{5} \tan \left (d x + c\right )^{2} - 20 \, a^{3} b^{2} \tan \left (d x + c\right )^{2} + 15 \, a^{4} b \tan \left (d x + c\right ) - 12 \, a^{5}}{a^{6} \tan \left (d x + c\right )^{5}}}{60 \, d} \]
-1/60*(60*(a^4*b + 2*a^2*b^3 + b^5)*log(abs(tan(d*x + c)))/a^6 - 60*(a^4*b ^2 + 2*a^2*b^4 + b^6)*log(abs(b*tan(d*x + c) + a))/(a^6*b) - (137*a^4*b*ta n(d*x + c)^5 + 274*a^2*b^3*tan(d*x + c)^5 + 137*b^5*tan(d*x + c)^5 - 60*a^ 5*tan(d*x + c)^4 - 120*a^3*b^2*tan(d*x + c)^4 - 60*a*b^4*tan(d*x + c)^4 + 60*a^4*b*tan(d*x + c)^3 + 30*a^2*b^3*tan(d*x + c)^3 - 40*a^5*tan(d*x + c)^ 2 - 20*a^3*b^2*tan(d*x + c)^2 + 15*a^4*b*tan(d*x + c) - 12*a^5)/(a^6*tan(d *x + c)^5))/d
Time = 5.69 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.99 \[ \int \frac {\csc ^6(c+d x)}{a+b \tan (c+d x)} \, dx=\frac {2\,b\,\mathrm {atanh}\left (\frac {b\,{\left (a^2+b^2\right )}^2\,\left (a+2\,b\,\mathrm {tan}\left (c+d\,x\right )\right )}{a\,\left (a^4\,b+2\,a^2\,b^3+b^5\right )}\right )\,{\left (a^2+b^2\right )}^2}{a^6\,d}-\frac {\frac {1}{5\,a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (2\,a^2+b^2\right )}{3\,a^3}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (a^4+2\,a^2\,b^2+b^4\right )}{a^5}-\frac {b\,\mathrm {tan}\left (c+d\,x\right )}{4\,a^2}-\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (2\,a^2+b^2\right )}{2\,a^4}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^5} \]
(2*b*atanh((b*(a^2 + b^2)^2*(a + 2*b*tan(c + d*x)))/(a*(a^4*b + b^5 + 2*a^ 2*b^3)))*(a^2 + b^2)^2)/(a^6*d) - (1/(5*a) + (tan(c + d*x)^2*(2*a^2 + b^2) )/(3*a^3) + (tan(c + d*x)^4*(a^4 + b^4 + 2*a^2*b^2))/a^5 - (b*tan(c + d*x) )/(4*a^2) - (b*tan(c + d*x)^3*(2*a^2 + b^2))/(2*a^4))/(d*tan(c + d*x)^5)