Integrand size = 19, antiderivative size = 75 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x)) \, dx=b x+\frac {b \cot (c+d x)}{d}+\frac {a \cot ^2(c+d x)}{2 d}-\frac {b \cot ^3(c+d x)}{3 d}-\frac {a \cot ^4(c+d x)}{4 d}+\frac {a \log (\sin (c+d x))}{d} \]
b*x+b*cot(d*x+c)/d+1/2*a*cot(d*x+c)^2/d-1/3*b*cot(d*x+c)^3/d-1/4*a*cot(d*x +c)^4/d+a*ln(sin(d*x+c))/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.21 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x)) \, dx=\frac {a \cot ^2(c+d x)}{2 d}-\frac {a \cot ^4(c+d x)}{4 d}-\frac {b \cot ^3(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\tan ^2(c+d x)\right )}{3 d}+\frac {a \log (\cos (c+d x))}{d}+\frac {a \log (\tan (c+d x))}{d} \]
(a*Cot[c + d*x]^2)/(2*d) - (a*Cot[c + d*x]^4)/(4*d) - (b*Cot[c + d*x]^3*Hy pergeometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/(3*d) + (a*Log[Cos[c + d* x]])/d + (a*Log[Tan[c + d*x]])/d
Time = 0.65 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.789, Rules used = {3042, 4012, 3042, 4012, 25, 3042, 4012, 3042, 4012, 25, 3042, 4014, 3042, 25, 3956}
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.
\(\displaystyle \int \cot ^5(c+d x) (a+b \tan (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a+b \tan (c+d x)}{\tan (c+d x)^5}dx\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle \int \cot ^4(c+d x) (b-a \tan (c+d x))dx-\frac {a \cot ^4(c+d x)}{4 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {b-a \tan (c+d x)}{\tan (c+d x)^4}dx-\frac {a \cot ^4(c+d x)}{4 d}\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle \int -\cot ^3(c+d x) (a+b \tan (c+d x))dx-\frac {a \cot ^4(c+d x)}{4 d}-\frac {b \cot ^3(c+d x)}{3 d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \cot ^3(c+d x) (a+b \tan (c+d x))dx-\frac {a \cot ^4(c+d x)}{4 d}-\frac {b \cot ^3(c+d x)}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int \frac {a+b \tan (c+d x)}{\tan (c+d x)^3}dx-\frac {a \cot ^4(c+d x)}{4 d}-\frac {b \cot ^3(c+d x)}{3 d}\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle -\int \cot ^2(c+d x) (b-a \tan (c+d x))dx-\frac {a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^2(c+d x)}{2 d}-\frac {b \cot ^3(c+d x)}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int \frac {b-a \tan (c+d x)}{\tan (c+d x)^2}dx-\frac {a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^2(c+d x)}{2 d}-\frac {b \cot ^3(c+d x)}{3 d}\) |
\(\Big \downarrow \) 4012 |
\(\displaystyle -\int -\cot (c+d x) (a+b \tan (c+d x))dx-\frac {a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^2(c+d x)}{2 d}-\frac {b \cot ^3(c+d x)}{3 d}+\frac {b \cot (c+d x)}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \cot (c+d x) (a+b \tan (c+d x))dx-\frac {a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^2(c+d x)}{2 d}-\frac {b \cot ^3(c+d x)}{3 d}+\frac {b \cot (c+d x)}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a+b \tan (c+d x)}{\tan (c+d x)}dx-\frac {a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^2(c+d x)}{2 d}-\frac {b \cot ^3(c+d x)}{3 d}+\frac {b \cot (c+d x)}{d}\) |
\(\Big \downarrow \) 4014 |
\(\displaystyle a \int \cot (c+d x)dx-\frac {a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^2(c+d x)}{2 d}-\frac {b \cot ^3(c+d x)}{3 d}+\frac {b \cot (c+d x)}{d}+b x\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx-\frac {a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^2(c+d x)}{2 d}-\frac {b \cot ^3(c+d x)}{3 d}+\frac {b \cot (c+d x)}{d}+b x\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -a \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx-\frac {a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^2(c+d x)}{2 d}-\frac {b \cot ^3(c+d x)}{3 d}+\frac {b \cot (c+d x)}{d}+b x\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle -\frac {a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^2(c+d x)}{2 d}+\frac {a \log (-\sin (c+d x))}{d}-\frac {b \cot ^3(c+d x)}{3 d}+\frac {b \cot (c+d x)}{d}+b x\) |
b*x + (b*Cot[c + d*x])/d + (a*Cot[c + d*x]^2)/(2*d) - (b*Cot[c + d*x]^3)/( 3*d) - (a*Cot[c + d*x]^4)/(4*d) + (a*Log[-Sin[c + d*x]])/d
3.5.22.3.1 Defintions of rubi rules used
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[(a*c + b*d)*(x/(a^2 + b^2)), x] + Simp[(b*c - a *d)/(a^2 + b^2) Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && N eQ[a*c + b*d, 0]
Time = 0.52 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.01
method | result | size |
parallelrisch | \(\frac {-3 \left (\cot ^{4}\left (d x +c \right )\right ) a -4 \left (\cot ^{3}\left (d x +c \right )\right ) b +6 \left (\cot ^{2}\left (d x +c \right )\right ) a +12 b d x +12 a \ln \left (\tan \left (d x +c \right )\right )-6 a \ln \left (\sec ^{2}\left (d x +c \right )\right )+12 \cot \left (d x +c \right ) b}{12 d}\) | \(76\) |
derivativedivides | \(\frac {-\frac {a}{4 \tan \left (d x +c \right )^{4}}-\frac {b}{3 \tan \left (d x +c \right )^{3}}+a \ln \left (\tan \left (d x +c \right )\right )+\frac {a}{2 \tan \left (d x +c \right )^{2}}+\frac {b}{\tan \left (d x +c \right )}-\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+b \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(81\) |
default | \(\frac {-\frac {a}{4 \tan \left (d x +c \right )^{4}}-\frac {b}{3 \tan \left (d x +c \right )^{3}}+a \ln \left (\tan \left (d x +c \right )\right )+\frac {a}{2 \tan \left (d x +c \right )^{2}}+\frac {b}{\tan \left (d x +c \right )}-\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+b \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(81\) |
norman | \(\frac {b x \left (\tan ^{4}\left (d x +c \right )\right )+\frac {b \left (\tan ^{3}\left (d x +c \right )\right )}{d}-\frac {a}{4 d}+\frac {a \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {b \tan \left (d x +c \right )}{3 d}}{\tan \left (d x +c \right )^{4}}+\frac {a \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(97\) |
risch | \(b x -i a x -\frac {2 i a c}{d}+\frac {4 i \left (3 i a \,{\mathrm e}^{6 i \left (d x +c \right )}+3 b \,{\mathrm e}^{6 i \left (d x +c \right )}-3 i a \,{\mathrm e}^{4 i \left (d x +c \right )}-6 b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 i a \,{\mathrm e}^{2 i \left (d x +c \right )}+5 b \,{\mathrm e}^{2 i \left (d x +c \right )}-2 b \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(133\) |
1/12*(-3*cot(d*x+c)^4*a-4*cot(d*x+c)^3*b+6*cot(d*x+c)^2*a+12*b*d*x+12*a*ln (tan(d*x+c))-6*a*ln(sec(d*x+c)^2)+12*cot(d*x+c)*b)/d
Time = 0.25 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.33 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x)) \, dx=\frac {6 \, a \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{4} + 3 \, {\left (4 \, b d x + 3 \, a\right )} \tan \left (d x + c\right )^{4} + 12 \, b \tan \left (d x + c\right )^{3} + 6 \, a \tan \left (d x + c\right )^{2} - 4 \, b \tan \left (d x + c\right ) - 3 \, a}{12 \, d \tan \left (d x + c\right )^{4}} \]
1/12*(6*a*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x + c)^4 + 3*(4*b *d*x + 3*a)*tan(d*x + c)^4 + 12*b*tan(d*x + c)^3 + 6*a*tan(d*x + c)^2 - 4* b*tan(d*x + c) - 3*a)/(d*tan(d*x + c)^4)
Time = 0.86 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.43 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x)) \, dx=\begin {cases} \tilde {\infty } a x & \text {for}\: c = 0 \wedge d = 0 \\x \left (a + b \tan {\left (c \right )}\right ) \cot ^{5}{\left (c \right )} & \text {for}\: d = 0 \\\tilde {\infty } a x & \text {for}\: c = - d x \\- \frac {a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {a}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {a}{4 d \tan ^{4}{\left (c + d x \right )}} + b x + \frac {b}{d \tan {\left (c + d x \right )}} - \frac {b}{3 d \tan ^{3}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*a*x, Eq(c, 0) & Eq(d, 0)), (x*(a + b*tan(c))*cot(c)**5, Eq( d, 0)), (zoo*a*x, Eq(c, -d*x)), (-a*log(tan(c + d*x)**2 + 1)/(2*d) + a*log (tan(c + d*x))/d + a/(2*d*tan(c + d*x)**2) - a/(4*d*tan(c + d*x)**4) + b*x + b/(d*tan(c + d*x)) - b/(3*d*tan(c + d*x)**3), True))
Time = 0.37 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.09 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x)) \, dx=\frac {12 \, {\left (d x + c\right )} b - 6 \, a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 12 \, a \log \left (\tan \left (d x + c\right )\right ) + \frac {12 \, b \tan \left (d x + c\right )^{3} + 6 \, a \tan \left (d x + c\right )^{2} - 4 \, b \tan \left (d x + c\right ) - 3 \, a}{\tan \left (d x + c\right )^{4}}}{12 \, d} \]
1/12*(12*(d*x + c)*b - 6*a*log(tan(d*x + c)^2 + 1) + 12*a*log(tan(d*x + c) ) + (12*b*tan(d*x + c)^3 + 6*a*tan(d*x + c)^2 - 4*b*tan(d*x + c) - 3*a)/ta n(d*x + c)^4)/d
Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (69) = 138\).
Time = 0.59 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.25 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x)) \, dx=-\frac {3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 192 \, {\left (d x + c\right )} b + 192 \, a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 192 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 120 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {400 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
-1/192*(3*a*tan(1/2*d*x + 1/2*c)^4 - 8*b*tan(1/2*d*x + 1/2*c)^3 - 36*a*tan (1/2*d*x + 1/2*c)^2 - 192*(d*x + c)*b + 192*a*log(tan(1/2*d*x + 1/2*c)^2 + 1) - 192*a*log(abs(tan(1/2*d*x + 1/2*c))) + 120*b*tan(1/2*d*x + 1/2*c) + (400*a*tan(1/2*d*x + 1/2*c)^4 - 120*b*tan(1/2*d*x + 1/2*c)^3 - 36*a*tan(1/ 2*d*x + 1/2*c)^2 + 8*b*tan(1/2*d*x + 1/2*c) + 3*a)/tan(1/2*d*x + 1/2*c)^4) /d
Time = 5.04 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.43 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x)) \, dx=\frac {a\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (\frac {a}{2}-\frac {b\,1{}\mathrm {i}}{2}\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^4\,\left (-b\,{\mathrm {tan}\left (c+d\,x\right )}^3-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+\frac {b\,\mathrm {tan}\left (c+d\,x\right )}{3}+\frac {a}{4}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (\frac {a}{2}+\frac {b\,1{}\mathrm {i}}{2}\right )}{d} \]