Integrand size = 19, antiderivative size = 75 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x)) \, dx=b x+\frac {b \cot (c+d x)}{d}+\frac {a \cot ^2(c+d x)}{2 d}-\frac {b \cot ^3(c+d x)}{3 d}-\frac {a \cot ^4(c+d x)}{4 d}+\frac {a \log (\sin (c+d x))}{d} \] Output:
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 = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.01 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x)) \, dx=\frac {a \csc ^2(c+d x)}{d}-\frac {a \csc ^4(c+d x)}{4 d}-\frac {b \cot ^3(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\tan ^2(c+d x)\right )}{3 d}+\frac {a \log (\sin (c+d x))}{d} \] Input:
Integrate[Cot[c + d*x]^5*(a + b*Tan[c + d*x]),x]
Output:
(a*Csc[c + d*x]^2)/d - (a*Csc[c + d*x]^4)/(4*d) - (b*Cot[c + d*x]^3*Hyperg eometric2F1[-3/2, 1, -1/2, -Tan[c + d*x]^2])/(3*d) + (a*Log[Sin[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 |
| \(\displaystyle -\frac {a \cot ^4(c+d x)}{4 d}+\frac {a \cot ^2(c+d x)}{2 d}+\frac {a \log (-\sin (c+d x))}{d}-\frac {b \cot ^3(c+d x)}{3 d}+\frac {b \cot (c+d x)}{d}+b x\) |
Input:
Int[Cot[c + d*x]^5*(a + b*Tan[c + d*x]),x]
Output:
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
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 = 0.93 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.04
| method | result | size |
| parallelrisch | \(-\frac {-6 a \left (2 \ln \left (\tan \left (d x +c \right )\right )-\ln \left (\sec \left (d x +c \right )^{2}\right )\right )+3 \cot \left (d x +c \right )^{4} a +4 \cot \left (d x +c \right )^{3} b -6 \cot \left (d x +c \right )^{2} a -12 b d x -12 \cot \left (d x +c \right ) b}{12 d}\) | \(78\) |
| derivativedivides | \(\frac {-\frac {a \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+b \arctan \left (\tan \left (d x +c \right )\right )-\frac {a}{4 \tan \left (d x +c \right )^{4}}+a \ln \left (\tan \left (d x +c \right )\right )-\frac {b}{3 \tan \left (d x +c \right )^{3}}+\frac {a}{2 \tan \left (d x +c \right )^{2}}+\frac {b}{\tan \left (d x +c \right )}}{d}\) | \(81\) |
| default | \(\frac {-\frac {a \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+b \arctan \left (\tan \left (d x +c \right )\right )-\frac {a}{4 \tan \left (d x +c \right )^{4}}+a \ln \left (\tan \left (d x +c \right )\right )-\frac {b}{3 \tan \left (d x +c \right )^{3}}+\frac {a}{2 \tan \left (d x +c \right )^{2}}+\frac {b}{\tan \left (d x +c \right )}}{d}\) | \(81\) |
| norman | \(\frac {b x \tan \left (d x +c \right )^{4}+\frac {b \tan \left (d x +c \right )^{3}}{d}-\frac {a}{4 d}+\frac {a \tan \left (d x +c \right )^{2}}{2 d}-\frac {b \tan \left (d x +c \right )}{3 d}}{\tan \left (d x +c \right )^{4}}+\frac {a \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {a \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d}\) | \(97\) |
| risch | \(b x -i a x -\frac {2 i a c}{d}+\frac {4 i \left (3 i a \,{\mathrm e}^{6 i \left (d x +c \right )}+3 b \,{\mathrm e}^{6 i \left (d x +c \right )}-3 i a \,{\mathrm e}^{4 i \left (d x +c \right )}-6 b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 i a \,{\mathrm e}^{2 i \left (d x +c \right )}+5 b \,{\mathrm e}^{2 i \left (d x +c \right )}-2 b \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(133\) |
Input:
int(cot(d*x+c)^5*(a+b*tan(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-1/12/d*(-6*a*(2*ln(tan(d*x+c))-ln(sec(d*x+c)^2))+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*cot(d*x+c)*b)
Time = 0.09 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.33 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x)) \, dx=\frac {6 \, a \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{4} + 3 \, {\left (4 \, b d x + 3 \, a\right )} \tan \left (d x + c\right )^{4} + 12 \, b \tan \left (d x + c\right )^{3} + 6 \, a \tan \left (d x + c\right )^{2} - 4 \, b \tan \left (d x + c\right ) - 3 \, a}{12 \, d \tan \left (d x + c\right )^{4}} \] Input:
integrate(cot(d*x+c)^5*(a+b*tan(d*x+c)),x, algorithm="fricas")
Output:
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 = 0.83 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.43 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x)) \, dx=\begin {cases} \tilde {\infty } a x & \text {for}\: c = 0 \wedge d = 0 \\x \left (a + b \tan {\left (c \right )}\right ) \cot ^{5}{\left (c \right )} & \text {for}\: d = 0 \\\tilde {\infty } a x & \text {for}\: c = - d x \\- \frac {a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {a}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {a}{4 d \tan ^{4}{\left (c + d x \right )}} + b x + \frac {b}{d \tan {\left (c + d x \right )}} - \frac {b}{3 d \tan ^{3}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \] Input:
integrate(cot(d*x+c)**5*(a+b*tan(d*x+c)),x)
Output:
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 = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.09 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x)) \, dx=\frac {12 \, {\left (d x + c\right )} b - 6 \, a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 12 \, a \log \left (\tan \left (d x + c\right )\right ) + \frac {12 \, b \tan \left (d x + c\right )^{3} + 6 \, a \tan \left (d x + c\right )^{2} - 4 \, b \tan \left (d x + c\right ) - 3 \, a}{\tan \left (d x + c\right )^{4}}}{12 \, d} \] Input:
integrate(cot(d*x+c)^5*(a+b*tan(d*x+c)),x, algorithm="maxima")
Output:
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
Time = 0.22 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.19 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x)) \, dx=\frac {{\left (d x + c\right )} b}{d} - \frac {a \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} + \frac {a \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{d} + \frac {12 \, b \tan \left (d x + c\right )^{3} + 6 \, a \tan \left (d x + c\right )^{2} - 4 \, b \tan \left (d x + c\right ) - 3 \, a}{12 \, d \tan \left (d x + c\right )^{4}} \] Input:
integrate(cot(d*x+c)^5*(a+b*tan(d*x+c)),x, algorithm="giac")
Output:
(d*x + c)*b/d - 1/2*a*log(tan(d*x + c)^2 + 1)/d + a*log(abs(tan(d*x + c))) /d + 1/12*(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 = 1.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.43 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x)) \, dx=\frac {a\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (\frac {a}{2}-\frac {b\,1{}\mathrm {i}}{2}\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^4\,\left (-b\,{\mathrm {tan}\left (c+d\,x\right )}^3-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+\frac {b\,\mathrm {tan}\left (c+d\,x\right )}{3}+\frac {a}{4}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (\frac {a}{2}+\frac {b\,1{}\mathrm {i}}{2}\right )}{d} \] Input:
int(cot(c + d*x)^5*(a + b*tan(c + d*x)),x)
Output:
(a*log(tan(c + d*x)))/d - (log(tan(c + d*x) + 1i)*(a/2 - (b*1i)/2))/d - (c ot(c + d*x)^4*(a/4 + (b*tan(c + d*x))/3 - (a*tan(c + d*x)^2)/2 - b*tan(c + d*x)^3))/d - (log(tan(c + d*x) - 1i)*(a/2 + (b*1i)/2))/d
Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.73 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x)) \, dx=\frac {128 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b -32 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b -96 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{4} a +96 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4} a -39 \sin \left (d x +c \right )^{4} a +96 \sin \left (d x +c \right )^{4} b d x +96 \sin \left (d x +c \right )^{2} a -24 a}{96 \sin \left (d x +c \right )^{4} d} \] Input:
int(cot(d*x+c)^5*(a+b*tan(d*x+c)),x)
Output:
(128*cos(c + d*x)*sin(c + d*x)**3*b - 32*cos(c + d*x)*sin(c + d*x)*b - 96* log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**4*a + 96*log(tan((c + d*x)/2))* sin(c + d*x)**4*a - 39*sin(c + d*x)**4*a + 96*sin(c + d*x)**4*b*d*x + 96*s in(c + d*x)**2*a - 24*a)/(96*sin(c + d*x)**4*d)