\(\int \cot ^6(c+d x) (a+b \tan (c+d x)) \, dx\) [423]

Optimal result

Integrand size = 19, antiderivative size = 93 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x)) \, dx=-a x-\frac {a \cot (c+d x)}{d}+\frac {b \cot ^2(c+d x)}{2 d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {b \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+\frac {b \log (\sin (c+d x))}{d} \]

[Out]

-a*x-a*cot(d*x+c)/d+1/2*b*cot(d*x+c)^2/d+1/3*a*cot(d*x+c)^3/d-1/4*b*cot(d*x+c)^4/d-1/5*a*cot(d*x+c)^5/d+b*ln(s
in(d*x+c))/d

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3610, 3612, 3556} \[ \int \cot ^6(c+d x) (a+b \tan (c+d x)) \, dx=-\frac {a \cot ^5(c+d x)}{5 d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {a \cot (c+d x)}{d}-a x-\frac {b \cot ^4(c+d x)}{4 d}+\frac {b \cot ^2(c+d x)}{2 d}+\frac {b \log (\sin (c+d x))}{d} \]

[In]

Int[Cot[c + d*x]^6*(a + b*Tan[c + d*x]),x]

[Out]

-(a*x) - (a*Cot[c + d*x])/d + (b*Cot[c + d*x]^2)/(2*d) + (a*Cot[c + d*x]^3)/(3*d) - (b*Cot[c + d*x]^4)/(4*d) -
(a*Cot[c + d*x]^5)/(5*d) + (b*Log[Sin[c + d*x]])/d

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3610

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] + Dist[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]

Rule 3612

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] + Dist[(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] && NeQ[a*c + b*d, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {a \cot ^5(c+d x)}{5 d}+\int \cot ^5(c+d x) (b-a \tan (c+d x)) \, dx \\ & = -\frac {b \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+\int \cot ^4(c+d x) (-a-b \tan (c+d x)) \, dx \\ & = \frac {a \cot ^3(c+d x)}{3 d}-\frac {b \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+\int \cot ^3(c+d x) (-b+a \tan (c+d x)) \, dx \\ & = \frac {b \cot ^2(c+d x)}{2 d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {b \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+\int \cot ^2(c+d x) (a+b \tan (c+d x)) \, dx \\ & = -\frac {a \cot (c+d x)}{d}+\frac {b \cot ^2(c+d x)}{2 d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {b \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+\int \cot (c+d x) (b-a \tan (c+d x)) \, dx \\ & = -a x-\frac {a \cot (c+d x)}{d}+\frac {b \cot ^2(c+d x)}{2 d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {b \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+b \int \cot (c+d x) \, dx \\ & = -a x-\frac {a \cot (c+d x)}{d}+\frac {b \cot ^2(c+d x)}{2 d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {b \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x)}{5 d}+\frac {b \log (\sin (c+d x))}{d} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.98 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x)) \, dx=\frac {b \cot ^2(c+d x)}{2 d}-\frac {b \cot ^4(c+d x)}{4 d}-\frac {a \cot ^5(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},-\tan ^2(c+d x)\right )}{5 d}+\frac {b \log (\cos (c+d x))}{d}+\frac {b \log (\tan (c+d x))}{d} \]

[In]

Integrate[Cot[c + d*x]^6*(a + b*Tan[c + d*x]),x]

[Out]

(b*Cot[c + d*x]^2)/(2*d) - (b*Cot[c + d*x]^4)/(4*d) - (a*Cot[c + d*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[
c + d*x]^2])/(5*d) + (b*Log[Cos[c + d*x]])/d + (b*Log[Tan[c + d*x]])/d

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.94

method	result	size
parallelrisch	\(\frac {-12 \left (\cot ^{5}\left (d x +c \right )\right ) a -15 \left (\cot ^{4}\left (d x +c \right )\right ) b +20 \left (\cot ^{3}\left (d x +c \right )\right ) a +30 \left (\cot ^{2}\left (d x +c \right )\right ) b -60 a d x +60 b \ln \left (\tan \left (d x +c \right )\right )-30 b \ln \left (\sec ^{2}\left (d x +c \right )\right )-60 \cot \left (d x +c \right ) a}{60 d}\)	\(87\)
derivativedivides	\(\frac {-\frac {a}{5 \tan \left (d x +c \right )^{5}}-\frac {a}{\tan \left (d x +c \right )}-\frac {b}{4 \tan \left (d x +c \right )^{4}}+b \ln \left (\tan \left (d x +c \right )\right )+\frac {a}{3 \tan \left (d x +c \right )^{3}}+\frac {b}{2 \tan \left (d x +c \right )^{2}}-\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-a \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(94\)
default	\(\frac {-\frac {a}{5 \tan \left (d x +c \right )^{5}}-\frac {a}{\tan \left (d x +c \right )}-\frac {b}{4 \tan \left (d x +c \right )^{4}}+b \ln \left (\tan \left (d x +c \right )\right )+\frac {a}{3 \tan \left (d x +c \right )^{3}}+\frac {b}{2 \tan \left (d x +c \right )^{2}}-\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-a \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(94\)
norman	\(\frac {-\frac {a}{5 d}-a x \left (\tan ^{5}\left (d x +c \right )\right )+\frac {a \left (\tan ^{2}\left (d x +c \right )\right )}{3 d}-\frac {a \left (\tan ^{4}\left (d x +c \right )\right )}{d}-\frac {b \tan \left (d x +c \right )}{4 d}+\frac {b \left (\tan ^{3}\left (d x +c \right )\right )}{2 d}}{\tan \left (d x +c \right )^{5}}+\frac {b \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(113\)
risch	\(-i b x -a x -\frac {2 i b c}{d}-\frac {2 \left (45 i a \,{\mathrm e}^{8 i \left (d x +c \right )}+30 b \,{\mathrm e}^{8 i \left (d x +c \right )}-90 i a \,{\mathrm e}^{6 i \left (d x +c \right )}-60 b \,{\mathrm e}^{6 i \left (d x +c \right )}+140 i a \,{\mathrm e}^{4 i \left (d x +c \right )}+60 b \,{\mathrm e}^{4 i \left (d x +c \right )}-70 i a \,{\mathrm e}^{2 i \left (d x +c \right )}-30 b \,{\mathrm e}^{2 i \left (d x +c \right )}+23 i a \right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(159\)

[In]

int(cot(d*x+c)^6*(a+b*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/60*(-12*cot(d*x+c)^5*a-15*cot(d*x+c)^4*b+20*cot(d*x+c)^3*a+30*cot(d*x+c)^2*b-60*a*d*x+60*b*ln(tan(d*x+c))-30
*b*ln(sec(d*x+c)^2)-60*cot(d*x+c)*a)/d

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.19 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x)) \, dx=\frac {30 \, b \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{5} - 15 \, {\left (4 \, a d x - 3 \, b\right )} \tan \left (d x + c\right )^{5} - 60 \, a \tan \left (d x + c\right )^{4} + 30 \, b \tan \left (d x + c\right )^{3} + 20 \, a \tan \left (d x + c\right )^{2} - 15 \, b \tan \left (d x + c\right ) - 12 \, a}{60 \, d \tan \left (d x + c\right )^{5}} \]

[In]

integrate(cot(d*x+c)^6*(a+b*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/60*(30*b*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x + c)^5 - 15*(4*a*d*x - 3*b)*tan(d*x + c)^5 - 60*a*
tan(d*x + c)^4 + 30*b*tan(d*x + c)^3 + 20*a*tan(d*x + c)^2 - 15*b*tan(d*x + c) - 12*a)/(d*tan(d*x + c)^5)

Sympy [A] (verification not implemented)

Time = 1.36 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.30 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x)) \, dx=\begin {cases} \tilde {\infty } a x & \text {for}\: c = 0 \wedge d = 0 \\x \left (a + b \tan {\left (c \right )}\right ) \cot ^{6}{\left (c \right )} & \text {for}\: d = 0 \\\tilde {\infty } a x & \text {for}\: c = - d x \\- a x - \frac {a}{d \tan {\left (c + d x \right )}} + \frac {a}{3 d \tan ^{3}{\left (c + d x \right )}} - \frac {a}{5 d \tan ^{5}{\left (c + d x \right )}} - \frac {b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {b}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {b}{4 d \tan ^{4}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \]

[In]

integrate(cot(d*x+c)**6*(a+b*tan(d*x+c)),x)

[Out]

Piecewise((zoo*a*x, Eq(c, 0) & Eq(d, 0)), (x*(a + b*tan(c))*cot(c)**6, Eq(d, 0)), (zoo*a*x, Eq(c, -d*x)), (-a*
x - a/(d*tan(c + d*x)) + a/(3*d*tan(c + d*x)**3) - a/(5*d*tan(c + d*x)**5) - b*log(tan(c + d*x)**2 + 1)/(2*d)
+ b*log(tan(c + d*x))/d + b/(2*d*tan(c + d*x)**2) - b/(4*d*tan(c + d*x)**4), True))

Maxima [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x)) \, dx=-\frac {60 \, {\left (d x + c\right )} a + 30 \, b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 \, b \log \left (\tan \left (d x + c\right )\right ) + \frac {60 \, a \tan \left (d x + c\right )^{4} - 30 \, b \tan \left (d x + c\right )^{3} - 20 \, a \tan \left (d x + c\right )^{2} + 15 \, b \tan \left (d x + c\right ) + 12 \, a}{\tan \left (d x + c\right )^{5}}}{60 \, d} \]

[In]

integrate(cot(d*x+c)^6*(a+b*tan(d*x+c)),x, algorithm="maxima")

[Out]

-1/60*(60*(d*x + c)*a + 30*b*log(tan(d*x + c)^2 + 1) - 60*b*log(tan(d*x + c)) + (60*a*tan(d*x + c)^4 - 30*b*ta
n(d*x + c)^3 - 20*a*tan(d*x + c)^2 + 15*b*tan(d*x + c) + 12*a)/tan(d*x + c)^5)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (85) = 170\).

Time = 0.62 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.13 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x)) \, dx=\frac {6 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 70 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 180 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 960 \, {\left (d x + c\right )} a - 960 \, b \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) + 960 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 660 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {2192 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 660 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 180 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 70 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \]

[In]

integrate(cot(d*x+c)^6*(a+b*tan(d*x+c)),x, algorithm="giac")

[Out]

1/960*(6*a*tan(1/2*d*x + 1/2*c)^5 - 15*b*tan(1/2*d*x + 1/2*c)^4 - 70*a*tan(1/2*d*x + 1/2*c)^3 + 180*b*tan(1/2*
d*x + 1/2*c)^2 - 960*(d*x + c)*a - 960*b*log(tan(1/2*d*x + 1/2*c)^2 + 1) + 960*b*log(abs(tan(1/2*d*x + 1/2*c))
) + 660*a*tan(1/2*d*x + 1/2*c) - (2192*b*tan(1/2*d*x + 1/2*c)^5 + 660*a*tan(1/2*d*x + 1/2*c)^4 - 180*b*tan(1/2
*d*x + 1/2*c)^3 - 70*a*tan(1/2*d*x + 1/2*c)^2 + 15*b*tan(1/2*d*x + 1/2*c) + 6*a)/tan(1/2*d*x + 1/2*c)^5)/d

Mupad [B] (verification not implemented)

Time = 4.96 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.25 \[ \int \cot ^6(c+d x) (a+b \tan (c+d x)) \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-\frac {b}{2}+\frac {a\,1{}\mathrm {i}}{2}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (\frac {b}{2}+\frac {a\,1{}\mathrm {i}}{2}\right )}{d}+\frac {b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^5\,\left (a\,{\mathrm {tan}\left (c+d\,x\right )}^4-\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^3}{2}-\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{3}+\frac {b\,\mathrm {tan}\left (c+d\,x\right )}{4}+\frac {a}{5}\right )}{d} \]

[In]

int(cot(c + d*x)^6*(a + b*tan(c + d*x)),x)

[Out]

(log(tan(c + d*x) - 1i)*((a*1i)/2 - b/2))/d - (log(tan(c + d*x) + 1i)*((a*1i)/2 + b/2))/d + (b*log(tan(c + d*x
)))/d - (cot(c + d*x)^5*(a/5 + (b*tan(c + d*x))/4 - (a*tan(c + d*x)^2)/3 + a*tan(c + d*x)^4 - (b*tan(c + d*x)^
3)/2))/d