\(\int \cot ^7(a+b x) \, dx\) [7]

Optimal result

Integrand size = 8, antiderivative size = 58 \[ \int \cot ^7(a+b x) \, dx=-\frac {\cot ^2(a+b x)}{2 b}+\frac {\cot ^4(a+b x)}{4 b}-\frac {\cot ^6(a+b x)}{6 b}-\frac {\log (\sin (a+b x))}{b} \]

[Out]

-1/2*cot(b*x+a)^2/b+1/4*cot(b*x+a)^4/b-1/6*cot(b*x+a)^6/b-ln(sin(b*x+a))/b

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3554, 3556} \[ \int \cot ^7(a+b x) \, dx=-\frac {\cot ^6(a+b x)}{6 b}+\frac {\cot ^4(a+b x)}{4 b}-\frac {\cot ^2(a+b x)}{2 b}-\frac {\log (\sin (a+b x))}{b} \]

[In]

Int[Cot[a + b*x]^7,x]

[Out]

-1/2*Cot[a + b*x]^2/b + Cot[a + b*x]^4/(4*b) - Cot[a + b*x]^6/(6*b) - Log[Sin[a + b*x]]/b

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3556

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\cot ^6(a+b x)}{6 b}-\int \cot ^5(a+b x) \, dx \\ & = \frac {\cot ^4(a+b x)}{4 b}-\frac {\cot ^6(a+b x)}{6 b}+\int \cot ^3(a+b x) \, dx \\ & = -\frac {\cot ^2(a+b x)}{2 b}+\frac {\cot ^4(a+b x)}{4 b}-\frac {\cot ^6(a+b x)}{6 b}-\int \cot (a+b x) \, dx \\ & = -\frac {\cot ^2(a+b x)}{2 b}+\frac {\cot ^4(a+b x)}{4 b}-\frac {\cot ^6(a+b x)}{6 b}-\frac {\log (\sin (a+b x))}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.97 \[ \int \cot ^7(a+b x) \, dx=-\frac {6 \cot ^2(a+b x)-3 \cot ^4(a+b x)+2 \cot ^6(a+b x)+12 \log (\cos (a+b x))+12 \log (\tan (a+b x))}{12 b} \]

[In]

Integrate[Cot[a + b*x]^7,x]

[Out]

-1/12*(6*Cot[a + b*x]^2 - 3*Cot[a + b*x]^4 + 2*Cot[a + b*x]^6 + 12*Log[Cos[a + b*x]] + 12*Log[Tan[a + b*x]])/b

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.84

method	result	size
derivativedivides	\(\frac {-\frac {\cot \left (b x +a \right )^{6}}{6}+\frac {\cot \left (b x +a \right )^{4}}{4}-\frac {\cot \left (b x +a \right )^{2}}{2}+\frac {\ln \left (\cot \left (b x +a \right )^{2}+1\right )}{2}}{b}\)	\(49\)
default	\(\frac {-\frac {\cot \left (b x +a \right )^{6}}{6}+\frac {\cot \left (b x +a \right )^{4}}{4}-\frac {\cot \left (b x +a \right )^{2}}{2}+\frac {\ln \left (\cot \left (b x +a \right )^{2}+1\right )}{2}}{b}\)	\(49\)
parallelrisch	\(\frac {-2 \cot \left (b x +a \right )^{6}+3 \cot \left (b x +a \right )^{4}-6 \cot \left (b x +a \right )^{2}-12 \ln \left (\tan \left (b x +a \right )\right )+6 \ln \left (\sec \left (b x +a \right )^{2}\right )}{12 b}\)	\(57\)
norman	\(\frac {-\frac {1}{6 b}+\frac {\tan \left (b x +a \right )^{2}}{4 b}-\frac {\tan \left (b x +a \right )^{4}}{2 b}}{\tan \left (b x +a \right )^{6}}-\frac {\ln \left (\tan \left (b x +a \right )\right )}{b}+\frac {\ln \left (1+\tan \left (b x +a \right )^{2}\right )}{2 b}\)	\(71\)
risch	\(i x +\frac {2 i a}{b}+\frac {6 \,{\mathrm e}^{10 i \left (b x +a \right )}-12 \,{\mathrm e}^{8 i \left (b x +a \right )}+\frac {68 \,{\mathrm e}^{6 i \left (b x +a \right )}}{3}-12 \,{\mathrm e}^{4 i \left (b x +a \right )}+6 \,{\mathrm e}^{2 i \left (b x +a \right )}}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{6}}-\frac {\ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}{b}\)	\(104\)

[In]

int(cot(b*x+a)^7,x,method=_RETURNVERBOSE)

[Out]

1/b*(-1/6*cot(b*x+a)^6+1/4*cot(b*x+a)^4-1/2*cot(b*x+a)^2+1/2*ln(cot(b*x+a)^2+1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (52) = 104\).

Time = 0.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 2.17 \[ \int \cot ^7(a+b x) \, dx=\frac {18 \, \cos \left (2 \, b x + 2 \, a\right )^{2} - 3 \, {\left (\cos \left (2 \, b x + 2 \, a\right )^{3} - 3 \, \cos \left (2 \, b x + 2 \, a\right )^{2} + 3 \, \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) - 18 \, \cos \left (2 \, b x + 2 \, a\right ) + 8}{6 \, {\left (b \cos \left (2 \, b x + 2 \, a\right )^{3} - 3 \, b \cos \left (2 \, b x + 2 \, a\right )^{2} + 3 \, b \cos \left (2 \, b x + 2 \, a\right ) - b\right )}} \]

[In]

integrate(cot(b*x+a)^7,x, algorithm="fricas")

[Out]

1/6*(18*cos(2*b*x + 2*a)^2 - 3*(cos(2*b*x + 2*a)^3 - 3*cos(2*b*x + 2*a)^2 + 3*cos(2*b*x + 2*a) - 1)*log(-1/2*c
os(2*b*x + 2*a) + 1/2) - 18*cos(2*b*x + 2*a) + 8)/(b*cos(2*b*x + 2*a)^3 - 3*b*cos(2*b*x + 2*a)^2 + 3*b*cos(2*b
*x + 2*a) - b)

Sympy [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.38 \[ \int \cot ^7(a+b x) \, dx=\begin {cases} \tilde {\infty } x & \text {for}\: a = 0 \wedge b = 0 \\x \cot ^{7}{\left (a \right )} & \text {for}\: b = 0 \\\tilde {\infty } x & \text {for}\: a = - b x \\\frac {\log {\left (\tan ^{2}{\left (a + b x \right )} + 1 \right )}}{2 b} - \frac {\log {\left (\tan {\left (a + b x \right )} \right )}}{b} - \frac {1}{2 b \tan ^{2}{\left (a + b x \right )}} + \frac {1}{4 b \tan ^{4}{\left (a + b x \right )}} - \frac {1}{6 b \tan ^{6}{\left (a + b x \right )}} & \text {otherwise} \end {cases} \]

[In]

integrate(cot(b*x+a)**7,x)

[Out]

Piecewise((zoo*x, Eq(a, 0) & Eq(b, 0)), (x*cot(a)**7, Eq(b, 0)), (zoo*x, Eq(a, -b*x)), (log(tan(a + b*x)**2 +
1)/(2*b) - log(tan(a + b*x))/b - 1/(2*b*tan(a + b*x)**2) + 1/(4*b*tan(a + b*x)**4) - 1/(6*b*tan(a + b*x)**6),
True))

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.83 \[ \int \cot ^7(a+b x) \, dx=-\frac {\frac {18 \, \sin \left (b x + a\right )^{4} - 9 \, \sin \left (b x + a\right )^{2} + 2}{\sin \left (b x + a\right )^{6}} + 6 \, \log \left (\sin \left (b x + a\right )^{2}\right )}{12 \, b} \]

[In]

integrate(cot(b*x+a)^7,x, algorithm="maxima")

[Out]

-1/12*((18*sin(b*x + a)^4 - 9*sin(b*x + a)^2 + 2)/sin(b*x + a)^6 + 6*log(sin(b*x + a)^2))/b

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (52) = 104\).

Time = 0.28 (sec) , antiderivative size = 208, normalized size of antiderivative = 3.59 \[ \int \cot ^7(a+b x) \, dx=\frac {\frac {{\left (\frac {12 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {87 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {352 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) - 1\right )}^{3}} + \frac {87 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {12 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {{\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} - 192 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) + 384 \, \log \left ({\left | -\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1 \right |}\right )}{384 \, b} \]

[In]

integrate(cot(b*x+a)^7,x, algorithm="giac")

[Out]

1/384*((12*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 87*(cos(b*x + a) - 1)^2/(cos(b*x + a) + 1)^2 + 352*(cos(b*x
+ a) - 1)^3/(cos(b*x + a) + 1)^3 + 1)*(cos(b*x + a) + 1)^3/(cos(b*x + a) - 1)^3 + 87*(cos(b*x + a) - 1)/(cos(
b*x + a) + 1) + 12*(cos(b*x + a) - 1)^2/(cos(b*x + a) + 1)^2 + (cos(b*x + a) - 1)^3/(cos(b*x + a) + 1)^3 - 192
*log(abs(-cos(b*x + a) + 1)/abs(cos(b*x + a) + 1)) + 384*log(abs(-(cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 1)))
/b

Mupad [B] (verification not implemented)

Time = 19.37 (sec) , antiderivative size = 340, normalized size of antiderivative = 5.86 \[ \int \cot ^7(a+b x) \, dx=x\,1{}\mathrm {i}-\frac {\ln \left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}-1\right )}{b}+\frac {32}{b\,\left (5\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-10\,{\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}+10\,{\mathrm {e}}^{a\,6{}\mathrm {i}+b\,x\,6{}\mathrm {i}}-5\,{\mathrm {e}}^{a\,8{}\mathrm {i}+b\,x\,8{}\mathrm {i}}+{\mathrm {e}}^{a\,10{}\mathrm {i}+b\,x\,10{}\mathrm {i}}-1\right )}+\frac {32}{3\,b\,\left (1+15\,{\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}-20\,{\mathrm {e}}^{a\,6{}\mathrm {i}+b\,x\,6{}\mathrm {i}}+15\,{\mathrm {e}}^{a\,8{}\mathrm {i}+b\,x\,8{}\mathrm {i}}-6\,{\mathrm {e}}^{a\,10{}\mathrm {i}+b\,x\,10{}\mathrm {i}}+{\mathrm {e}}^{a\,12{}\mathrm {i}+b\,x\,12{}\mathrm {i}}-6\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )}+\frac {6}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}+\frac {18}{b\,\left (1+{\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}-2\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )}+\frac {104}{3\,b\,\left (3\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-3\,{\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}+{\mathrm {e}}^{a\,6{}\mathrm {i}+b\,x\,6{}\mathrm {i}}-1\right )}+\frac {44}{b\,\left (1+6\,{\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}-4\,{\mathrm {e}}^{a\,6{}\mathrm {i}+b\,x\,6{}\mathrm {i}}+{\mathrm {e}}^{a\,8{}\mathrm {i}+b\,x\,8{}\mathrm {i}}-4\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )} \]

[In]

int(cot(a + b*x)^7,x)

[Out]

x*1i - log(exp(a*2i)*exp(b*x*2i) - 1)/b + 32/(b*(5*exp(a*2i + b*x*2i) - 10*exp(a*4i + b*x*4i) + 10*exp(a*6i +
b*x*6i) - 5*exp(a*8i + b*x*8i) + exp(a*10i + b*x*10i) - 1)) + 32/(3*b*(15*exp(a*4i + b*x*4i) - 6*exp(a*2i + b*
x*2i) - 20*exp(a*6i + b*x*6i) + 15*exp(a*8i + b*x*8i) - 6*exp(a*10i + b*x*10i) + exp(a*12i + b*x*12i) + 1)) +
6/(b*(exp(a*2i + b*x*2i) - 1)) + 18/(b*(exp(a*4i + b*x*4i) - 2*exp(a*2i + b*x*2i) + 1)) + 104/(3*b*(3*exp(a*2i
+ b*x*2i) - 3*exp(a*4i + b*x*4i) + exp(a*6i + b*x*6i) - 1)) + 44/(b*(6*exp(a*4i + b*x*4i) - 4*exp(a*2i + b*x*
2i) - 4*exp(a*6i + b*x*6i) + exp(a*8i + b*x*8i) + 1))