Integrand size = 8, antiderivative size = 58 \[ \int \cot ^7(a+b x) \, dx=-\frac {\cot ^2(a+b x)}{2 b}+\frac {\cot ^4(a+b x)}{4 b}-\frac {\cot ^6(a+b x)}{6 b}-\frac {\log (\sin (a+b x))}{b} \] Output:
-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
Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00 \[ \int \cot ^7(a+b x) \, dx=-\frac {3 \csc ^2(a+b x)}{2 b}+\frac {3 \csc ^4(a+b x)}{4 b}-\frac {\csc ^6(a+b x)}{6 b}-\frac {\log (\sin (a+b x))}{b} \] Input:
Integrate[Cot[a + b*x]^7,x]
Output:
(-3*Csc[a + b*x]^2)/(2*b) + (3*Csc[a + b*x]^4)/(4*b) - Csc[a + b*x]^6/(6*b ) - Log[Sin[a + b*x]]/b
Time = 0.40 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.875, Rules used = {3042, 25, 3954, 25, 3042, 25, 3954, 25, 3042, 25, 3954, 25, 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.
\(\displaystyle \int \cot ^7(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\tan \left (a+b x+\frac {\pi }{2}\right )^7dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \tan \left (\frac {1}{2} (2 a+\pi )+b x\right )^7dx\) |
\(\Big \downarrow \) 3954 |
\(\displaystyle \int -\cot ^5(a+b x)dx-\frac {\cot ^6(a+b x)}{6 b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \cot ^5(a+b x)dx-\frac {\cot ^6(a+b x)}{6 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int -\tan \left (a+b x+\frac {\pi }{2}\right )^5dx-\frac {\cot ^6(a+b x)}{6 b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \tan \left (\frac {1}{2} (2 a+\pi )+b x\right )^5dx-\frac {\cot ^6(a+b x)}{6 b}\) |
\(\Big \downarrow \) 3954 |
\(\displaystyle -\int -\cot ^3(a+b x)dx-\frac {\cot ^6(a+b x)}{6 b}+\frac {\cot ^4(a+b x)}{4 b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \cot ^3(a+b x)dx-\frac {\cot ^6(a+b x)}{6 b}+\frac {\cot ^4(a+b x)}{4 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\tan \left (a+b x+\frac {\pi }{2}\right )^3dx-\frac {\cot ^6(a+b x)}{6 b}+\frac {\cot ^4(a+b x)}{4 b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \tan \left (\frac {1}{2} (2 a+\pi )+b x\right )^3dx-\frac {\cot ^6(a+b x)}{6 b}+\frac {\cot ^4(a+b x)}{4 b}\) |
\(\Big \downarrow \) 3954 |
\(\displaystyle \int -\cot (a+b x)dx-\frac {\cot ^6(a+b x)}{6 b}+\frac {\cot ^4(a+b x)}{4 b}-\frac {\cot ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \cot (a+b x)dx-\frac {\cot ^6(a+b x)}{6 b}+\frac {\cot ^4(a+b x)}{4 b}-\frac {\cot ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int -\tan \left (a+b x+\frac {\pi }{2}\right )dx-\frac {\cot ^6(a+b x)}{6 b}+\frac {\cot ^4(a+b x)}{4 b}-\frac {\cot ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \tan \left (\frac {1}{2} (2 a+\pi )+b x\right )dx-\frac {\cot ^6(a+b x)}{6 b}+\frac {\cot ^4(a+b x)}{4 b}-\frac {\cot ^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle -\frac {\cot ^6(a+b x)}{6 b}+\frac {\cot ^4(a+b x)}{4 b}-\frac {\cot ^2(a+b x)}{2 b}-\frac {\log (-\sin (a+b x))}{b}\) |
Input:
Int[Cot[a + b*x]^7,x]
Output:
-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
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.19 (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 | \(i x +\frac {2 i a}{b}+\frac {6 \,{\mathrm e}^{10 i \left (b x +a \right )}-12 \,{\mathrm e}^{8 i \left (b x +a \right )}+\frac {68 \,{\mathrm e}^{6 i \left (b x +a \right )}}{3}-12 \,{\mathrm e}^{4 i \left (b x +a \right )}+6 \,{\mathrm e}^{2 i \left (b x +a \right )}}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{6}}-\frac {\ln \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}{b}\) | \(104\) |
Input:
int(cot(b*x+a)^7,x,method=_RETURNVERBOSE)
Output:
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))
Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (52) = 104\).
Time = 0.10 (sec) , antiderivative size = 126, normalized size of antiderivative = 2.17 \[ \int \cot ^7(a+b x) \, dx=\frac {18 \, \cos \left (2 \, b x + 2 \, a\right )^{2} - 3 \, {\left (\cos \left (2 \, b x + 2 \, a\right )^{3} - 3 \, \cos \left (2 \, b x + 2 \, a\right )^{2} + 3 \, \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2}\right ) - 18 \, \cos \left (2 \, b x + 2 \, a\right ) + 8}{6 \, {\left (b \cos \left (2 \, b x + 2 \, a\right )^{3} - 3 \, b \cos \left (2 \, b x + 2 \, a\right )^{2} + 3 \, b \cos \left (2 \, b x + 2 \, a\right ) - b\right )}} \] Input:
integrate(cot(b*x+a)^7,x, algorithm="fricas")
Output:
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*cos(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)
Time = 0.47 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.38 \[ \int \cot ^7(a+b x) \, dx=\begin {cases} \tilde {\infty } x & \text {for}\: a = 0 \wedge b = 0 \\x \cot ^{7}{\left (a \right )} & \text {for}\: b = 0 \\\tilde {\infty } x & \text {for}\: a = - b x \\\frac {\log {\left (\tan ^{2}{\left (a + b x \right )} + 1 \right )}}{2 b} - \frac {\log {\left (\tan {\left (a + b x \right )} \right )}}{b} - \frac {1}{2 b \tan ^{2}{\left (a + b x \right )}} + \frac {1}{4 b \tan ^{4}{\left (a + b x \right )}} - \frac {1}{6 b \tan ^{6}{\left (a + b x \right )}} & \text {otherwise} \end {cases} \] Input:
integrate(cot(b*x+a)**7,x)
Output:
Piecewise((zoo*x, Eq(a, 0) & Eq(b, 0)), (x*cot(a)**7, Eq(b, 0)), (zoo*x, E q(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), Tr ue))
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.83 \[ \int \cot ^7(a+b x) \, dx=-\frac {\frac {18 \, \sin \left (b x + a\right )^{4} - 9 \, \sin \left (b x + a\right )^{2} + 2}{\sin \left (b x + a\right )^{6}} + 6 \, \log \left (\sin \left (b x + a\right )^{2}\right )}{12 \, b} \] Input:
integrate(cot(b*x+a)^7,x, algorithm="maxima")
Output:
-1/12*((18*sin(b*x + a)^4 - 9*sin(b*x + a)^2 + 2)/sin(b*x + a)^6 + 6*log(s in(b*x + a)^2))/b
Time = 0.15 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.84 \[ \int \cot ^7(a+b x) \, dx=-\frac {\log \left ({\left | \sin \left (b x + a\right ) \right |}\right )}{b} - \frac {18 \, \sin \left (b x + a\right )^{4} - 9 \, \sin \left (b x + a\right )^{2} + 2}{12 \, b \sin \left (b x + a\right )^{6}} \] Input:
integrate(cot(b*x+a)^7,x, algorithm="giac")
Output:
-log(abs(sin(b*x + a)))/b - 1/12*(18*sin(b*x + a)^4 - 9*sin(b*x + a)^2 + 2 )/(b*sin(b*x + a)^6)
Time = 15.65 (sec) , antiderivative size = 340, normalized size of antiderivative = 5.86 \[ \int \cot ^7(a+b x) \, dx=x\,1{}\mathrm {i}-\frac {\ln \left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}-1\right )}{b}+\frac {32}{b\,\left (5\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-10\,{\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}+10\,{\mathrm {e}}^{a\,6{}\mathrm {i}+b\,x\,6{}\mathrm {i}}-5\,{\mathrm {e}}^{a\,8{}\mathrm {i}+b\,x\,8{}\mathrm {i}}+{\mathrm {e}}^{a\,10{}\mathrm {i}+b\,x\,10{}\mathrm {i}}-1\right )}+\frac {32}{3\,b\,\left (1+15\,{\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}-20\,{\mathrm {e}}^{a\,6{}\mathrm {i}+b\,x\,6{}\mathrm {i}}+15\,{\mathrm {e}}^{a\,8{}\mathrm {i}+b\,x\,8{}\mathrm {i}}-6\,{\mathrm {e}}^{a\,10{}\mathrm {i}+b\,x\,10{}\mathrm {i}}+{\mathrm {e}}^{a\,12{}\mathrm {i}+b\,x\,12{}\mathrm {i}}-6\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )}+\frac {6}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}+\frac {18}{b\,\left (1+{\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}-2\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )}+\frac {104}{3\,b\,\left (3\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-3\,{\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}+{\mathrm {e}}^{a\,6{}\mathrm {i}+b\,x\,6{}\mathrm {i}}-1\right )}+\frac {44}{b\,\left (1+6\,{\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}-4\,{\mathrm {e}}^{a\,6{}\mathrm {i}+b\,x\,6{}\mathrm {i}}+{\mathrm {e}}^{a\,8{}\mathrm {i}+b\,x\,8{}\mathrm {i}}-4\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )} \] Input:
int(cot(a + b*x)^7,x)
Output:
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*10 i) + 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))
Time = 0.18 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.53 \[ \int \cot ^7(a+b x) \, dx=\frac {48 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1\right ) \sin \left (b x +a \right )^{6}-48 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sin \left (b x +a \right )^{6}+25 \sin \left (b x +a \right )^{6}-72 \sin \left (b x +a \right )^{4}+36 \sin \left (b x +a \right )^{2}-8}{48 \sin \left (b x +a \right )^{6} b} \] Input:
int(cot(b*x+a)^7,x)
Output:
(48*log(tan((a + b*x)/2)**2 + 1)*sin(a + b*x)**6 - 48*log(tan((a + b*x)/2) )*sin(a + b*x)**6 + 25*sin(a + b*x)**6 - 72*sin(a + b*x)**4 + 36*sin(a + b *x)**2 - 8)/(48*sin(a + b*x)**6*b)