Integrand size = 17, antiderivative size = 55 \[ \int \cot ^2(a+b x) \csc ^3(a+b x) \, dx=\frac {\text {arctanh}(\cos (a+b x))}{8 b}+\frac {\cot (a+b x) \csc (a+b x)}{8 b}-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b} \] Output:
1/8*arctanh(cos(b*x+a))/b+1/8*cot(b*x+a)*csc(b*x+a)/b-1/4*cot(b*x+a)*csc(b *x+a)^3/b
Leaf count is larger than twice the leaf count of optimal. \(113\) vs. \(2(55)=110\).
Time = 0.14 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.05 \[ \int \cot ^2(a+b x) \csc ^3(a+b x) \, dx=\frac {\csc ^2\left (\frac {1}{2} (a+b x)\right )}{32 b}-\frac {\csc ^4\left (\frac {1}{2} (a+b x)\right )}{64 b}+\frac {\log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{8 b}-\frac {\log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{8 b}-\frac {\sec ^2\left (\frac {1}{2} (a+b x)\right )}{32 b}+\frac {\sec ^4\left (\frac {1}{2} (a+b x)\right )}{64 b} \] Input:
Integrate[Cot[a + b*x]^2*Csc[a + b*x]^3,x]
Output:
Csc[(a + b*x)/2]^2/(32*b) - Csc[(a + b*x)/2]^4/(64*b) + Log[Cos[(a + b*x)/ 2]]/(8*b) - Log[Sin[(a + b*x)/2]]/(8*b) - Sec[(a + b*x)/2]^2/(32*b) + Sec[ (a + b*x)/2]^4/(64*b)
Time = 0.58 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3042, 3091, 3042, 4255, 3042, 4257}
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.
\(\displaystyle \int \cot ^2(a+b x) \csc ^3(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \tan \left (a+b x-\frac {\pi }{2}\right )^2 \sec \left (a+b x-\frac {\pi }{2}\right )^3dx\) |
\(\Big \downarrow \) 3091 |
\(\displaystyle -\frac {1}{4} \int \csc ^3(a+b x)dx-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{4} \int \csc (a+b x)^3dx-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {1}{4} \left (\frac {\cot (a+b x) \csc (a+b x)}{2 b}-\frac {1}{2} \int \csc (a+b x)dx\right )-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{4} \left (\frac {\cot (a+b x) \csc (a+b x)}{2 b}-\frac {1}{2} \int \csc (a+b x)dx\right )-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle \frac {1}{4} \left (\frac {\text {arctanh}(\cos (a+b x))}{2 b}+\frac {\cot (a+b x) \csc (a+b x)}{2 b}\right )-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b}\) |
Input:
Int[Cot[a + b*x]^2*Csc[a + b*x]^3,x]
Output:
-1/4*(Cot[a + b*x]*Csc[a + b*x]^3)/b + (ArcTanh[Cos[a + b*x]]/(2*b) + (Cot [a + b*x]*Csc[a + b*x])/(2*b))/4
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1)) Int[(a*Sec[e + f*x])^m*( b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & & NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 1.35 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.24
method | result | size |
derivativedivides | \(\frac {-\frac {\cos \left (b x +a \right )^{3}}{4 \sin \left (b x +a \right )^{4}}-\frac {\cos \left (b x +a \right )^{3}}{8 \sin \left (b x +a \right )^{2}}-\frac {\cos \left (b x +a \right )}{8}-\frac {\ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{8}}{b}\) | \(68\) |
default | \(\frac {-\frac {\cos \left (b x +a \right )^{3}}{4 \sin \left (b x +a \right )^{4}}-\frac {\cos \left (b x +a \right )^{3}}{8 \sin \left (b x +a \right )^{2}}-\frac {\cos \left (b x +a \right )}{8}-\frac {\ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{8}}{b}\) | \(68\) |
risch | \(-\frac {{\mathrm e}^{7 i \left (b x +a \right )}+7 \,{\mathrm e}^{5 i \left (b x +a \right )}+7 \,{\mathrm e}^{3 i \left (b x +a \right )}+{\mathrm e}^{i \left (b x +a \right )}}{4 b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{8 b}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{8 b}\) | \(95\) |
Input:
int(cot(b*x+a)^2*csc(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/b*(-1/4/sin(b*x+a)^4*cos(b*x+a)^3-1/8/sin(b*x+a)^2*cos(b*x+a)^3-1/8*cos( b*x+a)-1/8*ln(csc(b*x+a)-cot(b*x+a)))
Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (49) = 98\).
Time = 0.09 (sec) , antiderivative size = 111, normalized size of antiderivative = 2.02 \[ \int \cot ^2(a+b x) \csc ^3(a+b x) \, dx=-\frac {2 \, \cos \left (b x + a\right )^{3} - {\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + {\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 2 \, \cos \left (b x + a\right )}{16 \, {\left (b \cos \left (b x + a\right )^{4} - 2 \, b \cos \left (b x + a\right )^{2} + b\right )}} \] Input:
integrate(cot(b*x+a)^2*csc(b*x+a)^3,x, algorithm="fricas")
Output:
-1/16*(2*cos(b*x + a)^3 - (cos(b*x + a)^4 - 2*cos(b*x + a)^2 + 1)*log(1/2* cos(b*x + a) + 1/2) + (cos(b*x + a)^4 - 2*cos(b*x + a)^2 + 1)*log(-1/2*cos (b*x + a) + 1/2) + 2*cos(b*x + a))/(b*cos(b*x + a)^4 - 2*b*cos(b*x + a)^2 + b)
\[ \int \cot ^2(a+b x) \csc ^3(a+b x) \, dx=\int \cot ^{2}{\left (a + b x \right )} \csc ^{3}{\left (a + b x \right )}\, dx \] Input:
integrate(cot(b*x+a)**2*csc(b*x+a)**3,x)
Output:
Integral(cot(a + b*x)**2*csc(a + b*x)**3, x)
Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.18 \[ \int \cot ^2(a+b x) \csc ^3(a+b x) \, dx=-\frac {\frac {2 \, {\left (\cos \left (b x + a\right )^{3} + \cos \left (b x + a\right )\right )}}{\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1} - \log \left (\cos \left (b x + a\right ) + 1\right ) + \log \left (\cos \left (b x + a\right ) - 1\right )}{16 \, b} \] Input:
integrate(cot(b*x+a)^2*csc(b*x+a)^3,x, algorithm="maxima")
Output:
-1/16*(2*(cos(b*x + a)^3 + cos(b*x + a))/(cos(b*x + a)^4 - 2*cos(b*x + a)^ 2 + 1) - log(cos(b*x + a) + 1) + log(cos(b*x + a) - 1))/b
Time = 0.13 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.60 \[ \int \cot ^2(a+b x) \csc ^3(a+b x) \, dx=\frac {\log \left ({\left | \frac {1}{\cos \left (b x + a\right )} + \cos \left (b x + a\right ) + 2 \right |}\right )}{32 \, b} - \frac {\log \left ({\left | \frac {1}{\cos \left (b x + a\right )} + \cos \left (b x + a\right ) - 2 \right |}\right )}{32 \, b} - \frac {\frac {1}{\cos \left (b x + a\right )} + \cos \left (b x + a\right )}{8 \, {\left ({\left (\frac {1}{\cos \left (b x + a\right )} + \cos \left (b x + a\right )\right )}^{2} - 4\right )} b} \] Input:
integrate(cot(b*x+a)^2*csc(b*x+a)^3,x, algorithm="giac")
Output:
1/32*log(abs(1/cos(b*x + a) + cos(b*x + a) + 2))/b - 1/32*log(abs(1/cos(b* x + a) + cos(b*x + a) - 2))/b - 1/8*(1/cos(b*x + a) + cos(b*x + a))/(((1/c os(b*x + a) + cos(b*x + a))^2 - 4)*b)
Time = 25.63 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87 \[ \int \cot ^2(a+b x) \csc ^3(a+b x) \, dx=\frac {{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^4}{64\,b}-\frac {1}{64\,b\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^4}-\frac {\ln \left (\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )\right )}{8\,b} \] Input:
int(cot(a + b*x)^2/sin(a + b*x)^3,x)
Output:
tan(a/2 + (b*x)/2)^4/(64*b) - 1/(64*b*tan(a/2 + (b*x)/2)^4) - log(tan(a/2 + (b*x)/2))/(8*b)
Time = 0.16 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.95 \[ \int \cot ^2(a+b x) \csc ^3(a+b x) \, dx=\frac {-8 \,\mathrm {log}\left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{8}-1}{64 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4} b} \] Input:
int(cot(b*x+a)^2*csc(b*x+a)^3,x)
Output:
( - 8*log(tan((a + b*x)/2))*tan((a + b*x)/2)**4 + tan((a + b*x)/2)**8 - 1) /(64*tan((a + b*x)/2)**4*b)