Integrand size = 11, antiderivative size = 79 \[ \int (a \cot (x)+b \csc (x))^3 \, dx=-\frac {(a+b)^3}{4 (1-\cos (x))}-\frac {(a-b)^3}{4 (1+\cos (x))}-\frac {1}{4} (2 a-b) (a+b)^2 \log (1-\cos (x))-\frac {1}{4} (a-b)^2 (2 a+b) \log (1+\cos (x)) \] Output:
-1/4*(a+b)^3/(1-cos(x))-(a-b)^3/(4+4*cos(x))-1/4*(2*a-b)*(a+b)^2*ln(1-cos( x))-1/4*(a-b)^2*(2*a+b)*ln(1+cos(x))
Time = 0.26 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00 \[ \int (a \cot (x)+b \csc (x))^3 \, dx=\frac {1}{8} \left (-(a+b)^3 \csc ^2\left (\frac {x}{2}\right )-4 (a-b)^2 (2 a+b) \log \left (\cos \left (\frac {x}{2}\right )\right )-4 (2 a-b) (a+b)^2 \log \left (\sin \left (\frac {x}{2}\right )\right )-(a-b)^3 \sec ^2\left (\frac {x}{2}\right )\right ) \] Input:
Integrate[(a*Cot[x] + b*Csc[x])^3,x]
Output:
(-((a + b)^3*Csc[x/2]^2) - 4*(a - b)^2*(2*a + b)*Log[Cos[x/2]] - 4*(2*a - b)*(a + b)^2*Log[Sin[x/2]] - (a - b)^3*Sec[x/2]^2)/8
Time = 0.37 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.24, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {3042, 4892, 3042, 3147, 477, 2009}
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.
\(\displaystyle \int (a \cot (x)+b \csc (x))^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a \cot (x)+b \csc (x))^3dx\) |
\(\Big \downarrow \) 4892 |
\(\displaystyle \int \csc ^3(x) (a \cos (x)+b)^3dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (b-a \sin \left (x-\frac {\pi }{2}\right )\right )^3}{\cos \left (x-\frac {\pi }{2}\right )^3}dx\) |
\(\Big \downarrow \) 3147 |
\(\displaystyle -a^3 \int \frac {(b+a \cos (x))^3}{\left (a^2-a^2 \cos ^2(x)\right )^2}d(a \cos (x))\) |
\(\Big \downarrow \) 477 |
\(\displaystyle -\frac {\int \left (-\frac {a^2 (a-b)^3}{4 (\cos (x) a+a)^2}+\frac {a (2 a+b) (a-b)^2}{4 (\cos (x) a+a)}-\frac {a (2 a-b) (a+b)^2}{4 (a-a \cos (x))}+\frac {a^2 (a+b)^3}{4 (a-a \cos (x))^2}\right )d(a \cos (x))}{a}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\frac {a^2 (a-b)^3}{4 (a \cos (x)+a)}+\frac {a^2 (a+b)^3}{4 (a-a \cos (x))}+\frac {1}{4} a (2 a+b) (a-b)^2 \log (a \cos (x)+a)+\frac {1}{4} a (2 a-b) (a+b)^2 \log (a-a \cos (x))}{a}\) |
Input:
Int[(a*Cot[x] + b*Csc[x])^3,x]
Output:
-(((a^2*(a + b)^3)/(4*(a - a*Cos[x])) + (a^2*(a - b)^3)/(4*(a + a*Cos[x])) + (a*(2*a - b)*(a + b)^2*Log[a - a*Cos[x]])/4 + (a*(a - b)^2*(2*a + b)*Lo g[a + a*Cos[x]])/4)/a)
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ a^p Int[ExpandIntegrand[(c + d*x)^n*(1 - Rt[-b/a, 2]*x)^p*(1 + Rt[-b/a, 2 ]*x)^p, x], x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[n] & & NiceSqrtQ[-b/a] && !FractionalPowerFactorQ[Rt[-b/a, 2]]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Int[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b _.))^(p_)*(u_.), x_Symbol] :> Int[ActivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a *Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]
Time = 2.88 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.01
method | result | size |
default | \(b^{3} \left (-\frac {\csc \left (x \right ) \cot \left (x \right )}{2}+\frac {\ln \left (-\cot \left (x \right )+\csc \left (x \right )\right )}{2}\right )-\frac {3 a \,b^{2}}{2 \sin \left (x \right )^{2}}+3 a^{2} b \left (-\frac {\cos \left (x \right )^{3}}{2 \sin \left (x \right )^{2}}-\frac {\cos \left (x \right )}{2}-\frac {\ln \left (-\cot \left (x \right )+\csc \left (x \right )\right )}{2}\right )+a^{3} \left (-\frac {\cot \left (x \right )^{2}}{2}-\ln \left (\sin \left (x \right )\right )\right )\) | \(80\) |
parts | \(a^{3} \left (-\frac {\cot \left (x \right )^{2}}{2}+\frac {\ln \left (1+\cot \left (x \right )^{2}\right )}{2}\right )+b^{3} \left (-\frac {\csc \left (x \right ) \cot \left (x \right )}{2}+\frac {\ln \left (-\cot \left (x \right )+\csc \left (x \right )\right )}{2}\right )+3 a^{2} b \left (-\frac {\cos \left (x \right )^{3}}{2 \sin \left (x \right )^{2}}-\frac {\cos \left (x \right )}{2}-\frac {\ln \left (-\cot \left (x \right )+\csc \left (x \right )\right )}{2}\right )-\frac {3 a \,b^{2} \cot \left (x \right )^{2}}{2}\) | \(84\) |
risch | \(i a^{3} x +\frac {{\mathrm e}^{i x} \left (3 a^{2} b \,{\mathrm e}^{2 i x}+b^{3} {\mathrm e}^{2 i x}+2 a^{3} {\mathrm e}^{i x}+6 a \,b^{2} {\mathrm e}^{i x}+3 a^{2} b +b^{3}\right )}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}-\ln \left ({\mathrm e}^{i x}+1\right ) a^{3}+\frac {3 \ln \left ({\mathrm e}^{i x}+1\right ) a^{2} b}{2}-\frac {\ln \left ({\mathrm e}^{i x}+1\right ) b^{3}}{2}-\ln \left ({\mathrm e}^{i x}-1\right ) a^{3}-\frac {3 \ln \left ({\mathrm e}^{i x}-1\right ) a^{2} b}{2}+\frac {\ln \left ({\mathrm e}^{i x}-1\right ) b^{3}}{2}\) | \(161\) |
Input:
int((a*cot(x)+b*csc(x))^3,x,method=_RETURNVERBOSE)
Output:
b^3*(-1/2*csc(x)*cot(x)+1/2*ln(-cot(x)+csc(x)))-3/2*a*b^2/sin(x)^2+3*a^2*b *(-1/2/sin(x)^2*cos(x)^3-1/2*cos(x)-1/2*ln(-cot(x)+csc(x)))+a^3*(-1/2*cot( x)^2-ln(sin(x)))
Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.62 \[ \int (a \cot (x)+b \csc (x))^3 \, dx=\frac {2 \, a^{3} + 6 \, a b^{2} + 2 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (x\right ) + {\left (2 \, a^{3} - 3 \, a^{2} b + b^{3} - {\left (2 \, a^{3} - 3 \, a^{2} b + b^{3}\right )} \cos \left (x\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + {\left (2 \, a^{3} + 3 \, a^{2} b - b^{3} - {\left (2 \, a^{3} + 3 \, a^{2} b - b^{3}\right )} \cos \left (x\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right )}{4 \, {\left (\cos \left (x\right )^{2} - 1\right )}} \] Input:
integrate((a*cot(x)+b*csc(x))^3,x, algorithm="fricas")
Output:
1/4*(2*a^3 + 6*a*b^2 + 2*(3*a^2*b + b^3)*cos(x) + (2*a^3 - 3*a^2*b + b^3 - (2*a^3 - 3*a^2*b + b^3)*cos(x)^2)*log(1/2*cos(x) + 1/2) + (2*a^3 + 3*a^2* b - b^3 - (2*a^3 + 3*a^2*b - b^3)*cos(x)^2)*log(-1/2*cos(x) + 1/2))/(cos(x )^2 - 1)
Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (61) = 122\).
Time = 5.65 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.57 \[ \int (a \cot (x)+b \csc (x))^3 \, dx=\frac {a^{3} \log {\left (- \csc ^{2}{\left (x \right )} \right )}}{2} - \frac {a^{3} \csc ^{2}{\left (x \right )}}{2} - \frac {3 a^{2} b \log {\left (\cos {\left (x \right )} - 1 \right )}}{4} + \frac {3 a^{2} b \log {\left (\cos {\left (x \right )} + 1 \right )}}{4} + \frac {3 a^{2} b \cos {\left (x \right )}}{2 \cos ^{2}{\left (x \right )} - 2} - \frac {3 a b^{2} \csc ^{2}{\left (x \right )}}{2} + \frac {b^{3} \log {\left (\cos {\left (x \right )} - 1 \right )}}{4} - \frac {b^{3} \log {\left (\cos {\left (x \right )} + 1 \right )}}{4} + \frac {b^{3} \cos {\left (x \right )}}{2 \cos ^{2}{\left (x \right )} - 2} \] Input:
integrate((a*cot(x)+b*csc(x))**3,x)
Output:
a**3*log(-csc(x)**2)/2 - a**3*csc(x)**2/2 - 3*a**2*b*log(cos(x) - 1)/4 + 3 *a**2*b*log(cos(x) + 1)/4 + 3*a**2*b*cos(x)/(2*cos(x)**2 - 2) - 3*a*b**2*c sc(x)**2/2 + b**3*log(cos(x) - 1)/4 - b**3*log(cos(x) + 1)/4 + b**3*cos(x) /(2*cos(x)**2 - 2)
Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.10 \[ \int (a \cot (x)+b \csc (x))^3 \, dx=-\frac {3}{2} \, a b^{2} \cot \left (x\right )^{2} + \frac {3}{4} \, a^{2} b {\left (\frac {2 \, \cos \left (x\right )}{\cos \left (x\right )^{2} - 1} + \log \left (\cos \left (x\right ) + 1\right ) - \log \left (\cos \left (x\right ) - 1\right )\right )} + \frac {1}{4} \, b^{3} {\left (\frac {2 \, \cos \left (x\right )}{\cos \left (x\right )^{2} - 1} - \log \left (\cos \left (x\right ) + 1\right ) + \log \left (\cos \left (x\right ) - 1\right )\right )} - \frac {1}{2} \, a^{3} {\left (\frac {1}{\sin \left (x\right )^{2}} + \log \left (\sin \left (x\right )^{2}\right )\right )} \] Input:
integrate((a*cot(x)+b*csc(x))^3,x, algorithm="maxima")
Output:
-3/2*a*b^2*cot(x)^2 + 3/4*a^2*b*(2*cos(x)/(cos(x)^2 - 1) + log(cos(x) + 1) - log(cos(x) - 1)) + 1/4*b^3*(2*cos(x)/(cos(x)^2 - 1) - log(cos(x) + 1) + log(cos(x) - 1)) - 1/2*a^3*(1/sin(x)^2 + log(sin(x)^2))
Time = 0.13 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.09 \[ \int (a \cot (x)+b \csc (x))^3 \, dx=-\frac {1}{4} \, {\left (2 \, a^{3} - 3 \, a^{2} b + b^{3}\right )} \log \left (\cos \left (x\right ) + 1\right ) - \frac {1}{4} \, {\left (2 \, a^{3} + 3 \, a^{2} b - b^{3}\right )} \log \left (-\cos \left (x\right ) + 1\right ) + \frac {a^{3} + 3 \, a b^{2} + {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (x\right )}{2 \, {\left (\cos \left (x\right ) + 1\right )} {\left (\cos \left (x\right ) - 1\right )}} \] Input:
integrate((a*cot(x)+b*csc(x))^3,x, algorithm="giac")
Output:
-1/4*(2*a^3 - 3*a^2*b + b^3)*log(cos(x) + 1) - 1/4*(2*a^3 + 3*a^2*b - b^3) *log(-cos(x) + 1) + 1/2*(a^3 + 3*a*b^2 + (3*a^2*b + b^3)*cos(x))/((cos(x) + 1)*(cos(x) - 1))
Time = 16.38 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.04 \[ \int (a \cot (x)+b \csc (x))^3 \, dx=a^3\,\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )-\frac {\frac {a^3}{8}+\frac {3\,a^2\,b}{8}+\frac {3\,a\,b^2}{8}+\frac {b^3}{8}}{{\mathrm {tan}\left (\frac {x}{2}\right )}^2}-\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )\,\left (a^3+\frac {3\,a^2\,b}{2}-\frac {b^3}{2}\right )-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,{\left (a-b\right )}^3}{8} \] Input:
int((b/sin(x) + a*cot(x))^3,x)
Output:
a^3*log(tan(x/2)^2 + 1) - ((3*a*b^2)/8 + (3*a^2*b)/8 + a^3/8 + b^3/8)/tan( x/2)^2 - log(tan(x/2))*((3*a^2*b)/2 + a^3 - b^3/2) - (tan(x/2)^2*(a - b)^3 )/8
Time = 0.17 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.42 \[ \int (a \cot (x)+b \csc (x))^3 \, dx=\frac {-6 \cos \left (x \right ) a^{2} b -2 \cos \left (x \right ) b^{3}+4 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2}+1\right ) \sin \left (x \right )^{2} a^{3}-4 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{2} a^{3}-6 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{2} a^{2} b +2 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{2} b^{3}+\sin \left (x \right )^{2} a^{3}+3 \sin \left (x \right )^{2} a \,b^{2}-2 a^{3}-6 a \,b^{2}}{4 \sin \left (x \right )^{2}} \] Input:
int((a*cot(x)+b*csc(x))^3,x)
Output:
( - 6*cos(x)*a**2*b - 2*cos(x)*b**3 + 4*log(tan(x/2)**2 + 1)*sin(x)**2*a** 3 - 4*log(tan(x/2))*sin(x)**2*a**3 - 6*log(tan(x/2))*sin(x)**2*a**2*b + 2* log(tan(x/2))*sin(x)**2*b**3 + sin(x)**2*a**3 + 3*sin(x)**2*a*b**2 - 2*a** 3 - 6*a*b**2)/(4*sin(x)**2)