Integrand size = 16, antiderivative size = 55 \[ \int \frac {\csc ^3(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {b \cot (x)}{a^2}-\frac {\csc ^2(x)}{2 a}+\frac {\left (a^2+b^2\right ) \log (\sin (x))}{a^3}-\frac {\left (a^2+b^2\right ) \log (a \cos (x)+b \sin (x))}{a^3} \] Output:
b*cot(x)/a^2-1/2*csc(x)^2/a+(a^2+b^2)*ln(sin(x))/a^3-(a^2+b^2)*ln(a*cos(x) +b*sin(x))/a^3
Time = 0.85 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87 \[ \int \frac {\csc ^3(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {2 a b \cot (x)-a^2 \csc ^2(x)+2 \left (a^2+b^2\right ) (\log (\sin (x))-\log (a \cos (x)+b \sin (x)))}{2 a^3} \] Input:
Integrate[Csc[x]^3/(a*Cos[x] + b*Sin[x]),x]
Output:
(2*a*b*Cot[x] - a^2*Csc[x]^2 + 2*(a^2 + b^2)*(Log[Sin[x]] - Log[a*Cos[x] + b*Sin[x]]))/(2*a^3)
Time = 0.49 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {3042, 3582, 3042, 3580, 3042, 25, 3612, 3956, 4254, 24}
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.
\(\displaystyle \int \frac {\csc ^3(x)}{a \cos (x)+b \sin (x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin (x)^3 (a \cos (x)+b \sin (x))}dx\) |
\(\Big \downarrow \) 3582 |
\(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {\csc (x)}{a \cos (x)+b \sin (x)}dx}{a^2}-\frac {b \int \csc ^2(x)dx}{a^2}-\frac {\csc ^2(x)}{2 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\left (a^2+b^2\right ) \int \frac {1}{\sin (x) (a \cos (x)+b \sin (x))}dx}{a^2}-\frac {b \int \csc (x)^2dx}{a^2}-\frac {\csc ^2(x)}{2 a}\) |
\(\Big \downarrow \) 3580 |
\(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\int \cot (x)dx}{a}-\frac {\int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a}\right )}{a^2}-\frac {b \int \csc (x)^2dx}{a^2}-\frac {\csc ^2(x)}{2 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\int -\tan \left (x+\frac {\pi }{2}\right )dx}{a}-\frac {\int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a}\right )}{a^2}-\frac {b \int \csc (x)^2dx}{a^2}-\frac {\csc ^2(x)}{2 a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\left (a^2+b^2\right ) \left (-\frac {\int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)}dx}{a}-\frac {\int \tan \left (x+\frac {\pi }{2}\right )dx}{a}\right )}{a^2}-\frac {b \int \csc (x)^2dx}{a^2}-\frac {\csc ^2(x)}{2 a}\) |
\(\Big \downarrow \) 3612 |
\(\displaystyle \frac {\left (a^2+b^2\right ) \left (-\frac {\int \tan \left (x+\frac {\pi }{2}\right )dx}{a}-\frac {\log (a \cos (x)+b \sin (x))}{a}\right )}{a^2}-\frac {b \int \csc (x)^2dx}{a^2}-\frac {\csc ^2(x)}{2 a}\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle -\frac {b \int \csc (x)^2dx}{a^2}+\frac {\left (a^2+b^2\right ) \left (\frac {\log (\sin (x))}{a}-\frac {\log (a \cos (x)+b \sin (x))}{a}\right )}{a^2}-\frac {\csc ^2(x)}{2 a}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle \frac {b \int 1d\cot (x)}{a^2}+\frac {\left (a^2+b^2\right ) \left (\frac {\log (\sin (x))}{a}-\frac {\log (a \cos (x)+b \sin (x))}{a}\right )}{a^2}-\frac {\csc ^2(x)}{2 a}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\left (a^2+b^2\right ) \left (\frac {\log (\sin (x))}{a}-\frac {\log (a \cos (x)+b \sin (x))}{a}\right )}{a^2}+\frac {b \cot (x)}{a^2}-\frac {\csc ^2(x)}{2 a}\) |
Input:
Int[Csc[x]^3/(a*Cos[x] + b*Sin[x]),x]
Output:
(b*Cot[x])/a^2 - Csc[x]^2/(2*a) + ((a^2 + b^2)*(Log[Sin[x]]/a - Log[a*Cos[ x] + b*Sin[x]]/a))/a^2
Int[1/(sin[(c_.) + (d_.)*(x_)]*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[( c_.) + (d_.)*(x_)])), x_Symbol] :> Simp[1/a Int[Cot[c + d*x], x], x] - Si mp[1/a Int[(b*Cos[c + d*x] - a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
Int[sin[(c_.) + (d_.)*(x_)]^(m_)/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin [(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[Sin[c + d*x]^(m + 1)/(a*d*(m + 1)) , x] + (-Simp[b/a^2 Int[Sin[c + d*x]^(m + 1), x], x] + Simp[(a^2 + b^2)/a ^2 Int[Sin[c + d*x]^(m + 2)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]
Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]) /((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]), x _Symbol] :> Simp[(b*B + c*C)*(x/(b^2 + c^2)), x] + Simp[(c*B - b*C)*(Log[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/(e*(b^2 + c^2))), x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[b^2 + c^2, 0] && EqQ[A*(b^2 + c^2) - a*(b*B + c*C ), 0]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Time = 0.22 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96
method | result | size |
default | \(-\frac {\left (a^{2}+b^{2}\right ) \ln \left (a +b \tan \left (x \right )\right )}{a^{3}}-\frac {1}{2 a \tan \left (x \right )^{2}}+\frac {\left (a^{2}+b^{2}\right ) \ln \left (\tan \left (x \right )\right )}{a^{3}}+\frac {b}{a^{2} \tan \left (x \right )}\) | \(53\) |
norman | \(\frac {-\frac {1}{8 a}-\frac {\tan \left (\frac {x}{2}\right )^{4}}{8 a}+\frac {b \tan \left (\frac {x}{2}\right )}{2 a^{2}}-\frac {b \tan \left (\frac {x}{2}\right )^{3}}{2 a^{2}}}{\tan \left (\frac {x}{2}\right )^{2}}+\frac {\left (a^{2}+b^{2}\right ) \ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{3}}-\frac {\left (a^{2}+b^{2}\right ) \ln \left (\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a \right )}{a^{3}}\) | \(96\) |
parallelrisch | \(-\frac {-8 a^{2} \left (\ln \left (\tan \left (\frac {x}{2}\right )\right )-\ln \left (\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a \right )\right )-8 b^{2} \left (\ln \left (\tan \left (\frac {x}{2}\right )\right )-\ln \left (\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a \right )\right )+a^{2} \cot \left (\frac {x}{2}\right )^{2}+a^{2} \tan \left (\frac {x}{2}\right )^{2}-4 a b \cot \left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right ) a b}{8 a^{3}}\) | \(109\) |
risch | \(\frac {2 i \left (-i a \,{\mathrm e}^{2 i x}+b \,{\mathrm e}^{2 i x}-b \right )}{\left ({\mathrm e}^{2 i x}-1\right )^{2} a^{2}}+\frac {\ln \left ({\mathrm e}^{2 i x}-1\right )}{a}+\frac {\ln \left ({\mathrm e}^{2 i x}-1\right ) b^{2}}{a^{3}}-\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {i b +a}{i b -a}\right )}{a}-\frac {\ln \left ({\mathrm e}^{2 i x}-\frac {i b +a}{i b -a}\right ) b^{2}}{a^{3}}\) | \(127\) |
Input:
int(csc(x)^3/(a*cos(x)+b*sin(x)),x,method=_RETURNVERBOSE)
Output:
-(a^2+b^2)/a^3*ln(a+b*tan(x))-1/2/a/tan(x)^2+(a^2+b^2)/a^3*ln(tan(x))+b/a^ 2/tan(x)
Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (53) = 106\).
Time = 0.09 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.13 \[ \int \frac {\csc ^3(x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {2 \, a b \cos \left (x\right ) \sin \left (x\right ) - a^{2} + {\left ({\left (a^{2} + b^{2}\right )} \cos \left (x\right )^{2} - a^{2} - b^{2}\right )} \log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}\right ) - {\left ({\left (a^{2} + b^{2}\right )} \cos \left (x\right )^{2} - a^{2} - b^{2}\right )} \log \left (-\frac {1}{4} \, \cos \left (x\right )^{2} + \frac {1}{4}\right )}{2 \, {\left (a^{3} \cos \left (x\right )^{2} - a^{3}\right )}} \] Input:
integrate(csc(x)^3/(a*cos(x)+b*sin(x)),x, algorithm="fricas")
Output:
-1/2*(2*a*b*cos(x)*sin(x) - a^2 + ((a^2 + b^2)*cos(x)^2 - a^2 - b^2)*log(2 *a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2) - ((a^2 + b^2)*cos(x)^2 - a^2 - b^2)*log(-1/4*cos(x)^2 + 1/4))/(a^3*cos(x)^2 - a^3)
\[ \int \frac {\csc ^3(x)}{a \cos (x)+b \sin (x)} \, dx=\int \frac {\csc ^{3}{\left (x \right )}}{a \cos {\left (x \right )} + b \sin {\left (x \right )}}\, dx \] Input:
integrate(csc(x)**3/(a*cos(x)+b*sin(x)),x)
Output:
Integral(csc(x)**3/(a*cos(x) + b*sin(x)), x)
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (53) = 106\).
Time = 0.03 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.16 \[ \int \frac {\csc ^3(x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {\frac {4 \, b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}}{8 \, a^{2}} - \frac {{\left (a^{2} + b^{2}\right )} \log \left (-a - \frac {2 \, b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{a^{3}} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{3}} - \frac {{\left (a - \frac {4 \, b \sin \left (x\right )}{\cos \left (x\right ) + 1}\right )} {\left (\cos \left (x\right ) + 1\right )}^{2}}{8 \, a^{2} \sin \left (x\right )^{2}} \] Input:
integrate(csc(x)^3/(a*cos(x)+b*sin(x)),x, algorithm="maxima")
Output:
-1/8*(4*b*sin(x)/(cos(x) + 1) + a*sin(x)^2/(cos(x) + 1)^2)/a^2 - (a^2 + b^ 2)*log(-a - 2*b*sin(x)/(cos(x) + 1) + a*sin(x)^2/(cos(x) + 1)^2)/a^3 + (a^ 2 + b^2)*log(sin(x)/(cos(x) + 1))/a^3 - 1/8*(a - 4*b*sin(x)/(cos(x) + 1))* (cos(x) + 1)^2/(a^2*sin(x)^2)
Time = 0.12 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.42 \[ \int \frac {\csc ^3(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {{\left (a^{2} + b^{2}\right )} \log \left ({\left | \tan \left (x\right ) \right |}\right )}{a^{3}} - \frac {{\left (a^{2} b + b^{3}\right )} \log \left ({\left | b \tan \left (x\right ) + a \right |}\right )}{a^{3} b} - \frac {3 \, a^{2} \tan \left (x\right )^{2} + 3 \, b^{2} \tan \left (x\right )^{2} - 2 \, a b \tan \left (x\right ) + a^{2}}{2 \, a^{3} \tan \left (x\right )^{2}} \] Input:
integrate(csc(x)^3/(a*cos(x)+b*sin(x)),x, algorithm="giac")
Output:
(a^2 + b^2)*log(abs(tan(x)))/a^3 - (a^2*b + b^3)*log(abs(b*tan(x) + a))/(a ^3*b) - 1/2*(3*a^2*tan(x)^2 + 3*b^2*tan(x)^2 - 2*a*b*tan(x) + a^2)/(a^3*ta n(x)^2)
Time = 16.96 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.65 \[ \int \frac {\csc ^3(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )\,\left (a^2+b^2\right )}{a^3}-\frac {\ln \left (-a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )+a\right )\,\left (a^2+b^2\right )}{a^3}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{8\,a}-\frac {b\,\mathrm {tan}\left (\frac {x}{2}\right )}{2\,a^2}-\frac {\frac {a}{2}-2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )}{4\,a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2} \] Input:
int(1/(sin(x)^3*(a*cos(x) + b*sin(x))),x)
Output:
(log(tan(x/2))*(a^2 + b^2))/a^3 - (log(a + 2*b*tan(x/2) - a*tan(x/2)^2)*(a ^2 + b^2))/a^3 - tan(x/2)^2/(8*a) - (b*tan(x/2))/(2*a^2) - (a/2 - 2*b*tan( x/2))/(4*a^2*tan(x/2)^2)
Time = 0.20 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.13 \[ \int \frac {\csc ^3(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {4 \cos \left (x \right ) \sin \left (x \right ) a b -4 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} a -2 \tan \left (\frac {x}{2}\right ) b -a \right ) \sin \left (x \right )^{2} a^{2}-4 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )^{2} a -2 \tan \left (\frac {x}{2}\right ) b -a \right ) \sin \left (x \right )^{2} b^{2}+4 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{2} a^{2}+4 \,\mathrm {log}\left (\tan \left (\frac {x}{2}\right )\right ) \sin \left (x \right )^{2} b^{2}+\sin \left (x \right )^{2} a^{2}-2 a^{2}}{4 \sin \left (x \right )^{2} a^{3}} \] Input:
int(csc(x)^3/(a*cos(x)+b*sin(x)),x)
Output:
(4*cos(x)*sin(x)*a*b - 4*log(tan(x/2)**2*a - 2*tan(x/2)*b - a)*sin(x)**2*a **2 - 4*log(tan(x/2)**2*a - 2*tan(x/2)*b - a)*sin(x)**2*b**2 + 4*log(tan(x /2))*sin(x)**2*a**2 + 4*log(tan(x/2))*sin(x)**2*b**2 + sin(x)**2*a**2 - 2* a**2)/(4*sin(x)**2*a**3)