Integrand size = 27, antiderivative size = 72 \[ \int \frac {\cot (c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {b \csc (c+d x)}{a^2 d}-\frac {\csc ^2(c+d x)}{2 a d}+\frac {b^2 \log (\sin (c+d x))}{a^3 d}-\frac {b^2 \log (a+b \sin (c+d x))}{a^3 d} \] Output:
b*csc(d*x+c)/a^2/d-1/2*csc(d*x+c)^2/a/d+b^2*ln(sin(d*x+c))/a^3/d-b^2*ln(a+ b*sin(d*x+c))/a^3/d
Time = 0.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {b \csc (c+d x)}{a^2 d}-\frac {\csc ^2(c+d x)}{2 a d}+\frac {b^2 \log (\sin (c+d x))}{a^3 d}-\frac {b^2 \log (a+b \sin (c+d x))}{a^3 d} \] Input:
Integrate[(Cot[c + d*x]*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]),x]
Output:
(b*Csc[c + d*x])/(a^2*d) - Csc[c + d*x]^2/(2*a*d) + (b^2*Log[Sin[c + d*x]] )/(a^3*d) - (b^2*Log[a + b*Sin[c + d*x]])/(a^3*d)
Time = 0.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 3312, 27, 54, 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.
\(\displaystyle \int \frac {\cot (c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)}{\sin (c+d x)^3 (a+b \sin (c+d x))}dx\) |
\(\Big \downarrow \) 3312 |
\(\displaystyle \frac {\int \frac {\csc ^3(c+d x)}{a+b \sin (c+d x)}d(b \sin (c+d x))}{b d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b^2 \int \frac {\csc ^3(c+d x)}{b^3 (a+b \sin (c+d x))}d(b \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {b^2 \int \left (\frac {\csc ^3(c+d x)}{a b^3}-\frac {\csc ^2(c+d x)}{a^2 b^2}+\frac {\csc (c+d x)}{a^3 b}-\frac {1}{a^3 (a+b \sin (c+d x))}\right )d(b \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b^2 \left (\frac {\log (b \sin (c+d x))}{a^3}-\frac {\log (a+b \sin (c+d x))}{a^3}+\frac {\csc (c+d x)}{a^2 b}-\frac {\csc ^2(c+d x)}{2 a b^2}\right )}{d}\) |
Input:
Int[(Cot[c + d*x]*Csc[c + d*x]^2)/(a + b*Sin[c + d*x]),x]
Output:
(b^2*(Csc[c + d*x]/(a^2*b) - Csc[c + d*x]^2/(2*a*b^2) + Log[b*Sin[c + d*x] ]/a^3 - Log[a + b*Sin[c + d*x]]/a^3))/d
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*(( c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b*f) Su bst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 0.35 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(-\frac {\csc \left (d x +c \right )^{2}}{2 a d}+\frac {b \csc \left (d x +c \right )}{a^{2} d}-\frac {b^{2} \ln \left (\csc \left (d x +c \right ) a +b \right )}{d \,a^{3}}\) | \(54\) |
default | \(-\frac {\csc \left (d x +c \right )^{2}}{2 a d}+\frac {b \csc \left (d x +c \right )}{a^{2} d}-\frac {b^{2} \ln \left (\csc \left (d x +c \right ) a +b \right )}{d \,a^{3}}\) | \(54\) |
risch | \(\frac {2 i \left (-i a \,{\mathrm e}^{2 i \left (d x +c \right )}+b \,{\mathrm e}^{3 i \left (d x +c \right )}-b \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{d \,a^{3}}+\frac {b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d \,a^{3}}\) | \(122\) |
Input:
int(cot(d*x+c)*csc(d*x+c)^2/(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-1/2*csc(d*x+c)^2/a/d+b*csc(d*x+c)/a^2/d-1/d*b^2/a^3*ln(csc(d*x+c)*a+b)
Time = 0.10 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.39 \[ \int \frac {\cot (c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 \, a b \sin \left (d x + c\right ) - a^{2} + 2 \, {\left (b^{2} \cos \left (d x + c\right )^{2} - b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 2 \, {\left (b^{2} \cos \left (d x + c\right )^{2} - b^{2}\right )} \log \left (-\frac {1}{2} \, \sin \left (d x + c\right )\right )}{2 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\right )}} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="fricas")
Output:
-1/2*(2*a*b*sin(d*x + c) - a^2 + 2*(b^2*cos(d*x + c)^2 - b^2)*log(b*sin(d* x + c) + a) - 2*(b^2*cos(d*x + c)^2 - b^2)*log(-1/2*sin(d*x + c)))/(a^3*d* cos(d*x + c)^2 - a^3*d)
\[ \int \frac {\cot (c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\cot {\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \] Input:
integrate(cot(d*x+c)*csc(d*x+c)**2/(a+b*sin(d*x+c)),x)
Output:
Integral(cot(c + d*x)*csc(c + d*x)**2/(a + b*sin(c + d*x)), x)
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.92 \[ \int \frac {\cot (c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\frac {2 \, b^{2} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{3}} - \frac {2 \, b^{2} \log \left (\sin \left (d x + c\right )\right )}{a^{3}} - \frac {2 \, b \sin \left (d x + c\right ) - a}{a^{2} \sin \left (d x + c\right )^{2}}}{2 \, d} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="maxima")
Output:
-1/2*(2*b^2*log(b*sin(d*x + c) + a)/a^3 - 2*b^2*log(sin(d*x + c))/a^3 - (2 *b*sin(d*x + c) - a)/(a^2*sin(d*x + c)^2))/d
Time = 0.16 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.03 \[ \int \frac {\cot (c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {b^{2} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{3} d} + \frac {b^{2} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{3} d} + \frac {2 \, a b \sin \left (d x + c\right ) - a^{2}}{2 \, a^{3} d \sin \left (d x + c\right )^{2}} \] Input:
integrate(cot(d*x+c)*csc(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="giac")
Output:
-b^2*log(abs(b*sin(d*x + c) + a))/(a^3*d) + b^2*log(abs(sin(d*x + c)))/(a^ 3*d) + 1/2*(2*a*b*sin(d*x + c) - a^2)/(a^3*d*sin(d*x + c)^2)
Time = 19.84 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.83 \[ \int \frac {\cot (c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^2\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a\,d}-\frac {b^2\,\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}{a^3\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a}{2}-2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,a^2\,d}+\frac {b^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d} \] Input:
int(cot(c + d*x)/(sin(c + d*x)^2*(a + b*sin(c + d*x))),x)
Output:
(b*tan(c/2 + (d*x)/2))/(2*a^2*d) - tan(c/2 + (d*x)/2)^2/(8*a*d) - (b^2*log (a + 2*b*tan(c/2 + (d*x)/2) + a*tan(c/2 + (d*x)/2)^2))/(a^3*d) - (cot(c/2 + (d*x)/2)^2*(a/2 - 2*b*tan(c/2 + (d*x)/2)))/(4*a^2*d) + (b^2*log(tan(c/2 + (d*x)/2)))/(a^3*d)
Time = 0.18 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.50 \[ \int \frac {\cot (c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-4 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right ) \sin \left (d x +c \right )^{2} b^{2}+4 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{2} b^{2}+\sin \left (d x +c \right )^{2} a^{2}+4 \sin \left (d x +c \right ) a b -2 a^{2}}{4 \sin \left (d x +c \right )^{2} a^{3} d} \] Input:
int(cot(d*x+c)*csc(d*x+c)^2/(a+b*sin(d*x+c)),x)
Output:
( - 4*log(tan((c + d*x)/2)**2*a + 2*tan((c + d*x)/2)*b + a)*sin(c + d*x)** 2*b**2 + 4*log(tan((c + d*x)/2))*sin(c + d*x)**2*b**2 + sin(c + d*x)**2*a* *2 + 4*sin(c + d*x)*a*b - 2*a**2)/(4*sin(c + d*x)**2*a**3*d)