Integrand size = 27, antiderivative size = 120 \[ \int \frac {\cos ^4(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\log (\sin (c+d x))}{a^2 d}+\frac {\left (a^2-b^2\right ) \left (3 a^2+b^2\right ) \log (a+b \sin (c+d x))}{a^2 b^4 d}-\frac {2 a \sin (c+d x)}{b^3 d}+\frac {\sin ^2(c+d x)}{2 b^2 d}+\frac {\left (a^2-b^2\right )^2}{a b^4 d (a+b \sin (c+d x))} \] Output:
ln(sin(d*x+c))/a^2/d+(a^2-b^2)*(3*a^2+b^2)*ln(a+b*sin(d*x+c))/a^2/b^4/d-2* a*sin(d*x+c)/b^3/d+1/2*sin(d*x+c)^2/b^2/d+(a^2-b^2)^2/a/b^4/d/(a+b*sin(d*x +c))
Time = 0.55 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^4(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {2 \log (\sin (c+d x))}{a^2}+\frac {2 (a-b) (a+b) \left (3 a^2+b^2\right ) \log (a+b \sin (c+d x))}{a^2 b^4}-\frac {4 a \sin (c+d x)}{b^3}+\frac {\sin ^2(c+d x)}{b^2}+\frac {2 \left (a^2-b^2\right )^2}{a b^4 (a+b \sin (c+d x))}}{2 d} \] Input:
Integrate[(Cos[c + d*x]^4*Cot[c + d*x])/(a + b*Sin[c + d*x])^2,x]
Output:
((2*Log[Sin[c + d*x]])/a^2 + (2*(a - b)*(a + b)*(3*a^2 + b^2)*Log[a + b*Si n[c + d*x]])/(a^2*b^4) - (4*a*Sin[c + d*x])/b^3 + Sin[c + d*x]^2/b^2 + (2* (a^2 - b^2)^2)/(a*b^4*(a + b*Sin[c + d*x])))/(2*d)
Time = 0.36 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.91, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 3316, 27, 522, 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.
| \(\displaystyle \int \frac {\cos ^4(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^5}{\sin (c+d x) (a+b \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle \frac {\int \frac {\csc (c+d x) \left (b^2-b^2 \sin ^2(c+d x)\right )^2}{(a+b \sin (c+d x))^2}d(b \sin (c+d x))}{b^5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\csc (c+d x) \left (b^2-b^2 \sin ^2(c+d x)\right )^2}{b (a+b \sin (c+d x))^2}d(b \sin (c+d x))}{b^4 d}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \frac {\int \left (\frac {\csc (c+d x) b^3}{a^2}+\sin (c+d x) b-2 a+\frac {\left (a^2-b^2\right ) \left (3 a^2+b^2\right )}{a^2 (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right )^2}{a (a+b \sin (c+d x))^2}\right )d(b \sin (c+d x))}{b^4 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {b^4 \log (b \sin (c+d x))}{a^2}+\frac {\left (a^2-b^2\right )^2}{a (a+b \sin (c+d x))}+\frac {\left (a^2-b^2\right ) \left (3 a^2+b^2\right ) \log (a+b \sin (c+d x))}{a^2}-2 a b \sin (c+d x)+\frac {1}{2} b^2 \sin ^2(c+d x)}{b^4 d}\) |
Input:
Int[(Cos[c + d*x]^4*Cot[c + d*x])/(a + b*Sin[c + d*x])^2,x]
Output:
((b^4*Log[b*Sin[c + d*x]])/a^2 + ((a^2 - b^2)*(3*a^2 + b^2)*Log[a + b*Sin[ c + d*x]])/a^2 - 2*a*b*Sin[c + d*x] + (b^2*Sin[c + d*x]^2)/2 + (a^2 - b^2) ^2/(a*(a + b*Sin[c + d*x])))/(b^4*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^m*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x, b* Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1) /2] && NeQ[a^2 - b^2, 0]
Time = 5.66 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.99
| method | result | size |
| derivativedivides | \(\frac {-\frac {-\frac {\sin \left (d x +c \right )^{2} b}{2}+2 a \sin \left (d x +c \right )}{b^{3}}-\frac {-a^{4}+2 a^{2} b^{2}-b^{4}}{a \,b^{4} \left (a +b \sin \left (d x +c \right )\right )}+\frac {\left (3 a^{4}-2 a^{2} b^{2}-b^{4}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{4} a^{2}}+\frac {\ln \left (\sin \left (d x +c \right )\right )}{a^{2}}}{d}\) | \(119\) |
| default | \(\frac {-\frac {-\frac {\sin \left (d x +c \right )^{2} b}{2}+2 a \sin \left (d x +c \right )}{b^{3}}-\frac {-a^{4}+2 a^{2} b^{2}-b^{4}}{a \,b^{4} \left (a +b \sin \left (d x +c \right )\right )}+\frac {\left (3 a^{4}-2 a^{2} b^{2}-b^{4}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{4} a^{2}}+\frac {\ln \left (\sin \left (d x +c \right )\right )}{a^{2}}}{d}\) | \(119\) |
| risch | \(-\frac {3 i a^{2} x}{b^{4}}+\frac {2 i x}{b^{2}}-\frac {{\mathrm e}^{2 i \left (d x +c \right )}}{8 b^{2} d}+\frac {i a \,{\mathrm e}^{i \left (d x +c \right )}}{b^{3} d}-\frac {i a \,{\mathrm e}^{-i \left (d x +c \right )}}{b^{3} d}-\frac {{\mathrm e}^{-2 i \left (d x +c \right )}}{8 b^{2} d}-\frac {6 i a^{2} c}{b^{4} d}+\frac {4 i c}{b^{2} d}+\frac {2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) {\mathrm e}^{i \left (d x +c \right )}}{b^{4} d a \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )}+\frac {3 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{b^{4} d}-\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{b^{2} d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{a^{2} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d \,a^{2}}\) | \(311\) |
Input:
int(cos(d*x+c)^4*cot(d*x+c)/(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(-1/b^3*(-1/2*sin(d*x+c)^2*b+2*a*sin(d*x+c))-(-a^4+2*a^2*b^2-b^4)/a/b^ 4/(a+b*sin(d*x+c))+1/b^4*(3*a^4-2*a^2*b^2-b^4)/a^2*ln(a+b*sin(d*x+c))+1/a^ 2*ln(sin(d*x+c)))
Time = 0.13 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.58 \[ \int \frac {\cos ^4(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {6 \, a^{3} b^{2} \cos \left (d x + c\right )^{2} + 4 \, a^{5} - 15 \, a^{3} b^{2} + 4 \, a b^{4} + 4 \, {\left (3 \, a^{5} - 2 \, a^{3} b^{2} - a b^{4} + {\left (3 \, a^{4} b - 2 \, a^{2} b^{3} - b^{5}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + 4 \, {\left (b^{5} \sin \left (d x + c\right ) + a b^{4}\right )} \log \left (-\frac {1}{2} \, \sin \left (d x + c\right )\right ) - {\left (2 \, a^{2} b^{3} \cos \left (d x + c\right )^{2} + 8 \, a^{4} b - a^{2} b^{3}\right )} \sin \left (d x + c\right )}{4 \, {\left (a^{2} b^{5} d \sin \left (d x + c\right ) + a^{3} b^{4} d\right )}} \] Input:
integrate(cos(d*x+c)^4*cot(d*x+c)/(a+b*sin(d*x+c))^2,x, algorithm="fricas" )
Output:
1/4*(6*a^3*b^2*cos(d*x + c)^2 + 4*a^5 - 15*a^3*b^2 + 4*a*b^4 + 4*(3*a^5 - 2*a^3*b^2 - a*b^4 + (3*a^4*b - 2*a^2*b^3 - b^5)*sin(d*x + c))*log(b*sin(d* x + c) + a) + 4*(b^5*sin(d*x + c) + a*b^4)*log(-1/2*sin(d*x + c)) - (2*a^2 *b^3*cos(d*x + c)^2 + 8*a^4*b - a^2*b^3)*sin(d*x + c))/(a^2*b^5*d*sin(d*x + c) + a^3*b^4*d)
\[ \int \frac {\cos ^4(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\int \frac {\cos ^{4}{\left (c + d x \right )} \cot {\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{2}}\, dx \] Input:
integrate(cos(d*x+c)**4*cot(d*x+c)/(a+b*sin(d*x+c))**2,x)
Output:
Integral(cos(c + d*x)**4*cot(c + d*x)/(a + b*sin(c + d*x))**2, x)
Time = 0.03 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.98 \[ \int \frac {\cos ^4(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )}}{a b^{5} \sin \left (d x + c\right ) + a^{2} b^{4}} + \frac {2 \, \log \left (\sin \left (d x + c\right )\right )}{a^{2}} + \frac {b \sin \left (d x + c\right )^{2} - 4 \, a \sin \left (d x + c\right )}{b^{3}} + \frac {2 \, {\left (3 \, a^{4} - 2 \, a^{2} b^{2} - b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{2} b^{4}}}{2 \, d} \] Input:
integrate(cos(d*x+c)^4*cot(d*x+c)/(a+b*sin(d*x+c))^2,x, algorithm="maxima" )
Output:
1/2*(2*(a^4 - 2*a^2*b^2 + b^4)/(a*b^5*sin(d*x + c) + a^2*b^4) + 2*log(sin( d*x + c))/a^2 + (b*sin(d*x + c)^2 - 4*a*sin(d*x + c))/b^3 + 2*(3*a^4 - 2*a ^2*b^2 - b^4)*log(b*sin(d*x + c) + a)/(a^2*b^4))/d
Time = 0.24 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.08 \[ \int \frac {\cos ^4(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{2} d} + \frac {b^{2} d \sin \left (d x + c\right )^{2} - 4 \, a b d \sin \left (d x + c\right )}{2 \, b^{4} d^{2}} + \frac {{\left (3 \, a^{4} - 2 \, a^{2} b^{2} - b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{2} b^{4} d} + \frac {a^{5} - 2 \, a^{3} b^{2} + a b^{4}}{{\left (b \sin \left (d x + c\right ) + a\right )} a^{2} b^{4} d} \] Input:
integrate(cos(d*x+c)^4*cot(d*x+c)/(a+b*sin(d*x+c))^2,x, algorithm="giac")
Output:
log(abs(sin(d*x + c)))/(a^2*d) + 1/2*(b^2*d*sin(d*x + c)^2 - 4*a*b*d*sin(d *x + c))/(b^4*d^2) + (3*a^4 - 2*a^2*b^2 - b^4)*log(abs(b*sin(d*x + c) + a) )/(a^2*b^4*d) + (a^5 - 2*a^3*b^2 + a*b^4)/((b*sin(d*x + c) + a)*a^2*b^4*d)
Time = 23.74 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.82 \[ \int \frac {\cos ^4(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d}-\frac {\frac {6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{b^2}+\frac {6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{b^2}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (3\,a^4-3\,a^2\,b^2+b^4\right )}{a^2\,b^3}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (3\,a^4-2\,a^2\,b^2+b^4\right )}{a^2\,b^3}+\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,a^4-2\,a^2\,b^2+b^4\right )}{a^2\,b^3}}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )\,\left (3\,a^2-2\,b^2\right )}{b^4\,d}-\frac {\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )\,\left (-3\,a^4+2\,a^2\,b^2+b^4\right )}{a^2\,b^4\,d} \] Input:
int((cos(c + d*x)^4*cot(c + d*x))/(a + b*sin(c + d*x))^2,x)
Output:
log(tan(c/2 + (d*x)/2))/(a^2*d) - ((6*a*tan(c/2 + (d*x)/2)^2)/b^2 + (6*a*t an(c/2 + (d*x)/2)^4)/b^2 + (4*tan(c/2 + (d*x)/2)^3*(3*a^4 + b^4 - 3*a^2*b^ 2))/(a^2*b^3) + (2*tan(c/2 + (d*x)/2)^5*(3*a^4 + b^4 - 2*a^2*b^2))/(a^2*b^ 3) + (2*tan(c/2 + (d*x)/2)*(3*a^4 + b^4 - 2*a^2*b^2))/(a^2*b^3))/(d*(a + 2 *b*tan(c/2 + (d*x)/2) + 3*a*tan(c/2 + (d*x)/2)^2 + 3*a*tan(c/2 + (d*x)/2)^ 4 + a*tan(c/2 + (d*x)/2)^6 + 4*b*tan(c/2 + (d*x)/2)^3 + 2*b*tan(c/2 + (d*x )/2)^5)) - (log(tan(c/2 + (d*x)/2)^2 + 1)*(3*a^2 - 2*b^2))/(b^4*d) - (log( a + 2*b*tan(c/2 + (d*x)/2) + a*tan(c/2 + (d*x)/2)^2)*(b^4 - 3*a^4 + 2*a^2* b^2))/(a^2*b^4*d)
Time = 0.19 (sec) , antiderivative size = 430, normalized size of antiderivative = 3.58 \[ \int \frac {\cos ^4(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {-6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right ) a^{4} b +4 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right ) a^{2} b^{3}-6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) a^{5}+4 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) a^{3} b^{2}+6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right ) \sin \left (d x +c \right ) a^{4} b -4 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right ) \sin \left (d x +c \right ) a^{2} b^{3}-2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right ) \sin \left (d x +c \right ) b^{5}+6 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right ) a^{5}-4 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right ) a^{3} b^{2}-2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right ) a \,b^{4}+2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right ) b^{5}+2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{4}+\sin \left (d x +c \right )^{3} a^{2} b^{3}-3 \sin \left (d x +c \right )^{2} a^{3} b^{2}+6 a^{5}-4 a^{3} b^{2}+2 a \,b^{4}}{2 a^{2} b^{4} d \left (\sin \left (d x +c \right ) b +a \right )} \] Input:
int(cos(d*x+c)^4*cot(d*x+c)/(a+b*sin(d*x+c))^2,x)
Output:
( - 6*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)*a**4*b + 4*log(tan((c + d* x)/2)**2 + 1)*sin(c + d*x)*a**2*b**3 - 6*log(tan((c + d*x)/2)**2 + 1)*a**5 + 4*log(tan((c + d*x)/2)**2 + 1)*a**3*b**2 + 6*log(tan((c + d*x)/2)**2*a + 2*tan((c + d*x)/2)*b + a)*sin(c + d*x)*a**4*b - 4*log(tan((c + d*x)/2)** 2*a + 2*tan((c + d*x)/2)*b + a)*sin(c + d*x)*a**2*b**3 - 2*log(tan((c + d* x)/2)**2*a + 2*tan((c + d*x)/2)*b + a)*sin(c + d*x)*b**5 + 6*log(tan((c + d*x)/2)**2*a + 2*tan((c + d*x)/2)*b + a)*a**5 - 4*log(tan((c + d*x)/2)**2* a + 2*tan((c + d*x)/2)*b + a)*a**3*b**2 - 2*log(tan((c + d*x)/2)**2*a + 2* tan((c + d*x)/2)*b + a)*a*b**4 + 2*log(tan((c + d*x)/2))*sin(c + d*x)*b**5 + 2*log(tan((c + d*x)/2))*a*b**4 + sin(c + d*x)**3*a**2*b**3 - 3*sin(c + d*x)**2*a**3*b**2 + 6*a**5 - 4*a**3*b**2 + 2*a*b**4)/(2*a**2*b**4*d*(sin(c + d*x)*b + a))