Integrand size = 19, antiderivative size = 81 \[ \int \cot ^5(c+d x) (a+b \sin (c+d x)) \, dx=\frac {2 b \csc (c+d x)}{d}+\frac {a \csc ^2(c+d x)}{d}-\frac {b \csc ^3(c+d x)}{3 d}-\frac {a \csc ^4(c+d x)}{4 d}+\frac {a \log (\sin (c+d x))}{d}+\frac {b \sin (c+d x)}{d} \] Output:
2*b*csc(d*x+c)/d+a*csc(d*x+c)^2/d-1/3*b*csc(d*x+c)^3/d-1/4*a*csc(d*x+c)^4/ d+a*ln(sin(d*x+c))/d+b*sin(d*x+c)/d
Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00 \[ \int \cot ^5(c+d x) (a+b \sin (c+d x)) \, dx=\frac {2 b \csc (c+d x)}{d}+\frac {a \csc ^2(c+d x)}{d}-\frac {b \csc ^3(c+d x)}{3 d}-\frac {a \csc ^4(c+d x)}{4 d}+\frac {a \log (\sin (c+d x))}{d}+\frac {b \sin (c+d x)}{d} \] Input:
Integrate[Cot[c + d*x]^5*(a + b*Sin[c + d*x]),x]
Output:
(2*b*Csc[c + d*x])/d + (a*Csc[c + d*x]^2)/d - (b*Csc[c + d*x]^3)/(3*d) - ( a*Csc[c + d*x]^4)/(4*d) + (a*Log[Sin[c + d*x]])/d + (b*Sin[c + d*x])/d
Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.85, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3042, 3200, 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.
\(\displaystyle \int \cot ^5(c+d x) (a+b \sin (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a+b \sin (c+d x)}{\tan (c+d x)^5}dx\) |
\(\Big \downarrow \) 3200 |
\(\displaystyle \frac {\int \frac {\csc ^5(c+d x) (a+b \sin (c+d x)) \left (b^2-b^2 \sin ^2(c+d x)\right )^2}{b^5}d(b \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 522 |
\(\displaystyle \frac {\int \left (\frac {a \csc ^5(c+d x)}{b}+\csc ^4(c+d x)-\frac {2 a \csc ^3(c+d x)}{b}-2 \csc ^2(c+d x)+\frac {a \csc (c+d x)}{b}+1\right )d(b \sin (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a \log (b \sin (c+d x))-\frac {1}{4} a \csc ^4(c+d x)+a \csc ^2(c+d x)+b \sin (c+d x)-\frac {1}{3} b \csc ^3(c+d x)+2 b \csc (c+d x)}{d}\) |
Input:
Int[Cot[c + d*x]^5*(a + b*Sin[c + d*x]),x]
Output:
(2*b*Csc[c + d*x] + a*Csc[c + d*x]^2 - (b*Csc[c + d*x]^3)/3 - (a*Csc[c + d *x]^4)/4 + a*Log[b*Sin[c + d*x]] + b*Sin[c + d*x])/d
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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p _.), x_Symbol] :> Simp[1/f Subst[Int[(x^p*(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] && NeQ[a^2 - b ^2, 0] && IntegerQ[(p + 1)/2]
Time = 1.06 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.25
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {\cot \left (d x +c \right )^{4}}{4}+\frac {\cot \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+b \left (-\frac {\cos \left (d x +c \right )^{6}}{3 \sin \left (d x +c \right )^{3}}+\frac {\cos \left (d x +c \right )^{6}}{\sin \left (d x +c \right )}+\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )\right )}{d}\) | \(101\) |
default | \(\frac {a \left (-\frac {\cot \left (d x +c \right )^{4}}{4}+\frac {\cot \left (d x +c \right )^{2}}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+b \left (-\frac {\cos \left (d x +c \right )^{6}}{3 \sin \left (d x +c \right )^{3}}+\frac {\cos \left (d x +c \right )^{6}}{\sin \left (d x +c \right )}+\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )\right )}{d}\) | \(101\) |
risch | \(-i a x -\frac {i b \,{\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i b \,{\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {2 i a c}{d}+\frac {4 i \left (3 i a \,{\mathrm e}^{6 i \left (d x +c \right )}+3 b \,{\mathrm e}^{7 i \left (d x +c \right )}-3 i a \,{\mathrm e}^{4 i \left (d x +c \right )}-7 b \,{\mathrm e}^{5 i \left (d x +c \right )}+3 i a \,{\mathrm e}^{2 i \left (d x +c \right )}+7 b \,{\mathrm e}^{3 i \left (d x +c \right )}-3 \,{\mathrm e}^{i \left (d x +c \right )} b \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(171\) |
Input:
int(cot(d*x+c)^5*(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(a*(-1/4*cot(d*x+c)^4+1/2*cot(d*x+c)^2+ln(sin(d*x+c)))+b*(-1/3/sin(d*x +c)^3*cos(d*x+c)^6+1/sin(d*x+c)*cos(d*x+c)^6+(8/3+cos(d*x+c)^4+4/3*cos(d*x +c)^2)*sin(d*x+c)))
Time = 0.10 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.36 \[ \int \cot ^5(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {12 \, a \cos \left (d x + c\right )^{2} - 12 \, {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 4 \, {\left (3 \, b \cos \left (d x + c\right )^{4} - 12 \, b \cos \left (d x + c\right )^{2} + 8 \, b\right )} \sin \left (d x + c\right ) - 9 \, a}{12 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \] Input:
integrate(cot(d*x+c)^5*(a+b*sin(d*x+c)),x, algorithm="fricas")
Output:
-1/12*(12*a*cos(d*x + c)^2 - 12*(a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^2 + a )*log(1/2*sin(d*x + c)) - 4*(3*b*cos(d*x + c)^4 - 12*b*cos(d*x + c)^2 + 8* b)*sin(d*x + c) - 9*a)/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)
\[ \int \cot ^5(c+d x) (a+b \sin (c+d x)) \, dx=\int \left (a + b \sin {\left (c + d x \right )}\right ) \cot ^{5}{\left (c + d x \right )}\, dx \] Input:
integrate(cot(d*x+c)**5*(a+b*sin(d*x+c)),x)
Output:
Integral((a + b*sin(c + d*x))*cot(c + d*x)**5, x)
Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.85 \[ \int \cot ^5(c+d x) (a+b \sin (c+d x)) \, dx=\frac {12 \, a \log \left (\sin \left (d x + c\right )\right ) + 12 \, b \sin \left (d x + c\right ) + \frac {24 \, b \sin \left (d x + c\right )^{3} + 12 \, a \sin \left (d x + c\right )^{2} - 4 \, b \sin \left (d x + c\right ) - 3 \, a}{\sin \left (d x + c\right )^{4}}}{12 \, d} \] Input:
integrate(cot(d*x+c)^5*(a+b*sin(d*x+c)),x, algorithm="maxima")
Output:
1/12*(12*a*log(sin(d*x + c)) + 12*b*sin(d*x + c) + (24*b*sin(d*x + c)^3 + 12*a*sin(d*x + c)^2 - 4*b*sin(d*x + c) - 3*a)/sin(d*x + c)^4)/d
Time = 0.13 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \cot ^5(c+d x) (a+b \sin (c+d x)) \, dx=\frac {12 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 12 \, b \sin \left (d x + c\right ) + \frac {24 \, b \sin \left (d x + c\right )^{3} + 12 \, a \sin \left (d x + c\right )^{2} - 4 \, b \sin \left (d x + c\right ) - 3 \, a}{\sin \left (d x + c\right )^{4}}}{12 \, d} \] Input:
integrate(cot(d*x+c)^5*(a+b*sin(d*x+c)),x, algorithm="giac")
Output:
1/12*(12*a*log(abs(sin(d*x + c))) + 12*b*sin(d*x + c) + (24*b*sin(d*x + c) ^3 + 12*a*sin(d*x + c)^2 - 4*b*sin(d*x + c) - 3*a)/sin(d*x + c)^4)/d
Time = 17.50 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.56 \[ \int \cot ^5(c+d x) (a+b \sin (c+d x)) \, dx=\frac {7\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}+\frac {46\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {40\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {11\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4}-\frac {2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}-\frac {a}{4}}{d\,\left (16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}-\frac {a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}+\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}+\frac {a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \] Input:
int(cot(c + d*x)^5*(a + b*sin(c + d*x)),x)
Output:
(7*b*tan(c/2 + (d*x)/2))/(8*d) + ((11*a*tan(c/2 + (d*x)/2)^2)/4 - (2*b*tan (c/2 + (d*x)/2))/3 - a/4 + 3*a*tan(c/2 + (d*x)/2)^4 + (40*b*tan(c/2 + (d*x )/2)^3)/3 + 46*b*tan(c/2 + (d*x)/2)^5)/(d*(16*tan(c/2 + (d*x)/2)^4 + 16*ta n(c/2 + (d*x)/2)^6)) - (a*log(tan(c/2 + (d*x)/2)^2 + 1))/d + (3*a*tan(c/2 + (d*x)/2)^2)/(16*d) - (a*tan(c/2 + (d*x)/2)^4)/(64*d) - (b*tan(c/2 + (d*x )/2)^3)/(24*d) + (a*log(tan(c/2 + (d*x)/2)))/d
Time = 0.18 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.43 \[ \int \cot ^5(c+d x) (a+b \sin (c+d x)) \, dx=\frac {-96 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{4} a +96 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4} a +96 \sin \left (d x +c \right )^{5} b -57 \sin \left (d x +c \right )^{4} a +192 \sin \left (d x +c \right )^{3} b +96 \sin \left (d x +c \right )^{2} a -32 \sin \left (d x +c \right ) b -24 a}{96 \sin \left (d x +c \right )^{4} d} \] Input:
int(cot(d*x+c)^5*(a+b*sin(d*x+c)),x)
Output:
( - 96*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**4*a + 96*log(tan((c + d* x)/2))*sin(c + d*x)**4*a + 96*sin(c + d*x)**5*b - 57*sin(c + d*x)**4*a + 1 92*sin(c + d*x)**3*b + 96*sin(c + d*x)**2*a - 32*sin(c + d*x)*b - 24*a)/(9 6*sin(c + d*x)**4*d)