\(\int \cot ^7(c+d x) (a+a \sin (c+d x))^2 \, dx\) [18]

Optimal result

Integrand size = 21, antiderivative size = 132 \[ \int \cot ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {6 a^2 \csc (c+d x)}{d}+\frac {2 a^2 \csc ^3(c+d x)}{d}+\frac {a^2 \csc ^4(c+d x)}{2 d}-\frac {2 a^2 \csc ^5(c+d x)}{5 d}-\frac {a^2 \csc ^6(c+d x)}{6 d}+\frac {2 a^2 \log (\sin (c+d x))}{d}-\frac {2 a^2 \sin (c+d x)}{d}-\frac {a^2 \sin ^2(c+d x)}{2 d} \] Output:

-6*a^2*csc(d*x+c)/d+2*a^2*csc(d*x+c)^3/d+1/2*a^2*csc(d*x+c)^4/d-2/5*a^2*cs 
c(d*x+c)^5/d-1/6*a^2*csc(d*x+c)^6/d+2*a^2*ln(sin(d*x+c))/d-2*a^2*sin(d*x+c 
)/d-1/2*a^2*sin(d*x+c)^2/d
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.65 \[ \int \cot ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \left (180 \csc (c+d x)-60 \csc ^3(c+d x)-15 \csc ^4(c+d x)+12 \csc ^5(c+d x)+5 \csc ^6(c+d x)-60 \log (\sin (c+d x))+60 \sin (c+d x)+15 \sin ^2(c+d x)\right )}{30 d} \] Input:

Integrate[Cot[c + d*x]^7*(a + a*Sin[c + d*x])^2,x]
 

Output:

-1/30*(a^2*(180*Csc[c + d*x] - 60*Csc[c + d*x]^3 - 15*Csc[c + d*x]^4 + 12* 
Csc[c + d*x]^5 + 5*Csc[c + d*x]^6 - 60*Log[Sin[c + d*x]] + 60*Sin[c + d*x] 
 + 15*Sin[c + d*x]^2))/d
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.86, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3186, 99, 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.

\(\displaystyle \int \cot ^7(c+d x) (a \sin (c+d x)+a)^2 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a \sin (c+d x)+a)^2}{\tan (c+d x)^7}dx\)

\(\Big \downarrow \) 3186

\(\displaystyle \frac {\int \frac {\csc ^7(c+d x) (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^5}{a^7}d(a \sin (c+d x))}{d}\)

\(\Big \downarrow \) 99

\(\displaystyle \frac {\int \left (a \csc ^7(c+d x)+2 a \csc ^6(c+d x)-2 a \csc ^5(c+d x)-6 a \csc ^4(c+d x)+6 a \csc ^2(c+d x)+2 a \csc (c+d x)-2 a-a \sin (c+d x)\right )d(a \sin (c+d x))}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {1}{2} a^2 \sin ^2(c+d x)-2 a^2 \sin (c+d x)-\frac {1}{6} a^2 \csc ^6(c+d x)-\frac {2}{5} a^2 \csc ^5(c+d x)+\frac {1}{2} a^2 \csc ^4(c+d x)+2 a^2 \csc ^3(c+d x)-6 a^2 \csc (c+d x)+2 a^2 \log (a \sin (c+d x))}{d}\)

Input:

Int[Cot[c + d*x]^7*(a + a*Sin[c + d*x])^2,x]
 

Output:

(-6*a^2*Csc[c + d*x] + 2*a^2*Csc[c + d*x]^3 + (a^2*Csc[c + d*x]^4)/2 - (2* 
a^2*Csc[c + d*x]^5)/5 - (a^2*Csc[c + d*x]^6)/6 + 2*a^2*Log[a*Sin[c + d*x]] 
 - 2*a^2*Sin[c + d*x] - (a^2*Sin[c + d*x]^2)/2)/d
 

Defintions of rubi rules used

rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | 
| (GtQ[m, 0] && GeQ[n, -1]))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3186

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 - (p + 1)/2)/(a - x) 
^((p + 1)/2)), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && E 
qQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]
 

Maple [A] (verified)

Time = 3.93 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.73

method	result	size
derivativedivides	\(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{8}}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{8}}{2 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{6}}{2}+\frac {3 \cos \left (d x +c \right )^{4}}{4}+\frac {3 \cos \left (d x +c \right )^{2}}{2}+3 \ln \left (\sin \left (d x +c \right )\right )\right )+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{8}}{5 \sin \left (d x +c \right )^{5}}+\frac {\cos \left (d x +c \right )^{8}}{5 \sin \left (d x +c \right )^{3}}-\frac {\cos \left (d x +c \right )^{8}}{\sin \left (d x +c \right )}-\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )\right )+a^{2} \left (-\frac {\cot \left (d x +c \right )^{6}}{6}+\frac {\cot \left (d x +c \right )^{4}}{4}-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\)	\(228\)
default	\(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{8}}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{8}}{2 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{6}}{2}+\frac {3 \cos \left (d x +c \right )^{4}}{4}+\frac {3 \cos \left (d x +c \right )^{2}}{2}+3 \ln \left (\sin \left (d x +c \right )\right )\right )+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{8}}{5 \sin \left (d x +c \right )^{5}}+\frac {\cos \left (d x +c \right )^{8}}{5 \sin \left (d x +c \right )^{3}}-\frac {\cos \left (d x +c \right )^{8}}{\sin \left (d x +c \right )}-\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )\right )+a^{2} \left (-\frac {\cot \left (d x +c \right )^{6}}{6}+\frac {\cot \left (d x +c \right )^{4}}{4}-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\)	\(228\)
risch	\(-2 i a^{2} x +\frac {a^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {i a^{2} {\mathrm e}^{i \left (d x +c \right )}}{d}-\frac {i a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{d}+\frac {a^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {4 i a^{2} c}{d}-\frac {4 i a^{2} \left (45 \,{\mathrm e}^{11 i \left (d x +c \right )}+30 i {\mathrm e}^{8 i \left (d x +c \right )}-165 \,{\mathrm e}^{9 i \left (d x +c \right )}-20 i {\mathrm e}^{6 i \left (d x +c \right )}+318 \,{\mathrm e}^{7 i \left (d x +c \right )}+30 i {\mathrm e}^{4 i \left (d x +c \right )}-318 \,{\mathrm e}^{5 i \left (d x +c \right )}+165 \,{\mathrm e}^{3 i \left (d x +c \right )}-45 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {2 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(234\)

Input:

int(cot(d*x+c)^7*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

1/d*(a^2*(-1/4/sin(d*x+c)^4*cos(d*x+c)^8+1/2/sin(d*x+c)^2*cos(d*x+c)^8+1/2 
*cos(d*x+c)^6+3/4*cos(d*x+c)^4+3/2*cos(d*x+c)^2+3*ln(sin(d*x+c)))+2*a^2*(- 
1/5/sin(d*x+c)^5*cos(d*x+c)^8+1/5/sin(d*x+c)^3*cos(d*x+c)^8-1/sin(d*x+c)*c 
os(d*x+c)^8-(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x+ 
c))+a^2*(-1/6*cot(d*x+c)^6+1/4*cot(d*x+c)^4-1/2*cot(d*x+c)^2-ln(sin(d*x+c) 
)))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.56 \[ \int \cot ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {30 \, a^{2} \cos \left (d x + c\right )^{8} - 105 \, a^{2} \cos \left (d x + c\right )^{6} + 135 \, a^{2} \cos \left (d x + c\right )^{4} - 45 \, a^{2} \cos \left (d x + c\right )^{2} - 5 \, a^{2} + 120 \, {\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 24 \, {\left (5 \, a^{2} \cos \left (d x + c\right )^{6} - 30 \, a^{2} \cos \left (d x + c\right )^{4} + 40 \, a^{2} \cos \left (d x + c\right )^{2} - 16 \, a^{2}\right )} \sin \left (d x + c\right )}{60 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:

integrate(cot(d*x+c)^7*(a+a*sin(d*x+c))^2,x, algorithm="fricas")
 

Output:

1/60*(30*a^2*cos(d*x + c)^8 - 105*a^2*cos(d*x + c)^6 + 135*a^2*cos(d*x + c 
)^4 - 45*a^2*cos(d*x + c)^2 - 5*a^2 + 120*(a^2*cos(d*x + c)^6 - 3*a^2*cos( 
d*x + c)^4 + 3*a^2*cos(d*x + c)^2 - a^2)*log(1/2*sin(d*x + c)) - 24*(5*a^2 
*cos(d*x + c)^6 - 30*a^2*cos(d*x + c)^4 + 40*a^2*cos(d*x + c)^2 - 16*a^2)* 
sin(d*x + c))/(d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 
- d)
 

Sympy [F]

\[ \int \cot ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=a^{2} \left (\int 2 \sin {\left (c + d x \right )} \cot ^{7}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \cot ^{7}{\left (c + d x \right )}\, dx + \int \cot ^{7}{\left (c + d x \right )}\, dx\right ) \] Input:

integrate(cot(d*x+c)**7*(a+a*sin(d*x+c))**2,x)
 

Output:

a**2*(Integral(2*sin(c + d*x)*cot(c + d*x)**7, x) + Integral(sin(c + d*x)* 
*2*cot(c + d*x)**7, x) + Integral(cot(c + d*x)**7, x))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.81 \[ \int \cot ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {15 \, a^{2} \sin \left (d x + c\right )^{2} - 60 \, a^{2} \log \left (\sin \left (d x + c\right )\right ) + 60 \, a^{2} \sin \left (d x + c\right ) + \frac {180 \, a^{2} \sin \left (d x + c\right )^{5} - 60 \, a^{2} \sin \left (d x + c\right )^{3} - 15 \, a^{2} \sin \left (d x + c\right )^{2} + 12 \, a^{2} \sin \left (d x + c\right ) + 5 \, a^{2}}{\sin \left (d x + c\right )^{6}}}{30 \, d} \] Input:

integrate(cot(d*x+c)^7*(a+a*sin(d*x+c))^2,x, algorithm="maxima")
 

Output:

-1/30*(15*a^2*sin(d*x + c)^2 - 60*a^2*log(sin(d*x + c)) + 60*a^2*sin(d*x + 
 c) + (180*a^2*sin(d*x + c)^5 - 60*a^2*sin(d*x + c)^3 - 15*a^2*sin(d*x + c 
)^2 + 12*a^2*sin(d*x + c) + 5*a^2)/sin(d*x + c)^6)/d
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.82 \[ \int \cot ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {15 \, a^{2} \sin \left (d x + c\right )^{2} - 60 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 60 \, a^{2} \sin \left (d x + c\right ) + \frac {180 \, a^{2} \sin \left (d x + c\right )^{5} - 60 \, a^{2} \sin \left (d x + c\right )^{3} - 15 \, a^{2} \sin \left (d x + c\right )^{2} + 12 \, a^{2} \sin \left (d x + c\right ) + 5 \, a^{2}}{\sin \left (d x + c\right )^{6}}}{30 \, d} \] Input:

integrate(cot(d*x+c)^7*(a+a*sin(d*x+c))^2,x, algorithm="giac")
 

Output:

-1/30*(15*a^2*sin(d*x + c)^2 - 60*a^2*log(abs(sin(d*x + c))) + 60*a^2*sin( 
d*x + c) + (180*a^2*sin(d*x + c)^5 - 60*a^2*sin(d*x + c)^3 - 15*a^2*sin(d* 
x + c)^2 + 12*a^2*sin(d*x + c) + 5*a^2)/sin(d*x + c)^6)/d
 

Mupad [B] (verification not implemented)

Time = 20.31 (sec) , antiderivative size = 392, normalized size of antiderivative = 2.97 \[ \int \cot ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2\,\left (24\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-312\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-220\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3864\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-360\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21000\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+3510\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+21000\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-360\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+3864\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-220\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-312\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+3840\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )+7680\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )+3840\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )-3840\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-7680\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-3840\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\right )}{1920\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2} \] Input:

int(cot(c + d*x)^7*(a + a*sin(c + d*x))^2,x)
 

Output:

-(a^2*(24*tan(c/2 + (d*x)/2) - 20*tan(c/2 + (d*x)/2)^2 - 312*tan(c/2 + (d* 
x)/2)^3 - 220*tan(c/2 + (d*x)/2)^4 + 3864*tan(c/2 + (d*x)/2)^5 - 360*tan(c 
/2 + (d*x)/2)^6 + 21000*tan(c/2 + (d*x)/2)^7 + 3510*tan(c/2 + (d*x)/2)^8 + 
 21000*tan(c/2 + (d*x)/2)^9 - 360*tan(c/2 + (d*x)/2)^10 + 3864*tan(c/2 + ( 
d*x)/2)^11 - 220*tan(c/2 + (d*x)/2)^12 - 312*tan(c/2 + (d*x)/2)^13 - 20*ta 
n(c/2 + (d*x)/2)^14 + 24*tan(c/2 + (d*x)/2)^15 + 5*tan(c/2 + (d*x)/2)^16 + 
 3840*tan(c/2 + (d*x)/2)^6*log(tan(c/2 + (d*x)/2)^2 + 1) + 7680*tan(c/2 + 
(d*x)/2)^8*log(tan(c/2 + (d*x)/2)^2 + 1) + 3840*tan(c/2 + (d*x)/2)^10*log( 
tan(c/2 + (d*x)/2)^2 + 1) - 3840*log(tan(c/2 + (d*x)/2))*tan(c/2 + (d*x)/2 
)^6 - 7680*log(tan(c/2 + (d*x)/2))*tan(c/2 + (d*x)/2)^8 - 3840*log(tan(c/2 
 + (d*x)/2))*tan(c/2 + (d*x)/2)^10 + 5))/(1920*d*tan(c/2 + (d*x)/2)^6*(tan 
(c/2 + (d*x)/2)^2 + 1)^2)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.98 \[ \int \cot ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \left (-960 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right )^{6}+960 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6}-240 \sin \left (d x +c \right )^{8}-960 \sin \left (d x +c \right )^{7}-155 \sin \left (d x +c \right )^{6}-2880 \sin \left (d x +c \right )^{5}+960 \sin \left (d x +c \right )^{3}+240 \sin \left (d x +c \right )^{2}-192 \sin \left (d x +c \right )-80\right )}{480 \sin \left (d x +c \right )^{6} d} \] Input:

int(cot(d*x+c)^7*(a+a*sin(d*x+c))^2,x)
 

Output:

(a**2*( - 960*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**6 + 960*log(tan(( 
c + d*x)/2))*sin(c + d*x)**6 - 240*sin(c + d*x)**8 - 960*sin(c + d*x)**7 - 
 155*sin(c + d*x)**6 - 2880*sin(c + d*x)**5 + 960*sin(c + d*x)**3 + 240*si 
n(c + d*x)**2 - 192*sin(c + d*x) - 80))/(480*sin(c + d*x)**6*d)