Integrand size = 21, antiderivative size = 132 \[ \int \cot ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {6 a^2 \csc (c+d x)}{d}+\frac {2 a^2 \csc ^3(c+d x)}{d}+\frac {a^2 \csc ^4(c+d x)}{2 d}-\frac {2 a^2 \csc ^5(c+d x)}{5 d}-\frac {a^2 \csc ^6(c+d x)}{6 d}+\frac {2 a^2 \log (\sin (c+d x))}{d}-\frac {2 a^2 \sin (c+d x)}{d}-\frac {a^2 \sin ^2(c+d x)}{2 d} \]
-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
Time = 0.15 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.65 \[ \int \cot ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \left (180 \csc (c+d x)-60 \csc ^3(c+d x)-15 \csc ^4(c+d x)+12 \csc ^5(c+d x)+5 \csc ^6(c+d x)-60 \log (\sin (c+d x))+60 \sin (c+d x)+15 \sin ^2(c+d x)\right )}{30 d} \]
-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
Time = 0.26 (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 |
\(\displaystyle \frac {-\frac {1}{2} a^2 \sin ^2(c+d x)-2 a^2 \sin (c+d x)-\frac {1}{6} a^2 \csc ^6(c+d x)-\frac {2}{5} a^2 \csc ^5(c+d x)+\frac {1}{2} a^2 \csc ^4(c+d x)+2 a^2 \csc ^3(c+d x)-6 a^2 \csc (c+d x)+2 a^2 \log (a \sin (c+d x))}{d}\) |
(-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
3.1.18.3.1 Defintions of rubi rules used
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]))
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]
Time = 4.64 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.73
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {\cos ^{8}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{8}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\cos ^{4}\left (d x +c \right )\right )}{4}+\frac {3 \left (\cos ^{2}\left (d x +c \right )\right )}{2}+3 \ln \left (\sin \left (d x +c \right )\right )\right )+2 a^{2} \left (-\frac {\cos ^{8}\left (d x +c \right )}{5 \sin \left (d x +c \right )^{5}}+\frac {\cos ^{8}\left (d x +c \right )}{5 \sin \left (d x +c \right )^{3}}-\frac {\cos ^{8}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )\right )+a^{2} \left (-\frac {\left (\cot ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(228\) |
default | \(\frac {a^{2} \left (-\frac {\cos ^{8}\left (d x +c \right )}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos ^{8}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}+\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{2}+\frac {3 \left (\cos ^{4}\left (d x +c \right )\right )}{4}+\frac {3 \left (\cos ^{2}\left (d x +c \right )\right )}{2}+3 \ln \left (\sin \left (d x +c \right )\right )\right )+2 a^{2} \left (-\frac {\cos ^{8}\left (d x +c \right )}{5 \sin \left (d x +c \right )^{5}}+\frac {\cos ^{8}\left (d x +c \right )}{5 \sin \left (d x +c \right )^{3}}-\frac {\cos ^{8}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )\right )+a^{2} \left (-\frac {\left (\cot ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(228\) |
risch | \(-2 i a^{2} x +\frac {a^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {i a^{2} {\mathrm e}^{i \left (d x +c \right )}}{d}-\frac {i a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{d}+\frac {a^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {4 i a^{2} c}{d}-\frac {4 i a^{2} \left (45 \,{\mathrm e}^{11 i \left (d x +c \right )}+30 i {\mathrm e}^{8 i \left (d x +c \right )}-165 \,{\mathrm e}^{9 i \left (d x +c \right )}-20 i {\mathrm e}^{6 i \left (d x +c \right )}+318 \,{\mathrm e}^{7 i \left (d x +c \right )}+30 i {\mathrm e}^{4 i \left (d x +c \right )}-318 \,{\mathrm e}^{5 i \left (d x +c \right )}+165 \,{\mathrm e}^{3 i \left (d x +c \right )}-45 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {2 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(234\) |
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) )))
Time = 0.32 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.56 \[ \int \cot ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {30 \, a^{2} \cos \left (d x + c\right )^{8} - 105 \, a^{2} \cos \left (d x + c\right )^{6} + 135 \, a^{2} \cos \left (d x + c\right )^{4} - 45 \, a^{2} \cos \left (d x + c\right )^{2} - 5 \, a^{2} + 120 \, {\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 24 \, {\left (5 \, a^{2} \cos \left (d x + c\right )^{6} - 30 \, a^{2} \cos \left (d x + c\right )^{4} + 40 \, a^{2} \cos \left (d x + c\right )^{2} - 16 \, a^{2}\right )} \sin \left (d x + c\right )}{60 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
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)
\[ \int \cot ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=a^{2} \left (\int 2 \sin {\left (c + d x \right )} \cot ^{7}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \cot ^{7}{\left (c + d x \right )}\, dx + \int \cot ^{7}{\left (c + d x \right )}\, dx\right ) \]
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))
Time = 0.20 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.81 \[ \int \cot ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {15 \, a^{2} \sin \left (d x + c\right )^{2} - 60 \, a^{2} \log \left (\sin \left (d x + c\right )\right ) + 60 \, a^{2} \sin \left (d x + c\right ) + \frac {180 \, a^{2} \sin \left (d x + c\right )^{5} - 60 \, a^{2} \sin \left (d x + c\right )^{3} - 15 \, a^{2} \sin \left (d x + c\right )^{2} + 12 \, a^{2} \sin \left (d x + c\right ) + 5 \, a^{2}}{\sin \left (d x + c\right )^{6}}}{30 \, d} \]
-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
Time = 0.75 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.92 \[ \int \cot ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {15 \, a^{2} \sin \left (d x + c\right )^{2} - 60 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 60 \, a^{2} \sin \left (d x + c\right ) + \frac {147 \, a^{2} \sin \left (d x + c\right )^{6} + 180 \, a^{2} \sin \left (d x + c\right )^{5} - 60 \, a^{2} \sin \left (d x + c\right )^{3} - 15 \, a^{2} \sin \left (d x + c\right )^{2} + 12 \, a^{2} \sin \left (d x + c\right ) + 5 \, a^{2}}{\sin \left (d x + c\right )^{6}}}{30 \, d} \]
-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) + (147*a^2*sin(d*x + c)^6 + 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
Time = 10.76 (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} \]
-(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)