Integrand size = 31, antiderivative size = 209 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {31 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{128 d}-\frac {31 a \cot (c+d x)}{128 d \sqrt {a+a \sin (c+d x)}}+\frac {97 a \cot (c+d x) \csc (c+d x)}{192 d \sqrt {a+a \sin (c+d x)}}+\frac {97 a \cot (c+d x) \csc ^2(c+d x)}{240 d \sqrt {a+a \sin (c+d x)}}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{5 d} \] Output:
-31/128*a^(1/2)*arctanh(a^(1/2)*cos(d*x+c)/(a+a*sin(d*x+c))^(1/2))/d-31/12 8*a*cot(d*x+c)/d/(a+a*sin(d*x+c))^(1/2)+97/192*a*cot(d*x+c)*csc(d*x+c)/d/( a+a*sin(d*x+c))^(1/2)+97/240*a*cot(d*x+c)*csc(d*x+c)^2/d/(a+a*sin(d*x+c))^ (1/2)-1/40*a*cot(d*x+c)*csc(d*x+c)^3/d/(a+a*sin(d*x+c))^(1/2)-1/5*cot(d*x+ c)*csc(d*x+c)^4*(a+a*sin(d*x+c))^(1/2)/d
Time = 3.88 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.93 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {\csc ^{16}\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sin (c+d x))} \left (10180 \cos \left (\frac {1}{2} (c+d x)\right )-2240 \cos \left (\frac {3}{2} (c+d x)\right )-1392 \cos \left (\frac {5}{2} (c+d x)\right )+4810 \cos \left (\frac {7}{2} (c+d x)\right )+930 \cos \left (\frac {9}{2} (c+d x)\right )-10180 \sin \left (\frac {1}{2} (c+d x)\right )+4650 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-4650 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-2240 \sin \left (\frac {3}{2} (c+d x)\right )+1392 \sin \left (\frac {5}{2} (c+d x)\right )-2325 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+2325 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+4810 \sin \left (\frac {7}{2} (c+d x)\right )-930 \sin \left (\frac {9}{2} (c+d x)\right )+465 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-465 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))\right )}{1920 d \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right ) \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^5} \] Input:
Integrate[Cot[c + d*x]^4*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]
Output:
-1/1920*(Csc[(c + d*x)/2]^16*Sqrt[a*(1 + Sin[c + d*x])]*(10180*Cos[(c + d* x)/2] - 2240*Cos[(3*(c + d*x))/2] - 1392*Cos[(5*(c + d*x))/2] + 4810*Cos[( 7*(c + d*x))/2] + 930*Cos[(9*(c + d*x))/2] - 10180*Sin[(c + d*x)/2] + 4650 *Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] - 4650*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] - 2240*Sin[(3*(c + d*x)) /2] + 1392*Sin[(5*(c + d*x))/2] - 2325*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 2325*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x) /2]]*Sin[3*(c + d*x)] + 4810*Sin[(7*(c + d*x))/2] - 930*Sin[(9*(c + d*x))/ 2] + 465*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 4 65*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/(d*(1 + Cot[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^5)
Time = 1.86 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.41, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.645, Rules used = {3042, 3360, 3042, 3251, 3042, 3252, 219, 3523, 27, 3042, 3459, 3042, 3251, 3042, 3251, 3042, 3251, 3042, 3252, 219}
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.
| \(\displaystyle \int \cot ^4(c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 \sqrt {a \sin (c+d x)+a}}{\sin (c+d x)^6}dx\) |
| \(\Big \downarrow \) 3360 |
| \(\displaystyle \int \csc ^6(c+d x) \sqrt {\sin (c+d x) a+a} \left (1-2 \sin ^2(c+d x)\right )dx+\int \csc ^2(c+d x) \sqrt {\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^2}dx+\int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^6}dx\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^6}dx+\frac {1}{2} \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx+\int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^6}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle -\frac {a \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}+\int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^6}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \int \frac {\sqrt {\sin (c+d x) a+a} \left (1-2 \sin (c+d x)^2\right )}{\sin (c+d x)^6}dx-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \frac {\int \frac {1}{2} \csc ^5(c+d x) (a-13 a \sin (c+d x)) \sqrt {\sin (c+d x) a+a}dx}{5 a}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x) \sqrt {a \sin (c+d x)+a}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \csc ^5(c+d x) (a-13 a \sin (c+d x)) \sqrt {\sin (c+d x) a+a}dx}{10 a}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x) \sqrt {a \sin (c+d x)+a}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(a-13 a \sin (c+d x)) \sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^5}dx}{10 a}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x) \sqrt {a \sin (c+d x)+a}}{5 d}\) |
| \(\Big \downarrow \) 3459 |
| \(\displaystyle \frac {-\frac {97}{8} a \int \csc ^4(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x) \sqrt {a \sin (c+d x)+a}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {97}{8} a \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^4}dx-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x) \sqrt {a \sin (c+d x)+a}}{5 d}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {-\frac {97}{8} a \left (\frac {5}{6} \int \csc ^3(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x) \sqrt {a \sin (c+d x)+a}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {97}{8} a \left (\frac {5}{6} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^3}dx-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x) \sqrt {a \sin (c+d x)+a}}{5 d}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {-\frac {97}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \int \csc ^2(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x) \sqrt {a \sin (c+d x)+a}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {97}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^2}dx-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x) \sqrt {a \sin (c+d x)+a}}{5 d}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {-\frac {97}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x) \sqrt {a \sin (c+d x)+a}}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {97}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x) \sqrt {a \sin (c+d x)+a}}{5 d}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {-\frac {97}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {a \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}}{10 a}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x) \sqrt {a \sin (c+d x)+a}}{5 d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}-\frac {97}{8} a \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}\right )}{10 a}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^4(c+d x) \sqrt {a \sin (c+d x)+a}}{5 d}\) |
Input:
Int[Cot[c + d*x]^4*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]
Output:
-((Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) - (a*Cot[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) - (Cot[c + d*x]*Csc[c + d*x] ^4*Sqrt[a + a*Sin[c + d*x]])/(5*d) + (-1/4*(a^2*Cot[c + d*x]*Csc[c + d*x]^ 3)/(d*Sqrt[a + a*Sin[c + d*x]]) - (97*a*(-1/3*(a*Cot[c + d*x]*Csc[c + d*x] ^2)/(d*Sqrt[a + a*Sin[c + d*x]]) + (5*(-1/2*(a*Cot[c + d*x]*Csc[c + d*x])/ (d*Sqrt[a + a*Sin[c + d*x]]) + (3*(-((Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x ])/Sqrt[a + a*Sin[c + d*x]]])/d) - (a*Cot[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]])))/4))/6))/8)/(10*a)
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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]])), x] + Sim p[(2*n + 3)*((b*c - a*d)/(2*b*(n + 1)*(c^2 - d^2))) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x ] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)]), x_Symbol] :> Simp[-2*(b/f) Subst[Int[1/(b*c + a*d - d*x^2), x], x, b*(Cos[e + f*x]/Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/d^4 Int[(d*Sin[e + f*x])^(n + 4)*(a + b*Sin[e + f*x])^m, x], x] + Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - 2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && !IGtQ[m, 0]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + ( f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp [(-b^2)*(B*c - A*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1) *(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b* c - 2*a*d*(n + 1)))/(2*d*(n + 1)*(b*c + a*d)) Int[Sqrt[a + b*Sin[e + f*x] ]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x ] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a *d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n + 2) + C* (c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
Time = 0.44 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.86
| method | result | size |
| default | \(\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (465 a^{\frac {11}{2}} \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}-2170 a^{\frac {9}{2}} \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}}+896 a^{\frac {7}{2}} \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}}+890 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {7}{2}} a^{\frac {5}{2}}-465 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {9}{2}} a^{\frac {3}{2}}-465 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) \sin \left (d x +c \right )^{5} a^{6}\right )}{1920 a^{\frac {11}{2}} \sin \left (d x +c \right )^{5} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(180\) |
Input:
int(cot(d*x+c)^4*csc(d*x+c)^2*(a+a*sin(d*x+c))^(1/2),x,method=_RETURNVERBO SE)
Output:
1/1920*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(465*a^(11/2)*(-a*(sin(d*x +c)-1))^(1/2)-2170*a^(9/2)*(-a*(sin(d*x+c)-1))^(3/2)+896*a^(7/2)*(-a*(sin( d*x+c)-1))^(5/2)+890*(-a*(sin(d*x+c)-1))^(7/2)*a^(5/2)-465*(-a*(sin(d*x+c) -1))^(9/2)*a^(3/2)-465*arctanh((-a*(sin(d*x+c)-1))^(1/2)/a^(1/2))*sin(d*x+ c)^5*a^6)/a^(11/2)/sin(d*x+c)^5/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d
Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (181) = 362\).
Time = 0.10 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.21 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {465 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right ) - 1\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (465 \, \cos \left (d x + c\right )^{5} + 1435 \, \cos \left (d x + c\right )^{4} - 154 \, \cos \left (d x + c\right )^{3} - 1662 \, \cos \left (d x + c\right )^{2} - {\left (465 \, \cos \left (d x + c\right )^{4} - 970 \, \cos \left (d x + c\right )^{3} - 1124 \, \cos \left (d x + c\right )^{2} + 538 \, \cos \left (d x + c\right ) + 611\right )} \sin \left (d x + c\right ) + 73 \, \cos \left (d x + c\right ) + 611\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{7680 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} - 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right ) - d\right )}} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^2*(a+a*sin(d*x+c))^(1/2),x, algorithm="f ricas")
Output:
1/7680*(465*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - (cos(d *x + c)^5 + cos(d*x + c)^4 - 2*cos(d*x + c)^3 - 2*cos(d*x + c)^2 + cos(d*x + c) + 1)*sin(d*x + c) - 1)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) + 4*(465*c os(d*x + c)^5 + 1435*cos(d*x + c)^4 - 154*cos(d*x + c)^3 - 1662*cos(d*x + c)^2 - (465*cos(d*x + c)^4 - 970*cos(d*x + c)^3 - 1124*cos(d*x + c)^2 + 53 8*cos(d*x + c) + 611)*sin(d*x + c) + 73*cos(d*x + c) + 611)*sqrt(a*sin(d*x + c) + a))/(d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - (d*cos(d*x + c)^5 + d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^3 - 2*d*cos(d*x + c)^2 + d*cos(d*x + c) + d)*sin(d*x + c) - d)
Timed out. \[ \int \cot ^4(c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)**2*(a+a*sin(d*x+c))**(1/2),x)
Output:
Timed out
\[ \int \cot ^4(c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \cot \left (d x + c\right )^{4} \csc \left (d x + c\right )^{2} \,d x } \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^2*(a+a*sin(d*x+c))^(1/2),x, algorithm="m axima")
Output:
integrate(sqrt(a*sin(d*x + c) + a)*cot(d*x + c)^4*csc(d*x + c)^2, x)
Time = 0.20 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.16 \[ \int \cot ^4(c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {\sqrt {2} {\left (465 \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + \frac {4 \, {\left (7440 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 7120 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3584 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4340 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 465 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}\right )} \sqrt {a}}{7680 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^2*(a+a*sin(d*x+c))^(1/2),x, algorithm="g iac")
Output:
-1/7680*sqrt(2)*(465*sqrt(2)*log(abs(-2*sqrt(2) + 4*sin(-1/4*pi + 1/2*d*x + 1/2*c))/abs(2*sqrt(2) + 4*sin(-1/4*pi + 1/2*d*x + 1/2*c)))*sgn(cos(-1/4* pi + 1/2*d*x + 1/2*c)) + 4*(7440*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin(- 1/4*pi + 1/2*d*x + 1/2*c)^9 - 7120*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*sin (-1/4*pi + 1/2*d*x + 1/2*c)^7 - 3584*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))*s in(-1/4*pi + 1/2*d*x + 1/2*c)^5 + 4340*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c)) *sin(-1/4*pi + 1/2*d*x + 1/2*c)^3 - 465*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c) )*sin(-1/4*pi + 1/2*d*x + 1/2*c))/(2*sin(-1/4*pi + 1/2*d*x + 1/2*c)^2 - 1) ^5)*sqrt(a)/d
Timed out. \[ \int \cot ^4(c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \frac {{\left ({\sin \left (c+d\,x\right )}^2-1\right )}^2\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{{\sin \left (c+d\,x\right )}^6} \,d x \] Input:
int((cot(c + d*x)^4*(a + a*sin(c + d*x))^(1/2))/sin(c + d*x)^2,x)
Output:
int(((sin(c + d*x)^2 - 1)^2*(a + a*sin(c + d*x))^(1/2))/sin(c + d*x)^6, x)
\[ \int \cot ^4(c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\sqrt {a}\, \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{4} \csc \left (d x +c \right )^{2}d x \right ) \] Input:
int(cot(d*x+c)^4*csc(d*x+c)^2*(a+a*sin(d*x+c))^(1/2),x)
Output:
sqrt(a)*int(sqrt(sin(c + d*x) + 1)*cot(c + d*x)**4*csc(c + d*x)**2,x)