Integrand size = 23, antiderivative size = 227 \[ \int \cot ^4(e+f x) (a+a \sin (e+f x))^{5/2} \, dx=\frac {55 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{8 f}-\frac {9 a^3 \cos (e+f x)}{40 f \sqrt {a+a \sin (e+f x)}}-\frac {16 a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 f}+\frac {17 a^2 \cot (e+f x) \sqrt {a+a \sin (e+f x)}}{24 f}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 f}-\frac {5 a \cot (e+f x) \csc (e+f x) (a+a \sin (e+f x))^{3/2}}{12 f}-\frac {\cot (e+f x) \csc ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{3 f} \] Output:
55/8*a^(5/2)*arctanh(a^(1/2)*cos(f*x+e)/(a+a*sin(f*x+e))^(1/2))/f-9/40*a^3 *cos(f*x+e)/f/(a+a*sin(f*x+e))^(1/2)-16/15*a^2*cos(f*x+e)*(a+a*sin(f*x+e)) ^(1/2)/f+17/24*a^2*cot(f*x+e)*(a+a*sin(f*x+e))^(1/2)/f-2/5*a*cos(f*x+e)*(a +a*sin(f*x+e))^(3/2)/f-5/12*a*cot(f*x+e)*csc(f*x+e)*(a+a*sin(f*x+e))^(3/2) /f-1/3*cot(f*x+e)*csc(f*x+e)^2*(a+a*sin(f*x+e))^(5/2)/f
Time = 7.80 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.59 \[ \int \cot ^4(e+f x) (a+a \sin (e+f x))^{5/2} \, dx=-\frac {a^2 \csc ^{10}\left (\frac {1}{2} (e+f x)\right ) \sqrt {a (1+\sin (e+f x))} \left (108 \cos \left (\frac {1}{2} (e+f x)\right )+706 \cos \left (\frac {3}{2} (e+f x)\right )-450 \cos \left (\frac {5}{2} (e+f x)\right )-156 \cos \left (\frac {7}{2} (e+f x)\right )+100 \cos \left (\frac {9}{2} (e+f x)\right )+12 \cos \left (\frac {11}{2} (e+f x)\right )-108 \sin \left (\frac {1}{2} (e+f x)\right )-2475 \log \left (1+\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x)+2475 \log \left (1-\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x)+706 \sin \left (\frac {3}{2} (e+f x)\right )+450 \sin \left (\frac {5}{2} (e+f x)\right )+825 \log \left (1+\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (3 (e+f x))-825 \log \left (1-\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (3 (e+f x))-156 \sin \left (\frac {7}{2} (e+f x)\right )-100 \sin \left (\frac {9}{2} (e+f x)\right )+12 \sin \left (\frac {11}{2} (e+f x)\right )\right )}{120 f \left (1+\cot \left (\frac {1}{2} (e+f x)\right )\right ) \left (\csc ^2\left (\frac {1}{4} (e+f x)\right )-\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )^3} \] Input:
Integrate[Cot[e + f*x]^4*(a + a*Sin[e + f*x])^(5/2),x]
Output:
-1/120*(a^2*Csc[(e + f*x)/2]^10*Sqrt[a*(1 + Sin[e + f*x])]*(108*Cos[(e + f *x)/2] + 706*Cos[(3*(e + f*x))/2] - 450*Cos[(5*(e + f*x))/2] - 156*Cos[(7* (e + f*x))/2] + 100*Cos[(9*(e + f*x))/2] + 12*Cos[(11*(e + f*x))/2] - 108* Sin[(e + f*x)/2] - 2475*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[e + f*x] + 2475*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[e + f*x] + 706*Sin[(3*(e + f*x))/2] + 450*Sin[(5*(e + f*x))/2] + 825*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[3*(e + f*x)] - 825*Log[1 - Cos[(e + f*x) /2] + Sin[(e + f*x)/2]]*Sin[3*(e + f*x)] - 156*Sin[(7*(e + f*x))/2] - 100* Sin[(9*(e + f*x))/2] + 12*Sin[(11*(e + f*x))/2]))/(f*(1 + Cot[(e + f*x)/2] )*(Csc[(e + f*x)/4]^2 - Sec[(e + f*x)/4]^2)^3)
Time = 1.91 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.21, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.913, Rules used = {3042, 3197, 3042, 3126, 3042, 3126, 3042, 3125, 3523, 27, 3042, 3454, 27, 3042, 3454, 27, 3042, 3460, 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.
| \(\displaystyle \int \cot ^4(e+f x) (a \sin (e+f x)+a)^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{5/2}}{\tan (e+f x)^4}dx\) |
| \(\Big \downarrow \) 3197 |
| \(\displaystyle \int (\sin (e+f x) a+a)^{5/2}dx+\int \csc ^4(e+f x) (\sin (e+f x) a+a)^{5/2} \left (1-2 \sin ^2(e+f x)\right )dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (\sin (e+f x) a+a)^{5/2}dx+\int \frac {(\sin (e+f x) a+a)^{5/2} \left (1-2 \sin (e+f x)^2\right )}{\sin (e+f x)^4}dx\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle \int \frac {(\sin (e+f x) a+a)^{5/2} \left (1-2 \sin (e+f x)^2\right )}{\sin (e+f x)^4}dx+\frac {8}{5} a \int (\sin (e+f x) a+a)^{3/2}dx-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {8}{5} a \int (\sin (e+f x) a+a)^{3/2}dx+\int \frac {(\sin (e+f x) a+a)^{5/2} \left (1-2 \sin (e+f x)^2\right )}{\sin (e+f x)^4}dx-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle \int \frac {(\sin (e+f x) a+a)^{5/2} \left (1-2 \sin (e+f x)^2\right )}{\sin (e+f x)^4}dx+\frac {8}{5} a \left (\frac {4}{3} a \int \sqrt {\sin (e+f x) a+a}dx-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(\sin (e+f x) a+a)^{5/2} \left (1-2 \sin (e+f x)^2\right )}{\sin (e+f x)^4}dx+\frac {8}{5} a \left (\frac {4}{3} a \int \sqrt {\sin (e+f x) a+a}dx-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}\) |
| \(\Big \downarrow \) 3125 |
| \(\displaystyle \int \frac {(\sin (e+f x) a+a)^{5/2} \left (1-2 \sin (e+f x)^2\right )}{\sin (e+f x)^4}dx+\frac {8}{5} a \left (-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \frac {\int \frac {1}{2} \csc ^3(e+f x) (5 a-13 a \sin (e+f x)) (\sin (e+f x) a+a)^{5/2}dx}{3 a}+\frac {8}{5} a \left (-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}-\frac {\cot (e+f x) \csc ^2(e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \csc ^3(e+f x) (5 a-13 a \sin (e+f x)) (\sin (e+f x) a+a)^{5/2}dx}{6 a}+\frac {8}{5} a \left (-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}-\frac {\cot (e+f x) \csc ^2(e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(5 a-13 a \sin (e+f x)) (\sin (e+f x) a+a)^{5/2}}{\sin (e+f x)^3}dx}{6 a}+\frac {8}{5} a \left (-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}-\frac {\cot (e+f x) \csc ^2(e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{2} \int -\frac {1}{2} \csc ^2(e+f x) (\sin (e+f x) a+a)^{3/2} \left (57 \sin (e+f x) a^2+17 a^2\right )dx-\frac {5 a^2 \cot (e+f x) \csc (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f}}{6 a}+\frac {8}{5} a \left (-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}-\frac {\cot (e+f x) \csc ^2(e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {1}{4} \int \csc ^2(e+f x) (\sin (e+f x) a+a)^{3/2} \left (57 \sin (e+f x) a^2+17 a^2\right )dx-\frac {5 a^2 \cot (e+f x) \csc (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f}}{6 a}+\frac {8}{5} a \left (-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}-\frac {\cot (e+f x) \csc ^2(e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {1}{4} \int \frac {(\sin (e+f x) a+a)^{3/2} \left (57 \sin (e+f x) a^2+17 a^2\right )}{\sin (e+f x)^2}dx-\frac {5 a^2 \cot (e+f x) \csc (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f}}{6 a}+\frac {8}{5} a \left (-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}-\frac {\cot (e+f x) \csc ^2(e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {17 a^3 \cot (e+f x) \sqrt {a \sin (e+f x)+a}}{f}-\int \frac {1}{2} \csc (e+f x) \sqrt {\sin (e+f x) a+a} \left (97 \sin (e+f x) a^3+165 a^3\right )dx\right )-\frac {5 a^2 \cot (e+f x) \csc (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f}}{6 a}+\frac {8}{5} a \left (-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}-\frac {\cot (e+f x) \csc ^2(e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {17 a^3 \cot (e+f x) \sqrt {a \sin (e+f x)+a}}{f}-\frac {1}{2} \int \csc (e+f x) \sqrt {\sin (e+f x) a+a} \left (97 \sin (e+f x) a^3+165 a^3\right )dx\right )-\frac {5 a^2 \cot (e+f x) \csc (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f}}{6 a}+\frac {8}{5} a \left (-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}-\frac {\cot (e+f x) \csc ^2(e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {17 a^3 \cot (e+f x) \sqrt {a \sin (e+f x)+a}}{f}-\frac {1}{2} \int \frac {\sqrt {\sin (e+f x) a+a} \left (97 \sin (e+f x) a^3+165 a^3\right )}{\sin (e+f x)}dx\right )-\frac {5 a^2 \cot (e+f x) \csc (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f}}{6 a}+\frac {8}{5} a \left (-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}-\frac {\cot (e+f x) \csc ^2(e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f}\) |
| \(\Big \downarrow \) 3460 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} \left (\frac {194 a^4 \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}-165 a^3 \int \csc (e+f x) \sqrt {\sin (e+f x) a+a}dx\right )+\frac {17 a^3 \cot (e+f x) \sqrt {a \sin (e+f x)+a}}{f}\right )-\frac {5 a^2 \cot (e+f x) \csc (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f}}{6 a}+\frac {8}{5} a \left (-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}-\frac {\cot (e+f x) \csc ^2(e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} \left (\frac {194 a^4 \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}-165 a^3 \int \frac {\sqrt {\sin (e+f x) a+a}}{\sin (e+f x)}dx\right )+\frac {17 a^3 \cot (e+f x) \sqrt {a \sin (e+f x)+a}}{f}\right )-\frac {5 a^2 \cot (e+f x) \csc (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f}}{6 a}+\frac {8}{5} a \left (-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}-\frac {\cot (e+f x) \csc ^2(e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} \left (\frac {330 a^4 \int \frac {1}{a-\frac {a^2 \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f}+\frac {194 a^4 \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )+\frac {17 a^3 \cot (e+f x) \sqrt {a \sin (e+f x)+a}}{f}\right )-\frac {5 a^2 \cot (e+f x) \csc (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f}}{6 a}+\frac {8}{5} a \left (-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}-\frac {\cot (e+f x) \csc ^2(e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {8}{5} a \left (-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}\right )+\frac {\frac {1}{4} \left (\frac {17 a^3 \cot (e+f x) \sqrt {a \sin (e+f x)+a}}{f}+\frac {1}{2} \left (\frac {330 a^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}+\frac {194 a^4 \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )\right )-\frac {5 a^2 \cot (e+f x) \csc (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f}}{6 a}-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}-\frac {\cot (e+f x) \csc ^2(e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f}\) |
Input:
Int[Cot[e + f*x]^4*(a + a*Sin[e + f*x])^(5/2),x]
Output:
(-2*a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(5*f) - (Cot[e + f*x]*Csc[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/(3*f) + (8*a*((-8*a^2*Cos[e + f*x])/ (3*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x] ])/(3*f)))/5 + ((-5*a^2*Cot[e + f*x]*Csc[e + f*x]*(a + a*Sin[e + f*x])^(3/ 2))/(2*f) + ((17*a^3*Cot[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/f + ((330*a^(7 /2)*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/f + (194*a^4 *Cos[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]))/2)/4)/(6*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[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*b*(Cos [c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]])), x] /; FreeQ[{a, b, c, d}, x] && Eq Q[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos [c + d*x]*((a + b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[a*((2*n - 1)/n) Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[ a^2 - b^2, 0] && IGtQ[n - 1/2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^4, x_Symbol] :> Int[(a + b*Sin[e + f*x])^m, x] + Int[(a + b*Sin[e + f*x])^m*(( 1 - 2*Sin[e + f*x]^2)/Sin[e + f*x]^4), x] /; FreeQ[{a, b, e, f, m}, x] && E qQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] && !LtQ[m, -1]
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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[ e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Simp[b/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp [a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B *(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], 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] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 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 [-2*b*B*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(2*n + 3)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b *d*(2*n + 3)) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, 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 = 6.65 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.98
| method | result | size |
| default | \(\frac {\left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (-1+\sin \left (f x +e \right )\right )}\, \left (-480 \sqrt {-a \left (-1+\sin \left (f x +e \right )\right )}\, a^{\frac {5}{2}} \sin \left (f x +e \right )^{3}+320 \left (-a \left (-1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}} a^{\frac {3}{2}} \sin \left (f x +e \right )^{3}-48 \left (-a \left (-1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}} \sin \left (f x +e \right )^{3} \sqrt {a}+825 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (-1+\sin \left (f x +e \right )\right )}}{\sqrt {a}}\right ) a^{3} \sin \left (f x +e \right )^{3}-345 \sqrt {-a \left (-1+\sin \left (f x +e \right )\right )}\, a^{\frac {5}{2}}+440 \left (-a \left (-1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}} a^{\frac {3}{2}}-135 \left (-a \left (-1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}} \sqrt {a}\right )}{120 \sin \left (f x +e \right )^{3} \sqrt {a}\, \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}\) | \(222\) |
Input:
int(cot(f*x+e)^4*(a+sin(f*x+e)*a)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/120*(1+sin(f*x+e))*(-a*(-1+sin(f*x+e)))^(1/2)*(-480*(-a*(-1+sin(f*x+e))) ^(1/2)*a^(5/2)*sin(f*x+e)^3+320*(-a*(-1+sin(f*x+e)))^(3/2)*a^(3/2)*sin(f*x +e)^3-48*(-a*(-1+sin(f*x+e)))^(5/2)*sin(f*x+e)^3*a^(1/2)+825*arctanh((-a*( -1+sin(f*x+e)))^(1/2)/a^(1/2))*a^3*sin(f*x+e)^3-345*(-a*(-1+sin(f*x+e)))^( 1/2)*a^(5/2)+440*(-a*(-1+sin(f*x+e)))^(3/2)*a^(3/2)-135*(-a*(-1+sin(f*x+e) ))^(5/2)*a^(1/2))/sin(f*x+e)^3/a^(1/2)/cos(f*x+e)/(a+sin(f*x+e)*a)^(1/2)/f
Leaf count of result is larger than twice the leaf count of optimal. 485 vs. \(2 (195) = 390\).
Time = 0.11 (sec) , antiderivative size = 485, normalized size of antiderivative = 2.14 \[ \int \cot ^4(e+f x) (a+a \sin (e+f x))^{5/2} \, dx=\frac {825 \, {\left (a^{2} \cos \left (f x + e\right )^{4} - 2 \, a^{2} \cos \left (f x + e\right )^{2} + a^{2} - {\left (a^{2} \cos \left (f x + e\right )^{3} + a^{2} \cos \left (f x + e\right )^{2} - a^{2} \cos \left (f x + e\right ) - a^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {a} \log \left (\frac {a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} + 4 \, {\left (\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 3\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) - 3\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} - 9 \, a \cos \left (f x + e\right ) + {\left (a \cos \left (f x + e\right )^{2} + 8 \, a \cos \left (f x + e\right ) - a\right )} \sin \left (f x + e\right ) - a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 1}\right ) - 4 \, {\left (48 \, a^{2} \cos \left (f x + e\right )^{6} + 224 \, a^{2} \cos \left (f x + e\right )^{5} - 128 \, a^{2} \cos \left (f x + e\right )^{4} - 583 \, a^{2} \cos \left (f x + e\right )^{3} + 147 \, a^{2} \cos \left (f x + e\right )^{2} + 399 \, a^{2} \cos \left (f x + e\right ) - 27 \, a^{2} + {\left (48 \, a^{2} \cos \left (f x + e\right )^{5} - 176 \, a^{2} \cos \left (f x + e\right )^{4} - 304 \, a^{2} \cos \left (f x + e\right )^{3} + 279 \, a^{2} \cos \left (f x + e\right )^{2} + 426 \, a^{2} \cos \left (f x + e\right ) + 27 \, a^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{480 \, {\left (f \cos \left (f x + e\right )^{4} - 2 \, f \cos \left (f x + e\right )^{2} - {\left (f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )^{2} - f \cos \left (f x + e\right ) - f\right )} \sin \left (f x + e\right ) + f\right )}} \] Input:
integrate(cot(f*x+e)^4*(a+a*sin(f*x+e))^(5/2),x, algorithm="fricas")
Output:
1/480*(825*(a^2*cos(f*x + e)^4 - 2*a^2*cos(f*x + e)^2 + a^2 - (a^2*cos(f*x + e)^3 + a^2*cos(f*x + e)^2 - a^2*cos(f*x + e) - a^2)*sin(f*x + e))*sqrt( a)*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 + 4*(cos(f*x + e)^2 + (cos(f *x + e) + 3)*sin(f*x + e) - 2*cos(f*x + e) - 3)*sqrt(a*sin(f*x + e) + a)*s qrt(a) - 9*a*cos(f*x + e) + (a*cos(f*x + e)^2 + 8*a*cos(f*x + e) - a)*sin( f*x + e) - a)/(cos(f*x + e)^3 + cos(f*x + e)^2 + (cos(f*x + e)^2 - 1)*sin( f*x + e) - cos(f*x + e) - 1)) - 4*(48*a^2*cos(f*x + e)^6 + 224*a^2*cos(f*x + e)^5 - 128*a^2*cos(f*x + e)^4 - 583*a^2*cos(f*x + e)^3 + 147*a^2*cos(f* x + e)^2 + 399*a^2*cos(f*x + e) - 27*a^2 + (48*a^2*cos(f*x + e)^5 - 176*a^ 2*cos(f*x + e)^4 - 304*a^2*cos(f*x + e)^3 + 279*a^2*cos(f*x + e)^2 + 426*a ^2*cos(f*x + e) + 27*a^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a))/(f*cos(f *x + e)^4 - 2*f*cos(f*x + e)^2 - (f*cos(f*x + e)^3 + f*cos(f*x + e)^2 - f* cos(f*x + e) - f)*sin(f*x + e) + f)
Timed out. \[ \int \cot ^4(e+f x) (a+a \sin (e+f x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate(cot(f*x+e)**4*(a+a*sin(f*x+e))**(5/2),x)
Output:
Timed out
\[ \int \cot ^4(e+f x) (a+a \sin (e+f x))^{5/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \cot \left (f x + e\right )^{4} \,d x } \] Input:
integrate(cot(f*x+e)^4*(a+a*sin(f*x+e))^(5/2),x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e) + a)^(5/2)*cot(f*x + e)^4, x)
Time = 0.18 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.28 \[ \int \cot ^4(e+f x) (a+a \sin (e+f x))^{5/2} \, dx=\frac {\sqrt {2} {\left (768 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 2560 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 825 \, \sqrt {2} a^{2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 1920 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \frac {20 \, {\left (108 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 176 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 69 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3}}\right )} \sqrt {a}}{480 \, f} \] Input:
integrate(cot(f*x+e)^4*(a+a*sin(f*x+e))^(5/2),x, algorithm="giac")
Output:
1/480*sqrt(2)*(768*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1 /2*f*x + 1/2*e)^5 - 2560*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4* pi + 1/2*f*x + 1/2*e)^3 + 825*sqrt(2)*a^2*log(abs(-2*sqrt(2) + 4*sin(-1/4* pi + 1/2*f*x + 1/2*e))/abs(2*sqrt(2) + 4*sin(-1/4*pi + 1/2*f*x + 1/2*e)))* sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 1920*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e) - 20*(108*a^2*sgn(cos(-1/4*pi + 1/ 2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^5 - 176*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 + 69*a^2*sgn(cos(-1/ 4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e))/(2*sin(-1/4*pi + 1/2*f*x + 1/2*e)^2 - 1)^3)*sqrt(a)/f
Timed out. \[ \int \cot ^4(e+f x) (a+a \sin (e+f x))^{5/2} \, dx=\int {\mathrm {cot}\left (e+f\,x\right )}^4\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \] Input:
int(cot(e + f*x)^4*(a + a*sin(e + f*x))^(5/2),x)
Output:
int(cot(e + f*x)^4*(a + a*sin(e + f*x))^(5/2), x)
\[ \int \cot ^4(e+f x) (a+a \sin (e+f x))^{5/2} \, dx=\sqrt {a}\, a^{2} \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \cot \left (f x +e \right )^{4} \sin \left (f x +e \right )^{2}d x +2 \left (\int \sqrt {\sin \left (f x +e \right )+1}\, \cot \left (f x +e \right )^{4} \sin \left (f x +e \right )d x \right )+\int \sqrt {\sin \left (f x +e \right )+1}\, \cot \left (f x +e \right )^{4}d x \right ) \] Input:
int(cot(f*x+e)^4*(a+a*sin(f*x+e))^(5/2),x)
Output:
sqrt(a)*a**2*(int(sqrt(sin(e + f*x) + 1)*cot(e + f*x)**4*sin(e + f*x)**2,x ) + 2*int(sqrt(sin(e + f*x) + 1)*cot(e + f*x)**4*sin(e + f*x),x) + int(sqr t(sin(e + f*x) + 1)*cot(e + f*x)**4,x))