Integrand size = 31, antiderivative size = 222 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{a^{5/2} d}-\frac {2048 \cos (c+d x)}{315 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {92 \cos (c+d x) \sin ^2(c+d x)}{105 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {38 \cos (c+d x) \sin ^3(c+d x)}{63 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sin ^4(c+d x)}{9 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {472 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{315 a^3 d} \] Output:
4*arctanh(1/2*a^(1/2)*cos(d*x+c)*2^(1/2)/(a+a*sin(d*x+c))^(1/2))*2^(1/2)/a ^(5/2)/d-2048/315*cos(d*x+c)/a^2/d/(a+a*sin(d*x+c))^(1/2)-92/105*cos(d*x+c )*sin(d*x+c)^2/a^2/d/(a+a*sin(d*x+c))^(1/2)+38/63*cos(d*x+c)*sin(d*x+c)^3/ a^2/d/(a+a*sin(d*x+c))^(1/2)-2/9*cos(d*x+c)*sin(d*x+c)^4/a^2/d/(a+a*sin(d* x+c))^(1/2)+472/315*cos(d*x+c)*(a+a*sin(d*x+c))^(1/2)/a^3/d
Result contains complex when optimal does not.
Time = 9.13 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.01 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {\sqrt {a (1+\sin (c+d x))} \left ((20160+20160 i) (-1)^{3/4} \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \sec \left (\frac {d x}{4}\right ) \left (\cos \left (\frac {1}{4} (2 c+d x)\right )-\sin \left (\frac {1}{4} (2 c+d x)\right )\right )\right )-16380 \cos \left (\frac {1}{2} (c+d x)\right )+3150 \cos \left (\frac {3}{2} (c+d x)\right )+882 \cos \left (\frac {5}{2} (c+d x)\right )-225 \cos \left (\frac {7}{2} (c+d x)\right )-35 \cos \left (\frac {9}{2} (c+d x)\right )+16380 \sin \left (\frac {1}{2} (c+d x)\right )+3150 \sin \left (\frac {3}{2} (c+d x)\right )-882 \sin \left (\frac {5}{2} (c+d x)\right )-225 \sin \left (\frac {7}{2} (c+d x)\right )+35 \sin \left (\frac {9}{2} (c+d x)\right )\right )}{2520 a^3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \] Input:
Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^(5/2),x]
Output:
(Sqrt[a*(1 + Sin[c + d*x])]*((20160 + 20160*I)*(-1)^(3/4)*ArcTanh[(1/2 + I /2)*(-1)^(3/4)*Sec[(d*x)/4]*(Cos[(2*c + d*x)/4] - Sin[(2*c + d*x)/4])] - 1 6380*Cos[(c + d*x)/2] + 3150*Cos[(3*(c + d*x))/2] + 882*Cos[(5*(c + d*x))/ 2] - 225*Cos[(7*(c + d*x))/2] - 35*Cos[(9*(c + d*x))/2] + 16380*Sin[(c + d *x)/2] + 3150*Sin[(3*(c + d*x))/2] - 882*Sin[(5*(c + d*x))/2] - 225*Sin[(7 *(c + d*x))/2] + 35*Sin[(9*(c + d*x))/2]))/(2520*a^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))
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.
| \(\displaystyle \int \frac {\sin ^3(c+d x) \cos ^4(c+d x)}{(a \sin (c+d x)+a)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^3 \cos (c+d x)^4}{(a \sin (c+d x)+a)^{5/2}}dx\) |
| \(\Big \downarrow \) 3359 |
| \(\displaystyle \frac {\int \frac {\sin ^3(c+d x) \left (\sin ^2(c+d x)+1\right )}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \int \frac {\sin ^4(c+d x)}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^3 \left (\sin (c+d x)^2+1\right )}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \int \frac {\sin (c+d x)^4}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}\) |
| \(\Big \downarrow \) 3257 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^3 \left (\sin (c+d x)^2+1\right )}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \left (-\frac {\int -\frac {\sin ^2(c+d x) (6 a-a \sin (c+d x))}{\sqrt {\sin (c+d x) a+a}}dx}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^3 \left (\sin (c+d x)^2+1\right )}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \left (\frac {\int \frac {\sin ^2(c+d x) (6 a-a \sin (c+d x))}{\sqrt {\sin (c+d x) a+a}}dx}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^3 \left (\sin (c+d x)^2+1\right )}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \left (\frac {\int \frac {\sin (c+d x)^2 (6 a-a \sin (c+d x))}{\sqrt {\sin (c+d x) a+a}}dx}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3462 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^3 \left (\sin (c+d x)^2+1\right )}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \left (\frac {\frac {2 \int -\frac {\sin (c+d x) \left (4 a^2-31 a^2 \sin (c+d x)\right )}{2 \sqrt {\sin (c+d x) a+a}}dx}{5 a}+\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^3 \left (\sin (c+d x)^2+1\right )}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \left (\frac {\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\int \frac {\sin (c+d x) \left (4 a^2-31 a^2 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{5 a}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^3 \left (\sin (c+d x)^2+1\right )}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \left (\frac {\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\int \frac {\sin (c+d x) \left (4 a^2-31 a^2 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{5 a}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^3 \left (\sin (c+d x)^2+1\right )}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \left (\frac {\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\int \frac {4 a^2 \sin (c+d x)-31 a^2 \sin ^2(c+d x)}{\sqrt {\sin (c+d x) a+a}}dx}{5 a}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^3 \left (\sin (c+d x)^2+1\right )}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \left (\frac {\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\int \frac {4 a^2 \sin (c+d x)-31 a^2 \sin (c+d x)^2}{\sqrt {\sin (c+d x) a+a}}dx}{5 a}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^3 \left (\sin (c+d x)^2+1\right )}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \left (\frac {\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {2 \int -\frac {31 a^3-74 a^3 \sin (c+d x)}{2 \sqrt {\sin (c+d x) a+a}}dx}{3 a}+\frac {62 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{5 a}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^3 \left (\sin (c+d x)^2+1\right )}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \left (\frac {\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {62 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\int \frac {31 a^3-74 a^3 \sin (c+d x)}{\sqrt {\sin (c+d x) a+a}}dx}{3 a}}{5 a}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^3 \left (\sin (c+d x)^2+1\right )}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \left (\frac {\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {62 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\int \frac {31 a^3-74 a^3 \sin (c+d x)}{\sqrt {\sin (c+d x) a+a}}dx}{3 a}}{5 a}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^3 \left (\sin (c+d x)^2+1\right )}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \left (\frac {\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {62 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {105 a^3 \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx+\frac {148 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{3 a}}{5 a}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^3 \left (\sin (c+d x)^2+1\right )}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \left (\frac {\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {62 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {105 a^3 \int \frac {1}{\sqrt {\sin (c+d x) a+a}}dx+\frac {148 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}}{3 a}}{5 a}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^3 \left (\sin (c+d x)^2+1\right )}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \left (\frac {\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {62 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {148 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {210 a^3 \int \frac {1}{2 a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}}{3 a}}{5 a}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^3 \left (\sin (c+d x)^2+1\right )}{\sqrt {\sin (c+d x) a+a}}dx}{a^2}-\frac {2 \left (\frac {\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {62 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {148 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {105 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3525 |
| \(\displaystyle \frac {\frac {2 \int \frac {\sin ^3(c+d x) (17 a-a \sin (c+d x))}{2 \sqrt {\sin (c+d x) a+a}}dx}{9 a}-\frac {2 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (\frac {\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {62 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {148 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {105 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sin ^3(c+d x) (17 a-a \sin (c+d x))}{\sqrt {\sin (c+d x) a+a}}dx}{9 a}-\frac {2 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (\frac {\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {62 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {148 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {105 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sin (c+d x)^3 (17 a-a \sin (c+d x))}{\sqrt {\sin (c+d x) a+a}}dx}{9 a}-\frac {2 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (\frac {\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {62 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {148 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {105 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3462 |
| \(\displaystyle \frac {\frac {\frac {2 \int -\frac {3 \sin ^2(c+d x) \left (a^2-20 a^2 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{7 a}+\frac {2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}}{9 a}-\frac {2 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (\frac {\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {62 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {148 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {105 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}-\frac {6 \int \frac {\sin ^2(c+d x) \left (a^2-20 a^2 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{7 a}}{9 a}-\frac {2 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (\frac {\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {62 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {148 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {105 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}-\frac {6 \int \frac {\sin (c+d x)^2 \left (a^2-20 a^2 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{7 a}}{9 a}-\frac {2 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (\frac {\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {62 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {148 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {105 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3462 |
| \(\displaystyle \frac {\frac {\frac {2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}-\frac {6 \left (\frac {2 \int -\frac {5 \sin (c+d x) \left (16 a^3-5 a^3 \sin (c+d x)\right )}{2 \sqrt {\sin (c+d x) a+a}}dx}{5 a}+\frac {8 a^2 \sin ^2(c+d x) \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )}{7 a}}{9 a}-\frac {2 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (\frac {\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {62 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {148 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {105 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}-\frac {6 \left (\frac {8 a^2 \sin ^2(c+d x) \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\int \frac {\sin (c+d x) \left (16 a^3-5 a^3 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{a}\right )}{7 a}}{9 a}-\frac {2 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (\frac {\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {62 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {148 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {105 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}-\frac {6 \left (\frac {8 a^2 \sin ^2(c+d x) \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\int \frac {\sin (c+d x) \left (16 a^3-5 a^3 \sin (c+d x)\right )}{\sqrt {\sin (c+d x) a+a}}dx}{a}\right )}{7 a}}{9 a}-\frac {2 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (\frac {\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {62 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {148 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {105 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {\frac {2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}-\frac {6 \left (\frac {8 a^2 \sin ^2(c+d x) \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\int \frac {16 a^3 \sin (c+d x)-5 a^3 \sin ^2(c+d x)}{\sqrt {\sin (c+d x) a+a}}dx}{a}\right )}{7 a}}{9 a}-\frac {2 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (\frac {\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {62 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {148 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {105 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}-\frac {6 \left (\frac {8 a^2 \sin ^2(c+d x) \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\int \frac {16 a^3 \sin (c+d x)-5 a^3 \sin (c+d x)^2}{\sqrt {\sin (c+d x) a+a}}dx}{a}\right )}{7 a}}{9 a}-\frac {2 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (\frac {\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {62 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {148 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {105 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {\frac {2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}-\frac {6 \left (\frac {8 a^2 \sin ^2(c+d x) \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {2 \int -\frac {5 a^4-58 a^4 \sin (c+d x)}{2 \sqrt {\sin (c+d x) a+a}}dx}{3 a}+\frac {10 a^2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{a}\right )}{7 a}}{9 a}-\frac {2 \sin ^4(c+d x) \cos (c+d x)}{9 d \sqrt {a \sin (c+d x)+a}}}{a^2}-\frac {2 \left (\frac {\frac {2 a \sin ^2(c+d x) \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {\frac {62 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}-\frac {\frac {148 a^3 \cos (c+d x)}{d \sqrt {a \sin (c+d x)+a}}-\frac {105 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{d}}{3 a}}{5 a}}{7 a}-\frac {2 \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\) |
Input:
Int[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^(5/2),x]
Output:
$Aborted
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[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/Sqrt[(a_) + (b_.)*sin[(e_. ) + (f_.)*(x_)]], x_Symbol] :> Simp[-2*d*Cos[e + f*x]*((c + d*Sin[e + f*x]) ^(n - 1)/(f*(2*n - 1)*Sqrt[a + b*Sin[e + f*x]])), x] - Simp[1/(b*(2*n - 1)) Int[((c + d*Sin[e + f*x])^(n - 2)/Sqrt[a + b*Sin[e + f*x]])*Simp[a*c*d - b*(2*d^2*(n - 1) + c^2*(2*n - 1)) + d*(a*d - b*c*(4*n - 3))*Sin[e + f*x], x], 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] && GtQ[n, 1] && IntegerQ[2*n]
Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[-2/(a*b*d) Int[(d* Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 2), x], x] + Simp[1/a^2 I nt[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^(m + 2)*(1 + Sin[e + f*x]^2), x] , x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 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)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/(f*(m + n + 1))), x] + Simp[1/(b*(m + n + 1)) Int[(a + b*Sin[e + f*x])^m*(c + d*S in[e + f*x])^(n - 1)*Simp[A*b*c*(m + n + 1) + B*(a*c*m + b*d*n) + (A*b*d*(m + n + 1) + B*(a*d*m + b*c*n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] && (IntegerQ[n] || EqQ[m + 1/2, 0])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -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)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1 )/(d*f*(m + n + 2))), x] + Simp[1/(b*d*(m + n + 2)) Int[(a + b*Sin[e + f* x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1 )) + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && NeQ[m + n + 2, 0]
Time = 0.51 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.75
| method | result | size |
| default | \(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (-630 a^{\frac {9}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )+35 \left (a -a \sin \left (d x +c \right )\right )^{\frac {9}{2}}-45 a \left (a -a \sin \left (d x +c \right )\right )^{\frac {7}{2}}+63 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} a^{2}+105 a^{3} \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}}+630 \sqrt {a -a \sin \left (d x +c \right )}\, a^{4}\right )}{315 a^{7} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(166\) |
Input:
int(cos(d*x+c)^4*sin(d*x+c)^3/(a+a*sin(d*x+c))^(5/2),x,method=_RETURNVERBO SE)
Output:
-2/315/a^7*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(-630*a^(9/2)*2^(1/2)* arctanh(1/2*(a-a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2))+35*(a-a*sin(d*x+c))^(9 /2)-45*a*(a-a*sin(d*x+c))^(7/2)+63*(a-a*sin(d*x+c))^(5/2)*a^2+105*a^3*(a-a *sin(d*x+c))^(3/2)+630*(a-a*sin(d*x+c))^(1/2)*a^4)/cos(d*x+c)/(a+a*sin(d*x +c))^(1/2)/d
Time = 0.09 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.26 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {2 \, {\left (\frac {315 \, \sqrt {2} {\left (a \cos \left (d x + c\right ) + a \sin \left (d x + c\right ) + a\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + \frac {2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}}{\sqrt {a}} + 3 \, \cos \left (d x + c\right ) + 2}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right )}{\sqrt {a}} - {\left (35 \, \cos \left (d x + c\right )^{5} + 130 \, \cos \left (d x + c\right )^{4} - 208 \, \cos \left (d x + c\right )^{3} - 634 \, \cos \left (d x + c\right )^{2} - {\left (35 \, \cos \left (d x + c\right )^{4} - 95 \, \cos \left (d x + c\right )^{3} - 303 \, \cos \left (d x + c\right )^{2} + 331 \, \cos \left (d x + c\right ) + 1292\right )} \sin \left (d x + c\right ) + 961 \, \cos \left (d x + c\right ) + 1292\right )} \sqrt {a \sin \left (d x + c\right ) + a}\right )}}{315 \, {\left (a^{3} d \cos \left (d x + c\right ) + a^{3} d \sin \left (d x + c\right ) + a^{3} d\right )}} \] Input:
integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+a*sin(d*x+c))^(5/2),x, algorithm="f ricas")
Output:
2/315*(315*sqrt(2)*(a*cos(d*x + c) + a*sin(d*x + c) + a)*log(-(cos(d*x + c )^2 - (cos(d*x + c) - 2)*sin(d*x + c) + 2*sqrt(2)*sqrt(a*sin(d*x + c) + a) *(cos(d*x + c) - sin(d*x + c) + 1)/sqrt(a) + 3*cos(d*x + c) + 2)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2))/sqrt(a) - (3 5*cos(d*x + c)^5 + 130*cos(d*x + c)^4 - 208*cos(d*x + c)^3 - 634*cos(d*x + c)^2 - (35*cos(d*x + c)^4 - 95*cos(d*x + c)^3 - 303*cos(d*x + c)^2 + 331* cos(d*x + c) + 1292)*sin(d*x + c) + 961*cos(d*x + c) + 1292)*sqrt(a*sin(d* x + c) + a))/(a^3*d*cos(d*x + c) + a^3*d*sin(d*x + c) + a^3*d)
Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**4*sin(d*x+c)**3/(a+a*sin(d*x+c))**(5/2),x)
Output:
Timed out
\[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{3}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+a*sin(d*x+c))^(5/2),x, algorithm="m axima")
Output:
integrate(cos(d*x + c)^4*sin(d*x + c)^3/(a*sin(d*x + c) + a)^(5/2), x)
Time = 0.18 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.91 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2 \, {\left (\frac {315 \, \sqrt {2} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {315 \, \sqrt {2} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {2 \, \sqrt {2} {\left (280 \, a^{\frac {49}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 180 \, a^{\frac {49}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 126 \, a^{\frac {49}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, a^{\frac {49}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 315 \, a^{\frac {49}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{27} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}\right )}}{315 \, d} \] Input:
integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+a*sin(d*x+c))^(5/2),x, algorithm="g iac")
Output:
-2/315*(315*sqrt(2)*log(sin(-1/4*pi + 1/2*d*x + 1/2*c) + 1)/(a^(5/2)*sgn(c os(-1/4*pi + 1/2*d*x + 1/2*c))) - 315*sqrt(2)*log(-sin(-1/4*pi + 1/2*d*x + 1/2*c) + 1)/(a^(5/2)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))) - 2*sqrt(2)*(28 0*a^(49/2)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^9 - 180*a^(49/2)*sin(-1/4*pi + 1 /2*d*x + 1/2*c)^7 + 126*a^(49/2)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^5 + 105*a^ (49/2)*sin(-1/4*pi + 1/2*d*x + 1/2*c)^3 + 315*a^(49/2)*sin(-1/4*pi + 1/2*d *x + 1/2*c))/(a^27*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))))/d
Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^3}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \] Input:
int((cos(c + d*x)^4*sin(c + d*x)^3)/(a + a*sin(c + d*x))^(5/2),x)
Output:
int((cos(c + d*x)^4*sin(c + d*x)^3)/(a + a*sin(c + d*x))^(5/2), x)
\[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )^{3}}{\sin \left (d x +c \right )^{3}+3 \sin \left (d x +c \right )^{2}+3 \sin \left (d x +c \right )+1}d x \right )}{a^{3}} \] Input:
int(cos(d*x+c)^4*sin(d*x+c)^3/(a+a*sin(d*x+c))^(5/2),x)
Output:
(sqrt(a)*int((sqrt(sin(c + d*x) + 1)*cos(c + d*x)**4*sin(c + d*x)**3)/(sin (c + d*x)**3 + 3*sin(c + d*x)**2 + 3*sin(c + d*x) + 1),x))/a**3