Integrand size = 31, antiderivative size = 374 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {\left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{35 b d}+\frac {\left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 a b d}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}-\frac {a \left (4 a^2+167 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{35 b^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (4 a^4+61 a^2 b^2+40 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{35 b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {3 a b \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{d \sqrt {a+b \sin (c+d x)}} \] Output:
1/35*(4*a^2+65*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b/d+1/35*(4*a^2+35*b ^2)*cos(d*x+c)*(a+b*sin(d*x+c))^(3/2)/a/b/d-2/7*cos(d*x+c)*(a+b*sin(d*x+c) )^(5/2)/b/d-cot(d*x+c)*(a+b*sin(d*x+c))^(5/2)/a/d+1/35*a*(4*a^2+167*b^2)*E llipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2)*(b/(a+b))^(1/2))*(a+b*sin(d*x+c ))^(1/2)/b^2/d/((a+b*sin(d*x+c))/(a+b))^(1/2)+1/35*(4*a^4+61*a^2*b^2+40*b^ 4)*InverseJacobiAM(1/2*c-1/4*Pi+1/2*d*x,2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin (d*x+c))/(a+b))^(1/2)/b^2/d/(a+b*sin(d*x+c))^(1/2)-3*a*b*EllipticPi(cos(1/ 2*c+1/4*Pi+1/2*d*x),2,2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin(d*x+c))/(a+b))^(1 /2)/d/(a+b*sin(d*x+c))^(1/2)
Result contains complex when optimal does not.
Time = 4.92 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.21 \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {\frac {2 i \left (4 a^2+167 b^2\right ) \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {\frac {b (1+\sin (c+d x))}{-a+b}}}{b^3 \sqrt {-\frac {1}{a+b}}}+\frac {8 \left (53 a^2-20 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a \left (4 a^2-43 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{b \sqrt {a+b \sin (c+d x)}}-\frac {2 \sqrt {a+b \sin (c+d x)} \left (\left (4 a^2-55 b^2\right ) \cos (c+d x)+b (-5 b \cos (3 (c+d x))+70 a \cot (c+d x)+16 a \sin (2 (c+d x)))\right )}{b}}{140 d} \] Input:
Integrate[Cos[c + d*x]^2*Cot[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2),x]
Output:
(((2*I)*(4*a^2 + 167*b^2)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^ (-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*Ar cSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b* EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x ]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b ))]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)])/(b^3*Sqrt[-(a + b)^(-1)]) + (8* (53*a^2 - 20*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] + (2*a*(4*a^2 - 43*b^ 2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(b*Sqrt[a + b*Sin[c + d*x]]) - (2*Sqrt[a + b*Sin[c + d*x] ]*((4*a^2 - 55*b^2)*Cos[c + d*x] + b*(-5*b*Cos[3*(c + d*x)] + 70*a*Cot[c + d*x] + 16*a*Sin[2*(c + d*x)])))/b)/(140*d)
Time = 3.04 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.05, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.774, Rules used = {3042, 3373, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3538, 25, 3042, 3134, 3042, 3132, 3481, 3042, 3142, 3042, 3140, 3286, 3042, 3284}
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.
| \(\displaystyle \int \cos ^2(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a+b \sin (c+d x))^{3/2}}{\sin (c+d x)^2}dx\) |
| \(\Big \downarrow \) 3373 |
| \(\displaystyle -\frac {2 \int -\frac {1}{4} \csc (c+d x) (a+b \sin (c+d x))^{3/2} \left (21 b^2-18 a \sin (c+d x) b-\left (4 a^2+35 b^2\right ) \sin ^2(c+d x)\right )dx}{7 a b}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \csc (c+d x) (a+b \sin (c+d x))^{3/2} \left (21 b^2-18 a \sin (c+d x) b-\left (4 a^2+35 b^2\right ) \sin ^2(c+d x)\right )dx}{14 a b}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(a+b \sin (c+d x))^{3/2} \left (21 b^2-18 a \sin (c+d x) b-\left (4 a^2+35 b^2\right ) \sin (c+d x)^2\right )}{\sin (c+d x)}dx}{14 a b}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {2}{5} \int \frac {3}{2} \csc (c+d x) \sqrt {a+b \sin (c+d x)} \left (-34 b \sin (c+d x) a^2+35 b^2 a-\left (4 a^2+65 b^2\right ) \sin ^2(c+d x) a\right )dx+\frac {2 \left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}}{14 a b}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3}{5} \int \csc (c+d x) \sqrt {a+b \sin (c+d x)} \left (-34 b \sin (c+d x) a^2+35 b^2 a-\left (4 a^2+65 b^2\right ) \sin ^2(c+d x) a\right )dx+\frac {2 \left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}}{14 a b}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{5} \int \frac {\sqrt {a+b \sin (c+d x)} \left (-34 b \sin (c+d x) a^2+35 b^2 a-\left (4 a^2+65 b^2\right ) \sin (c+d x)^2 a\right )}{\sin (c+d x)}dx+\frac {2 \left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}}{14 a b}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {3}{5} \left (\frac {2}{3} \int \frac {\csc (c+d x) \left (105 a^2 b^2-2 a \left (53 a^2-20 b^2\right ) \sin (c+d x) b-a^2 \left (4 a^2+167 b^2\right ) \sin ^2(c+d x)\right )}{2 \sqrt {a+b \sin (c+d x)}}dx+\frac {2 a \left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 \left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}}{14 a b}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3}{5} \left (\frac {1}{3} \int \frac {\csc (c+d x) \left (105 a^2 b^2-2 a \left (53 a^2-20 b^2\right ) \sin (c+d x) b-a^2 \left (4 a^2+167 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx+\frac {2 a \left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 \left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}}{14 a b}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{5} \left (\frac {1}{3} \int \frac {105 a^2 b^2-2 a \left (53 a^2-20 b^2\right ) \sin (c+d x) b-a^2 \left (4 a^2+167 b^2\right ) \sin (c+d x)^2}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {2 a \left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 \left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}}{14 a b}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {\frac {3}{5} \left (\frac {1}{3} \left (-\frac {a^2 \left (4 a^2+167 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {\int -\frac {\csc (c+d x) \left (105 a^2 b^3+a \left (4 a^4+61 b^2 a^2+40 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )+\frac {2 a \left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 \left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}}{14 a b}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\int \frac {\csc (c+d x) \left (105 a^2 b^3+a \left (4 a^4+61 b^2 a^2+40 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {a^2 \left (4 a^2+167 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}\right )+\frac {2 a \left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 \left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}}{14 a b}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\int \frac {105 a^2 b^3+a \left (4 a^4+61 b^2 a^2+40 b^4\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {a^2 \left (4 a^2+167 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}\right )+\frac {2 a \left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 \left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}}{14 a b}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\int \frac {105 a^2 b^3+a \left (4 a^4+61 b^2 a^2+40 b^4\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {a^2 \left (4 a^2+167 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 a \left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 \left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}}{14 a b}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\int \frac {105 a^2 b^3+a \left (4 a^4+61 b^2 a^2+40 b^4\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {a^2 \left (4 a^2+167 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 a \left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 \left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}}{14 a b}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\int \frac {105 a^2 b^3+a \left (4 a^4+61 b^2 a^2+40 b^4\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 a^2 \left (4 a^2+167 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 a \left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 \left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}}{14 a b}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {105 a^2 b^3 \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx+a \left (4 a^4+61 a^2 b^2+40 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 a^2 \left (4 a^2+167 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 a \left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 \left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}}{14 a b}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {105 a^2 b^3 \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+a \left (4 a^4+61 a^2 b^2+40 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 a^2 \left (4 a^2+167 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 a \left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 \left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}}{14 a b}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {105 a^2 b^3 \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {a \left (4 a^4+61 a^2 b^2+40 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 a^2 \left (4 a^2+167 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 a \left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 \left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}}{14 a b}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {105 a^2 b^3 \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {a \left (4 a^4+61 a^2 b^2+40 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 a^2 \left (4 a^2+167 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 a \left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 \left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}}{14 a b}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {105 a^2 b^3 \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {2 a \left (4 a^4+61 a^2 b^2+40 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 a^2 \left (4 a^2+167 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 a \left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 \left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}}{14 a b}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle \frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\frac {105 a^2 b^3 \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a \left (4 a^4+61 a^2 b^2+40 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 a^2 \left (4 a^2+167 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 a \left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 \left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}}{14 a b}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {\frac {105 a^2 b^3 \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sin (c+d x) \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a \left (4 a^4+61 a^2 b^2+40 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 a^2 \left (4 a^2+167 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 a \left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 \left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}}{14 a b}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {\frac {2 \left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}+\frac {3}{5} \left (\frac {2 a \left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}+\frac {1}{3} \left (\frac {\frac {210 a^2 b^3 \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}+\frac {2 a \left (4 a^4+61 a^2 b^2+40 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 a^2 \left (4 a^2+167 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )\right )}{14 a b}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}\) |
Input:
Int[Cos[c + d*x]^2*Cot[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2),x]
Output:
(-2*Cos[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(7*b*d) - (Cot[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(a*d) + ((2*(4*a^2 + 35*b^2)*Cos[c + d*x]*(a + b*Si n[c + d*x])^(3/2))/(5*d) + (3*((2*a*(4*a^2 + 65*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(3*d) + ((-2*a^2*(4*a^2 + 167*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(b*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((2*a*(4*a^4 + 61*a^2*b^2 + 40*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]]) + (210*a^2*b^3*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]]))/b )/3))/5)/(14*a*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt [c + d*Sin[e + f*x]] Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + d))*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a* d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[Cos[e + f*x]*(a + b* Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 1)/(a*d*f*(n + 1))), x] + (-Si mp[Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 2)/(b*d ^2*f*(m + n + 4))), x] + Simp[1/(a*b*d*(n + 1)*(m + n + 4)) Int[(a + b*Si n[e + f*x])^m*(d*Sin[e + f*x])^(n + 1)*Simp[a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 4) + a*b*(m + 3)*Sin[e + f*x] - (a^2*(n + 1)*(n + 3) - b^2* (m + n + 3)*(m + n + 4))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n]) && ! m < -1 && LtQ[n, -1] && NeQ[m + n + 4, 0]
Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ B/d Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d Int[(a + b* Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (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/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d) Int[Sqrt[a + b*Sin[e + f*x]], x] , x] - Simp[1/(b*d) Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre eQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0 ] && NeQ[c^2 - d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(725\) vs. \(2(352)=704\).
Time = 2.25 (sec) , antiderivative size = 726, normalized size of antiderivative = 1.94
| method | result | size |
| default | \(-\frac {10 b^{5} \cos \left (d x +c \right )^{6}-26 a \,b^{4} \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )+\left (-18 a^{2} b^{3}+10 b^{5}\right ) \cos \left (d x +c \right )^{4}+\left (2 a^{3} b^{2}+31 a \,b^{4}\right ) \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )+\left (53 a^{2} b^{3}-20 b^{5}\right ) \cos \left (d x +c \right )^{2}+\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}\, \sqrt {-\frac {b \sin \left (d x +c \right )}{a +b}+\frac {b}{a +b}}\, \sqrt {-\frac {b \sin \left (d x +c \right )}{a -b}-\frac {b}{a -b}}\, \left (105 \operatorname {EllipticPi}\left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \frac {a -b}{a}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{4}-105 \operatorname {EllipticPi}\left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \frac {a -b}{a}, \sqrt {\frac {a -b}{a +b}}\right ) b^{5}+4 \operatorname {EllipticF}\left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{4} b +102 \operatorname {EllipticF}\left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3} b^{2}+61 \operatorname {EllipticF}\left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{2} b^{3}-207 \operatorname {EllipticF}\left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{4}+40 \operatorname {EllipticF}\left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b^{5}-4 \operatorname {EllipticE}\left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{5}-163 \operatorname {EllipticE}\left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3} b^{2}+167 \operatorname {EllipticE}\left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{4}\right ) \sin \left (d x +c \right )}{35 b^{3} \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {a +b \sin \left (d x +c \right )}\, d}\) | \(726\) |
Input:
int(cos(d*x+c)^2*cot(d*x+c)^2*(a+b*sin(d*x+c))^(3/2),x,method=_RETURNVERBO SE)
Output:
-1/35*(10*b^5*cos(d*x+c)^6-26*a*b^4*cos(d*x+c)^4*sin(d*x+c)+(-18*a^2*b^3+1 0*b^5)*cos(d*x+c)^4+(2*a^3*b^2+31*a*b^4)*cos(d*x+c)^2*sin(d*x+c)+(53*a^2*b ^3-20*b^5)*cos(d*x+c)^2+(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*(-b/(a+b)*sin(d *x+c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*(105*EllipticPi(( b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a*b^4-105*E llipticPi((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))* b^5+4*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^ 4*b+102*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))* a^3*b^2+61*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2 ))*a^2*b^3-207*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^ (1/2))*a*b^4+40*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b)) ^(1/2))*b^5-4*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^( 1/2))*a^5-163*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^( 1/2))*a^3*b^2+167*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b ))^(1/2))*a*b^4)*sin(d*x+c))/b^3/sin(d*x+c)/cos(d*x+c)/(a+b*sin(d*x+c))^(1 /2)/d
Timed out. \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)^2*cot(d*x+c)^2*(a+b*sin(d*x+c))^(3/2),x, algorithm="f ricas")
Output:
Timed out
Timed out. \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**2*cot(d*x+c)**2*(a+b*sin(d*x+c))**(3/2),x)
Output:
Timed out
\[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{2} \cot \left (d x + c\right )^{2} \,d x } \] Input:
integrate(cos(d*x+c)^2*cot(d*x+c)^2*(a+b*sin(d*x+c))^(3/2),x, algorithm="m axima")
Output:
integrate((b*sin(d*x + c) + a)^(3/2)*cos(d*x + c)^2*cot(d*x + c)^2, x)
Timed out. \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)^2*cot(d*x+c)^2*(a+b*sin(d*x+c))^(3/2),x, algorithm="g iac")
Output:
Timed out
Timed out. \[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int {\cos \left (c+d\,x\right )}^2\,{\mathrm {cot}\left (c+d\,x\right )}^2\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \] Input:
int(cos(c + d*x)^2*cot(c + d*x)^2*(a + b*sin(c + d*x))^(3/2),x)
Output:
int(cos(c + d*x)^2*cot(c + d*x)^2*(a + b*sin(c + d*x))^(3/2), x)
\[ \int \cos ^2(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\left (\int \sqrt {\sin \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \cot \left (d x +c \right )^{2} \sin \left (d x +c \right )d x \right ) b +\left (\int \sqrt {\sin \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \cot \left (d x +c \right )^{2}d x \right ) a \] Input:
int(cos(d*x+c)^2*cot(d*x+c)^2*(a+b*sin(d*x+c))^(3/2),x)
Output:
int(sqrt(sin(c + d*x)*b + a)*cos(c + d*x)**2*cot(c + d*x)**2*sin(c + d*x), x)*b + int(sqrt(sin(c + d*x)*b + a)*cos(c + d*x)**2*cot(c + d*x)**2,x)*a