Integrand size = 29, antiderivative size = 366 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{4 a^3 b d}-\frac {\left (8 a^2-15 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{4 a^3 b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (8 a^2-5 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{4 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {3 \left (4 a^2-5 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{4 a^3 d \sqrt {a+b \sin (c+d x)}} \]
1/2*(4*a^2-5*b^2)*cot(d*x+c)/a^2/b/d/(a+b*sin(d*x+c))^(1/2)-1/2*cot(d*x+c) *csc(d*x+c)/a/d/(a+b*sin(d*x+c))^(1/2)-1/4*(8*a^2-15*b^2)*cot(d*x+c)*(a+b* sin(d*x+c))^(1/2)/a^3/b/d+1/4*(8*a^2-15*b^2)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2) ^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticE(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)/a^3/b/d/((a+b*sin(d*x+c))/(a+b) )^(1/2)-1/4*(8*a^2-5*b^2)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/ 4*Pi+1/2*d*x)*EllipticF(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))/(a+b))^(1/2)/a^2/b/d/(a+b*sin(d*x+c))^(1/2)+3/4*(4*a^2- 5*b^2)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*Ellip ticPi(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)/a^3/d/(a+b*sin(d*x+c))^(1/2)
Result contains complex when optimal does not.
Time = 3.51 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.19 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {\frac {\left (-32 a^2+60 b^2\right ) \cos (c+d x)+4 a \cot (c+d x) (5 b-2 a \csc (c+d x))}{a^3 \sqrt {a+b \sin (c+d x)}}+\frac {-\frac {2 i \left (-8 a^2+15 b^2\right ) \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}}{a b^2 \sqrt {-\frac {1}{a+b}}}-\frac {40 a b \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}+\frac {2 \left (32 a^2-45 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}}{a^3}}{16 d} \]
(((-32*a^2 + 60*b^2)*Cos[c + d*x] + 4*a*Cot[c + d*x]*(5*b - 2*a*Csc[c + d* x]))/(a^3*Sqrt[a + b*Sin[c + d*x]]) + (((-2*I)*(-8*a^2 + 15*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*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqr t[-(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))])/(a*b^2*Sqrt[-(a + b)^(-1)]) - (40*a*b*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*(32*a^2 - 45*b^2)*EllipticPi[2, (-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]])/a^ 3)/(16*d)
Time = 2.61 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.01, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.724, Rules used = {3042, 3369, 27, 3042, 3534, 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.
\(\displaystyle \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^4}{\sin (c+d x)^3 (a+b \sin (c+d x))^{3/2}}dx\) |
\(\Big \downarrow \) 3369 |
\(\displaystyle \frac {\int \frac {\csc ^2(c+d x) \left (8 a^2-2 b \sin (c+d x) a-15 b^2+5 b^2 \sin ^2(c+d x)\right )}{4 \sqrt {a+b \sin (c+d x)}}dx}{a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\csc ^2(c+d x) \left (8 a^2-2 b \sin (c+d x) a-15 b^2+5 b^2 \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {8 a^2-2 b \sin (c+d x) a-15 b^2+5 b^2 \sin (c+d x)^2}{\sin (c+d x)^2 \sqrt {a+b \sin (c+d x)}}dx}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\) |
\(\Big \downarrow \) 3534 |
\(\displaystyle \frac {\frac {\int -\frac {\csc (c+d x) \left (-10 a \sin (c+d x) b^2+\left (8 a^2-15 b^2\right ) \sin ^2(c+d x) b+3 \left (4 a^2-5 b^2\right ) b\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {-\frac {\int \frac {\csc (c+d x) \left (-10 a \sin (c+d x) b^2+\left (8 a^2-15 b^2\right ) \sin ^2(c+d x) b+3 \left (4 a^2-5 b^2\right ) b\right )}{\sqrt {a+b \sin (c+d x)}}dx}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {\int \frac {-10 a \sin (c+d x) b^2+\left (8 a^2-15 b^2\right ) \sin (c+d x)^2 b+3 \left (4 a^2-5 b^2\right ) b}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\) |
\(\Big \downarrow \) 3538 |
\(\displaystyle \frac {-\frac {\left (8 a^2-15 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx-\frac {\int -\frac {\csc (c+d x) \left (3 b^2 \left (4 a^2-5 b^2\right )-a b \left (8 a^2-5 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {-\frac {\left (8 a^2-15 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx+\frac {\int \frac {\csc (c+d x) \left (3 b^2 \left (4 a^2-5 b^2\right )-a b \left (8 a^2-5 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {\left (8 a^2-15 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx+\frac {\int \frac {3 b^2 \left (4 a^2-5 b^2\right )-a b \left (8 a^2-5 b^2\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\) |
\(\Big \downarrow \) 3134 |
\(\displaystyle \frac {-\frac {\frac {\int \frac {3 b^2 \left (4 a^2-5 b^2\right )-a b \left (8 a^2-5 b^2\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {\left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {\frac {\left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\int \frac {3 b^2 \left (4 a^2-5 b^2\right )-a b \left (8 a^2-5 b^2\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\) |
\(\Big \downarrow \) 3132 |
\(\displaystyle \frac {-\frac {\frac {\int \frac {3 b^2 \left (4 a^2-5 b^2\right )-a b \left (8 a^2-5 b^2\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\) |
\(\Big \downarrow \) 3481 |
\(\displaystyle \frac {-\frac {\frac {3 b^2 \left (4 a^2-5 b^2\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx-a b \left (8 a^2-5 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {\frac {3 b^2 \left (4 a^2-5 b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-a b \left (8 a^2-5 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\) |
\(\Big \downarrow \) 3142 |
\(\displaystyle \frac {-\frac {\frac {3 b^2 \left (4 a^2-5 b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {a b \left (8 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {\frac {3 b^2 \left (4 a^2-5 b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {a b \left (8 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\) |
\(\Big \downarrow \) 3140 |
\(\displaystyle \frac {-\frac {\frac {3 b^2 \left (4 a^2-5 b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {2 a b \left (8 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\) |
\(\Big \downarrow \) 3286 |
\(\displaystyle \frac {-\frac {\frac {\frac {3 b^2 \left (4 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}-\frac {2 a b \left (8 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {\frac {\frac {3 b^2 \left (4 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sin (c+d x) \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}-\frac {2 a b \left (8 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\) |
\(\Big \downarrow \) 3284 |
\(\displaystyle \frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}+\frac {-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\frac {6 b^2 \left (4 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}-\frac {2 a b \left (8 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}}{2 a}}{4 a^2 b}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\) |
((4*a^2 - 5*b^2)*Cot[c + d*x])/(2*a^2*b*d*Sqrt[a + b*Sin[c + d*x]]) - (Cot [c + d*x]*Csc[c + d*x])/(2*a*d*Sqrt[a + b*Sin[c + d*x]]) + (-(((8*a^2 - 15 *b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(a*d)) - ((2*(8*a^2 - 15*b^2) *EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(d *Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((-2*a*b*(8*a^2 - 5*b^2)*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]]) + (6*b^2*(4*a^2 - 5*b^2)*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)/(2*a))/(4*a^2*b)
3.12.81.3.1 Defintions of rubi rules used
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]*(d*Sin[ e + f*x])^(n + 1)*((a + b*Sin[e + f*x])^(m + 1)/(a*d*f*(n + 1))), x] + (-Si mp[(a^2*(n + 1) - b^2*(m + n + 2))*Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*(( a + b*Sin[e + f*x])^(m + 1)/(a^2*b*d^2*f*(n + 1)*(m + 1))), x] + Simp[1/(a^ 2*b*d*(n + 1)*(m + 1)) Int[(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^ (m + 1)*Simp[a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 3) + a*b*(m + 1 )*Sin[e + f*x] - (a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] && LtQ[n, -1]
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[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*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] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 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. \(1348\) vs. \(2(435)=870\).
Time = 1.43 (sec) , antiderivative size = 1349, normalized size of antiderivative = 3.69
-1/4*(8*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-( 1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a- b)/(a+b))^(1/2))*a^4*b*sin(d*x+c)^2-18*b^2*((a+b*sin(d*x+c))/(a-b))^(1/2)* (-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF( ((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*sin(d*x+c)^2-5*b^3 *((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d *x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b ))^(1/2))*a^2*sin(d*x+c)^2+15*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c) -1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x +c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^4*sin(d*x+c)^2-8*((a+b*sin(d*x+ c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b)) ^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*s in(d*x+c)^2+23*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1 /2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/ 2),((a-b)/(a+b))^(1/2))*a^3*b^2*sin(d*x+c)^2-15*((a+b*sin(d*x+c))/(a-b))^( 1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*Ellip ticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^4*sin(d*x+c)^ 2-12*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+s in(d*x+c))*b/(a-b))^(1/2)*b^2*EllipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a -b)/a,((a-b)/(a+b))^(1/2))*a^3*sin(d*x+c)^2+12*((a+b*sin(d*x+c))/(a-b))...
Timed out. \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int \frac {\cos {\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Timed out. \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\int \frac {\cos \left (c+d\,x\right )\,{\mathrm {cot}\left (c+d\,x\right )}^3}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]