Integrand size = 23, antiderivative size = 386 \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=-\frac {b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{8 a^2 d}+\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{24 a^2 d}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}+\frac {\left (32 a^2-b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{8 a d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (16 a^2+21 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{8 d \sqrt {a+b \sin (c+d x)}}-\frac {b \left (36 a^2+b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{8 a d \sqrt {a+b \sin (c+d x)}} \]
1/24*(32*a^2+b^2)*cot(d*x+c)*(a+b*sin(d*x+c))^(3/2)/a^2/d+1/12*b*cot(d*x+c )*csc(d*x+c)*(a+b*sin(d*x+c))^(5/2)/a^2/d-1/3*cot(d*x+c)*csc(d*x+c)^2*(a+b *sin(d*x+c))^(5/2)/a/d-1/8*b*(16*a^2+b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2 )/a^2/d-1/8*(32*a^2-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/d/((a+b*sin(d*x+c))/(a+b))^(1/2)+1/8*(16*a^2+21*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)*Elliptic F(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)/d/(a+b*sin(d*x+c))^(1/2)+1/8*b*(36*a^2+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)*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)/a/d/(a+b*si n(d*x+c))^(1/2)
Result contains complex when optimal does not.
Time = 4.50 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.26 \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {\frac {2 i \left (32 a^2-b^2\right ) \cos (2 (c+d x)) \csc ^2(c+d x) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}}{a^2 b \sqrt {-\frac {1}{a+b}} \left (-2+\csc ^2(c+d x)\right )}-\frac {4 \left (16 a b \cos (c+d x)+\cot (c+d x) \left (-32 a^2+3 b^2+14 a b \csc (c+d x)+8 a^2 \csc ^2(c+d x)\right )\right ) \sqrt {a+b \sin (c+d x)}}{3 a}-\frac {8 \left (8 a^2-11 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}+\frac {2 b \left (40 a^2+3 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{a \sqrt {a+b \sin (c+d x)}}}{32 d} \]
(((2*I)*(32*a^2 - b^2)*Cos[2*(c + d*x)]*Csc[c + d*x]^2*(2*a*(a - b)*Ellipt icE[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[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))])/(a ^2*b*Sqrt[-(a + b)^(-1)]*(-2 + Csc[c + d*x]^2)) - (4*(16*a*b*Cos[c + d*x] + Cot[c + d*x]*(-32*a^2 + 3*b^2 + 14*a*b*Csc[c + d*x] + 8*a^2*Csc[c + d*x] ^2))*Sqrt[a + b*Sin[c + d*x]])/(3*a) - (8*(8*a^2 - 11*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*b*(40*a^2 + 3*b^2)*EllipticPi[2, (-2*c + Pi - 2*d *x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(a*Sqrt[a + b*Si n[c + d*x]]))/(32*d)
Time = 2.82 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.01, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.043, Rules used = {3042, 3204, 27, 3042, 3526, 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.
| \(\displaystyle \int \cot ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (c+d x))^{3/2}}{\tan (c+d x)^4}dx\) |
| \(\Big \downarrow \) 3204 |
| \(\displaystyle -\frac {\int \frac {1}{4} \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \left (32 a^2+6 b \sin (c+d x) a+b^2-3 \left (8 a^2+b^2\right ) \sin ^2(c+d x)\right )dx}{6 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \left (32 a^2+6 b \sin (c+d x) a+b^2-3 \left (8 a^2+b^2\right ) \sin ^2(c+d x)\right )dx}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {(a+b \sin (c+d x))^{3/2} \left (32 a^2+6 b \sin (c+d x) a+b^2-3 \left (8 a^2+b^2\right ) \sin (c+d x)^2\right )}{\sin (c+d x)^2}dx}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {\int \frac {3}{2} \csc (c+d x) \sqrt {a+b \sin (c+d x)} \left (-3 b \left (16 a^2+b^2\right ) \sin ^2(c+d x)-2 a \left (8 a^2-b^2\right ) \sin (c+d x)+b \left (36 a^2+b^2\right )\right )dx-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {3}{2} \int \csc (c+d x) \sqrt {a+b \sin (c+d x)} \left (-3 b \left (16 a^2+b^2\right ) \sin ^2(c+d x)-2 a \left (8 a^2-b^2\right ) \sin (c+d x)+b \left (36 a^2+b^2\right )\right )dx-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3}{2} \int \frac {\sqrt {a+b \sin (c+d x)} \left (-3 b \left (16 a^2+b^2\right ) \sin (c+d x)^2-2 a \left (8 a^2-b^2\right ) \sin (c+d x)+b \left (36 a^2+b^2\right )\right )}{\sin (c+d x)}dx-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {\frac {3}{2} \left (\frac {2}{3} \int \frac {3 \csc (c+d x) \left (-2 \left (8 a^2-11 b^2\right ) \sin (c+d x) a^2-b \left (32 a^2-b^2\right ) \sin ^2(c+d x) a+b \left (36 a^2+b^2\right ) a\right )}{2 \sqrt {a+b \sin (c+d x)}}dx+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {3}{2} \left (\int \frac {\csc (c+d x) \left (-2 \left (8 a^2-11 b^2\right ) \sin (c+d x) a^2-b \left (32 a^2-b^2\right ) \sin ^2(c+d x) a+b \left (36 a^2+b^2\right ) a\right )}{\sqrt {a+b \sin (c+d x)}}dx+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3}{2} \left (\int \frac {-2 \left (8 a^2-11 b^2\right ) \sin (c+d x) a^2-b \left (32 a^2-b^2\right ) \sin (c+d x)^2 a+b \left (36 a^2+b^2\right ) a}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle -\frac {\frac {3}{2} \left (-a \left (32 a^2-b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx-\frac {\int -\frac {\csc (c+d x) \left (b \left (16 a^2+21 b^2\right ) \sin (c+d x) a^2+b^2 \left (36 a^2+b^2\right ) a\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {3}{2} \left (-a \left (32 a^2-b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx+\frac {\int \frac {\csc (c+d x) \left (b \left (16 a^2+21 b^2\right ) \sin (c+d x) a^2+b^2 \left (36 a^2+b^2\right ) a\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3}{2} \left (-a \left (32 a^2-b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx+\frac {\int \frac {b \left (16 a^2+21 b^2\right ) \sin (c+d x) a^2+b^2 \left (36 a^2+b^2\right ) a}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle -\frac {\frac {3}{2} \left (\frac {\int \frac {b \left (16 a^2+21 b^2\right ) \sin (c+d x) a^2+b^2 \left (36 a^2+b^2\right ) a}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {a \left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3}{2} \left (-\frac {a \left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\int \frac {b \left (16 a^2+21 b^2\right ) \sin (c+d x) a^2+b^2 \left (36 a^2+b^2\right ) a}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle -\frac {\frac {3}{2} \left (\frac {\int \frac {b \left (16 a^2+21 b^2\right ) \sin (c+d x) a^2+b^2 \left (36 a^2+b^2\right ) a}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a \left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle -\frac {\frac {3}{2} \left (\frac {a^2 b \left (16 a^2+21 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx+a b^2 \left (36 a^2+b^2\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a \left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3}{2} \left (\frac {a^2 b \left (16 a^2+21 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx+a b^2 \left (36 a^2+b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a \left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle -\frac {\frac {3}{2} \left (\frac {\frac {a^2 b \left (16 a^2+21 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+a b^2 \left (36 a^2+b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a \left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3}{2} \left (\frac {\frac {a^2 b \left (16 a^2+21 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+a b^2 \left (36 a^2+b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a \left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle -\frac {\frac {3}{2} \left (\frac {a b^2 \left (36 a^2+b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {2 a^2 b \left (16 a^2+21 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a \left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle -\frac {\frac {3}{2} \left (\frac {\frac {a b^2 \left (36 a^2+b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a^2 b \left (16 a^2+21 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a \left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3}{2} \left (\frac {\frac {a b^2 \left (36 a^2+b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sin (c+d x) \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a^2 b \left (16 a^2+21 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a \left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle -\frac {\frac {3}{2} \left (\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a \left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\frac {2 a^2 b \left (16 a^2+21 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}+\frac {2 a b^2 \left (36 a^2+b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\) |
(b*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(12*a^2*d) - (Cot [c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2))/(3*a*d) - (-(((32*a^2 + b^2)*Cot[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/d) + (3*((2*b*(16*a^2 + b ^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/d - (2*a*(32*a^2 - b^2)*Ellipti cE[(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^2*b*(16*a^2 + 21*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]]) + (2*a*b^2*(36*a^2 + 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*Si n[c + d*x]]))/b))/2)/(24*a^2)
3.12.56.3.1 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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^4, x_Symbol] :> Simp[(-Cos[e + f*x])*((a + b*Sin[e + f*x])^(m + 1)/(3*a*f*Sin[ e + f*x]^3)), x] + (-Simp[b*(m - 2)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(6*a^2*f*Sin[e + f*x]^2)), x] - Simp[1/(6*a^2) Int[((a + b*Sin[e + f* x])^m/Sin[e + f*x]^2)*Simp[8*a^2 - b^2*(m - 1)*(m - 2) + a*b*m*Sin[e + f*x] - (6*a^2 - b^2*m*(m - 2))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, e, f , m}, x] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1] && IntegerQ[2*m]
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[(((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^2*C - B*c*d + 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/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{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] && GtQ[m, 0] && LtQ[n, -1]
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. \(1510\) vs. \(2(451)=902\).
Time = 1.83 (sec) , antiderivative size = 1511, normalized size of antiderivative = 3.91
1/24*(48*a^5*((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))*sin(d*x+c)^3+48*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(si n(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)^3-162*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)^3+63*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)^3+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*b^ 4*sin(d*x+c)^3-96*((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*sin(d*x+c)^3+99*((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)*Ellipt icE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^2*sin(d*x+c) ^3-3*((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)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/ (a+b))^(1/2))*a*b^4*sin(d*x+c)^3+108*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(...
Timed out. \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
\[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right )^{4} \,d x } \]
Timed out. \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^4\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]