Integrand size = 29, antiderivative size = 430 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=-\frac {\left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{28 d}-\frac {\left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{28 a d}-\frac {\left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{28 a^2 d}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}+\frac {a \left (8 a^2-247 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{28 b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (8 a^4+3 a^2 b^2-32 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{28 b d \sqrt {a+b \sin (c+d x)}}-\frac {3 a \left (4 a^2-5 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{4 d \sqrt {a+b \sin (c+d x)}} \] Output:
-1/28*(8*a^2-73*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)/d-1/28*(8*a^2-35*b^ 2)*cos(d*x+c)*(a+b*sin(d*x+c))^(3/2)/a/d-1/28*(8*a^2-21*b^2)*cos(d*x+c)*(a +b*sin(d*x+c))^(5/2)/a^2/d-3/4*b*cot(d*x+c)*(a+b*sin(d*x+c))^(7/2)/a^2/d-1 /2*cot(d*x+c)*csc(d*x+c)*(a+b*sin(d*x+c))^(7/2)/a/d-1/28*a*(8*a^2-247*b^2) *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)/b/d/((a+b*sin(d*x+c))/(a+b))^(1/2)-1/28*(8*a^4+3*a^2*b^2-32*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/d/(a+b*sin(d*x+c))^(1/2)+3/4*a*(4*a^2-5*b^2)*Ellipt icPi(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 = 6.12 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.07 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {\frac {2 i \left (-8 a^2+247 b^2\right ) \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {\frac {b (1+\sin (c+d x))}{-a+b}}}{b^2 \sqrt {-\frac {1}{a+b}}}+\frac {8 b \left (125 a^2-16 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a \left (160 a^2+37 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}+4 \sqrt {a+b \sin (c+d x)} \left (\left (-24 a^2+22 b^2\right ) \cos (c+d x)+2 b^2 \cos (3 (c+d x))-7 a \cot (c+d x) (9 b+2 a \csc (c+d x))-12 a b \sin (2 (c+d x))\right )}{112 d} \] Input:
Integrate[Cos[c + d*x]*Cot[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2),x]
Output:
(((2*I)*(-8*a^2 + 247*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*A rcSinh[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^2*Sqrt[-(a + b)^(-1)]) + (8 *b*(125*a^2 - 16*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*(160*a^2 + 37*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*S in[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] + 4*Sqrt[a + b*Sin[c + d*x ]]*((-24*a^2 + 22*b^2)*Cos[c + d*x] + 2*b^2*Cos[3*(c + d*x)] - 7*a*Cot[c + d*x]*(9*b + 2*a*Csc[c + d*x]) - 12*a*b*Sin[2*(c + d*x)]))/(112*d)
Time = 3.67 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.05, number of steps used = 27, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.931, Rules used = {3042, 3372, 27, 3042, 3528, 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.
| \(\displaystyle \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a+b \sin (c+d x))^{5/2}}{\sin (c+d x)^3}dx\) |
| \(\Big \downarrow \) 3372 |
| \(\displaystyle -\frac {\int \frac {1}{4} \csc (c+d x) (a+b \sin (c+d x))^{5/2} \left (-\left (\left (8 a^2-21 b^2\right ) \sin ^2(c+d x)\right )+10 a b \sin (c+d x)+3 \left (4 a^2-5 b^2\right )\right )dx}{2 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \csc (c+d x) (a+b \sin (c+d x))^{5/2} \left (-\left (\left (8 a^2-21 b^2\right ) \sin ^2(c+d x)\right )+10 a b \sin (c+d x)+3 \left (4 a^2-5 b^2\right )\right )dx}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {(a+b \sin (c+d x))^{5/2} \left (-\left (\left (8 a^2-21 b^2\right ) \sin (c+d x)^2\right )+10 a b \sin (c+d x)+3 \left (4 a^2-5 b^2\right )\right )}{\sin (c+d x)}dx}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {\frac {2}{7} \int \frac {1}{2} \csc (c+d x) (a+b \sin (c+d x))^{3/2} \left (114 b \sin (c+d x) a^2-5 \left (8 a^2-35 b^2\right ) \sin ^2(c+d x) a+21 \left (4 a^2-5 b^2\right ) a\right )dx+\frac {2 \left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{7} \int \csc (c+d x) (a+b \sin (c+d x))^{3/2} \left (114 b \sin (c+d x) a^2-5 \left (8 a^2-35 b^2\right ) \sin ^2(c+d x) a+21 \left (4 a^2-5 b^2\right ) a\right )dx+\frac {2 \left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{7} \int \frac {(a+b \sin (c+d x))^{3/2} \left (114 b \sin (c+d x) a^2-5 \left (8 a^2-35 b^2\right ) \sin (c+d x)^2 a+21 \left (4 a^2-5 b^2\right ) a\right )}{\sin (c+d x)}dx+\frac {2 \left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {\frac {1}{7} \left (\frac {2}{5} \int \frac {15}{2} \csc (c+d x) \sqrt {a+b \sin (c+d x)} \left (58 b \sin (c+d x) a^3-\left (8 a^2-73 b^2\right ) \sin ^2(c+d x) a^2+7 \left (4 a^2-5 b^2\right ) a^2\right )dx+\frac {2 a \left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )+\frac {2 \left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{7} \left (3 \int \csc (c+d x) \sqrt {a+b \sin (c+d x)} \left (58 b \sin (c+d x) a^3-\left (8 a^2-73 b^2\right ) \sin ^2(c+d x) a^2+7 \left (4 a^2-5 b^2\right ) a^2\right )dx+\frac {2 a \left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )+\frac {2 \left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{7} \left (3 \int \frac {\sqrt {a+b \sin (c+d x)} \left (58 b \sin (c+d x) a^3-\left (8 a^2-73 b^2\right ) \sin (c+d x)^2 a^2+7 \left (4 a^2-5 b^2\right ) a^2\right )}{\sin (c+d x)}dx+\frac {2 a \left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )+\frac {2 \left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {\frac {1}{7} \left (3 \left (\frac {2}{3} \int \frac {\csc (c+d x) \left (-\left (\left (8 a^2-247 b^2\right ) \sin ^2(c+d x) a^3\right )+21 \left (4 a^2-5 b^2\right ) a^3+2 b \left (125 a^2-16 b^2\right ) \sin (c+d x) a^2\right )}{2 \sqrt {a+b \sin (c+d x)}}dx+\frac {2 a^2 \left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 a \left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )+\frac {2 \left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \int \frac {\csc (c+d x) \left (-\left (\left (8 a^2-247 b^2\right ) \sin ^2(c+d x) a^3\right )+21 \left (4 a^2-5 b^2\right ) a^3+2 b \left (125 a^2-16 b^2\right ) \sin (c+d x) a^2\right )}{\sqrt {a+b \sin (c+d x)}}dx+\frac {2 a^2 \left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 a \left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )+\frac {2 \left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \int \frac {-\left (\left (8 a^2-247 b^2\right ) \sin (c+d x)^2 a^3\right )+21 \left (4 a^2-5 b^2\right ) a^3+2 b \left (125 a^2-16 b^2\right ) \sin (c+d x) a^2}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {2 a^2 \left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 a \left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )+\frac {2 \left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle -\frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (-\frac {a^3 \left (8 a^2-247 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {\int -\frac {\csc (c+d x) \left (21 b \left (4 a^2-5 b^2\right ) a^3+\left (8 a^4+3 b^2 a^2-32 b^4\right ) \sin (c+d x) a^2\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )+\frac {2 a^2 \left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 a \left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )+\frac {2 \left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {\int \frac {\csc (c+d x) \left (21 b \left (4 a^2-5 b^2\right ) a^3+\left (8 a^4+3 b^2 a^2-32 b^4\right ) \sin (c+d x) a^2\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {a^3 \left (8 a^2-247 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}\right )+\frac {2 a^2 \left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 a \left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )+\frac {2 \left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {\int \frac {21 b \left (4 a^2-5 b^2\right ) a^3+\left (8 a^4+3 b^2 a^2-32 b^4\right ) \sin (c+d x) a^2}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {a^3 \left (8 a^2-247 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}\right )+\frac {2 a^2 \left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 a \left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )+\frac {2 \left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle -\frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {\int \frac {21 b \left (4 a^2-5 b^2\right ) a^3+\left (8 a^4+3 b^2 a^2-32 b^4\right ) \sin (c+d x) a^2}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {a^3 \left (8 a^2-247 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 a^2 \left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 a \left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )+\frac {2 \left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {\int \frac {21 b \left (4 a^2-5 b^2\right ) a^3+\left (8 a^4+3 b^2 a^2-32 b^4\right ) \sin (c+d x) a^2}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {a^3 \left (8 a^2-247 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 a^2 \left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 a \left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )+\frac {2 \left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle -\frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {\int \frac {21 b \left (4 a^2-5 b^2\right ) a^3+\left (8 a^4+3 b^2 a^2-32 b^4\right ) \sin (c+d x) a^2}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 a^3 \left (8 a^2-247 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 a^2 \left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 a \left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )+\frac {2 \left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle -\frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {a^2 \left (8 a^4+3 a^2 b^2-32 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx+21 a^3 b \left (4 a^2-5 b^2\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 a^3 \left (8 a^2-247 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 a^2 \left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 a \left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )+\frac {2 \left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {a^2 \left (8 a^4+3 a^2 b^2-32 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx+21 a^3 b \left (4 a^2-5 b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 a^3 \left (8 a^2-247 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 a^2 \left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 a \left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )+\frac {2 \left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle -\frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {\frac {a^2 \left (8 a^4+3 a^2 b^2-32 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+21 a^3 b \left (4 a^2-5 b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 a^3 \left (8 a^2-247 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 a^2 \left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 a \left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )+\frac {2 \left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {\frac {a^2 \left (8 a^4+3 a^2 b^2-32 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+21 a^3 b \left (4 a^2-5 b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 a^3 \left (8 a^2-247 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 a^2 \left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 a \left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )+\frac {2 \left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle -\frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {21 a^3 b \left (4 a^2-5 b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {2 a^2 \left (8 a^4+3 a^2 b^2-32 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 a^3 \left (8 a^2-247 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 a^2 \left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 a \left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )+\frac {2 \left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle -\frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {\frac {21 a^3 b \left (4 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a^2 \left (8 a^4+3 a^2 b^2-32 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 a^3 \left (8 a^2-247 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 a^2 \left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 a \left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )+\frac {2 \left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{7} \left (3 \left (\frac {1}{3} \left (\frac {\frac {21 a^3 b \left (4 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sin (c+d x) \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a^2 \left (8 a^4+3 a^2 b^2-32 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 a^3 \left (8 a^2-247 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 a^2 \left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}\right )+\frac {2 a \left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )+\frac {2 \left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}}{8 a^2}-\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle -\frac {3 b \cot (c+d x) (a+b \sin (c+d x))^{7/2}}{4 a^2 d}-\frac {\frac {2 \left (8 a^2-21 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 d}+\frac {1}{7} \left (\frac {2 a \left (8 a^2-35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{d}+3 \left (\frac {2 a^2 \left (8 a^2-73 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}+\frac {1}{3} \left (\frac {\frac {2 a^2 \left (8 a^4+3 a^2 b^2-32 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}+\frac {42 a^3 b \left (4 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 a^3 \left (8 a^2-247 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )\right )\right )}{8 a^2}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{2 a d}\) |
Input:
Int[Cos[c + d*x]*Cot[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2),x]
Output:
(-3*b*Cot[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(4*a^2*d) - (Cot[c + d*x]*C sc[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(2*a*d) - ((2*(8*a^2 - 21*b^2)*Cos [c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(7*d) + ((2*a*(8*a^2 - 35*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/d + 3*((2*a^2*(8*a^2 - 73*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(3*d) + ((-2*a^3*(8*a^2 - 247*b^2)*Ellipti cE[(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^2*(8*a^4 + 3*a^2*b^2 - 32*b^4)*Elli pticF[(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]]) + (42*a^3*b*(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)/3))/7)/(8*a^2)
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[b*(m + n + 2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x] )^(n + 2)/(a^2*d^2*f*(n + 1)*(n + 2))), x] - Simp[1/(a^2*d^2*(n + 1)*(n + 2 )) Int[(a + b*Sin[e + f*x])^m*(d*Sin[e + f*x])^(n + 2)*Simp[a^2*n*(n + 2) - b^2*(m + n + 2)*(m + n + 3) + a*b*m*Sin[e + f*x] - (a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(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] && (LtQ[n, -2] || EqQ[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. \(1519\) vs. \(2(400)=800\).
Time = 19.39 (sec) , antiderivative size = 1520, normalized size of antiderivative = 3.53
Input:
int(cos(d*x+c)*cot(d*x+c)^3*(a+b*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE )
Output:
1/28*(8*b^5*sin(d*x+c)^7+8*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1) *b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(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)^2-258*b^2*((a+b*sin(d *x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(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+3*b^3*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+ b))^(1/2)*(-b*(1+sin(d*x+c))/(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+279*((a+b*sin(d*x+c))/(a-b) )^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*El lipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^4*sin(d*x+ c)^2-32*b^5*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2) *(-b*(1+sin(d*x+c))/(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)^2-8*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin( d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticE(((a+b*s in(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*sin(d*x+c)^2+255*((a+b*si n(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/ (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)^2-247*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)* b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(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)^2+84*((a+b*sin(d*x+...
Timed out. \[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^3*(a+b*sin(d*x+c))^(5/2),x, algorithm="fri cas")
Output:
Timed out
Timed out. \[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)**3*(a+b*sin(d*x+c))**(5/2),x)
Output:
Timed out
\[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right ) \cot \left (d x + c\right )^{3} \,d x } \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^3*(a+b*sin(d*x+c))^(5/2),x, algorithm="max ima")
Output:
integrate((b*sin(d*x + c) + a)^(5/2)*cos(d*x + c)*cot(d*x + c)^3, x)
Timed out. \[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^3*(a+b*sin(d*x+c))^(5/2),x, algorithm="gia c")
Output:
Timed out
Timed out. \[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int \cos \left (c+d\,x\right )\,{\mathrm {cot}\left (c+d\,x\right )}^3\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \] Input:
int(cos(c + d*x)*cot(c + d*x)^3*(a + b*sin(c + d*x))^(5/2),x)
Output:
int(cos(c + d*x)*cot(c + d*x)^3*(a + b*sin(c + d*x))^(5/2), x)
\[ \int \cos (c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\left (\int \sqrt {\sin \left (d x +c \right ) b +a}\, \cos \left (d x +c \right ) \cot \left (d x +c \right )^{3} \sin \left (d x +c \right )^{2}d x \right ) b^{2}+2 \left (\int \sqrt {\sin \left (d x +c \right ) b +a}\, \cos \left (d x +c \right ) \cot \left (d x +c \right )^{3} \sin \left (d x +c \right )d x \right ) a b +\left (\int \sqrt {\sin \left (d x +c \right ) b +a}\, \cos \left (d x +c \right ) \cot \left (d x +c \right )^{3}d x \right ) a^{2} \] Input:
int(cos(d*x+c)*cot(d*x+c)^3*(a+b*sin(d*x+c))^(5/2),x)
Output:
int(sqrt(sin(c + d*x)*b + a)*cos(c + d*x)*cot(c + d*x)**3*sin(c + d*x)**2, x)*b**2 + 2*int(sqrt(sin(c + d*x)*b + a)*cos(c + d*x)*cot(c + d*x)**3*sin( c + d*x),x)*a*b + int(sqrt(sin(c + d*x)*b + a)*cos(c + d*x)*cot(c + d*x)** 3,x)*a**2