Integrand size = 25, antiderivative size = 285 \[ \int \frac {a+b \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx=-\frac {2 (b c-a d) \cos (e+f x)}{3 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \cos (e+f x)}{3 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 d \left (c^2-d^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 (b c-a d) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 d \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}} \] Output:
-2/3*(-a*d+b*c)*cos(f*x+e)/(c^2-d^2)/f/(c+d*sin(f*x+e))^(3/2)+2/3*(4*a*c*d -b*(c^2+3*d^2))*cos(f*x+e)/(c^2-d^2)^2/f/(c+d*sin(f*x+e))^(1/2)-2/3*(4*a*c *d-b*(c^2+3*d^2))*EllipticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1 /2))*(c+d*sin(f*x+e))^(1/2)/d/(c^2-d^2)^2/f/((c+d*sin(f*x+e))/(c+d))^(1/2) +2/3*(-a*d+b*c)*InverseJacobiAM(1/2*e-1/4*Pi+1/2*f*x,2^(1/2)*(d/(c+d))^(1/ 2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/d/(c^2-d^2)/f/(c+d*sin(f*x+e))^(1/2)
Time = 1.28 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.70 \[ \int \frac {a+b \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx=\frac {2 \left (\frac {\left (\left (-4 a c d+b \left (c^2+3 d^2\right )\right ) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-(c-d) (b c-a d) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right )\right ) \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{3/2}}{(c-d)^2 d}-\frac {\cos (e+f x) \left (a d \left (-5 c^2+d^2\right )+2 b c \left (c^2+d^2\right )+d \left (-4 a c d+b \left (c^2+3 d^2\right )\right ) \sin (e+f x)\right )}{\left (c^2-d^2\right )^2}\right )}{3 f (c+d \sin (e+f x))^{3/2}} \] Input:
Integrate[(a + b*Sin[e + f*x])/(c + d*Sin[e + f*x])^(5/2),x]
Output:
(2*((((-4*a*c*d + b*(c^2 + 3*d^2))*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/ (c + d)] - (c - d)*(b*c - a*d)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)])*((c + d*Sin[e + f*x])/(c + d))^(3/2))/((c - d)^2*d) - (Cos[e + f*x]* (a*d*(-5*c^2 + d^2) + 2*b*c*(c^2 + d^2) + d*(-4*a*c*d + b*(c^2 + 3*d^2))*S in[e + f*x]))/(c^2 - d^2)^2))/(3*f*(c + d*Sin[e + f*x])^(3/2))
Time = 1.33 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.04, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 3233, 27, 3042, 3233, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}
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.
| \(\displaystyle \int \frac {a+b \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a+b \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle -\frac {2 \int -\frac {3 (a c-b d)+(b c-a d) \sin (e+f x)}{2 (c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {3 (a c-b d)+(b c-a d) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 (a c-b d)+(b c-a d) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {2 \int \frac {4 b c d-a \left (3 c^2+d^2\right )-\left (4 a c d-b \left (c^2+3 d^2\right )\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}}{3 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {\int \frac {4 b c d-a \left (3 c^2+d^2\right )-\left (4 a c d-b \left (c^2+3 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}}{3 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {\int \frac {4 b c d-a \left (3 c^2+d^2\right )-\left (4 a c d-b \left (c^2+3 d^2\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}}{3 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {-\frac {\left (c^2-d^2\right ) (b c-a d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (4 a c d-b \left (c^2+3 d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}}{c^2-d^2}}{3 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {-\frac {\left (c^2-d^2\right ) (b c-a d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (4 a c d-b \left (c^2+3 d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}}{c^2-d^2}}{3 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {-\frac {\left (c^2-d^2\right ) (b c-a d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (4 a c d-b \left (c^2+3 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}}{3 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {-\frac {\left (c^2-d^2\right ) (b c-a d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {\left (4 a c d-b \left (c^2+3 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}}{3 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {-\frac {\left (c^2-d^2\right ) (b c-a d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}}{3 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {-\frac {\left (c^2-d^2\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}}{3 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {-\frac {\left (c^2-d^2\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}}{3 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}}-\frac {-\frac {2 \left (c^2-d^2\right ) (b c-a d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}-\frac {2 \left (4 a c d-b \left (c^2+3 d^2\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}}{3 \left (c^2-d^2\right )}-\frac {2 (b c-a d) \cos (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{3/2}}\) |
Input:
Int[(a + b*Sin[e + f*x])/(c + d*Sin[e + f*x])^(5/2),x]
Output:
(-2*(b*c - a*d)*Cos[e + f*x])/(3*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^(3/2)) + ((2*(4*a*c*d - b*(c^2 + 3*d^2))*Cos[e + f*x])/((c^2 - d^2)*f*Sqrt[c + d *Sin[e + f*x]]) - ((-2*(4*a*c*d - b*(c^2 + 3*d^2))*EllipticE[(e - Pi/2 + f *x)/2, (2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sqrt[(c + d*Sin[e + f *x])/(c + d)]) - (2*(b*c - a*d)*(c^2 - d^2)*EllipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]]))/(c^2 - d^2))/(3*(c^2 - d^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[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*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]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*( m + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Leaf count of result is larger than twice the leaf count of optimal. \(882\) vs. \(2(270)=540\).
Time = 3.10 (sec) , antiderivative size = 883, normalized size of antiderivative = 3.10
| method | result | size |
| default | \(\frac {\sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}\, \left (\frac {b \left (\frac {2 d \cos \left (f x +e \right )^{2}}{\left (c^{2}-d^{2}\right ) \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}+\frac {2 c \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{\left (c^{2}-d^{2}\right ) \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}+\frac {2 d \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \left (\left (-\frac {c}{d}-1\right ) \operatorname {EllipticE}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )+\operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )\right )}{\left (c^{2}-d^{2}\right ) \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}\right )}{d}+\frac {\left (a d -b c \right ) \left (\frac {2 \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}{3 \left (c^{2}-d^{2}\right ) d \left (\sin \left (f x +e \right )+\frac {c}{d}\right )^{2}}+\frac {8 d \cos \left (f x +e \right )^{2} c}{3 \left (c^{2}-d^{2}\right )^{2} \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}+\frac {2 \left (3 c^{2}+d^{2}\right ) \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{\left (3 c^{4}-6 d^{2} c^{2}+3 d^{4}\right ) \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}+\frac {8 c d \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {-\frac {d \left (1+\sin \left (f x +e \right )\right )}{c -d}}\, \left (\left (-\frac {c}{d}-1\right ) \operatorname {EllipticE}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )+\operatorname {EllipticF}\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )\right )}{3 \left (c^{2}-d^{2}\right )^{2} \sqrt {-\left (-c -d \sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2}}}\right )}{d}\right )}{\cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\) | \(883\) |
| parts | \(\text {Expression too large to display}\) | \(1373\) |
Input:
int((a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x,method=_RETURNVERBOSE)
Output:
(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*(b/d*(2*d*cos(f*x+e)^2/(c^2-d^2)/( -(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)+2/(c^2-d^2)*c*(c/d-1)*((c+d*sin(f*x +e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d)) ^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e)) /(c-d))^(1/2),((c-d)/(c+d))^(1/2))+2/(c^2-d^2)*d*(c/d-1)*((c+d*sin(f*x+e)) /(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/ 2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f *x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d) )^(1/2),((c-d)/(c+d))^(1/2))))+(a*d-b*c)/d*(2/3/(c^2-d^2)/d*(-(-c-d*sin(f* x+e))*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+c/d)^2+8/3*d*cos(f*x+e)^2/(c^2-d^2)^ 2*c/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)+2*(3*c^2+d^2)/(3*c^4-6*c^2*d^2 +3*d^4)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1 /2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2 )*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+8/3*c*d/(c ^2-d^2)^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^ (1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1 /2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2) )+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2)))))/cos(f*x +e)/(c+d*sin(f*x+e))^(1/2)/f
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 981, normalized size of antiderivative = 3.44 \[ \int \frac {a+b \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate((a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas")
Output:
-2/9*((2*b*c^5 + a*c^4*d - 4*b*c^3*d^2 + 4*a*c^2*d^3 - 6*b*c*d^4 + 3*a*d^5 - (2*b*c^3*d^2 + a*c^2*d^3 - 6*b*c*d^4 + 3*a*d^5)*cos(f*x + e)^2 + 2*(2*b *c^4*d + a*c^3*d^2 - 6*b*c^2*d^3 + 3*a*c*d^4)*sin(f*x + e))*sqrt(1/2*I*d)* weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/ d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d) + (2*b*c^5 + a *c^4*d - 4*b*c^3*d^2 + 4*a*c^2*d^3 - 6*b*c*d^4 + 3*a*d^5 - (2*b*c^3*d^2 + a*c^2*d^3 - 6*b*c*d^4 + 3*a*d^5)*cos(f*x + e)^2 + 2*(2*b*c^4*d + a*c^3*d^2 - 6*b*c^2*d^3 + 3*a*c*d^4)*sin(f*x + e))*sqrt(-1/2*I*d)*weierstrassPInver se(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*co s(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d) - 3*(-I*b*c^4*d + 4*I*a*c^3*d^ 2 - 4*I*b*c^2*d^3 + 4*I*a*c*d^4 - 3*I*b*d^5 + (I*b*c^2*d^3 - 4*I*a*c*d^4 + 3*I*b*d^5)*cos(f*x + e)^2 + 2*(-I*b*c^3*d^2 + 4*I*a*c^2*d^3 - 3*I*b*c*d^4 )*sin(f*x + e))*sqrt(1/2*I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8 /27*(8*I*c^3 - 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^ 2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d)) - 3*(I*b*c^4*d - 4*I*a*c^3*d^2 + 4*I*b*c^2*d^3 - 4*I*a*c *d^4 + 3*I*b*d^5 + (-I*b*c^2*d^3 + 4*I*a*c*d^4 - 3*I*b*d^5)*cos(f*x + e)^2 + 2*(I*b*c^3*d^2 - 4*I*a*c^2*d^3 + 3*I*b*c*d^4)*sin(f*x + e))*sqrt(-1/2*I *d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2) /d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9...
Timed out. \[ \int \frac {a+b \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((a+b*sin(f*x+e))/(c+d*sin(f*x+e))**(5/2),x)
Output:
Timed out
\[ \int \frac {a+b \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {b \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x, algorithm="maxima")
Output:
integrate((b*sin(f*x + e) + a)/(d*sin(f*x + e) + c)^(5/2), x)
\[ \int \frac {a+b \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {b \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x, algorithm="giac")
Output:
integrate((b*sin(f*x + e) + a)/(d*sin(f*x + e) + c)^(5/2), x)
Timed out. \[ \int \frac {a+b \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {a+b\,\sin \left (e+f\,x\right )}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \] Input:
int((a + b*sin(e + f*x))/(c + d*sin(e + f*x))^(5/2),x)
Output:
int((a + b*sin(e + f*x))/(c + d*sin(e + f*x))^(5/2), x)
\[ \int \frac {a+b \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx=\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right )^{3} d^{3}+3 \sin \left (f x +e \right )^{2} c \,d^{2}+3 \sin \left (f x +e \right ) c^{2} d +c^{3}}d x \right ) a +\left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{3} d^{3}+3 \sin \left (f x +e \right )^{2} c \,d^{2}+3 \sin \left (f x +e \right ) c^{2} d +c^{3}}d x \right ) b \] Input:
int((a+b*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x)
Output:
int(sqrt(sin(e + f*x)*d + c)/(sin(e + f*x)**3*d**3 + 3*sin(e + f*x)**2*c*d **2 + 3*sin(e + f*x)*c**2*d + c**3),x)*a + int((sqrt(sin(e + f*x)*d + c)*s in(e + f*x))/(sin(e + f*x)**3*d**3 + 3*sin(e + f*x)**2*c*d**2 + 3*sin(e + f*x)*c**2*d + c**3),x)*b