Integrand size = 25, antiderivative size = 114 \[ \int \frac {(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{9/2}} \, dx=\frac {2 a^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{7 d e^4 \sqrt {e \cos (c+d x)}}+\frac {2 a^2 \sin (c+d x)}{7 d e^3 (e \cos (c+d x))^{3/2}}+\frac {4 \left (a^2+a^2 \sin (c+d x)\right )}{7 d e (e \cos (c+d x))^{7/2}} \]
2/7*a^2*sin(d*x+c)/d/e^3/(e*cos(d*x+c))^(3/2)+4/7*(a^2+a^2*sin(d*x+c))/d/e /(e*cos(d*x+c))^(7/2)+2/7*a^2*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2 *c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)/d/e^4/(e*cos(d* x+c))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.07 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.58 \[ \int \frac {(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{9/2}} \, dx=\frac {2 \sqrt [4]{2} a^2 \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},\frac {3}{4},-\frac {3}{4},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{7/4}}{7 d e (e \cos (c+d x))^{7/2}} \]
(2*2^(1/4)*a^2*Hypergeometric2F1[-7/4, 3/4, -3/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(7/4))/(7*d*e*(e*Cos[c + d*x])^(7/2))
Time = 0.48 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 3155, 3042, 3116, 3042, 3121, 3042, 3120}
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.
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^2}{(e \cos (c+d x))^{9/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^2}{(e \cos (c+d x))^{9/2}}dx\) |
| \(\Big \downarrow \) 3155 |
| \(\displaystyle \frac {3 a^2 \int \frac {1}{(e \cos (c+d x))^{5/2}}dx}{7 e^2}+\frac {4 \left (a^2 \sin (c+d x)+a^2\right )}{7 d e (e \cos (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 a^2 \int \frac {1}{\left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{7 e^2}+\frac {4 \left (a^2 \sin (c+d x)+a^2\right )}{7 d e (e \cos (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {3 a^2 \left (\frac {\int \frac {1}{\sqrt {e \cos (c+d x)}}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}\right )}{7 e^2}+\frac {4 \left (a^2 \sin (c+d x)+a^2\right )}{7 d e (e \cos (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 a^2 \left (\frac {\int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}\right )}{7 e^2}+\frac {4 \left (a^2 \sin (c+d x)+a^2\right )}{7 d e (e \cos (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {3 a^2 \left (\frac {\sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{3 e^2 \sqrt {e \cos (c+d x)}}+\frac {2 \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}\right )}{7 e^2}+\frac {4 \left (a^2 \sin (c+d x)+a^2\right )}{7 d e (e \cos (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 a^2 \left (\frac {\sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 e^2 \sqrt {e \cos (c+d x)}}+\frac {2 \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}\right )}{7 e^2}+\frac {4 \left (a^2 \sin (c+d x)+a^2\right )}{7 d e (e \cos (c+d x))^{7/2}}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {3 a^2 \left (\frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d e^2 \sqrt {e \cos (c+d x)}}+\frac {2 \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}\right )}{7 e^2}+\frac {4 \left (a^2 \sin (c+d x)+a^2\right )}{7 d e (e \cos (c+d x))^{7/2}}\) |
(4*(a^2 + a^2*Sin[c + d*x]))/(7*d*e*(e*Cos[c + d*x])^(7/2)) + (3*a^2*((2*S qrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*e^2*Sqrt[e*Cos[c + d*x]] ) + (2*Sin[c + d*x])/(3*d*e*(e*Cos[c + d*x])^(3/2))))/(7*e^2)
3.3.12.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[-2*b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(p + 1))), x] + Simp[b^2*((2*m + p - 1)/(g^2*(p + 1))) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2), x], x] /; FreeQ [{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && Int egersQ[2*m, 2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(374\) vs. \(2(126)=252\).
Time = 7.23 (sec) , antiderivative size = 375, normalized size of antiderivative = 3.29
| method | result | size |
| default | \(-\frac {2 \left (8 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-12 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+6 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{7 \left (8 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e^{4} d}\) | \(375\) |
| parts | \(\text {Expression too large to display}\) | \(821\) |
-2/7/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^ 2-1)/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)/e^4*(8*(sin(1/ 2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x +1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^6+8*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x +1/2*c)-12*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/ 2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^4-8*sin(1/2*d*x+1/ 2*c)^4*cos(1/2*d*x+1/2*c)+6*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2 *d*x+1/2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^2 +6*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2 *sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+2*sin (1/2*d*x+1/2*c))*a^2/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.64 \[ \int \frac {(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{9/2}} \, dx=\frac {{\left (-i \, \sqrt {2} a^{2} \cos \left (d x + c\right )^{2} - 2 i \, \sqrt {2} a^{2} \sin \left (d x + c\right ) + 2 i \, \sqrt {2} a^{2}\right )} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + {\left (i \, \sqrt {2} a^{2} \cos \left (d x + c\right )^{2} + 2 i \, \sqrt {2} a^{2} \sin \left (d x + c\right ) - 2 i \, \sqrt {2} a^{2}\right )} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}\right )} \sqrt {e \cos \left (d x + c\right )}}{7 \, {\left (d e^{5} \cos \left (d x + c\right )^{2} + 2 \, d e^{5} \sin \left (d x + c\right ) - 2 \, d e^{5}\right )}} \]
1/7*((-I*sqrt(2)*a^2*cos(d*x + c)^2 - 2*I*sqrt(2)*a^2*sin(d*x + c) + 2*I*s qrt(2)*a^2)*sqrt(e)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + (I*sqrt(2)*a^2*cos(d*x + c)^2 + 2*I*sqrt(2)*a^2*sin(d*x + c) - 2*I*s qrt(2)*a^2)*sqrt(e)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 2*(a^2*sin(d*x + c) - 2*a^2)*sqrt(e*cos(d*x + c)))/(d*e^5*cos(d*x + c)^2 + 2*d*e^5*sin(d*x + c) - 2*d*e^5)
Timed out. \[ \int \frac {(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{9/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{9/2}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{2}}{\left (e \cos \left (d x + c\right )\right )^{\frac {9}{2}}} \,d x } \]
\[ \int \frac {(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{9/2}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{2}}{\left (e \cos \left (d x + c\right )\right )^{\frac {9}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{9/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{9/2}} \,d x \]