Integrand size = 25, antiderivative size = 145 \[ \int \frac {(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{11/2}} \, dx=-\frac {2 a^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d e^6 \sqrt {\cos (c+d x)}}+\frac {2 a^2 \sin (c+d x)}{9 d e^3 (e \cos (c+d x))^{5/2}}+\frac {2 a^2 \sin (c+d x)}{3 d e^5 \sqrt {e \cos (c+d x)}}+\frac {4 \left (a^2+a^2 \sin (c+d x)\right )}{9 d e (e \cos (c+d x))^{9/2}} \]
2/9*a^2*sin(d*x+c)/d/e^3/(e*cos(d*x+c))^(5/2)+4/9*(a^2+a^2*sin(d*x+c))/d/e /(e*cos(d*x+c))^(9/2)+2/3*a^2*sin(d*x+c)/d/e^5/(e*cos(d*x+c))^(1/2)-2/3*a^ 2*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/ 2*c),2^(1/2))*(e*cos(d*x+c))^(1/2)/d/e^6/cos(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.46 \[ \int \frac {(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{11/2}} \, dx=\frac {2^{3/4} a^2 \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},\frac {5}{4},-\frac {5}{4},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{9/4}}{9 d e (e \cos (c+d x))^{9/2}} \]
(2^(3/4)*a^2*Hypergeometric2F1[-9/4, 5/4, -5/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(9/4))/(9*d*e*(e*Cos[c + d*x])^(9/2))
Time = 0.60 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3155, 3042, 3116, 3042, 3116, 3042, 3121, 3042, 3119}
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.
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^2}{(e \cos (c+d x))^{11/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^2}{(e \cos (c+d x))^{11/2}}dx\) |
| \(\Big \downarrow \) 3155 |
| \(\displaystyle \frac {5 a^2 \int \frac {1}{(e \cos (c+d x))^{7/2}}dx}{9 e^2}+\frac {4 \left (a^2 \sin (c+d x)+a^2\right )}{9 d e (e \cos (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 a^2 \int \frac {1}{\left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx}{9 e^2}+\frac {4 \left (a^2 \sin (c+d x)+a^2\right )}{9 d e (e \cos (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {5 a^2 \left (\frac {3 \int \frac {1}{(e \cos (c+d x))^{3/2}}dx}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )}{9 e^2}+\frac {4 \left (a^2 \sin (c+d x)+a^2\right )}{9 d e (e \cos (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 a^2 \left (\frac {3 \int \frac {1}{\left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )}{9 e^2}+\frac {4 \left (a^2 \sin (c+d x)+a^2\right )}{9 d e (e \cos (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {5 a^2 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\int \sqrt {e \cos (c+d x)}dx}{e^2}\right )}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )}{9 e^2}+\frac {4 \left (a^2 \sin (c+d x)+a^2\right )}{9 d e (e \cos (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 a^2 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2}\right )}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )}{9 e^2}+\frac {4 \left (a^2 \sin (c+d x)+a^2\right )}{9 d e (e \cos (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {5 a^2 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{e^2 \sqrt {\cos (c+d x)}}\right )}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )}{9 e^2}+\frac {4 \left (a^2 \sin (c+d x)+a^2\right )}{9 d e (e \cos (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 a^2 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \sqrt {\cos (c+d x)}}\right )}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )}{9 e^2}+\frac {4 \left (a^2 \sin (c+d x)+a^2\right )}{9 d e (e \cos (c+d x))^{9/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {5 a^2 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d e^2 \sqrt {\cos (c+d x)}}\right )}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )}{9 e^2}+\frac {4 \left (a^2 \sin (c+d x)+a^2\right )}{9 d e (e \cos (c+d x))^{9/2}}\) |
(4*(a^2 + a^2*Sin[c + d*x]))/(9*d*e*(e*Cos[c + d*x])^(9/2)) + (5*a^2*((2*S in[c + d*x])/(5*d*e*(e*Cos[c + d*x])^(5/2)) + (3*((-2*Sqrt[e*Cos[c + d*x]] *EllipticE[(c + d*x)/2, 2])/(d*e^2*Sqrt[Cos[c + d*x]]) + (2*Sin[c + d*x])/ (d*e*Sqrt[e*Cos[c + d*x]])))/(5*e^2)))/(9*e^2)
3.3.13.3.1 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[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(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. \(487\) vs. \(2(153)=306\).
Time = 11.89 (sec) , antiderivative size = 488, normalized size of antiderivative = 3.37
| method | result | size |
| default | \(\frac {2 \left (96 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-48 E\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}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-192 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+96 E\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}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+152 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-72 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-56 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+24 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+12 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \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}\, E\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}}{9 \left (16 \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-32 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \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^{5} d}\) | \(488\) |
| parts | \(\text {Expression too large to display}\) | \(855\) |
2/9/(16*sin(1/2*d*x+1/2*c)^8-32*sin(1/2*d*x+1/2*c)^6+24*sin(1/2*d*x+1/2*c) ^4-8*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^5*(96*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)-48*EllipticE(co s(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)^(1/2)*sin(1/2*d*x+1/2*c)^8-192*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2* c)^8+96*EllipticE(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)^(1/2)*sin(1/2*d*x+1/2*c)^6+152*sin(1/2*d*x+1/2 *c)^6*cos(1/2*d*x+1/2*c)-72*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos (1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^4 -56*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+24*(2*sin(1/2*d*x+1/2*c)^2-1)^ (1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*s in(1/2*d*x+1/2*c)^2+12*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-3*(sin(1/2* d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(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 = 261, normalized size of antiderivative = 1.80 \[ \int \frac {(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{11/2}} \, dx=-\frac {3 \, {\left (i \, \sqrt {2} a^{2} \cos \left (d x + c\right )^{3} + 2 i \, \sqrt {2} a^{2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 i \, \sqrt {2} a^{2} \cos \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, {\left (-i \, \sqrt {2} a^{2} \cos \left (d x + c\right )^{3} - 2 i \, \sqrt {2} a^{2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 2 i \, \sqrt {2} a^{2} \cos \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (6 \, a^{2} \cos \left (d x + c\right )^{2} - 4 \, a^{2} - {\left (3 \, a^{2} \cos \left (d x + c\right )^{2} - 5 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{9 \, {\left (d e^{6} \cos \left (d x + c\right )^{3} + 2 \, d e^{6} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, d e^{6} \cos \left (d x + c\right )\right )}} \]
-1/9*(3*(I*sqrt(2)*a^2*cos(d*x + c)^3 + 2*I*sqrt(2)*a^2*cos(d*x + c)*sin(d *x + c) - 2*I*sqrt(2)*a^2*cos(d*x + c))*sqrt(e)*weierstrassZeta(-4, 0, wei erstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 3*(-I*sqrt(2)*a^ 2*cos(d*x + c)^3 - 2*I*sqrt(2)*a^2*cos(d*x + c)*sin(d*x + c) + 2*I*sqrt(2) *a^2*cos(d*x + c))*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) + 2*(6*a^2*cos(d*x + c)^2 - 4*a^2 - (3* a^2*cos(d*x + c)^2 - 5*a^2)*sin(d*x + c))*sqrt(e*cos(d*x + c)))/(d*e^6*cos (d*x + c)^3 + 2*d*e^6*cos(d*x + c)*sin(d*x + c) - 2*d*e^6*cos(d*x + c))
Timed out. \[ \int \frac {(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{11/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{11/2}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{2}}{\left (e \cos \left (d x + c\right )\right )^{\frac {11}{2}}} \,d x } \]
\[ \int \frac {(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{11/2}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{2}}{\left (e \cos \left (d x + c\right )\right )^{\frac {11}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{11/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{11/2}} \,d x \]