Integrand size = 25, antiderivative size = 143 \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx=-\frac {14 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 a d e^4 \sqrt {\cos (c+d x)}}+\frac {14 \sin (c+d x)}{45 a d e (e \cos (c+d x))^{5/2}}+\frac {14 \sin (c+d x)}{15 a d e^3 \sqrt {e \cos (c+d x)}}-\frac {2}{9 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))} \]
14/45*sin(d*x+c)/a/d/e/(e*cos(d*x+c))^(5/2)-2/9/d/e/(e*cos(d*x+c))^(5/2)/( a+a*sin(d*x+c))+14/15*sin(d*x+c)/a/d/e^3/(e*cos(d*x+c))^(1/2)-14/15*(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)/a/d/e^4/cos(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.46 \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {13}{4},-\frac {1}{4},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{5/4}}{10 \sqrt [4]{2} a d e (e \cos (c+d x))^{5/2}} \]
(Hypergeometric2F1[-5/4, 13/4, -1/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d* x])^(5/4))/(10*2^(1/4)*a*d*e*(e*Cos[c + d*x])^(5/2))
Time = 0.60 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3162, 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.
| \(\displaystyle \int \frac {1}{(a \sin (c+d x)+a) (e \cos (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (c+d x)+a) (e \cos (c+d x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3162 |
| \(\displaystyle \frac {7 \int \frac {1}{(e \cos (c+d x))^{7/2}}dx}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \int \frac {1}{\left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {7 \left (\frac {3 \int \frac {1}{(e \cos (c+d x))^{3/2}}dx}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \left (\frac {3 \int \frac {1}{\left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {7 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\int \sqrt {e \cos (c+d x)}dx}{e^2}\right )}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2}\right )}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {7 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{e^2 \sqrt {\cos (c+d x)}}\right )}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \sqrt {\cos (c+d x)}}\right )}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {7 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d e^2 \sqrt {\cos (c+d x)}}\right )}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}}\) |
-2/(9*d*e*(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])) + (7*((2*Sin[c + d* x])/(5*d*e*(e*Cos[c + d*x])^(5/2)) + (3*((-2*Sqrt[e*Cos[c + d*x]]*Elliptic E[(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*a)
3.3.42.3.1 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 _)]), x_Symbol] :> Simp[b*((g*Cos[e + f*x])^(p + 1)/(a*f*g*(p - 1)*(a + b*S in[e + f*x]))), x] + Simp[p/(a*(p - 1)) Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && !GeQ[p, 1] && Intege rQ[2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(487\) vs. \(2(151)=302\).
Time = 6.22 (sec) , antiderivative size = 488, normalized size of antiderivative = 3.41
| method | result | size |
| default | \(\frac {\frac {448 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{15}-\frac {224 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 )}{15}-\frac {896 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15}+\frac {448 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 )}{15}+\frac {2128 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{45}-\frac {112 \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 )}{5}-\frac {784 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{45}+\frac {112 \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 )}{15}+\frac {44 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{15}-\frac {14 \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 )}{15}-\frac {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )}{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 ) a \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^{3} d}\) | \(488\) |
2/45/(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)/a/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^ 2*e+e)^(1/2)/e^3*(672*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)-336*Ellipti cE(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)^8-1344*cos(1/2*d*x+1/2*c)*sin(1/2*d* x+1/2*c)^8+672*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+1064*sin(1/2 *d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-504*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Elli pticE(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)^4-392*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+168*(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)^2+66*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)- 21*(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))-5*sin(1/2*d*x+1/2*c))/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.48 \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx=-\frac {21 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + i \, \sqrt {2} \cos \left (d x + c\right )^{3}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - i \, \sqrt {2} \cos \left (d x + c\right )^{3}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (21 \, \cos \left (d x + c\right )^{4} - 14 \, \cos \left (d x + c\right )^{2} - 7 \, {\left (3 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) - 2\right )} \sqrt {e \cos \left (d x + c\right )}}{45 \, {\left (a d e^{4} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + a d e^{4} \cos \left (d x + c\right )^{3}\right )}} \]
-1/45*(21*(I*sqrt(2)*cos(d*x + c)^3*sin(d*x + c) + I*sqrt(2)*cos(d*x + c)^ 3)*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*(-I*sqrt(2)*cos(d*x + c)^3*sin(d*x + c) - I*sqrt(2 )*cos(d*x + c)^3)*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0 , cos(d*x + c) - I*sin(d*x + c))) + 2*(21*cos(d*x + c)^4 - 14*cos(d*x + c) ^2 - 7*(3*cos(d*x + c)^2 + 1)*sin(d*x + c) - 2)*sqrt(e*cos(d*x + c)))/(a*d *e^4*cos(d*x + c)^3*sin(d*x + c) + a*d*e^4*cos(d*x + c)^3)
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]
\[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx=\int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]