Integrand size = 23, antiderivative size = 126 \[ \int \frac {a+a \sin (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx=\frac {2 a}{5 d e (e \cos (c+d x))^{5/2}}-\frac {6 a \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d e^4 \sqrt {\cos (c+d x)}}+\frac {2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}+\frac {6 a \sin (c+d x)}{5 d e^3 \sqrt {e \cos (c+d x)}} \]
2/5*a/d/e/(e*cos(d*x+c))^(5/2)+2/5*a*sin(d*x+c)/d/e/(e*cos(d*x+c))^(5/2)+6 /5*a*sin(d*x+c)/d/e^3/(e*cos(d*x+c))^(1/2)-6/5*a*(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^4/cos(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.43 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.51 \[ \int \frac {a+a \sin (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx=\frac {2^{3/4} a \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {5}{4},-\frac {1}{4},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{5/4}}{5 d e (e \cos (c+d x))^{5/2}} \]
(2^(3/4)*a*Hypergeometric2F1[-5/4, 5/4, -1/4, (1 - Sin[c + d*x])/2]*(1 + S in[c + d*x])^(5/4))/(5*d*e*(e*Cos[c + d*x])^(5/2))
Time = 0.53 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3042, 3148, 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.
| \(\displaystyle \int \frac {a \sin (c+d x)+a}{(e \cos (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a \sin (c+d x)+a}{(e \cos (c+d x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle a \int \frac {1}{(e \cos (c+d x))^{7/2}}dx+\frac {2 a}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \frac {1}{\left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx+\frac {2 a}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle a \left (\frac {3 \int \frac {1}{(e \cos (c+d x))^{3/2}}dx}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )+\frac {2 a}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {3 \int \frac {1}{\left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )+\frac {2 a}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle a \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\int \sqrt {e \cos (c+d x)}dx}{e^2}\right )}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )+\frac {2 a}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2}\right )}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )+\frac {2 a}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle a \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{e^2 \sqrt {\cos (c+d x)}}\right )}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )+\frac {2 a}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \sqrt {\cos (c+d x)}}\right )}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )+\frac {2 a}{5 d e (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle a \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d e^2 \sqrt {\cos (c+d x)}}\right )}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )+\frac {2 a}{5 d e (e \cos (c+d x))^{5/2}}\) |
(2*a)/(5*d*e*(e*Cos[c + d*x])^(5/2)) + a*((2*Sin[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))
3.3.3.3.1 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)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(301\) vs. \(2(134)=268\).
Time = 3.83 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.40
| method | result | size |
| default | \(\frac {2 \left (24 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-12 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+12 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{5 \left (4 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e^{3} d}\) | \(302\) |
| parts | \(-\frac {2 a \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (24 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-12 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+12 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e}}{5 e^{4} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (8 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}+\frac {2 a}{5 d e \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}}}\) | \(387\) |
2/5/(4*sin(1/2*d*x+1/2*c)^4-4*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^3*(24*sin(1/2*d*x+1/2*c)^6*cos(1/2*d* x+1/2*c)-12*(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)^4-24*sin(1/2*d*x+ 1/2*c)^4*cos(1/2*d*x+1/2*c)+12*(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+8*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))+ sin(1/2*d*x+1/2*c))*a/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.47 \[ \int \frac {a+a \sin (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx=-\frac {3 \, {\left (i \, \sqrt {2} a \cos \left (d x + c\right ) \sin \left (d x + c\right ) - i \, \sqrt {2} a \cos \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, {\left (-i \, \sqrt {2} a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + i \, \sqrt {2} a \cos \left (d x + c\right )\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (3 \, a \cos \left (d x + c\right )^{2} + 3 \, a \sin \left (d x + c\right ) - 2 \, a\right )} \sqrt {e \cos \left (d x + c\right )}}{5 \, {\left (d e^{4} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - d e^{4} \cos \left (d x + c\right )\right )}} \]
-1/5*(3*(I*sqrt(2)*a*cos(d*x + c)*sin(d*x + c) - I*sqrt(2)*a*cos(d*x + c)) *sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 3*(-I*sqrt(2)*a*cos(d*x + c)*sin(d*x + c) + I*sqrt(2)*a *cos(d*x + c))*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, c os(d*x + c) - I*sin(d*x + c))) + 2*(3*a*cos(d*x + c)^2 + 3*a*sin(d*x + c) - 2*a)*sqrt(e*cos(d*x + c)))/(d*e^4*cos(d*x + c)*sin(d*x + c) - d*e^4*cos( d*x + c))
Timed out. \[ \int \frac {a+a \sin (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {a+a \sin (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx=\int { \frac {a \sin \left (d x + c\right ) + a}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {a+a \sin (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx=\int { \frac {a \sin \left (d x + c\right ) + a}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+a \sin (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx=\int \frac {a+a\,\sin \left (c+d\,x\right )}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \]