Integrand size = 23, antiderivative size = 91 \[ \int \frac {a+a \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx=\frac {2 a}{d e \sqrt {e \cos (c+d x)}}-\frac {2 a \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d e^2 \sqrt {\cos (c+d x)}}+\frac {2 a \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}} \]
2*a/d/e/(e*cos(d*x+c))^(1/2)+2*a*sin(d*x+c)/d/e/(e*cos(d*x+c))^(1/2)-2*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^2/cos(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.31 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.68 \[ \int \frac {a+a \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx=\frac {2\ 2^{3/4} a \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{4},\frac {3}{4},\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt [4]{1+\sin (c+d x)}}{d e \sqrt {e \cos (c+d x)}} \]
(2*2^(3/4)*a*Hypergeometric2F1[-1/4, 1/4, 3/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(1/4))/(d*e*Sqrt[e*Cos[c + d*x]])
Time = 0.42 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 3148, 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.
| \(\displaystyle \int \frac {a \sin (c+d x)+a}{(e \cos (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a \sin (c+d x)+a}{(e \cos (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle a \int \frac {1}{(e \cos (c+d x))^{3/2}}dx+\frac {2 a}{d e \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \frac {1}{\left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx+\frac {2 a}{d e \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle a \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\int \sqrt {e \cos (c+d x)}dx}{e^2}\right )+\frac {2 a}{d e \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2}\right )+\frac {2 a}{d e \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle a \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{e^2 \sqrt {\cos (c+d x)}}\right )+\frac {2 a}{d e \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \sqrt {\cos (c+d x)}}\right )+\frac {2 a}{d e \sqrt {e \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle a \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d e^2 \sqrt {\cos (c+d x)}}\right )+\frac {2 a}{d e \sqrt {e \cos (c+d x)}}\) |
(2*a)/(d*e*Sqrt[e*Cos[c + d*x]]) + a*((-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]]))
3.3.1.3.1 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])
Time = 2.15 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.27
| method | result | size |
| default | \(\frac {2 \left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{e \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) d}\) | \(116\) |
| parts | \(-\frac {2 a \left (-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{e \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}+\frac {2 a}{d e \sqrt {e \cos \left (d x +c \right )}}\) | \(219\) |
2/e/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)/sin(1/2*d*x+1/2*c)*(2*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)*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 = 176, normalized size of antiderivative = 1.93 \[ \int \frac {a+a \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx=-\frac {{\left (i \, \sqrt {2} a \cos \left (d x + c\right ) - i \, \sqrt {2} a \sin \left (d x + c\right ) + i \, \sqrt {2} a\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + {\left (-i \, \sqrt {2} a \cos \left (d x + c\right ) + i \, \sqrt {2} a \sin \left (d x + c\right ) - i \, \sqrt {2} a\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (a \cos \left (d x + c\right ) + a \sin \left (d x + c\right ) + a\right )} \sqrt {e \cos \left (d x + c\right )}}{d e^{2} \cos \left (d x + c\right ) - d e^{2} \sin \left (d x + c\right ) + d e^{2}} \]
-((I*sqrt(2)*a*cos(d*x + c) - I*sqrt(2)*a*sin(d*x + c) + I*sqrt(2)*a)*sqrt (e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin (d*x + c))) + (-I*sqrt(2)*a*cos(d*x + c) + I*sqrt(2)*a*sin(d*x + c) - I*sq rt(2)*a)*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(a*cos(d*x + c) + a*sin(d*x + c) + a)*sqrt(e* cos(d*x + c)))/(d*e^2*cos(d*x + c) - d*e^2*sin(d*x + c) + d*e^2)
\[ \int \frac {a+a \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx=a \left (\int \frac {1}{\left (e \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {\sin {\left (c + d x \right )}}{\left (e \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx\right ) \]
a*(Integral((e*cos(c + d*x))**(-3/2), x) + Integral(sin(c + d*x)/(e*cos(c + d*x))**(3/2), x))
\[ \int \frac {a+a \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx=\int { \frac {a \sin \left (d x + c\right ) + a}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {a+a \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx=\int { \frac {a \sin \left (d x + c\right ) + a}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+a \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx=\int \frac {a+a\,\sin \left (c+d\,x\right )}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]