Integrand size = 23, antiderivative size = 97 \[ \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{5/2}} \, dx=\frac {2 b}{3 d e (e \cos (c+d x))^{3/2}}+\frac {2 a \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d e^2 \sqrt {e \cos (c+d x)}}+\frac {2 a \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}} \]
2/3*b/d/e/(e*cos(d*x+c))^(3/2)+2/3*a*sin(d*x+c)/d/e/(e*cos(d*x+c))^(3/2)+2 /3*a*(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^2/(e*cos(d*x+c))^(1/2)
Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.57 \[ \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{5/2}} \, dx=\frac {2 \left (b+a \cos ^{\frac {3}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+a \sin (c+d x)\right )}{3 d e (e \cos (c+d x))^{3/2}} \]
(2*(b + a*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + a*Sin[c + d*x]))/ (3*d*e*(e*Cos[c + d*x])^(3/2))
Time = 0.42 (sec) , antiderivative size = 98, 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, 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.
| \(\displaystyle \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle a \int \frac {1}{(e \cos (c+d x))^{5/2}}dx+\frac {2 b}{3 d e (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \frac {1}{\left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx+\frac {2 b}{3 d e (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle a \left (\frac {\int \frac {1}{\sqrt {e \cos (c+d x)}}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}\right )+\frac {2 b}{3 d e (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {\int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 e^2}+\frac {2 \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}\right )+\frac {2 b}{3 d e (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle a \left (\frac {\sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{3 e^2 \sqrt {e \cos (c+d x)}}+\frac {2 \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}\right )+\frac {2 b}{3 d e (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {\sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 e^2 \sqrt {e \cos (c+d x)}}+\frac {2 \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}\right )+\frac {2 b}{3 d e (e \cos (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle a \left (\frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d e^2 \sqrt {e \cos (c+d x)}}+\frac {2 \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}\right )+\frac {2 b}{3 d e (e \cos (c+d x))^{3/2}}\) |
(2*b)/(3*d*e*(e*Cos[c + d*x])^(3/2)) + a*((2*Sqrt[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)))
3.6.45.3.1 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 _)]), 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 = 2.32 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.99
| method | result | size |
| default | \(-\frac {2 \left (2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a +b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e^{2} d}\) | \(193\) |
| parts | \(-\frac {2 a \left (-2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{3 e^{2} \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}+\frac {2 b}{3 d e \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}\) | \(262\) |
-2/3/(2*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^2*(2*(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*a+2*cos (1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*a-(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))*a+b*sin(1 /2*d*x+1/2*c))/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{5/2}} \, dx=\frac {-i \, \sqrt {2} a \sqrt {e} \cos \left (d x + c\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + i \, \sqrt {2} a \sqrt {e} \cos \left (d x + c\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, \sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + b\right )}}{3 \, d e^{3} \cos \left (d x + c\right )^{2}} \]
1/3*(-I*sqrt(2)*a*sqrt(e)*cos(d*x + c)^2*weierstrassPInverse(-4, 0, cos(d* x + c) + I*sin(d*x + c)) + I*sqrt(2)*a*sqrt(e)*cos(d*x + c)^2*weierstrassP Inverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 2*sqrt(e*cos(d*x + c))*(a* sin(d*x + c) + b))/(d*e^3*cos(d*x + c)^2)
Timed out. \[ \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{5/2}} \, dx=\int { \frac {b \sin \left (d x + c\right ) + a}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{5/2}} \, dx=\int { \frac {b \sin \left (d x + c\right ) + a}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+b \sin (c+d x)}{(e \cos (c+d x))^{5/2}} \, dx=\int \frac {a+b\,\sin \left (c+d\,x\right )}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]