Integrand size = 25, antiderivative size = 106 \[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{3/2}} \, dx=\frac {14 a^3 (e \cos (c+d x))^{3/2}}{3 d e^3}-\frac {14 a^3 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d e^2 \sqrt {\cos (c+d x)}}+\frac {4 a^5 (e \cos (c+d x))^{7/2}}{d e^5 (a-a \sin (c+d x))^2} \]
14/3*a^3*(e*cos(d*x+c))^(3/2)/d/e^3+4*a^5*(e*cos(d*x+c))^(7/2)/d/e^5/(a-a* sin(d*x+c))^2-14*a^3*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Ellip ticE(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 = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.60 \[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{3/2}} \, dx=\frac {8\ 2^{3/4} a^3 \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},-\frac {1}{4},\frac {3}{4},\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt [4]{1+\sin (c+d x)}}{d e \sqrt {e \cos (c+d x)}} \]
(8*2^(3/4)*a^3*Hypergeometric2F1[-7/4, -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 = 0.66 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3149, 3042, 3159, 3042, 3161, 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.
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^3}{(e \cos (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^3}{(e \cos (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3149 |
| \(\displaystyle \frac {a^6 \int \frac {(e \cos (c+d x))^{9/2}}{(a-a \sin (c+d x))^3}dx}{e^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^6 \int \frac {(e \cos (c+d x))^{9/2}}{(a-a \sin (c+d x))^3}dx}{e^6}\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle \frac {a^6 \left (\frac {4 e (e \cos (c+d x))^{7/2}}{a d (a-a \sin (c+d x))^2}-\frac {7 e^2 \int \frac {(e \cos (c+d x))^{5/2}}{a-a \sin (c+d x)}dx}{a^2}\right )}{e^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^6 \left (\frac {4 e (e \cos (c+d x))^{7/2}}{a d (a-a \sin (c+d x))^2}-\frac {7 e^2 \int \frac {(e \cos (c+d x))^{5/2}}{a-a \sin (c+d x)}dx}{a^2}\right )}{e^6}\) |
| \(\Big \downarrow \) 3161 |
| \(\displaystyle \frac {a^6 \left (\frac {4 e (e \cos (c+d x))^{7/2}}{a d (a-a \sin (c+d x))^2}-\frac {7 e^2 \left (\frac {e^2 \int \sqrt {e \cos (c+d x)}dx}{a}-\frac {2 e (e \cos (c+d x))^{3/2}}{3 a d}\right )}{a^2}\right )}{e^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^6 \left (\frac {4 e (e \cos (c+d x))^{7/2}}{a d (a-a \sin (c+d x))^2}-\frac {7 e^2 \left (\frac {e^2 \int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}-\frac {2 e (e \cos (c+d x))^{3/2}}{3 a d}\right )}{a^2}\right )}{e^6}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {a^6 \left (\frac {4 e (e \cos (c+d x))^{7/2}}{a d (a-a \sin (c+d x))^2}-\frac {7 e^2 \left (\frac {e^2 \sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{a \sqrt {\cos (c+d x)}}-\frac {2 e (e \cos (c+d x))^{3/2}}{3 a d}\right )}{a^2}\right )}{e^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^6 \left (\frac {4 e (e \cos (c+d x))^{7/2}}{a d (a-a \sin (c+d x))^2}-\frac {7 e^2 \left (\frac {e^2 \sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a \sqrt {\cos (c+d x)}}-\frac {2 e (e \cos (c+d x))^{3/2}}{3 a d}\right )}{a^2}\right )}{e^6}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {a^6 \left (\frac {4 e (e \cos (c+d x))^{7/2}}{a d (a-a \sin (c+d x))^2}-\frac {7 e^2 \left (\frac {2 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{a d \sqrt {\cos (c+d x)}}-\frac {2 e (e \cos (c+d x))^{3/2}}{3 a d}\right )}{a^2}\right )}{e^6}\) |
(a^6*((-7*e^2*((-2*e*(e*Cos[c + d*x])^(3/2))/(3*a*d) + (2*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(a*d*Sqrt[Cos[c + d*x]])))/a^2 + (4*e* (e*Cos[c + d*x])^(7/2))/(a*d*(a - a*Sin[c + d*x])^2)))/e^6
3.3.19.3.1 Defintions of rubi rules used
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[(a/g)^(2*m) Int[(g*Cos[e + f*x])^(2*m + p)/( a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2 , 0] && IntegerQ[m] && LtQ[p, -1] && GeQ[2*m + p, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[2*g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f *x])^(m + 1)/(b*f*(2*m + p + 1))), x] + Simp[g^2*((p - 1)/(b^2*(2*m + 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] && LeQ[m, -2] && GtQ[p, 1] & & NeQ[2*m + p + 1, 0] && !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[g*((g*Cos[e + f*x])^(p - 1)/(b*f*(p - 1))), x] + Si mp[g^2/a Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g}, x ] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]
Time = 4.48 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.38
| method | result | size |
| default | \(\frac {2 \left (4 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-21 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-4 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+13 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{3 e \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) d}\) | \(146\) |
| parts | \(-\frac {2 a^{3} \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^{3} \left (\frac {\left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+\frac {e^{2}}{\sqrt {e \cos \left (d x +c \right )}}\right )}{d \,e^{3}}+\frac {6 a^{3}}{\sqrt {e \cos \left (d x +c \right )}\, e d}-\frac {12 a^{3} \left (-\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}\) | \(461\) |
2/3/e/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)/sin(1/2*d*x+1/2*c)*(4*sin(1/2*d* x+1/2*c)^5+24*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))-4*sin(1/2*d*x+1/2*c)^3+13*sin(1/2*d*x+1/2*c))*a^3/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.12 \[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{3/2}} \, dx=\frac {21 \, {\left (-i \, \sqrt {2} a^{3} \cos \left (d x + c\right ) + i \, \sqrt {2} a^{3} \sin \left (d x + c\right ) - i \, \sqrt {2} a^{3}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, {\left (i \, \sqrt {2} a^{3} \cos \left (d x + c\right ) - i \, \sqrt {2} a^{3} \sin \left (d x + c\right ) + i \, \sqrt {2} a^{3}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (a^{3} \cos \left (d x + c\right )^{2} + 13 \, a^{3} \cos \left (d x + c\right ) + 12 \, a^{3} - {\left (a^{3} \cos \left (d x + c\right ) - 12 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{3 \, {\left (d e^{2} \cos \left (d x + c\right ) - d e^{2} \sin \left (d x + c\right ) + d e^{2}\right )}} \]
1/3*(21*(-I*sqrt(2)*a^3*cos(d*x + c) + I*sqrt(2)*a^3*sin(d*x + c) - I*sqrt (2)*a^3)*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*(I*sqrt(2)*a^3*cos(d*x + c) - I*sqrt(2)*a^3* sin(d*x + c) + I*sqrt(2)*a^3)*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPI nverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) + 2*(a^3*cos(d*x + c)^2 + 13 *a^3*cos(d*x + c) + 12*a^3 - (a^3*cos(d*x + c) - 12*a^3)*sin(d*x + c))*sqr t(e*cos(d*x + c)))/(d*e^2*cos(d*x + c) - d*e^2*sin(d*x + c) + d*e^2)
Timed out. \[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{3/2}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{3/2}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{3/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]