Integrand size = 25, antiderivative size = 156 \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{3/2}} \, dx=-\frac {154 a^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d e^2 \sqrt {\cos (c+d x)}}-\frac {154 a^4 (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 d e^3}+\frac {4 a^7 (e \cos (c+d x))^{11/2}}{d e^7 (a-a \sin (c+d x))^3}+\frac {44 a^8 (e \cos (c+d x))^{7/2}}{3 d e^5 \left (a^4-a^4 \sin (c+d x)\right )} \]
-154/15*a^4*(e*cos(d*x+c))^(3/2)*sin(d*x+c)/d/e^3+4*a^7*(e*cos(d*x+c))^(11 /2)/d/e^7/(a-a*sin(d*x+c))^3+44/3*a^8*(e*cos(d*x+c))^(7/2)/d/e^5/(a^4-a^4* sin(d*x+c))-154/5*a^4*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Elli pticE(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 = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.41 \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{3/2}} \, dx=\frac {16\ 2^{3/4} a^4 \operatorname {Hypergeometric2F1}\left (-\frac {11}{4},-\frac {1}{4},\frac {3}{4},\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt [4]{1+\sin (c+d x)}}{d e \sqrt {e \cos (c+d x)}} \]
(16*2^(3/4)*a^4*Hypergeometric2F1[-11/4, -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 = 0.82 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 3149, 3042, 3159, 3042, 3159, 3042, 3115, 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.
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^4}{(e \cos (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (c+d x)+a)^4}{(e \cos (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3149 |
| \(\displaystyle \frac {a^8 \int \frac {(e \cos (c+d x))^{13/2}}{(a-a \sin (c+d x))^4}dx}{e^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^8 \int \frac {(e \cos (c+d x))^{13/2}}{(a-a \sin (c+d x))^4}dx}{e^8}\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle \frac {a^8 \left (\frac {4 e (e \cos (c+d x))^{11/2}}{a d (a-a \sin (c+d x))^3}-\frac {11 e^2 \int \frac {(e \cos (c+d x))^{9/2}}{(a-a \sin (c+d x))^2}dx}{a^2}\right )}{e^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^8 \left (\frac {4 e (e \cos (c+d x))^{11/2}}{a d (a-a \sin (c+d x))^3}-\frac {11 e^2 \int \frac {(e \cos (c+d x))^{9/2}}{(a-a \sin (c+d x))^2}dx}{a^2}\right )}{e^8}\) |
| \(\Big \downarrow \) 3159 |
| \(\displaystyle \frac {a^8 \left (\frac {4 e (e \cos (c+d x))^{11/2}}{a d (a-a \sin (c+d x))^3}-\frac {11 e^2 \left (\frac {7 e^2 \int (e \cos (c+d x))^{5/2}dx}{3 a^2}-\frac {4 e (e \cos (c+d x))^{7/2}}{3 d \left (a^2-a^2 \sin (c+d x)\right )}\right )}{a^2}\right )}{e^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^8 \left (\frac {4 e (e \cos (c+d x))^{11/2}}{a d (a-a \sin (c+d x))^3}-\frac {11 e^2 \left (\frac {7 e^2 \int \left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}dx}{3 a^2}-\frac {4 e (e \cos (c+d x))^{7/2}}{3 d \left (a^2-a^2 \sin (c+d x)\right )}\right )}{a^2}\right )}{e^8}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {a^8 \left (\frac {4 e (e \cos (c+d x))^{11/2}}{a d (a-a \sin (c+d x))^3}-\frac {11 e^2 \left (\frac {7 e^2 \left (\frac {3}{5} e^2 \int \sqrt {e \cos (c+d x)}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )}{3 a^2}-\frac {4 e (e \cos (c+d x))^{7/2}}{3 d \left (a^2-a^2 \sin (c+d x)\right )}\right )}{a^2}\right )}{e^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^8 \left (\frac {4 e (e \cos (c+d x))^{11/2}}{a d (a-a \sin (c+d x))^3}-\frac {11 e^2 \left (\frac {7 e^2 \left (\frac {3}{5} e^2 \int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )}{3 a^2}-\frac {4 e (e \cos (c+d x))^{7/2}}{3 d \left (a^2-a^2 \sin (c+d x)\right )}\right )}{a^2}\right )}{e^8}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {a^8 \left (\frac {4 e (e \cos (c+d x))^{11/2}}{a d (a-a \sin (c+d x))^3}-\frac {11 e^2 \left (\frac {7 e^2 \left (\frac {3 e^2 \sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{5 \sqrt {\cos (c+d x)}}+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )}{3 a^2}-\frac {4 e (e \cos (c+d x))^{7/2}}{3 d \left (a^2-a^2 \sin (c+d x)\right )}\right )}{a^2}\right )}{e^8}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a^8 \left (\frac {4 e (e \cos (c+d x))^{11/2}}{a d (a-a \sin (c+d x))^3}-\frac {11 e^2 \left (\frac {7 e^2 \left (\frac {3 e^2 \sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{5 \sqrt {\cos (c+d x)}}+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )}{3 a^2}-\frac {4 e (e \cos (c+d x))^{7/2}}{3 d \left (a^2-a^2 \sin (c+d x)\right )}\right )}{a^2}\right )}{e^8}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {a^8 \left (\frac {4 e (e \cos (c+d x))^{11/2}}{a d (a-a \sin (c+d x))^3}-\frac {11 e^2 \left (\frac {7 e^2 \left (\frac {6 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}\right )}{3 a^2}-\frac {4 e (e \cos (c+d x))^{7/2}}{3 d \left (a^2-a^2 \sin (c+d x)\right )}\right )}{a^2}\right )}{e^8}\) |
(a^8*((4*e*(e*Cos[c + d*x])^(11/2))/(a*d*(a - a*Sin[c + d*x])^3) - (11*e^2 *((-4*e*(e*Cos[c + d*x])^(7/2))/(3*d*(a^2 - a^2*Sin[c + d*x])) + (7*e^2*(( 6*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d* x]]) + (2*e*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)))/(3*a^2)))/a^2))/e ^8
3.3.27.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[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 _)])^(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]
Time = 6.29 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.22
| method | result | size |
| default | \(\frac {2 \left (24 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-24 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+80 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+246 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-231 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-80 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+140 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{15 e \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) d}\) | \(190\) |
| parts | \(-\frac {2 a^{4} \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 {8 a^{4} \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 (2 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+3 \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 )}{5 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 {8 a^{4}}{\sqrt {e \cos \left (d x +c \right )}\, e d}-\frac {24 a^{4} \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}+\frac {8 a^{4} \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}}\) | \(676\) |
2/15/e/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)/sin(1/2*d*x+1/2*c)*(24*sin(1/2* d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-24*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c) +80*sin(1/2*d*x+1/2*c)^5+246*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-231*( 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))-80*sin(1/2*d*x+1/2*c)^3+140*sin(1/2*d*x+1/2*c))*a ^4/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.62 \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{3/2}} \, dx=\frac {231 \, {\left (-i \, \sqrt {2} a^{4} \cos \left (d x + c\right ) + i \, \sqrt {2} a^{4} \sin \left (d x + c\right ) - i \, \sqrt {2} a^{4}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 \, {\left (i \, \sqrt {2} a^{4} \cos \left (d x + c\right ) - i \, \sqrt {2} a^{4} \sin \left (d x + c\right ) + i \, \sqrt {2} a^{4}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (3 \, a^{4} \cos \left (d x + c\right )^{3} + 20 \, a^{4} \cos \left (d x + c\right )^{2} + 137 \, a^{4} \cos \left (d x + c\right ) + 120 \, a^{4} + {\left (3 \, a^{4} \cos \left (d x + c\right )^{2} - 17 \, a^{4} \cos \left (d x + c\right ) + 120 \, a^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{15 \, {\left (d e^{2} \cos \left (d x + c\right ) - d e^{2} \sin \left (d x + c\right ) + d e^{2}\right )}} \]
1/15*(231*(-I*sqrt(2)*a^4*cos(d*x + c) + I*sqrt(2)*a^4*sin(d*x + c) - I*sq rt(2)*a^4)*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d *x + c) + I*sin(d*x + c))) + 231*(I*sqrt(2)*a^4*cos(d*x + c) - I*sqrt(2)*a ^4*sin(d*x + c) + I*sqrt(2)*a^4)*sqrt(e)*weierstrassZeta(-4, 0, weierstras sPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) + 2*(3*a^4*cos(d*x + c)^3 + 20*a^4*cos(d*x + c)^2 + 137*a^4*cos(d*x + c) + 120*a^4 + (3*a^4*cos(d*x + c)^2 - 17*a^4*cos(d*x + c) + 120*a^4)*sin(d*x + c))*sqrt(e*cos(d*x + c) ))/(d*e^2*cos(d*x + c) - d*e^2*sin(d*x + c) + d*e^2)
Timed out. \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{3/2}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{3/2}} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{3/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]