Integrand size = 25, antiderivative size = 178 \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4 \, dx=-\frac {22 a^4 (e \cos (c+d x))^{3/2}}{9 d e}+\frac {22 a^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d \sqrt {\cos (c+d x)}}-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3}{9 d e}-\frac {10 (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )^2}{21 d e}-\frac {22 (e \cos (c+d x))^{3/2} \left (a^4+a^4 \sin (c+d x)\right )}{21 d e} \]
-22/9*a^4*(e*cos(d*x+c))^(3/2)/d/e-2/9*a*(e*cos(d*x+c))^(3/2)*(a+a*sin(d*x +c))^3/d/e-10/21*(e*cos(d*x+c))^(3/2)*(a^2+a^2*sin(d*x+c))^2/d/e-22/21*(e* cos(d*x+c))^(3/2)*(a^4+a^4*sin(d*x+c))/d/e+22/3*a^4*(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/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 = 66, normalized size of antiderivative = 0.37 \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4 \, dx=-\frac {32\ 2^{3/4} a^4 (e \cos (c+d x))^{3/2} \operatorname {Hypergeometric2F1}\left (-\frac {15}{4},\frac {3}{4},\frac {7}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{3 d e (1+\sin (c+d x))^{3/4}} \]
(-32*2^(3/4)*a^4*(e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[-15/4, 3/4, 7/4, (1 - Sin[c + d*x])/2])/(3*d*e*(1 + Sin[c + d*x])^(3/4))
Time = 0.82 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 3157, 3042, 3157, 3042, 3157, 3042, 3148, 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.
| \(\displaystyle \int (a \sin (c+d x)+a)^4 \sqrt {e \cos (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (c+d x)+a)^4 \sqrt {e \cos (c+d x)}dx\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {5}{3} a \int \sqrt {e \cos (c+d x)} (\sin (c+d x) a+a)^3dx-\frac {2 a (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{3/2}}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{3} a \int \sqrt {e \cos (c+d x)} (\sin (c+d x) a+a)^3dx-\frac {2 a (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{3/2}}{9 d e}\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {5}{3} a \left (\frac {11}{7} a \int \sqrt {e \cos (c+d x)} (\sin (c+d x) a+a)^2dx-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{3/2}}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{3} a \left (\frac {11}{7} a \int \sqrt {e \cos (c+d x)} (\sin (c+d x) a+a)^2dx-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{3/2}}{9 d e}\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \int \sqrt {e \cos (c+d x)} (\sin (c+d x) a+a)dx-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{3/2}}{5 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{3/2}}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \int \sqrt {e \cos (c+d x)} (\sin (c+d x) a+a)dx-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{3/2}}{5 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{3/2}}{9 d e}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (a \int \sqrt {e \cos (c+d x)}dx-\frac {2 a (e \cos (c+d x))^{3/2}}{3 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{3/2}}{5 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{3/2}}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (a \int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {2 a (e \cos (c+d x))^{3/2}}{3 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{3/2}}{5 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{3/2}}{9 d e}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (\frac {a \sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{\sqrt {\cos (c+d x)}}-\frac {2 a (e \cos (c+d x))^{3/2}}{3 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{3/2}}{5 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{3/2}}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (\frac {a \sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\cos (c+d x)}}-\frac {2 a (e \cos (c+d x))^{3/2}}{3 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{3/2}}{5 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{3/2}}{9 d e}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {5}{3} a \left (\frac {11}{7} a \left (\frac {7}{5} a \left (\frac {2 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\frac {2 a (e \cos (c+d x))^{3/2}}{3 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{3/2}}{5 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{3/2}}{9 d e}\) |
(-2*a*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^3)/(9*d*e) + (5*a*((-2*a *(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^2)/(7*d*e) + (11*a*((7*a*((-2 *a*(e*Cos[c + d*x])^(3/2))/(3*d*e) + (2*a*Sqrt[e*Cos[c + d*x]]*EllipticE[( c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]])))/5 - (2*(e*Cos[c + d*x])^(3/2)*(a^ 2 + a^2*Sin[c + d*x]))/(5*d*e)))/7))/3
3.3.25.3.1 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 _)]), 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])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[a*((2*m + p - 1)/(m + p)) Int[(g* Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + p, 0] && Integers Q[2*m, 2*p]
Time = 10.03 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.45
| method | result | size |
| default | \(\frac {2 a^{4} e \left (224 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-448 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+576 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-392 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-1152 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+616 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+192 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-168 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+231 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+384 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-132 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(258\) |
| parts | \(\frac {2 a^{4} \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 )}\, e \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{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 {-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 \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 )}\, e \left (40 \left (\cos ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+118 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-36 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{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 )+3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{45 \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 (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d e}+\frac {24 a^{4} \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 )}\, e \left (4 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \left (\cos ^{3}\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 (\cos ^{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 )-\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 \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 {7}{2}}}{7}-\frac {e^{2} \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{3}\right )}{d \,e^{3}}\) | \(625\) |
2/63/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*a^4*e*(224*sin (1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)-448*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1 /2*c)^8+576*sin(1/2*d*x+1/2*c)^9-392*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2* c)-1152*sin(1/2*d*x+1/2*c)^7+616*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+1 92*sin(1/2*d*x+1/2*c)^5-168*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+231*(s in(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))+384*sin(1/2*d*x+1/2*c)^3-132*sin(1/2*d*x+1/2*c))/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.78 \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4 \, dx=\frac {231 i \, \sqrt {2} a^{4} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 i \, \sqrt {2} a^{4} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (36 \, a^{4} \cos \left (d x + c\right )^{3} - 168 \, a^{4} \cos \left (d x + c\right ) + 7 \, {\left (a^{4} \cos \left (d x + c\right )^{3} - 13 \, a^{4} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{63 \, d} \]
1/63*(231*I*sqrt(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*sqrt(e)*weiers trassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) ) + 2*(36*a^4*cos(d*x + c)^3 - 168*a^4*cos(d*x + c) + 7*(a^4*cos(d*x + c)^ 3 - 13*a^4*cos(d*x + c))*sin(d*x + c))*sqrt(e*cos(d*x + c)))/d
Timed out. \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4 \, dx=\text {Timed out} \]
\[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4 \, dx=\int { \sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{4} \,d x } \]
\[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4 \, dx=\int { \sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{4} \,d x } \]
Timed out. \[ \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4 \, dx=\int \sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4 \,d x \]