Integrand size = 25, antiderivative size = 161 \[ \int (a+b \cos (c+d x))^3 \sqrt {e \sin (c+d x)} \, dx=\frac {2 a \left (5 a^2+6 b^2\right ) E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 d \sqrt {\sin (c+d x)}}+\frac {2 b \left (57 a^2+20 b^2\right ) (e \sin (c+d x))^{3/2}}{105 d e}+\frac {22 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}{35 d e}+\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}}{7 d e} \]
2/105*b*(57*a^2+20*b^2)*(e*sin(d*x+c))^(3/2)/d/e+22/35*a*b*(a+b*cos(d*x+c) )*(e*sin(d*x+c))^(3/2)/d/e+2/7*b*(a+b*cos(d*x+c))^2*(e*sin(d*x+c))^(3/2)/d /e-2/5*a*(5*a^2+6*b^2)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*P i+1/2*d*x)*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*(e*sin(d*x+c))^(1/ 2)/d/sin(d*x+c)^(1/2)
Time = 1.23 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.65 \[ \int (a+b \cos (c+d x))^3 \sqrt {e \sin (c+d x)} \, dx=\frac {\sqrt {e \sin (c+d x)} \left (-42 \left (5 a^3+6 a b^2\right ) E\left (\left .\frac {1}{4} (-2 c+\pi -2 d x)\right |2\right )+b \left (210 a^2+55 b^2+126 a b \cos (c+d x)+15 b^2 \cos (2 (c+d x))\right ) \sin ^{\frac {3}{2}}(c+d x)\right )}{105 d \sqrt {\sin (c+d x)}} \]
(Sqrt[e*Sin[c + d*x]]*(-42*(5*a^3 + 6*a*b^2)*EllipticE[(-2*c + Pi - 2*d*x) /4, 2] + b*(210*a^2 + 55*b^2 + 126*a*b*Cos[c + d*x] + 15*b^2*Cos[2*(c + d* x)])*Sin[c + d*x]^(3/2)))/(105*d*Sqrt[Sin[c + d*x]])
Time = 0.82 (sec) , antiderivative size = 169, 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, 3171, 27, 3042, 3341, 27, 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.
| \(\displaystyle \int \sqrt {e \sin (c+d x)} (a+b \cos (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3dx\) |
| \(\Big \downarrow \) 3171 |
| \(\displaystyle \frac {2}{7} \int \frac {1}{2} (a+b \cos (c+d x)) \left (7 a^2+11 b \cos (c+d x) a+4 b^2\right ) \sqrt {e \sin (c+d x)}dx+\frac {2 b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))^2}{7 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int (a+b \cos (c+d x)) \left (7 a^2+11 b \cos (c+d x) a+4 b^2\right ) \sqrt {e \sin (c+d x)}dx+\frac {2 b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))^2}{7 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \sqrt {-e \cos \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (7 a^2+11 b \sin \left (c+d x+\frac {\pi }{2}\right ) a+4 b^2\right )dx+\frac {2 b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))^2}{7 d e}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {1}{2} \left (7 a \left (5 a^2+6 b^2\right )+b \left (57 a^2+20 b^2\right ) \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}dx+\frac {22 a b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}{5 d e}\right )+\frac {2 b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))^2}{7 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \left (7 a \left (5 a^2+6 b^2\right )+b \left (57 a^2+20 b^2\right ) \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}dx+\frac {22 a b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}{5 d e}\right )+\frac {2 b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))^2}{7 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {e \cos \left (c+d x-\frac {\pi }{2}\right )} \left (7 a \left (5 a^2+6 b^2\right )-b \left (57 a^2+20 b^2\right ) \sin \left (c+d x-\frac {\pi }{2}\right )\right )dx+\frac {22 a b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}{5 d e}\right )+\frac {2 b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))^2}{7 d e}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (7 a \left (5 a^2+6 b^2\right ) \int \sqrt {e \sin (c+d x)}dx+\frac {2 b \left (57 a^2+20 b^2\right ) (e \sin (c+d x))^{3/2}}{3 d e}\right )+\frac {22 a b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}{5 d e}\right )+\frac {2 b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))^2}{7 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (7 a \left (5 a^2+6 b^2\right ) \int \sqrt {e \sin (c+d x)}dx+\frac {2 b \left (57 a^2+20 b^2\right ) (e \sin (c+d x))^{3/2}}{3 d e}\right )+\frac {22 a b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}{5 d e}\right )+\frac {2 b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))^2}{7 d e}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {7 a \left (5 a^2+6 b^2\right ) \sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)}}+\frac {2 b \left (57 a^2+20 b^2\right ) (e \sin (c+d x))^{3/2}}{3 d e}\right )+\frac {22 a b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}{5 d e}\right )+\frac {2 b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))^2}{7 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {7 a \left (5 a^2+6 b^2\right ) \sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)}}+\frac {2 b \left (57 a^2+20 b^2\right ) (e \sin (c+d x))^{3/2}}{3 d e}\right )+\frac {22 a b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}{5 d e}\right )+\frac {2 b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))^2}{7 d e}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2 b \left (57 a^2+20 b^2\right ) (e \sin (c+d x))^{3/2}}{3 d e}+\frac {14 a \left (5 a^2+6 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}\right )+\frac {22 a b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}{5 d e}\right )+\frac {2 b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))^2}{7 d e}\) |
(2*b*(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(3/2))/(7*d*e) + ((22*a*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))/(5*d*e) + ((14*a*(5*a^2 + 6*b^2) *EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(d*Sqrt[Sin[c + d* x]]) + (2*b*(57*a^2 + 20*b^2)*(e*Sin[c + d*x])^(3/2))/(3*d*e))/5)/7
3.1.52.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[1/(m + p) Int[(g*Cos[e + f*x])^p* (a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(m + p) + a*b*(2*m + p - 1) *Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + p, 0] && (IntegersQ[2*m, 2*p] || IntegerQ[m])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)* (g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x] + S imp[1/(m + p + 1) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*Sim p[a*c*(m + p + 1) + b*d*m + (a*d*m + b*c*(m + p + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && !LtQ[p, -1] && IntegerQ[2*m] && !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && S implerQ[c + d*x, a + b*x])
Time = 4.06 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.95
| method | result | size |
| parts | \(-\frac {a^{3} e \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \left (2 E\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )\right )}{\cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}-\frac {2 b^{3} \left (\frac {\left (e \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{7}-\frac {e^{2} \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3}\right )}{d \,e^{3}}-\frac {6 a \,b^{2} e \left (2 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) E\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )+\cos ^{4}\left (d x +c \right )-\left (\cos ^{2}\left (d x +c \right )\right )\right )}{5 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}+\frac {2 a^{2} b \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{e d}\) | \(314\) |
| default | \(\frac {\frac {2 b \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}} \left (3 b^{2} \left (\cos ^{2}\left (d x +c \right )\right )+21 a^{2}+4 b^{2}\right )}{21 e}-\frac {a e \left (10 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) E\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) a^{2}+12 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) E\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) b^{2}-5 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) a^{2}-6 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) b^{2}+6 \left (\sin ^{4}\left (d x +c \right )\right ) b^{2}-6 b^{2} \left (\sin ^{2}\left (d x +c \right )\right )\right )}{5 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(315\) |
-a^3*e*(1-sin(d*x+c))^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*(2*Ell ipticE((1-sin(d*x+c))^(1/2),1/2*2^(1/2))-EllipticF((1-sin(d*x+c))^(1/2),1/ 2*2^(1/2)))/cos(d*x+c)/(e*sin(d*x+c))^(1/2)/d-2*b^3/d/e^3*(1/7*(e*sin(d*x+ c))^(7/2)-1/3*e^2*(e*sin(d*x+c))^(3/2))-6/5*a*b^2*e*(2*(1-sin(d*x+c))^(1/2 )*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*EllipticE((1-sin(d*x+c))^(1/2),1 /2*2^(1/2))-(1-sin(d*x+c))^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*E llipticF((1-sin(d*x+c))^(1/2),1/2*2^(1/2))+cos(d*x+c)^4-cos(d*x+c)^2)/cos( d*x+c)/(e*sin(d*x+c))^(1/2)/d+2*a^2*b*(e*sin(d*x+c))^(3/2)/e/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.91 \[ \int (a+b \cos (c+d x))^3 \sqrt {e \sin (c+d x)} \, dx=-\frac {21 \, \sqrt {2} {\left (-5 i \, a^{3} - 6 i \, a b^{2}\right )} \sqrt {-i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (5 i \, a^{3} + 6 i \, a b^{2}\right )} \sqrt {i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (15 \, b^{3} \cos \left (d x + c\right )^{2} + 63 \, a b^{2} \cos \left (d x + c\right ) + 105 \, a^{2} b + 20 \, b^{3}\right )} \sqrt {e \sin \left (d x + c\right )} \sin \left (d x + c\right )}{105 \, d} \]
-1/105*(21*sqrt(2)*(-5*I*a^3 - 6*I*a*b^2)*sqrt(-I*e)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(5 *I*a^3 + 6*I*a*b^2)*sqrt(I*e)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(15*b^3*cos(d*x + c)^2 + 63*a*b^2* cos(d*x + c) + 105*a^2*b + 20*b^3)*sqrt(e*sin(d*x + c))*sin(d*x + c))/d
\[ \int (a+b \cos (c+d x))^3 \sqrt {e \sin (c+d x)} \, dx=\int \sqrt {e \sin {\left (c + d x \right )}} \left (a + b \cos {\left (c + d x \right )}\right )^{3}\, dx \]
\[ \int (a+b \cos (c+d x))^3 \sqrt {e \sin (c+d x)} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sqrt {e \sin \left (d x + c\right )} \,d x } \]
\[ \int (a+b \cos (c+d x))^3 \sqrt {e \sin (c+d x)} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sqrt {e \sin \left (d x + c\right )} \,d x } \]
Timed out. \[ \int (a+b \cos (c+d x))^3 \sqrt {e \sin (c+d x)} \, dx=\int \sqrt {e\,\sin \left (c+d\,x\right )}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3 \,d x \]