Integrand size = 25, antiderivative size = 156 \[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3 \, dx=-\frac {2 b \left (57 a^2+20 b^2\right ) (e \cos (c+d x))^{3/2}}{105 d e}+\frac {2 a \left (5 a^2+6 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}-\frac {22 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{35 d e}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e} \]
-2/105*b*(57*a^2+20*b^2)*(e*cos(d*x+c))^(3/2)/d/e-22/35*a*b*(e*cos(d*x+c)) ^(3/2)*(a+b*sin(d*x+c))/d/e-2/7*b*(e*cos(d*x+c))^(3/2)*(a+b*sin(d*x+c))^2/ d/e+2/5*a*(5*a^2+6*b^2)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*El lipticE(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 )
Time = 0.82 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.65 \[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3 \, dx=\frac {\sqrt {e \cos (c+d x)} \left (42 \left (5 a^3+6 a b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+b \cos ^{\frac {3}{2}}(c+d x) \left (-210 a^2-55 b^2+15 b^2 \cos (2 (c+d x))-126 a b \sin (c+d x)\right )\right )}{105 d \sqrt {\cos (c+d x)}} \]
(Sqrt[e*Cos[c + d*x]]*(42*(5*a^3 + 6*a*b^2)*EllipticE[(c + d*x)/2, 2] + b* Cos[c + d*x]^(3/2)*(-210*a^2 - 55*b^2 + 15*b^2*Cos[2*(c + d*x)] - 126*a*b* Sin[c + d*x])))/(105*d*Sqrt[Cos[c + d*x]])
Time = 0.76 (sec) , antiderivative size = 164, 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.
| \(\displaystyle \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3dx\) |
| \(\Big \downarrow \) 3171 |
| \(\displaystyle \frac {2}{7} \int \frac {1}{2} \sqrt {e \cos (c+d x)} (a+b \sin (c+d x)) \left (7 a^2+11 b \sin (c+d x) a+4 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x)) \left (7 a^2+11 b \sin (c+d x) a+4 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x)) \left (7 a^2+11 b \sin (c+d x) a+4 b^2\right )dx-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {1}{2} \sqrt {e \cos (c+d x)} \left (7 a \left (5 a^2+6 b^2\right )+b \left (57 a^2+20 b^2\right ) \sin (c+d x)\right )dx-\frac {22 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}\right )-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {e \cos (c+d x)} \left (7 a \left (5 a^2+6 b^2\right )+b \left (57 a^2+20 b^2\right ) \sin (c+d x)\right )dx-\frac {22 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}\right )-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \sqrt {e \cos (c+d x)} \left (7 a \left (5 a^2+6 b^2\right )+b \left (57 a^2+20 b^2\right ) \sin (c+d x)\right )dx-\frac {22 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}\right )-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (7 a \left (5 a^2+6 b^2\right ) \int \sqrt {e \cos (c+d x)}dx-\frac {2 b \left (57 a^2+20 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e}\right )-\frac {22 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}\right )-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (7 a \left (5 a^2+6 b^2\right ) \int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {2 b \left (57 a^2+20 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e}\right )-\frac {22 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}\right )-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {7 a \left (5 a^2+6 b^2\right ) \sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{\sqrt {\cos (c+d x)}}-\frac {2 b \left (57 a^2+20 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e}\right )-\frac {22 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}\right )-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {7 a \left (5 a^2+6 b^2\right ) \sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\cos (c+d x)}}-\frac {2 b \left (57 a^2+20 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e}\right )-\frac {22 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}\right )-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {14 a \left (5 a^2+6 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\frac {2 b \left (57 a^2+20 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e}\right )-\frac {22 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}\right )-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}\) |
(-2*b*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^2)/(7*d*e) + (((-2*b*(57 *a^2 + 20*b^2)*(e*Cos[c + d*x])^(3/2))/(3*d*e) + (14*a*(5*a^2 + 6*b^2)*Sqr t[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]))/5 - ( 22*a*b*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x]))/(5*d*e))/7
3.6.58.3.1 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])
Leaf count of result is larger than twice the leaf count of optimal. \(338\) vs. \(2(164)=328\).
Time = 5.34 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.17
| method | result | size |
| default | \(-\frac {2 e \left (-240 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+504 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+480 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}-504 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+420 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -220 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+126 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}-105 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{3}-126 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a \,b^{2}-420 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -20 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+105 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b +20 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}\right )}{105 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(339\) |
| parts | \(\frac {2 a^{3} \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, e \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{\sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}+\frac {2 b^{3} \left (\frac {\left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}{7}-\frac {e^{2} \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{3}\right )}{d \,e^{3}}+\frac {12 a \,b^{2} \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, e \left (4 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}-\frac {2 a^{2} b \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{e d}\) | \(404\) |
-2/105/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e*(-240*sin( 1/2*d*x+1/2*c)^9*b^3+504*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6*a*b^2+480 *sin(1/2*d*x+1/2*c)^7*b^3-504*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4*a*b^ 2+420*sin(1/2*d*x+1/2*c)^5*a^2*b-220*sin(1/2*d*x+1/2*c)^5*b^3+126*cos(1/2* d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*a*b^2-105*(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))*a^3-12 6*(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))*a*b^2-420*sin(1/2*d*x+1/2*c)^3*a^2*b-20*sin(1/ 2*d*x+1/2*c)^3*b^3+105*sin(1/2*d*x+1/2*c)*a^2*b+20*sin(1/2*d*x+1/2*c)*b^3) /d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.96 \[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3 \, dx=-\frac {21 \, \sqrt {2} {\left (-5 i \, a^{3} - 6 i \, a b^{2}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (5 i \, a^{3} + 6 i \, a b^{2}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (15 \, b^{3} \cos \left (d x + c\right )^{3} - 63 \, a b^{2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 35 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{105 \, d} \]
-1/105*(21*sqrt(2)*(-5*I*a^3 - 6*I*a*b^2)*sqrt(e)*weierstrassZeta(-4, 0, w eierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(5* I*a^3 + 6*I*a*b^2)*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(15*b^3*cos(d*x + c)^3 - 63*a*b^2*c os(d*x + c)*sin(d*x + c) - 35*(3*a^2*b + b^3)*cos(d*x + c))*sqrt(e*cos(d*x + c)))/d
Timed out. \[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \]
\[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3 \, dx=\int { \sqrt {e \cos \left (d x + c\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{3} \,d x } \]
\[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3 \, dx=\int { \sqrt {e \cos \left (d x + c\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3 \, dx=\int \sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3 \,d x \]