Integrand size = 25, antiderivative size = 161 \[ \int (a+b \cos (c+d x))^3 \sqrt {e \sin (c+d x)} \, dx=\frac {2 a \left (5 a^2+6 b^2\right ) E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 d \sqrt {\sin (c+d x)}}+\frac {2 b \left (57 a^2+20 b^2\right ) (e \sin (c+d x))^{3/2}}{105 d e}+\frac {22 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}}{35 d e}+\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}}{7 d e} \] Output:
-2/5*a*(5*a^2+6*b^2)*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)+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
Time = 1.45 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.65 \[ \int (a+b \cos (c+d x))^3 \sqrt {e \sin (c+d x)} \, dx=\frac {\sqrt {e \sin (c+d x)} \left (-42 \left (5 a^3+6 a b^2\right ) E\left (\left .\frac {1}{4} (-2 c+\pi -2 d x)\right |2\right )+b \left (210 a^2+55 b^2+126 a b \cos (c+d x)+15 b^2 \cos (2 (c+d x))\right ) \sin ^{\frac {3}{2}}(c+d x)\right )}{105 d \sqrt {\sin (c+d x)}} \] Input:
Integrate[(a + b*Cos[c + d*x])^3*Sqrt[e*Sin[c + d*x]],x]
Output:
(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 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2 b \left (57 a^2+20 b^2\right ) (e \sin (c+d x))^{3/2}}{3 d e}+\frac {14 a \left (5 a^2+6 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d \sqrt {\sin (c+d x)}}\right )+\frac {22 a b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}{5 d e}\right )+\frac {2 b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))^2}{7 d e}\) |
Input:
Int[(a + b*Cos[c + d*x])^3*Sqrt[e*Sin[c + d*x]],x]
Output:
(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
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. \(313\) vs. \(2(143)=286\).
Time = 4.27 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.95
| method | result | size |
| parts | \(-\frac {a^{3} e \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \left (2 \operatorname {EllipticE}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-\operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )\right )}{\cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}-\frac {2 b^{3} \left (\frac {\left (e \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{7}-\frac {e^{2} \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3}\right )}{d \,e^{3}}-\frac {6 b^{2} a e \left (2 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )+\cos \left (d x +c \right )^{4}-\cos \left (d x +c \right )^{2}\right )}{5 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}+\frac {2 a^{2} b \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{e d}\) | \(314\) |
| default | \(\frac {\frac {2 b \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}} \left (3 \cos \left (d x +c \right )^{2} b^{2}+21 a^{2}+4 b^{2}\right )}{21 e}-\frac {a e \left (10 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) a^{2}+12 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) b^{2}-5 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) a^{2}-6 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) b^{2}+6 \sin \left (d x +c \right )^{4} b^{2}-6 b^{2} \sin \left (d x +c \right )^{2}\right )}{5 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(315\) |
Input:
int((a+cos(d*x+c)*b)^3*(e*sin(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
-a^3*e*(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c))^(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*b^2*a*e*(2*(1-sin(d*x+c))^(1/2 )*(2+2*sin(d*x+c))^(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+2*sin(d*x+c))^(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 complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.87 \[ \int (a+b \cos (c+d x))^3 \sqrt {e \sin (c+d x)} \, dx=\frac {2 \, {\left ({\left (15 \, b^{3} \cos \left (d x + c\right )^{2} + 63 \, a b^{2} \cos \left (d x + c\right ) + 105 \, a^{2} b + 20 \, b^{3}\right )} \sqrt {e \sin \left (d x + c\right )} \sin \left (d x + c\right ) - 21 \, {\left (-5 i \, a^{3} - 6 i \, a b^{2}\right )} \sqrt {-\frac {1}{2} i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 \, {\left (5 i \, a^{3} + 6 i \, a b^{2}\right )} \sqrt {\frac {1}{2} i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )\right )}}{105 \, d} \] Input:
integrate((a+b*cos(d*x+c))^3*(e*sin(d*x+c))^(1/2),x, algorithm="fricas")
Output:
2/105*((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) - 21*(-5*I*a^3 - 6*I*a*b^2)*sqrt(-1/2* I*e)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) + I*sin( d*x + c))) - 21*(5*I*a^3 + 6*I*a*b^2)*sqrt(1/2*I*e)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c))))/d
\[ \int (a+b \cos (c+d x))^3 \sqrt {e \sin (c+d x)} \, dx=\int \sqrt {e \sin {\left (c + d x \right )}} \left (a + b \cos {\left (c + d x \right )}\right )^{3}\, dx \] Input:
integrate((a+b*cos(d*x+c))**3*(e*sin(d*x+c))**(1/2),x)
Output:
Integral(sqrt(e*sin(c + d*x))*(a + b*cos(c + d*x))**3, x)
\[ \int (a+b \cos (c+d x))^3 \sqrt {e \sin (c+d x)} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sqrt {e \sin \left (d x + c\right )} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^3*(e*sin(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate((b*cos(d*x + c) + a)^3*sqrt(e*sin(d*x + c)), x)
\[ \int (a+b \cos (c+d x))^3 \sqrt {e \sin (c+d x)} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sqrt {e \sin \left (d x + c\right )} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^3*(e*sin(d*x+c))^(1/2),x, algorithm="giac")
Output:
integrate((b*cos(d*x + c) + a)^3*sqrt(e*sin(d*x + c)), x)
Timed out. \[ \int (a+b \cos (c+d x))^3 \sqrt {e \sin (c+d x)} \, dx=\int \sqrt {e\,\sin \left (c+d\,x\right )}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3 \,d x \] Input:
int((e*sin(c + d*x))^(1/2)*(a + b*cos(c + d*x))^3,x)
Output:
int((e*sin(c + d*x))^(1/2)*(a + b*cos(c + d*x))^3, x)
\[ \int (a+b \cos (c+d x))^3 \sqrt {e \sin (c+d x)} \, dx=\frac {\sqrt {e}\, \left (2 \sqrt {\sin \left (d x +c \right )}\, \sin \left (d x +c \right ) a^{2} b +\left (\int \sqrt {\sin \left (d x +c \right )}d x \right ) a^{3} d +\left (\int \sqrt {\sin \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) b^{3} d +3 \left (\int \sqrt {\sin \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a \,b^{2} d \right )}{d} \] Input:
int((a+b*cos(d*x+c))^3*(e*sin(d*x+c))^(1/2),x)
Output:
(sqrt(e)*(2*sqrt(sin(c + d*x))*sin(c + d*x)*a**2*b + int(sqrt(sin(c + d*x) ),x)*a**3*d + int(sqrt(sin(c + d*x))*cos(c + d*x)**3,x)*b**3*d + 3*int(sqr t(sin(c + d*x))*cos(c + d*x)**2,x)*a*b**2*d))/d