Integrand size = 25, antiderivative size = 172 \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3 \, dx=-\frac {26 a^3 (e \cos (c+d x))^{5/2}}{35 d e}+\frac {26 a^3 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \sqrt {e \cos (c+d x)}}+\frac {26 a^3 e \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}-\frac {2 a (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^2}{9 d e}-\frac {26 (e \cos (c+d x))^{5/2} \left (a^3+a^3 \sin (c+d x)\right )}{63 d e} \] Output:
-26/35*a^3*(e*cos(d*x+c))^(5/2)/d/e+26/21*a^3*e^2*cos(d*x+c)^(1/2)*Inverse JacobiAM(1/2*d*x+1/2*c,2^(1/2))/d/(e*cos(d*x+c))^(1/2)+26/21*a^3*e*(e*cos( d*x+c))^(1/2)*sin(d*x+c)/d-2/9*a*(e*cos(d*x+c))^(5/2)*(a+a*sin(d*x+c))^2/d /e-26/63*(e*cos(d*x+c))^(5/2)*(a^3+a^3*sin(d*x+c))/d/e
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.38 \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3 \, dx=-\frac {32 \sqrt [4]{2} a^3 (e \cos (c+d x))^{5/2} \operatorname {Hypergeometric2F1}\left (-\frac {13}{4},\frac {5}{4},\frac {9}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{5 d e (1+\sin (c+d x))^{5/4}} \] Input:
Integrate[(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^3,x]
Output:
(-32*2^(1/4)*a^3*(e*Cos[c + d*x])^(5/2)*Hypergeometric2F1[-13/4, 5/4, 9/4, (1 - Sin[c + d*x])/2])/(5*d*e*(1 + Sin[c + d*x])^(5/4))
Time = 0.79 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.04, 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, 3148, 3042, 3115, 3042, 3121, 3042, 3120}
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.
| \(\displaystyle \int (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{3/2}dx\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {13}{9} a \int (e \cos (c+d x))^{3/2} (\sin (c+d x) a+a)^2dx-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {13}{9} a \int (e \cos (c+d x))^{3/2} (\sin (c+d x) a+a)^2dx-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}{9 d e}\) |
| \(\Big \downarrow \) 3157 |
| \(\displaystyle \frac {13}{9} a \left (\frac {9}{7} a \int (e \cos (c+d x))^{3/2} (\sin (c+d x) a+a)dx-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{5/2}}{7 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {13}{9} a \left (\frac {9}{7} a \int (e \cos (c+d x))^{3/2} (\sin (c+d x) a+a)dx-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{5/2}}{7 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}{9 d e}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {13}{9} a \left (\frac {9}{7} a \left (a \int (e \cos (c+d x))^{3/2}dx-\frac {2 a (e \cos (c+d x))^{5/2}}{5 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{5/2}}{7 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {13}{9} a \left (\frac {9}{7} a \left (a \int \left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}dx-\frac {2 a (e \cos (c+d x))^{5/2}}{5 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{5/2}}{7 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}{9 d e}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {13}{9} a \left (\frac {9}{7} a \left (a \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \cos (c+d x)}}dx+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )-\frac {2 a (e \cos (c+d x))^{5/2}}{5 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{5/2}}{7 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {13}{9} a \left (\frac {9}{7} a \left (a \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )-\frac {2 a (e \cos (c+d x))^{5/2}}{5 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{5/2}}{7 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}{9 d e}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {13}{9} a \left (\frac {9}{7} a \left (a \left (\frac {e^2 \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{3 \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )-\frac {2 a (e \cos (c+d x))^{5/2}}{5 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{5/2}}{7 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {13}{9} a \left (\frac {9}{7} a \left (a \left (\frac {e^2 \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )-\frac {2 a (e \cos (c+d x))^{5/2}}{5 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{5/2}}{7 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}{9 d e}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {13}{9} a \left (\frac {9}{7} a \left (a \left (\frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )-\frac {2 a (e \cos (c+d x))^{5/2}}{5 d e}\right )-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{5/2}}{7 d e}\right )-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}{9 d e}\) |
Input:
Int[(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^3,x]
Output:
(-2*a*(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^2)/(9*d*e) + (13*a*((-2* (e*Cos[c + d*x])^(5/2)*(a^2 + a^2*Sin[c + d*x]))/(7*d*e) + (9*a*((-2*a*(e* Cos[c + d*x])^(5/2))/(5*d*e) + a*((2*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[e*Cos[c + d*x]]) + (2*e*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(3*d))))/7))/9
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[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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 = 1.10 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.46
| method | result | size |
| default | \(\frac {2 a^{3} e^{2} \left (-1120 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}+2160 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+2800 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}-3240 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-784 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+840 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-1624 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+120 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-195 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+1162 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-217 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} e +e}\, d}\) | \(251\) |
| parts | \(-\frac {2 a^{3} \sqrt {e \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, e^{2} \left (4 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{3 \sqrt {-e \left (2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\, d}+\frac {2 a^{3} \left (\frac {\left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}}}{9}-\frac {e^{2} \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}}}{5}\right )}{d \,e^{3}}-\frac {6 a^{3} \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}}}{5 d e}+\frac {4 a^{3} \sqrt {e \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, e^{2} \left (24 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}-60 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+50 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-15 \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 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticF}\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 )}{7 \sqrt {-e \left (2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\, d}\) | \(464\) |
Input:
int((e*cos(d*x+c))^(3/2)*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
2/315/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*a^3*e^2*(-112 0*sin(1/2*d*x+1/2*c)^11+2160*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+2800* sin(1/2*d*x+1/2*c)^9-3240*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-784*sin( 1/2*d*x+1/2*c)^7+840*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-1624*sin(1/2* d*x+1/2*c)^5+120*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-195*(sin(1/2*d*x+ 1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2 *c),2^(1/2))+1162*sin(1/2*d*x+1/2*c)^3-217*sin(1/2*d*x+1/2*c))/d
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.78 \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3 \, dx=-\frac {2 \, {\left (195 i \, \sqrt {\frac {1}{2}} a^{3} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 195 i \, \sqrt {\frac {1}{2}} a^{3} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - {\left (35 \, a^{3} e \cos \left (d x + c\right )^{4} - 252 \, a^{3} e \cos \left (d x + c\right )^{2} - 15 \, {\left (9 \, a^{3} e \cos \left (d x + c\right )^{2} - 13 \, a^{3} e\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}\right )}}{315 \, d} \] Input:
integrate((e*cos(d*x+c))^(3/2)*(a+a*sin(d*x+c))^3,x, algorithm="fricas")
Output:
-2/315*(195*I*sqrt(1/2)*a^3*e^(3/2)*weierstrassPInverse(-4, 0, cos(d*x + c ) + I*sin(d*x + c)) - 195*I*sqrt(1/2)*a^3*e^(3/2)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - (35*a^3*e*cos(d*x + c)^4 - 252*a^3*e*c os(d*x + c)^2 - 15*(9*a^3*e*cos(d*x + c)^2 - 13*a^3*e)*sin(d*x + c))*sqrt( e*cos(d*x + c)))/d
Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate((e*cos(d*x+c))**(3/2)*(a+a*sin(d*x+c))**3,x)
Output:
Timed out
\[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{3} \,d x } \] Input:
integrate((e*cos(d*x+c))^(3/2)*(a+a*sin(d*x+c))^3,x, algorithm="maxima")
Output:
integrate((e*cos(d*x + c))^(3/2)*(a*sin(d*x + c) + a)^3, x)
\[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3 \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{3} \,d x } \] Input:
integrate((e*cos(d*x+c))^(3/2)*(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
integrate((e*cos(d*x + c))^(3/2)*(a*sin(d*x + c) + a)^3, x)
Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3 \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3 \,d x \] Input:
int((e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^3,x)
Output:
int((e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^3, x)
\[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3 \, dx=\frac {\sqrt {e}\, a^{3} e \left (-6 \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}+5 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}d x \right ) d +15 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}d x \right ) d +5 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) d \right )}{5 d} \] Input:
int((e*cos(d*x+c))^(3/2)*(a+a*sin(d*x+c))^3,x)
Output:
(sqrt(e)*a**3*e*( - 6*sqrt(cos(c + d*x))*cos(c + d*x)**2 + 5*int(sqrt(cos( c + d*x))*cos(c + d*x)*sin(c + d*x)**3,x)*d + 15*int(sqrt(cos(c + d*x))*co s(c + d*x)*sin(c + d*x)**2,x)*d + 5*int(sqrt(cos(c + d*x))*cos(c + d*x),x) *d))/(5*d)