Integrand size = 25, antiderivative size = 202 \[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2} \, dx=\frac {2 a \left (7 a^2+6 b^2\right ) e^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{21 d \sqrt {e \sin (c+d x)}}-\frac {2 a \left (7 a^2+6 b^2\right ) e \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 d}+\frac {2 b \left (89 a^2+28 b^2\right ) (e \sin (c+d x))^{5/2}}{315 d e}+\frac {26 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{63 d e}+\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2}}{9 d e} \] Output:
2/21*a*(7*a^2+6*b^2)*e^2*InverseJacobiAM(1/2*c-1/4*Pi+1/2*d*x,2^(1/2))*sin (d*x+c)^(1/2)/d/(e*sin(d*x+c))^(1/2)-2/21*a*(7*a^2+6*b^2)*e*cos(d*x+c)*(e* sin(d*x+c))^(1/2)/d+2/315*b*(89*a^2+28*b^2)*(e*sin(d*x+c))^(5/2)/d/e+26/63 *a*b*(a+b*cos(d*x+c))*(e*sin(d*x+c))^(5/2)/d/e+2/9*b*(a+b*cos(d*x+c))^2*(e *sin(d*x+c))^(5/2)/d/e
Time = 2.17 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.73 \[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2} \, dx=\frac {\left (-20 a \left (28 a^2+15 b^2\right ) \cot (c+d x)-\frac {2}{3} b \left (-756 a^2-147 b^2+28 \left (27 a^2+4 b^2\right ) \cos (2 (c+d x))+270 a b \cos (3 (c+d x))+35 b^2 \cos (4 (c+d x))\right ) \csc (c+d x)-\frac {80 a \left (7 a^2+6 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right )}{\sin ^{\frac {3}{2}}(c+d x)}\right ) (e \sin (c+d x))^{3/2}}{840 d} \] Input:
Integrate[(a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(3/2),x]
Output:
((-20*a*(28*a^2 + 15*b^2)*Cot[c + d*x] - (2*b*(-756*a^2 - 147*b^2 + 28*(27 *a^2 + 4*b^2)*Cos[2*(c + d*x)] + 270*a*b*Cos[3*(c + d*x)] + 35*b^2*Cos[4*( c + d*x)])*Csc[c + d*x])/3 - (80*a*(7*a^2 + 6*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, 2])/Sin[c + d*x]^(3/2))*(e*Sin[c + d*x])^(3/2))/(840*d)
Time = 0.97 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {3042, 3171, 27, 3042, 3341, 27, 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.
| \(\displaystyle \int (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{3/2} \left (a-b \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3dx\) |
| \(\Big \downarrow \) 3171 |
| \(\displaystyle \frac {2}{9} \int \frac {1}{2} (a+b \cos (c+d x)) \left (9 a^2+13 b \cos (c+d x) a+4 b^2\right ) (e \sin (c+d x))^{3/2}dx+\frac {2 b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int (a+b \cos (c+d x)) \left (9 a^2+13 b \cos (c+d x) a+4 b^2\right ) (e \sin (c+d x))^{3/2}dx+\frac {2 b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \left (-e \cos \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (9 a^2+13 b \sin \left (c+d x+\frac {\pi }{2}\right ) a+4 b^2\right )dx+\frac {2 b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{7} \int \frac {1}{2} \left (9 a \left (7 a^2+6 b^2\right )+b \left (89 a^2+28 b^2\right ) \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}dx+\frac {26 a b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}{7 d e}\right )+\frac {2 b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \left (9 a \left (7 a^2+6 b^2\right )+b \left (89 a^2+28 b^2\right ) \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}dx+\frac {26 a b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}{7 d e}\right )+\frac {2 b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{3/2} \left (9 a \left (7 a^2+6 b^2\right )-b \left (89 a^2+28 b^2\right ) \sin \left (c+d x-\frac {\pi }{2}\right )\right )dx+\frac {26 a b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}{7 d e}\right )+\frac {2 b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (9 a \left (7 a^2+6 b^2\right ) \int (e \sin (c+d x))^{3/2}dx+\frac {2 b \left (89 a^2+28 b^2\right ) (e \sin (c+d x))^{5/2}}{5 d e}\right )+\frac {26 a b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}{7 d e}\right )+\frac {2 b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (9 a \left (7 a^2+6 b^2\right ) \int (e \sin (c+d x))^{3/2}dx+\frac {2 b \left (89 a^2+28 b^2\right ) (e \sin (c+d x))^{5/2}}{5 d e}\right )+\frac {26 a b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}{7 d e}\right )+\frac {2 b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (9 a \left (7 a^2+6 b^2\right ) \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \sin (c+d x)}}dx-\frac {2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}\right )+\frac {2 b \left (89 a^2+28 b^2\right ) (e \sin (c+d x))^{5/2}}{5 d e}\right )+\frac {26 a b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}{7 d e}\right )+\frac {2 b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (9 a \left (7 a^2+6 b^2\right ) \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \sin (c+d x)}}dx-\frac {2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}\right )+\frac {2 b \left (89 a^2+28 b^2\right ) (e \sin (c+d x))^{5/2}}{5 d e}\right )+\frac {26 a b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}{7 d e}\right )+\frac {2 b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (9 a \left (7 a^2+6 b^2\right ) \left (\frac {e^2 \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)}}dx}{3 \sqrt {e \sin (c+d x)}}-\frac {2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}\right )+\frac {2 b \left (89 a^2+28 b^2\right ) (e \sin (c+d x))^{5/2}}{5 d e}\right )+\frac {26 a b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}{7 d e}\right )+\frac {2 b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (9 a \left (7 a^2+6 b^2\right ) \left (\frac {e^2 \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\sin (c+d x)}}dx}{3 \sqrt {e \sin (c+d x)}}-\frac {2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}\right )+\frac {2 b \left (89 a^2+28 b^2\right ) (e \sin (c+d x))^{5/2}}{5 d e}\right )+\frac {26 a b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}{7 d e}\right )+\frac {2 b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))^2}{9 d e}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (9 a \left (7 a^2+6 b^2\right ) \left (\frac {2 e^2 \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{3 d \sqrt {e \sin (c+d x)}}-\frac {2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}\right )+\frac {2 b \left (89 a^2+28 b^2\right ) (e \sin (c+d x))^{5/2}}{5 d e}\right )+\frac {26 a b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}{7 d e}\right )+\frac {2 b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))^2}{9 d e}\) |
Input:
Int[(a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(3/2),x]
Output:
(2*b*(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(5/2))/(9*d*e) + ((26*a*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2))/(7*d*e) + ((2*b*(89*a^2 + 28*b^2 )*(e*Sin[c + d*x])^(5/2))/(5*d*e) + 9*a*(7*a^2 + 6*b^2)*((2*e^2*EllipticF[ (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*d*Sqrt[e*Sin[c + d*x]]) - (2 *e*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(3*d)))/7)/9
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[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 = 3.83 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.35
| method | result | size |
| parts | \(-\frac {a^{3} e^{2} \left (\sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-2 \sin \left (d x +c \right )^{3}+2 \sin \left (d x +c \right )\right )}{3 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}-\frac {2 b^{3} \left (\frac {\left (e \sin \left (d x +c \right )\right )^{\frac {9}{2}}}{9}-\frac {e^{2} \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{5}\right )}{d \,e^{3}}+\frac {6 a^{2} b \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{5 e d}-\frac {2 b^{2} a \,e^{2} \left (3 \sin \left (d x +c \right )^{5}+\sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-5 \sin \left (d x +c \right )^{3}+2 \sin \left (d x +c \right )\right )}{7 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}\) | \(272\) |
| default | \(-\frac {e^{2} \left (70 b^{3} \cos \left (d x +c \right )^{5} \sin \left (d x +c \right )+270 a \,b^{2} \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )+105 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) a^{3}+90 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) a \,b^{2}+378 a^{2} b \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )-14 b^{3} \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )+210 a^{3} \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )-90 a \,b^{2} \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )-378 a^{2} b \cos \left (d x +c \right ) \sin \left (d x +c \right )-56 b^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )\right )}{315 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}\) | \(291\) |
Input:
int((a+cos(d*x+c)*b)^3*(e*sin(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/3*a^3*e^2*((1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2) *EllipticF((1-sin(d*x+c))^(1/2),1/2*2^(1/2))-2*sin(d*x+c)^3+2*sin(d*x+c))/ cos(d*x+c)/(e*sin(d*x+c))^(1/2)/d-2*b^3/d/e^3*(1/9*(e*sin(d*x+c))^(9/2)-1/ 5*e^2*(e*sin(d*x+c))^(5/2))+6/5*a^2*b*(e*sin(d*x+c))^(5/2)/e/d-2/7*b^2*a*e ^2*(3*sin(d*x+c)^5+(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^ (1/2)*EllipticF((1-sin(d*x+c))^(1/2),1/2*2^(1/2))-5*sin(d*x+c)^3+2*sin(d*x +c))/cos(d*x+c)/(e*sin(d*x+c))^(1/2)/d
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.91 \[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2} \, dx=\frac {2 \, {\left (15 \, {\left (7 \, a^{3} + 6 \, a b^{2}\right )} \sqrt {-\frac {1}{2} i \, e} e {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, {\left (7 \, a^{3} + 6 \, a b^{2}\right )} \sqrt {\frac {1}{2} i \, e} e {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - {\left (35 \, b^{3} e \cos \left (d x + c\right )^{4} + 135 \, a b^{2} e \cos \left (d x + c\right )^{3} + 7 \, {\left (27 \, a^{2} b - b^{3}\right )} e \cos \left (d x + c\right )^{2} + 15 \, {\left (7 \, a^{3} - 3 \, a b^{2}\right )} e \cos \left (d x + c\right ) - 7 \, {\left (27 \, a^{2} b + 4 \, b^{3}\right )} e\right )} \sqrt {e \sin \left (d x + c\right )}\right )}}{315 \, d} \] Input:
integrate((a+b*cos(d*x+c))^3*(e*sin(d*x+c))^(3/2),x, algorithm="fricas")
Output:
2/315*(15*(7*a^3 + 6*a*b^2)*sqrt(-1/2*I*e)*e*weierstrassPInverse(4, 0, cos (d*x + c) + I*sin(d*x + c)) + 15*(7*a^3 + 6*a*b^2)*sqrt(1/2*I*e)*e*weierst rassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c)) - (35*b^3*e*cos(d*x + c) ^4 + 135*a*b^2*e*cos(d*x + c)^3 + 7*(27*a^2*b - b^3)*e*cos(d*x + c)^2 + 15 *(7*a^3 - 3*a*b^2)*e*cos(d*x + c) - 7*(27*a^2*b + 4*b^3)*e)*sqrt(e*sin(d*x + c)))/d
\[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2} \, dx=\int \left (e \sin {\left (c + d x \right )}\right )^{\frac {3}{2}} \left (a + b \cos {\left (c + d x \right )}\right )^{3}\, dx \] Input:
integrate((a+b*cos(d*x+c))**3*(e*sin(d*x+c))**(3/2),x)
Output:
Integral((e*sin(c + d*x))**(3/2)*(a + b*cos(c + d*x))**3, x)
\[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^3*(e*sin(d*x+c))^(3/2),x, algorithm="maxima")
Output:
integrate((b*cos(d*x + c) + a)^3*(e*sin(d*x + c))^(3/2), x)
\[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2} \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^3*(e*sin(d*x+c))^(3/2),x, algorithm="giac")
Output:
integrate((b*cos(d*x + c) + a)^3*(e*sin(d*x + c))^(3/2), x)
Timed out. \[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2} \, dx=\int {\left (e\,\sin \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3 \,d x \] Input:
int((e*sin(c + d*x))^(3/2)*(a + b*cos(c + d*x))^3,x)
Output:
int((e*sin(c + d*x))^(3/2)*(a + b*cos(c + d*x))^3, x)
\[ \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2} \, dx=\frac {\sqrt {e}\, e \left (6 \sqrt {\sin \left (d x +c \right )}\, \sin \left (d x +c \right )^{2} a^{2} b +5 \left (\int \sqrt {\sin \left (d x +c \right )}\, \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )d x \right ) b^{3} d +15 \left (\int \sqrt {\sin \left (d x +c \right )}\, \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )d x \right ) a \,b^{2} d +5 \left (\int \sqrt {\sin \left (d x +c \right )}\, \sin \left (d x +c \right )d x \right ) a^{3} d \right )}{5 d} \] Input:
int((a+b*cos(d*x+c))^3*(e*sin(d*x+c))^(3/2),x)
Output:
(sqrt(e)*e*(6*sqrt(sin(c + d*x))*sin(c + d*x)**2*a**2*b + 5*int(sqrt(sin(c + d*x))*cos(c + d*x)**3*sin(c + d*x),x)*b**3*d + 15*int(sqrt(sin(c + d*x) )*cos(c + d*x)**2*sin(c + d*x),x)*a*b**2*d + 5*int(sqrt(sin(c + d*x))*sin( c + d*x),x)*a**3*d))/(5*d)