\(\int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3 \, dx\) [529]

Optimal result

Integrand size = 27, antiderivative size = 219 \[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3 \, dx=\frac {4 (c-17 d) (c+d) \left (15 c^2+10 c d+7 d^2\right ) \cos (e+f x)}{35 d f \sqrt {3+3 \sin (e+f x)}}+\frac {8 (c-17 d) (5 c-d) (c+d) \cos (e+f x) \sqrt {3+3 \sin (e+f x)}}{105 f}+\frac {4 (c-17 d) d (c+d) \cos (e+f x) (3+3 \sin (e+f x))^{3/2}}{105 f}+\frac {2 (c-17 d) \cos (e+f x) (c+d \sin (e+f x))^3}{7 d f \sqrt {3+3 \sin (e+f x)}}-\frac {2 \cos (e+f x) (c+d \sin (e+f x))^4}{d f \sqrt {3+3 \sin (e+f x)}} \]

[Out]

4/105*(c-17*d)*d*(c+d)*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/f+4/315*a^2*(c-17*d)*(c+d)*(15*c^2+10*c*d+7*d^2)*cos(
f*x+e)/d/f/(a+a*sin(f*x+e))^(1/2)+2/63*a^2*(c-17*d)*cos(f*x+e)*(c+d*sin(f*x+e))^3/d/f/(a+a*sin(f*x+e))^(1/2)-2
/9*a^2*cos(f*x+e)*(c+d*sin(f*x+e))^4/d/f/(a+a*sin(f*x+e))^(1/2)+8/315*a*(c-17*d)*(5*c-d)*(c+d)*cos(f*x+e)*(a+a
*sin(f*x+e))^(1/2)/f

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2842, 21, 2849, 2840, 2830, 2725} \[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3 \, dx=\frac {4 a^2 (c-17 d) (c+d) \left (15 c^2+10 c d+7 d^2\right ) \cos (e+f x)}{315 d f \sqrt {a \sin (e+f x)+a}}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}+\frac {2 a^2 (c-17 d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 d f \sqrt {a \sin (e+f x)+a}}+\frac {4 d (c-17 d) (c+d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{105 f}+\frac {8 a (c-17 d) (5 c-d) (c+d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{315 f} \]

[In]

Int[(a + a*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^3,x]

[Out]

(4*a^2*(c - 17*d)*(c + d)*(15*c^2 + 10*c*d + 7*d^2)*Cos[e + f*x])/(315*d*f*Sqrt[a + a*Sin[e + f*x]]) + (8*a*(c
- 17*d)*(5*c - d)*(c + d)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(315*f) + (4*(c - 17*d)*d*(c + d)*Cos[e + f*
x]*(a + a*Sin[e + f*x])^(3/2))/(105*f) + (2*a^2*(c - 17*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(63*d*f*Sqrt[a
+ a*Sin[e + f*x]]) - (2*a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^4)/(9*d*f*Sqrt[a + a*Sin[e + f*x]])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 2725

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*b*(Cos[c + d*x]/(d*Sqrt[a + b*Sin[c + d*x
]])), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2830

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d
)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/(f*(m + 1))), x] + Dist[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*S
in[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&  !LtQ[m
, -2^(-1)]

Rule 2840

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(-
d^2)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(b*(m + 2)), Int[(a + b*Sin[e + f*
x])^m*Simp[b*(d^2*(m + 1) + c^2*(m + 2)) - d*(a*d - 2*b*c*(m + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c,
d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&  !LtQ[m, -1]

Rule 2842

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Dist[1/(
d*(m + n)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c*(m - 2) + b^2*d*(n + 1) + a^2*d
*(m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &
& NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] &&  !LtQ[n, -1] && (IntegersQ[2*m,
2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))

Rule 2849

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[-2*b*Cos[e + f*x]*((c + d*Sin[e + f*x])^n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]])), x] + Dist[2*n*((b*c + a*d)
/(b*(2*n + 1))), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f}
, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] && IntegerQ[2*n]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a+a \sin (e+f x)}}+\frac {2 \int \frac {\left (-\frac {1}{2} a^2 (c-17 d)-\frac {1}{2} a^2 (c-17 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^3}{\sqrt {a+a \sin (e+f x)}} \, dx}{9 d} \\ & = -\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a+a \sin (e+f x)}}-\frac {(a (c-17 d)) \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3 \, dx}{9 d} \\ & = \frac {2 a^2 (c-17 d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 d f \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a+a \sin (e+f x)}}-\frac {(2 a (c-17 d) (c+d)) \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \, dx}{21 d} \\ & = \frac {4 (c-17 d) d (c+d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{105 f}+\frac {2 a^2 (c-17 d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 d f \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a+a \sin (e+f x)}}-\frac {(4 (c-17 d) (c+d)) \int \sqrt {a+a \sin (e+f x)} \left (\frac {1}{2} a \left (5 c^2+3 d^2\right )+a (5 c-d) d \sin (e+f x)\right ) \, dx}{105 d} \\ & = \frac {8 a (c-17 d) (5 c-d) (c+d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{315 f}+\frac {4 (c-17 d) d (c+d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{105 f}+\frac {2 a^2 (c-17 d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 d f \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a+a \sin (e+f x)}}-\frac {\left (2 a (c-17 d) (c+d) \left (15 c^2+10 c d+7 d^2\right )\right ) \int \sqrt {a+a \sin (e+f x)} \, dx}{315 d} \\ & = \frac {4 a^2 (c-17 d) (c+d) \left (15 c^2+10 c d+7 d^2\right ) \cos (e+f x)}{315 d f \sqrt {a+a \sin (e+f x)}}+\frac {8 a (c-17 d) (5 c-d) (c+d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{315 f}+\frac {4 (c-17 d) d (c+d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{105 f}+\frac {2 a^2 (c-17 d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 d f \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a+a \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.51 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.94 \[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3 \, dx=-\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^{3/2} \left (4200 c^3+9828 c^2 d+8892 c d^2+2689 d^3-4 d \left (189 c^2+351 c d+137 d^2\right ) \cos (2 (e+f x))+35 d^3 \cos (4 (e+f x))+840 c^3 \sin (e+f x)+4536 c^2 d \sin (e+f x)+4554 c d^2 \sin (e+f x)+1598 d^3 \sin (e+f x)-270 c d^2 \sin (3 (e+f x))-170 d^3 \sin (3 (e+f x))\right )}{140 \sqrt {3} f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3} \]

[In]

Integrate[(3 + 3*Sin[e + f*x])^(3/2)*(c + d*Sin[e + f*x])^3,x]

[Out]

-1/140*((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^(3/2)*(4200*c^3 + 9828*c^2*d + 8892*c*d^2 + 2
689*d^3 - 4*d*(189*c^2 + 351*c*d + 137*d^2)*Cos[2*(e + f*x)] + 35*d^3*Cos[4*(e + f*x)] + 840*c^3*Sin[e + f*x]
+ 4536*c^2*d*Sin[e + f*x] + 4554*c*d^2*Sin[e + f*x] + 1598*d^3*Sin[e + f*x] - 270*c*d^2*Sin[3*(e + f*x)] - 170
*d^3*Sin[3*(e + f*x)]))/(Sqrt[3]*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)

Maple [A] (verified)

Time = 3.02 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.89

method	result	size
default	\(\frac {2 \left (\sin \left (f x +e \right )+1\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) \left (35 \left (\sin ^{4}\left (f x +e \right )\right ) d^{3}+135 \left (\sin ^{3}\left (f x +e \right )\right ) c \,d^{2}+85 \left (\sin ^{3}\left (f x +e \right )\right ) d^{3}+189 \left (\sin ^{2}\left (f x +e \right )\right ) c^{2} d +351 \left (\sin ^{2}\left (f x +e \right )\right ) c \,d^{2}+102 d^{3} \left (\sin ^{2}\left (f x +e \right )\right )+105 c^{3} \sin \left (f x +e \right )+567 c^{2} d \sin \left (f x +e \right )+468 \sin \left (f x +e \right ) c \,d^{2}+136 d^{3} \sin \left (f x +e \right )+525 c^{3}+1134 c^{2} d +936 c \,d^{2}+272 d^{3}\right )}{315 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(195\)
parts	\(\frac {2 c^{3} \left (\sin \left (f x +e \right )+1\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) \left (\sin \left (f x +e \right )+5\right )}{3 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 d^{3} \left (\sin \left (f x +e \right )+1\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) \left (35 \left (\sin ^{4}\left (f x +e \right )\right )+85 \left (\sin ^{3}\left (f x +e \right )\right )+102 \left (\sin ^{2}\left (f x +e \right )\right )+136 \sin \left (f x +e \right )+272\right )}{315 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {6 c^{2} d \left (\sin \left (f x +e \right )+1\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) \left (\sin ^{2}\left (f x +e \right )+3 \sin \left (f x +e \right )+6\right )}{5 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 c \,d^{2} \left (\sin \left (f x +e \right )+1\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) \left (15 \left (\sin ^{3}\left (f x +e \right )\right )+39 \left (\sin ^{2}\left (f x +e \right )\right )+52 \sin \left (f x +e \right )+104\right )}{35 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(288\)

[In]

int((a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

2/315*(sin(f*x+e)+1)*a^2*(sin(f*x+e)-1)*(35*sin(f*x+e)^4*d^3+135*sin(f*x+e)^3*c*d^2+85*sin(f*x+e)^3*d^3+189*si
n(f*x+e)^2*c^2*d+351*sin(f*x+e)^2*c*d^2+102*d^3*sin(f*x+e)^2+105*c^3*sin(f*x+e)+567*c^2*d*sin(f*x+e)+468*sin(f
*x+e)*c*d^2+136*d^3*sin(f*x+e)+525*c^3+1134*c^2*d+936*c*d^2+272*d^3)/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.55 \[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3 \, dx=-\frac {2 \, {\left (35 \, a d^{3} \cos \left (f x + e\right )^{5} - 5 \, {\left (27 \, a c d^{2} + 10 \, a d^{3}\right )} \cos \left (f x + e\right )^{4} + 420 \, a c^{3} + 756 \, a c^{2} d + 684 \, a c d^{2} + 188 \, a d^{3} - {\left (189 \, a c^{2} d + 351 \, a c d^{2} + 172 \, a d^{3}\right )} \cos \left (f x + e\right )^{3} + {\left (105 \, a c^{3} + 378 \, a c^{2} d + 387 \, a c d^{2} + 134 \, a d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (525 \, a c^{3} + 1323 \, a c^{2} d + 1287 \, a c d^{2} + 409 \, a d^{3}\right )} \cos \left (f x + e\right ) - {\left (35 \, a d^{3} \cos \left (f x + e\right )^{4} + 420 \, a c^{3} + 756 \, a c^{2} d + 684 \, a c d^{2} + 188 \, a d^{3} + 5 \, {\left (27 \, a c d^{2} + 17 \, a d^{3}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (63 \, a c^{2} d + 72 \, a c d^{2} + 29 \, a d^{3}\right )} \cos \left (f x + e\right )^{2} - {\left (105 \, a c^{3} + 567 \, a c^{2} d + 603 \, a c d^{2} + 221 \, a d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{315 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \]

[In]

integrate((a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

-2/315*(35*a*d^3*cos(f*x + e)^5 - 5*(27*a*c*d^2 + 10*a*d^3)*cos(f*x + e)^4 + 420*a*c^3 + 756*a*c^2*d + 684*a*c
*d^2 + 188*a*d^3 - (189*a*c^2*d + 351*a*c*d^2 + 172*a*d^3)*cos(f*x + e)^3 + (105*a*c^3 + 378*a*c^2*d + 387*a*c
*d^2 + 134*a*d^3)*cos(f*x + e)^2 + (525*a*c^3 + 1323*a*c^2*d + 1287*a*c*d^2 + 409*a*d^3)*cos(f*x + e) - (35*a*
d^3*cos(f*x + e)^4 + 420*a*c^3 + 756*a*c^2*d + 684*a*c*d^2 + 188*a*d^3 + 5*(27*a*c*d^2 + 17*a*d^3)*cos(f*x + e
)^3 - 3*(63*a*c^2*d + 72*a*c*d^2 + 29*a*d^3)*cos(f*x + e)^2 - (105*a*c^3 + 567*a*c^2*d + 603*a*c*d^2 + 221*a*d
^3)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x + e) + f*sin(f*x + e) + f)

Sympy [F]

\[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3 \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \left (c + d \sin {\left (e + f x \right )}\right )^{3}\, dx \]

[In]

integrate((a+a*sin(f*x+e))**(3/2)*(c+d*sin(f*x+e))**3,x)

[Out]

Integral((a*(sin(e + f*x) + 1))**(3/2)*(c + d*sin(e + f*x))**3, x)

Maxima [F]

\[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3 \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{3} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^(3/2)*(d*sin(f*x + e) + c)^3, x)

Giac [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.62 \[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3 \, dx=\frac {\sqrt {2} {\left (35 \, a d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, f x + \frac {9}{2} \, e\right ) + 1890 \, {\left (4 \, a c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 8 \, a c^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 7 \, a c d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 2 \, a d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 210 \, {\left (4 \, a c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 18 \, a c^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 15 \, a c d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 5 \, a d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) + 378 \, {\left (2 \, a c^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, a c d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + a d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) + 135 \, {\left (2 \, a c d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + a d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right )\right )} \sqrt {a}}{2520 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^(3/2)*(c+d*sin(f*x+e))^3,x, algorithm="giac")

[Out]

1/2520*sqrt(2)*(35*a*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-9/4*pi + 9/2*f*x + 9/2*e) + 1890*(4*a*c^3*sg
n(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 8*a*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 7*a*c*d^2*sgn(cos(-1/4*pi
+ 1/2*f*x + 1/2*e)) + 2*a*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-1/4*pi + 1/2*f*x + 1/2*e) + 210*(4*a*c
^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 18*a*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 15*a*c*d^2*sgn(cos(-
1/4*pi + 1/2*f*x + 1/2*e)) + 5*a*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-3/4*pi + 3/2*f*x + 3/2*e) + 378
*(2*a*c^2*d*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 3*a*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + a*d^3*sgn(co
s(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-5/4*pi + 5/2*f*x + 5/2*e) + 135*(2*a*c*d^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2
*e)) + a*d^3*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sin(-7/4*pi + 7/2*f*x + 7/2*e))*sqrt(a)/f

Mupad [F(-1)]

Timed out. \[ \int (3+3 \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3 \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3 \,d x \]

[In]

int((a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^3,x)

[Out]

int((a + a*sin(e + f*x))^(3/2)*(c + d*sin(e + f*x))^3, x)