3.293 \(\int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx\)

Optimal. Leaf size=374 \[ \frac{2 a^2 \left (11 A d (c-17 d)-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt{a \sin (e+f x)+a}}+\frac{4 a^2 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (11 A d (c-17 d)-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x)}{3465 d^2 f \sqrt{a \sin (e+f x)+a}}+\frac{2 a^2 (3 B (c-4 d)-11 A d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt{a \sin (e+f x)+a}}+\frac{4 (c+d) \left (11 A d (c-17 d)-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{1155 f}+\frac{8 a (5 c-d) (c+d) \left (11 A d (c-17 d)-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3465 d f}-\frac{2 a B \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{11 d f} \]

[Out]

(4*a^2*(c + d)*(15*c^2 + 10*c*d + 7*d^2)*(11*A*(c - 17*d)*d - 3*B*(c^2 - 9*c*d + 56*d^2))*Cos[e + f*x])/(3465*
d^2*f*Sqrt[a + a*Sin[e + f*x]]) + (8*a*(5*c - d)*(c + d)*(11*A*(c - 17*d)*d - 3*B*(c^2 - 9*c*d + 56*d^2))*Cos[
e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3465*d*f) + (4*(c + d)*(11*A*(c - 17*d)*d - 3*B*(c^2 - 9*c*d + 56*d^2))*Co
s[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(1155*f) + (2*a^2*(11*A*(c - 17*d)*d - 3*B*(c^2 - 9*c*d + 56*d^2))*Cos[
e + f*x]*(c + d*Sin[e + f*x])^3)/(693*d^2*f*Sqrt[a + a*Sin[e + f*x]]) + (2*a^2*(3*B*(c - 4*d) - 11*A*d)*Cos[e
+ f*x]*(c + d*Sin[e + f*x])^4)/(99*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*B*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*
x]]*(c + d*Sin[e + f*x])^4)/(11*d*f)

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Rubi [A]  time = 0.920242, antiderivative size = 374, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162, Rules used = {2976, 2981, 2770, 2761, 2751, 2646} \[ \frac{2 a^2 \left (11 A d (c-17 d)-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt{a \sin (e+f x)+a}}+\frac{4 a^2 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (11 A d (c-17 d)-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x)}{3465 d^2 f \sqrt{a \sin (e+f x)+a}}+\frac{2 a^2 (3 B (c-4 d)-11 A d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt{a \sin (e+f x)+a}}+\frac{4 (c+d) \left (11 A d (c-17 d)-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{1155 f}+\frac{8 a (5 c-d) (c+d) \left (11 A d (c-17 d)-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{3465 d f}-\frac{2 a B \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^4}{11 d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a^2*(c + d)*(15*c^2 + 10*c*d + 7*d^2)*(11*A*(c - 17*d)*d - 3*B*(c^2 - 9*c*d + 56*d^2))*Cos[e + f*x])/(3465*
d^2*f*Sqrt[a + a*Sin[e + f*x]]) + (8*a*(5*c - d)*(c + d)*(11*A*(c - 17*d)*d - 3*B*(c^2 - 9*c*d + 56*d^2))*Cos[
e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3465*d*f) + (4*(c + d)*(11*A*(c - 17*d)*d - 3*B*(c^2 - 9*c*d + 56*d^2))*Co
s[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(1155*f) + (2*a^2*(11*A*(c - 17*d)*d - 3*B*(c^2 - 9*c*d + 56*d^2))*Cos[
e + f*x]*(c + d*Sin[e + f*x])^3)/(693*d^2*f*Sqrt[a + a*Sin[e + f*x]]) + (2*a^2*(3*B*(c - 4*d) - 11*A*d)*Cos[e
+ f*x]*(c + d*Sin[e + f*x])^4)/(99*d^2*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*B*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*
x]]*(c + d*Sin[e + f*x])^4)/(11*d*f)

Rule 2976

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

Rule 2981

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2*b*B*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(2*n + 3)*Sqr
t[a + b*Sin[e + f*x]]), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b*d*(2*n + 3)), Int[Sqrt[a + b*
Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] &&
EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[n, -1]

Rule 2770

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]

Rule 2761

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 2751

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*Sin
[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 2646

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]

Rubi steps

\begin{align*} \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^3 \, dx &=-\frac{2 a B \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f}+\frac{2 \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^3 \left (\frac{1}{2} a (11 A d+B (c+8 d))-\frac{1}{2} a (3 B (c-4 d)-11 A d) \sin (e+f x)\right ) \, dx}{11 d}\\ &=\frac{2 a^2 (3 B (c-4 d)-11 A d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{2 a B \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f}-\frac{\left (a \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right )\right ) \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^3 \, dx}{99 d^2}\\ &=\frac{2 a^2 \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt{a+a \sin (e+f x)}}+\frac{2 a^2 (3 B (c-4 d)-11 A d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{2 a B \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f}-\frac{\left (2 a (c+d) \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right )\right ) \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \, dx}{231 d^2}\\ &=\frac{4 (c+d) \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 f}+\frac{2 a^2 \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt{a+a \sin (e+f x)}}+\frac{2 a^2 (3 B (c-4 d)-11 A d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{2 a B \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f}-\frac{\left (4 (c+d) \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right )\right ) \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}{1155 d^2}\\ &=\frac{8 a (5 c-d) (c+d) \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{3465 d f}+\frac{4 (c+d) \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 f}+\frac{2 a^2 \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt{a+a \sin (e+f x)}}+\frac{2 a^2 (3 B (c-4 d)-11 A d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{2 a B \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f}-\frac{\left (2 a (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right )\right ) \int \sqrt{a+a \sin (e+f x)} \, dx}{3465 d^2}\\ &=\frac{4 a^2 (c+d) \left (15 c^2+10 c d+7 d^2\right ) \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x)}{3465 d^2 f \sqrt{a+a \sin (e+f x)}}+\frac{8 a (5 c-d) (c+d) \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{3465 d f}+\frac{4 (c+d) \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 f}+\frac{2 a^2 \left (11 A (c-17 d) d-3 B \left (c^2-9 c d+56 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^2 f \sqrt{a+a \sin (e+f x)}}+\frac{2 a^2 (3 B (c-4 d)-11 A d) \cos (e+f x) (c+d \sin (e+f x))^4}{99 d^2 f \sqrt{a+a \sin (e+f x)}}-\frac{2 a B \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^4}{11 d f}\\ \end{align*}

Mathematica [A]  time = 4.59295, size = 390, normalized size = 1.04 \[ -\frac{a \sqrt{a (\sin (e+f x)+1)} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (-8 \left (11 A d \left (189 c^2+351 c d+137 d^2\right )+3 B \left (1287 c^2 d+231 c^3+1507 c d^2+581 d^3\right )\right ) \cos (2 (e+f x))+70 d^2 (11 A d+33 B c+21 B d) \cos (4 (e+f x))+99792 A c^2 d \sin (e+f x)+216216 A c^2 d+18480 A c^3 \sin (e+f x)+92400 A c^3+100188 A c d^2 \sin (e+f x)-5940 A c d^2 \sin (3 (e+f x))+195624 A c d^2+35156 A d^3 \sin (e+f x)-3740 A d^3 \sin (3 (e+f x))+59158 A d^3+100188 B c^2 d \sin (e+f x)-5940 B c^2 d \sin (3 (e+f x))+195624 B c^2 d+33264 B c^3 \sin (e+f x)+72072 B c^3+105468 B c d^2 \sin (e+f x)-11220 B c d^2 \sin (3 (e+f x))+177474 B c d^2+34734 B d^3 \sin (e+f x)-4935 B d^3 \sin (3 (e+f x))+315 B d^3 \sin (5 (e+f x))+55482 B d^3\right )}{27720 f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(92400*A*c^3 + 72072*B*c^3 + 216216*A*c^2
*d + 195624*B*c^2*d + 195624*A*c*d^2 + 177474*B*c*d^2 + 59158*A*d^3 + 55482*B*d^3 - 8*(11*A*d*(189*c^2 + 351*c
*d + 137*d^2) + 3*B*(231*c^3 + 1287*c^2*d + 1507*c*d^2 + 581*d^3))*Cos[2*(e + f*x)] + 70*d^2*(33*B*c + 11*A*d
+ 21*B*d)*Cos[4*(e + f*x)] + 18480*A*c^3*Sin[e + f*x] + 33264*B*c^3*Sin[e + f*x] + 99792*A*c^2*d*Sin[e + f*x]
+ 100188*B*c^2*d*Sin[e + f*x] + 100188*A*c*d^2*Sin[e + f*x] + 105468*B*c*d^2*Sin[e + f*x] + 35156*A*d^3*Sin[e
+ f*x] + 34734*B*d^3*Sin[e + f*x] - 5940*B*c^2*d*Sin[3*(e + f*x)] - 5940*A*c*d^2*Sin[3*(e + f*x)] - 11220*B*c*
d^2*Sin[3*(e + f*x)] - 3740*A*d^3*Sin[3*(e + f*x)] - 4935*B*d^3*Sin[3*(e + f*x)] + 315*B*d^3*Sin[5*(e + f*x)])
)/(27720*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))

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Maple [A]  time = 1.015, size = 312, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( fx+e \right ) \right ){a}^{2} \left ( -1+\sin \left ( fx+e \right ) \right ) \left ( 315\,B \left ( \cos \left ( fx+e \right ) \right ) ^{4}\sin \left ( fx+e \right ){d}^{3}+ \left ( -1485\,Ac{d}^{2}-935\,A{d}^{3}-1485\,B{c}^{2}d-2805\,Bc{d}^{2}-1470\,B{d}^{3} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) + \left ( 1155\,A{c}^{3}+6237\,A{c}^{2}d+6633\,Ac{d}^{2}+2431\,A{d}^{3}+2079\,B{c}^{3}+6633\,B{c}^{2}d+7293\,Bc{d}^{2}+2499\,B{d}^{3} \right ) \sin \left ( fx+e \right ) + \left ( 385\,A{d}^{3}+1155\,Bc{d}^{2}+735\,B{d}^{3} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( -2079\,A{c}^{2}d-3861\,Ac{d}^{2}-1892\,A{d}^{3}-693\,B{c}^{3}-3861\,B{c}^{2}d-5676\,Bc{d}^{2}-2478\,B{d}^{3} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+5775\,A{c}^{3}+14553\,A{c}^{2}d+14157\,Ac{d}^{2}+4499\,A{d}^{3}+4851\,B{c}^{3}+14157\,B{c}^{2}d+13497\,Bc{d}^{2}+4431\,B{d}^{3} \right ) }{3465\,f\cos \left ( fx+e \right ) }{\frac{1}{\sqrt{a+a\sin \left ( fx+e \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(1+sin(f*x+e))*a^2*(-1+sin(f*x+e))*(315*B*cos(f*x+e)^4*sin(f*x+e)*d^3+(-1485*A*c*d^2-935*A*d^3-1485*B*c
^2*d-2805*B*c*d^2-1470*B*d^3)*cos(f*x+e)^2*sin(f*x+e)+(1155*A*c^3+6237*A*c^2*d+6633*A*c*d^2+2431*A*d^3+2079*B*
c^3+6633*B*c^2*d+7293*B*c*d^2+2499*B*d^3)*sin(f*x+e)+(385*A*d^3+1155*B*c*d^2+735*B*d^3)*cos(f*x+e)^4+(-2079*A*
c^2*d-3861*A*c*d^2-1892*A*d^3-693*B*c^3-3861*B*c^2*d-5676*B*c*d^2-2478*B*d^3)*cos(f*x+e)^2+5775*A*c^3+14553*A*
c^2*d+14157*A*c*d^2+4499*A*d^3+4851*B*c^3+14157*B*c^2*d+13497*B*c*d^2+4431*B*d^3)/cos(f*x+e)/(a+a*sin(f*x+e))^
(1/2)/f

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \sin \left (f x + e\right ) + A\right )}{\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.98949, size = 1667, normalized size = 4.46 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3465*(315*B*a*d^3*cos(f*x + e)^6 + 35*(33*B*a*c*d^2 + (11*A + 21*B)*a*d^3)*cos(f*x + e)^5 + 924*(5*A + 3*B)
*a*c^3 + 396*(21*A + 19*B)*a*c^2*d + 132*(57*A + 47*B)*a*c*d^2 + 4*(517*A + 483*B)*a*d^3 - 5*(297*B*a*c^2*d +
33*(9*A + 10*B)*a*c*d^2 + 10*(11*A + 21*B)*a*d^3)*cos(f*x + e)^4 - (693*B*a*c^3 + 297*(7*A + 13*B)*a*c^2*d + 3
3*(117*A + 172*B)*a*c*d^2 + 2*(946*A + 1239*B)*a*d^3)*cos(f*x + e)^3 + (231*(5*A + 6*B)*a*c^3 + 99*(42*A + 43*
B)*a*c^2*d + 33*(129*A + 134*B)*a*c*d^2 + (1474*A + 1491*B)*a*d^3)*cos(f*x + e)^2 + (231*(25*A + 21*B)*a*c^3 +
99*(147*A + 143*B)*a*c^2*d + 33*(429*A + 409*B)*a*c*d^2 + (4499*A + 4431*B)*a*d^3)*cos(f*x + e) + (315*B*a*d^
3*cos(f*x + e)^5 - 924*(5*A + 3*B)*a*c^3 - 396*(21*A + 19*B)*a*c^2*d - 132*(57*A + 47*B)*a*c*d^2 - 4*(517*A +
483*B)*a*d^3 - 35*(33*B*a*c*d^2 + (11*A + 12*B)*a*d^3)*cos(f*x + e)^4 - 5*(297*B*a*c^2*d + 33*(9*A + 17*B)*a*c
*d^2 + (187*A + 294*B)*a*d^3)*cos(f*x + e)^3 + 3*(231*B*a*c^3 + 99*(7*A + 8*B)*a*c^2*d + 33*(24*A + 29*B)*a*c*
d^2 + (319*A + 336*B)*a*d^3)*cos(f*x + e)^2 + (231*(5*A + 9*B)*a*c^3 + 99*(63*A + 67*B)*a*c^2*d + 33*(201*A +
221*B)*a*c*d^2 + 17*(143*A + 147*B)*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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out