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

Optimal. Leaf size=427 \[ \frac{2 a^2 (A-B) (c-d (4 n+5)) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d}\right )}{d f (2 n+3) \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 (A-B) \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 B \left (3 c^2-2 c d (4 n+7)+d^2 \left (16 n^2+56 n+43\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d}\right )}{d^2 f (2 n+3) (2 n+5) \sqrt{a \sin (e+f x)+a}}+\frac{2 a^2 B (3 c-d (4 n+11)) \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d^2 f (2 n+3) (2 n+5) \sqrt{a \sin (e+f x)+a}}-\frac{2 a B \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{n+1}}{d f (2 n+5)} \]

[Out]

(-2*a^2*(A - B)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]]) + (2*a^2*B
*(3*c - d*(11 + 4*n))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d^2*f*(3 + 2*n)*(5 + 2*n)*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])^(1 + n))/(d*f*(5 + 2*n)) + (2*a^2*
(A - B)*(c - d*(5 + 4*n))*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, (d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*
Sin[e + f*x])^n)/(d*f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n) - (2*a^2*B*(3*c^2 -
2*c*d*(7 + 4*n) + d^2*(43 + 56*n + 16*n^2))*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, (d*(1 - Sin[e + f*x]
))/(c + d)]*(c + d*Sin[e + f*x])^n)/(d^2*f*(3 + 2*n)*(5 + 2*n)*Sqrt[a + a*Sin[e + f*x]]*((c + d*Sin[e + f*x])/
(c + d))^n)

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Rubi [A]  time = 0.914932, antiderivative size = 427, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.189, Rules used = {2987, 2763, 21, 2776, 70, 69, 2981} \[ \frac{2 a^2 (A-B) (c-d (4 n+5)) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d}\right )}{d f (2 n+3) \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 (A-B) \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d f (2 n+3) \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 B \left (3 c^2-2 c d (4 n+7)+d^2 \left (16 n^2+56 n+43\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d}\right )}{d^2 f (2 n+3) (2 n+5) \sqrt{a \sin (e+f x)+a}}+\frac{2 a^2 B (3 c-d (4 n+11)) \cos (e+f x) (c+d \sin (e+f x))^{n+1}}{d^2 f (2 n+3) (2 n+5) \sqrt{a \sin (e+f x)+a}}-\frac{2 a B \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{n+1}}{d f (2 n+5)} \]

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])^n,x]

[Out]

(-2*a^2*(A - B)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d*f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]]) + (2*a^2*B
*(3*c - d*(11 + 4*n))*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1 + n))/(d^2*f*(3 + 2*n)*(5 + 2*n)*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])^(1 + n))/(d*f*(5 + 2*n)) + (2*a^2*
(A - B)*(c - d*(5 + 4*n))*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, (d*(1 - Sin[e + f*x]))/(c + d)]*(c + d*
Sin[e + f*x])^n)/(d*f*(3 + 2*n)*Sqrt[a + a*Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^n) - (2*a^2*B*(3*c^2 -
2*c*d*(7 + 4*n) + d^2*(43 + 56*n + 16*n^2))*Cos[e + f*x]*Hypergeometric2F1[1/2, -n, 3/2, (d*(1 - Sin[e + f*x]
))/(c + d)]*(c + d*Sin[e + f*x])^n)/(d^2*f*(3 + 2*n)*(5 + 2*n)*Sqrt[a + a*Sin[e + f*x]]*((c + d*Sin[e + f*x])/
(c + d))^n)

Rule 2987

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

Rule 2763

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[(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 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 2776

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist
[(a^2*Cos[e + f*x])/(f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]]), Subst[Int[(c + d*x)^n/Sqrt[a - b*x]
, x], x, Sin[e + f*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] &&  !IntegerQ[2*n]

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*Simp[(b*c)/(b*c - a*d) + (b*d*x)/(b*c -
a*d), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 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]

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))^n \, dx &=(A-B) \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^n \, dx+\frac{B \int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^n \, dx}{a}\\ &=-\frac{2 a^2 (A-B) \cos (e+f x) (c+d \sin (e+f x))^{1+n}}{d f (3+2 n) \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))^{1+n}}{d f (5+2 n)}+\frac{(2 (A-B)) \int \frac{(c+d \sin (e+f x))^n \left (-\frac{1}{2} a^2 (c-5 d-4 d n)-\frac{1}{2} a^2 (c-5 d-4 d n) \sin (e+f x)\right )}{\sqrt{a+a \sin (e+f x)}} \, dx}{d (3+2 n)}+\frac{(2 B) \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^n \left (\frac{1}{2} a^2 (c+d (7+4 n))-\frac{1}{2} a^2 (3 c-11 d-4 d n) \sin (e+f x)\right ) \, dx}{a d (5+2 n)}\\ &=-\frac{2 a^2 (A-B) \cos (e+f x) (c+d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt{a+a \sin (e+f x)}}+\frac{2 a^2 B (3 c-d (11+4 n)) \cos (e+f x) (c+d \sin (e+f x))^{1+n}}{d^2 f (3+2 n) (5+2 n) \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))^{1+n}}{d f (5+2 n)}-\frac{(a (A-B) (c-d (5+4 n))) \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^n \, dx}{d (3+2 n)}+\frac{\left (a B \left (3 c^2-2 c d (7+4 n)+d^2 \left (43+56 n+16 n^2\right )\right )\right ) \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^n \, dx}{d^2 (3+2 n) (5+2 n)}\\ &=-\frac{2 a^2 (A-B) \cos (e+f x) (c+d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt{a+a \sin (e+f x)}}+\frac{2 a^2 B (3 c-d (11+4 n)) \cos (e+f x) (c+d \sin (e+f x))^{1+n}}{d^2 f (3+2 n) (5+2 n) \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))^{1+n}}{d f (5+2 n)}-\frac{\left (a^3 (A-B) (c-d (5+4 n)) \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(c+d x)^n}{\sqrt{a-a x}} \, dx,x,\sin (e+f x)\right )}{d f (3+2 n) \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}+\frac{\left (a^3 B \left (3 c^2-2 c d (7+4 n)+d^2 \left (43+56 n+16 n^2\right )\right ) \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(c+d x)^n}{\sqrt{a-a x}} \, dx,x,\sin (e+f x)\right )}{d^2 f (3+2 n) (5+2 n) \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\\ &=-\frac{2 a^2 (A-B) \cos (e+f x) (c+d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt{a+a \sin (e+f x)}}+\frac{2 a^2 B (3 c-d (11+4 n)) \cos (e+f x) (c+d \sin (e+f x))^{1+n}}{d^2 f (3+2 n) (5+2 n) \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))^{1+n}}{d f (5+2 n)}-\frac{\left (a^3 (A-B) (c-d (5+4 n)) \cos (e+f x) (c+d \sin (e+f x))^n \left (-\frac{a (c+d \sin (e+f x))}{-a c-a d}\right )^{-n}\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{c}{c+d}+\frac{d x}{c+d}\right )^n}{\sqrt{a-a x}} \, dx,x,\sin (e+f x)\right )}{d f (3+2 n) \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}+\frac{\left (a^3 B \left (3 c^2-2 c d (7+4 n)+d^2 \left (43+56 n+16 n^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^n \left (-\frac{a (c+d \sin (e+f x))}{-a c-a d}\right )^{-n}\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{c}{c+d}+\frac{d x}{c+d}\right )^n}{\sqrt{a-a x}} \, dx,x,\sin (e+f x)\right )}{d^2 f (3+2 n) (5+2 n) \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\\ &=-\frac{2 a^2 (A-B) \cos (e+f x) (c+d \sin (e+f x))^{1+n}}{d f (3+2 n) \sqrt{a+a \sin (e+f x)}}+\frac{2 a^2 B (3 c-d (11+4 n)) \cos (e+f x) (c+d \sin (e+f x))^{1+n}}{d^2 f (3+2 n) (5+2 n) \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))^{1+n}}{d f (5+2 n)}+\frac{2 a^2 (A-B) (c-d (5+4 n)) \cos (e+f x) \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d}\right ) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n}}{d f (3+2 n) \sqrt{a+a \sin (e+f x)}}-\frac{2 a^2 B \left (3 c^2-2 c d (7+4 n)+d^2 \left (43+56 n+16 n^2\right )\right ) \cos (e+f x) \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d}\right ) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n}}{d^2 f (3+2 n) (5+2 n) \sqrt{a+a \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 26.581, size = 245, normalized size = 0.57 \[ -\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} \left (-30 (A+B) (c-d (4 n+5)) \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};-\frac{d (\sin (e+f x)-1)}{c+d}\right )+30 (A+B) (c+d) \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{n+1}+20 B d (2 n+3) (\sin (e+f x)-1) \, _2F_1\left (\frac{3}{2},-n;\frac{5}{2};-\frac{d (\sin (e+f x)-1)}{c+d}\right )+6 B d (2 n+3) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^4 \, _2F_1\left (\frac{5}{2},-n;\frac{7}{2};-\frac{d (\sin (e+f x)-1)}{c+d}\right )\right )}{15 d f (2 n+3) \sqrt{a (\sin (e+f x)+1)}} \]

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])^n,x]

[Out]

-(a^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^n*(-30*(A + B)*(c - d*(5 + 4*n))*Hypergeometric2F1[1/2, -n, 3/2, -((d*
(-1 + Sin[e + f*x]))/(c + d))] + 6*B*d*(3 + 2*n)*Hypergeometric2F1[5/2, -n, 7/2, -((d*(-1 + Sin[e + f*x]))/(c
+ d))]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4 + 20*B*d*(3 + 2*n)*Hypergeometric2F1[3/2, -n, 5/2, -((d*(-1 + S
in[e + f*x]))/(c + d))]*(-1 + Sin[e + f*x]) + 30*(A + B)*(c + d)*((c + d*Sin[e + f*x])/(c + d))^(1 + n)))/(15*
d*f*(3 + 2*n)*Sqrt[a*(1 + Sin[e + f*x])]*((c + d*Sin[e + f*x])/(c + d))^n)

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Maple [F]  time = 0.431, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sin \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}} \left ( A+B\sin \left ( fx+e \right ) \right ) \left ( c+d\sin \left ( fx+e \right ) \right ) ^{n}\, dx \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))^n,x)

[Out]

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

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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 )}^{n}\,{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))^n,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)^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (B a \cos \left (f x + e\right )^{2} -{\left (A + B\right )} a \sin \left (f x + e\right ) -{\left (A + B\right )} a\right )} \sqrt{a \sin \left (f x + e\right ) + a}{\left (d \sin \left (f x + e\right ) + c\right )}^{n}, x\right ) \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))^n,x, algorithm="fricas")

[Out]

integral(-(B*a*cos(f*x + e)^2 - (A + B)*a*sin(f*x + e) - (A + B)*a)*sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) +
c)^n, x)

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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))**n,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))^n,x, algorithm="giac")

[Out]

Timed out