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

Optimal. Leaf size=294 \[ \frac{2 a^2 \left (15 c^2+10 c d+7 d^2\right ) \left (3 A d (c-13 d)-B \left (c^2-7 c d+34 d^2\right )\right ) \cos (e+f x)}{315 d^2 f \sqrt{a \sin (e+f x)+a}}+\frac{2 a^2 (-9 A d+3 B c-10 B d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 d^2 f \sqrt{a \sin (e+f x)+a}}+\frac{2 \left (3 A d (c-13 d)-B \left (c^2-7 c d+34 d^2\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{105 f}+\frac{4 a (5 c-d) \left (3 A d (c-13 d)-B \left (c^2-7 c d+34 d^2\right )\right ) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{315 d f}-\frac{2 a B \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f} \]

[Out]

(2*a^2*(15*c^2 + 10*c*d + 7*d^2)*(3*A*(c - 13*d)*d - B*(c^2 - 7*c*d + 34*d^2))*Cos[e + f*x])/(315*d^2*f*Sqrt[a
+ a*Sin[e + f*x]]) + (4*a*(5*c - d)*(3*A*(c - 13*d)*d - B*(c^2 - 7*c*d + 34*d^2))*Cos[e + f*x]*Sqrt[a + a*Sin
[e + f*x]])/(315*d*f) + (2*(3*A*(c - 13*d)*d - B*(c^2 - 7*c*d + 34*d^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/
2))/(105*f) + (2*a^2*(3*B*c - 9*A*d - 10*B*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(63*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])^3)/(9*d*f)

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Rubi [A]  time = 0.711904, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {2976, 2981, 2761, 2751, 2646} \[ \frac{2 a^2 \left (15 c^2+10 c d+7 d^2\right ) \left (3 A d (c-13 d)-B \left (c^2-7 c d+34 d^2\right )\right ) \cos (e+f x)}{315 d^2 f \sqrt{a \sin (e+f x)+a}}+\frac{2 a^2 (-9 A d+3 B c-10 B d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 d^2 f \sqrt{a \sin (e+f x)+a}}+\frac{2 \left (3 A d (c-13 d)-B \left (c^2-7 c d+34 d^2\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{105 f}+\frac{4 a (5 c-d) \left (3 A d (c-13 d)-B \left (c^2-7 c d+34 d^2\right )\right ) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{315 d f}-\frac{2 a B \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{9 d f} \]

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])^2,x]

[Out]

(2*a^2*(15*c^2 + 10*c*d + 7*d^2)*(3*A*(c - 13*d)*d - B*(c^2 - 7*c*d + 34*d^2))*Cos[e + f*x])/(315*d^2*f*Sqrt[a
+ a*Sin[e + f*x]]) + (4*a*(5*c - d)*(3*A*(c - 13*d)*d - B*(c^2 - 7*c*d + 34*d^2))*Cos[e + f*x]*Sqrt[a + a*Sin
[e + f*x]])/(315*d*f) + (2*(3*A*(c - 13*d)*d - B*(c^2 - 7*c*d + 34*d^2))*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/
2))/(105*f) + (2*a^2*(3*B*c - 9*A*d - 10*B*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(63*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])^3)/(9*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 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))^2 \, dx &=-\frac{2 a B \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^3}{9 d f}+\frac{2 \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \left (\frac{1}{2} a (9 A d+B (c+6 d))-\frac{1}{2} a (3 B c-9 A d-10 B d) \sin (e+f x)\right ) \, dx}{9 d}\\ &=\frac{2 a^2 (3 B c-9 A d-10 B d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 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))^3}{9 d f}-\frac{\left (a \left (3 A (c-13 d) d-B \left (c^2-7 c d+34 d^2\right )\right )\right ) \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \, dx}{21 d^2}\\ &=\frac{2 \left (3 A (c-13 d) d-B \left (c^2-7 c d+34 d^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{105 f}+\frac{2 a^2 (3 B c-9 A d-10 B d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 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))^3}{9 d f}-\frac{\left (2 \left (3 A (c-13 d) d-B \left (c^2-7 c d+34 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}{105 d^2}\\ &=\frac{4 a (5 c-d) \left (3 A (c-13 d) d-B \left (c^2-7 c d+34 d^2\right )\right ) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{315 d f}+\frac{2 \left (3 A (c-13 d) d-B \left (c^2-7 c d+34 d^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{105 f}+\frac{2 a^2 (3 B c-9 A d-10 B d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 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))^3}{9 d f}-\frac{\left (a \left (15 c^2+10 c d+7 d^2\right ) \left (3 A (c-13 d) d-B \left (c^2-7 c d+34 d^2\right )\right )\right ) \int \sqrt{a+a \sin (e+f x)} \, dx}{315 d^2}\\ &=\frac{2 a^2 \left (15 c^2+10 c d+7 d^2\right ) \left (3 A (c-13 d) d-B \left (c^2-7 c d+34 d^2\right )\right ) \cos (e+f x)}{315 d^2 f \sqrt{a+a \sin (e+f x)}}+\frac{4 a (5 c-d) \left (3 A (c-13 d) d-B \left (c^2-7 c d+34 d^2\right )\right ) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{315 d f}+\frac{2 \left (3 A (c-13 d) d-B \left (c^2-7 c d+34 d^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{105 f}+\frac{2 a^2 (3 B c-9 A d-10 B d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 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))^3}{9 d f}\\ \end{align*}

Mathematica [A]  time = 2.2475, size = 267, normalized size = 0.91 \[ -\frac{a \sqrt{a (\sin (e+f x)+1)} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (-4 \left (9 A d (14 c+13 d)+B \left (63 c^2+234 c d+137 d^2\right )\right ) \cos (2 (e+f x))+840 A c^2 \sin (e+f x)+4200 A c^2+3024 A c d \sin (e+f x)+6552 A c d+1518 A d^2 \sin (e+f x)-90 A d^2 \sin (3 (e+f x))+2964 A d^2+1512 B c^2 \sin (e+f x)+3276 B c^2+3036 B c d \sin (e+f x)-180 B c d \sin (3 (e+f x))+5928 B c d+1598 B d^2 \sin (e+f x)-170 B d^2 \sin (3 (e+f x))+35 B d^2 \cos (4 (e+f x))+2689 B d^2\right )}{1260 f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]

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])^2,x]

[Out]

-(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(4200*A*c^2 + 3276*B*c^2 + 6552*A*c*d + 5
928*B*c*d + 2964*A*d^2 + 2689*B*d^2 - 4*(9*A*d*(14*c + 13*d) + B*(63*c^2 + 234*c*d + 137*d^2))*Cos[2*(e + f*x)
] + 35*B*d^2*Cos[4*(e + f*x)] + 840*A*c^2*Sin[e + f*x] + 1512*B*c^2*Sin[e + f*x] + 3024*A*c*d*Sin[e + f*x] + 3
036*B*c*d*Sin[e + f*x] + 1518*A*d^2*Sin[e + f*x] + 1598*B*d^2*Sin[e + f*x] - 180*B*c*d*Sin[3*(e + f*x)] - 90*A
*d^2*Sin[3*(e + f*x)] - 170*B*d^2*Sin[3*(e + f*x)]))/(1260*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))

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Maple [A]  time = 1.083, size = 207, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( fx+e \right ) \right ){a}^{2} \left ( -1+\sin \left ( fx+e \right ) \right ) \left ( \left ( -45\,A{d}^{2}-90\,Bcd-85\,B{d}^{2} \right ) \sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+ \left ( 105\,A{c}^{2}+378\,Acd+201\,A{d}^{2}+189\,B{c}^{2}+402\,Bcd+221\,B{d}^{2} \right ) \sin \left ( fx+e \right ) +35\,B \left ( \cos \left ( fx+e \right ) \right ) ^{4}{d}^{2}+ \left ( -126\,Acd-117\,A{d}^{2}-63\,B{c}^{2}-234\,Bcd-172\,B{d}^{2} \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+525\,A{c}^{2}+882\,Acd+429\,A{d}^{2}+441\,B{c}^{2}+858\,Bcd+409\,B{d}^{2} \right ) }{315\,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))^2,x)

[Out]

2/315*(1+sin(f*x+e))*a^2*(-1+sin(f*x+e))*((-45*A*d^2-90*B*c*d-85*B*d^2)*sin(f*x+e)*cos(f*x+e)^2+(105*A*c^2+378
*A*c*d+201*A*d^2+189*B*c^2+402*B*c*d+221*B*d^2)*sin(f*x+e)+35*B*cos(f*x+e)^4*d^2+(-126*A*c*d-117*A*d^2-63*B*c^
2-234*B*c*d-172*B*d^2)*cos(f*x+e)^2+525*A*c^2+882*A*c*d+429*A*d^2+441*B*c^2+858*B*c*d+409*B*d^2)/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 )}^{2}\,{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))^2,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)^2, x)

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Fricas [A]  time = 1.84799, size = 1094, normalized size = 3.72 \begin{align*} -\frac{2 \,{\left (35 \, B a d^{2} \cos \left (f x + e\right )^{5} - 5 \,{\left (18 \, B a c d +{\left (9 \, A + 10 \, B\right )} a d^{2}\right )} \cos \left (f x + e\right )^{4} + 84 \,{\left (5 \, A + 3 \, B\right )} a c^{2} + 24 \,{\left (21 \, A + 19 \, B\right )} a c d + 4 \,{\left (57 \, A + 47 \, B\right )} a d^{2} -{\left (63 \, B a c^{2} + 18 \,{\left (7 \, A + 13 \, B\right )} a c d +{\left (117 \, A + 172 \, B\right )} a d^{2}\right )} \cos \left (f x + e\right )^{3} +{\left (21 \,{\left (5 \, A + 6 \, B\right )} a c^{2} + 6 \,{\left (42 \, A + 43 \, B\right )} a c d +{\left (129 \, A + 134 \, B\right )} a d^{2}\right )} \cos \left (f x + e\right )^{2} +{\left (21 \,{\left (25 \, A + 21 \, B\right )} a c^{2} + 6 \,{\left (147 \, A + 143 \, B\right )} a c d +{\left (429 \, A + 409 \, B\right )} a d^{2}\right )} \cos \left (f x + e\right ) -{\left (35 \, B a d^{2} \cos \left (f x + e\right )^{4} + 84 \,{\left (5 \, A + 3 \, B\right )} a c^{2} + 24 \,{\left (21 \, A + 19 \, B\right )} a c d + 4 \,{\left (57 \, A + 47 \, B\right )} a d^{2} + 5 \,{\left (18 \, B a c d +{\left (9 \, A + 17 \, B\right )} a d^{2}\right )} \cos \left (f x + e\right )^{3} - 3 \,{\left (21 \, B a c^{2} + 6 \,{\left (7 \, A + 8 \, B\right )} a c d +{\left (24 \, A + 29 \, B\right )} a d^{2}\right )} \cos \left (f x + e\right )^{2} -{\left (21 \,{\left (5 \, A + 9 \, B\right )} a c^{2} + 6 \,{\left (63 \, A + 67 \, B\right )} a c d +{\left (201 \, A + 221 \, B\right )} a d^{2}\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 )}} \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))^2,x, algorithm="fricas")

[Out]

-2/315*(35*B*a*d^2*cos(f*x + e)^5 - 5*(18*B*a*c*d + (9*A + 10*B)*a*d^2)*cos(f*x + e)^4 + 84*(5*A + 3*B)*a*c^2
+ 24*(21*A + 19*B)*a*c*d + 4*(57*A + 47*B)*a*d^2 - (63*B*a*c^2 + 18*(7*A + 13*B)*a*c*d + (117*A + 172*B)*a*d^2
)*cos(f*x + e)^3 + (21*(5*A + 6*B)*a*c^2 + 6*(42*A + 43*B)*a*c*d + (129*A + 134*B)*a*d^2)*cos(f*x + e)^2 + (21
*(25*A + 21*B)*a*c^2 + 6*(147*A + 143*B)*a*c*d + (429*A + 409*B)*a*d^2)*cos(f*x + e) - (35*B*a*d^2*cos(f*x + e
)^4 + 84*(5*A + 3*B)*a*c^2 + 24*(21*A + 19*B)*a*c*d + 4*(57*A + 47*B)*a*d^2 + 5*(18*B*a*c*d + (9*A + 17*B)*a*d
^2)*cos(f*x + e)^3 - 3*(21*B*a*c^2 + 6*(7*A + 8*B)*a*c*d + (24*A + 29*B)*a*d^2)*cos(f*x + e)^2 - (21*(5*A + 9*
B)*a*c^2 + 6*(63*A + 67*B)*a*c*d + (201*A + 221*B)*a*d^2)*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(-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))**2,x)

[Out]

Timed out

________________________________________________________________________________________

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))^2,x, algorithm="giac")

[Out]

Timed out