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

Optimal. Leaf size=231 \[ \frac{4 a^2 (c-17 d) (c+d) \left (15 c^2+10 c d+7 d^2\right ) \cos (e+f x)}{315 d f \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt{a \sin (e+f x)+a}}+\frac{2 a^2 (c-17 d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 d f \sqrt{a \sin (e+f x)+a}}+\frac{4 d (c-17 d) (c+d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{105 f}+\frac{8 a (c-17 d) (5 c-d) (c+d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{315 f} \]

[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]])

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Rubi [A]  time = 0.384428, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2763, 21, 2770, 2761, 2751, 2646} \[ \frac{4 a^2 (c-17 d) (c+d) \left (15 c^2+10 c d+7 d^2\right ) \cos (e+f x)}{315 d f \sqrt{a \sin (e+f x)+a}}-\frac{2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt{a \sin (e+f x)+a}}+\frac{2 a^2 (c-17 d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 d f \sqrt{a \sin (e+f x)+a}}+\frac{4 d (c-17 d) (c+d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{105 f}+\frac{8 a (c-17 d) (5 c-d) (c+d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{315 f} \]

Antiderivative was successfully verified.

[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 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 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} (c+d \sin (e+f x))^3 \, dx &=-\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]  time = 1.67798, size = 203, normalized size = 0.88 \[ -\frac{a \sqrt{a (\sin (e+f x)+1)} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (-4 d \left (189 c^2+351 c d+137 d^2\right ) \cos (2 (e+f x))+4536 c^2 d \sin (e+f x)+9828 c^2 d+840 c^3 \sin (e+f x)+4200 c^3+4554 c d^2 \sin (e+f x)-270 c d^2 \sin (3 (e+f x))+8892 c d^2+1598 d^3 \sin (e+f x)-170 d^3 \sin (3 (e+f x))+35 d^3 \cos (4 (e+f x))+2689 d^3\right )}{1260 f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^(3/2)*(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])]*(4200*c^3 + 9828*c^2*d + 8892*c*d^2 + 268
9*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)]))/(1260*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))

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Maple [A]  time = 0.596, size = 195, 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 ( 35\,{d}^{3} \left ( \sin \left ( fx+e \right ) \right ) ^{4}+135\,c{d}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{3}+85\,{d}^{3} \left ( \sin \left ( fx+e \right ) \right ) ^{3}+189\,{c}^{2}d \left ( \sin \left ( fx+e \right ) \right ) ^{2}+351\,c{d}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{2}+102\,{d}^{3} \left ( \sin \left ( fx+e \right ) \right ) ^{2}+105\,{c}^{3}\sin \left ( fx+e \right ) +567\,{c}^{2}d\sin \left ( fx+e \right ) +468\,\sin \left ( fx+e \right ){d}^{2}c+136\,{d}^{3}\sin \left ( fx+e \right ) +525\,{c}^{3}+1134\,{c}^{2}d+936\,c{d}^{2}+272\,{d}^{3} \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)*(c+d*sin(f*x+e))^3,x)

[Out]

2/315*(1+sin(f*x+e))*a^2*(-1+sin(f*x+e))*(35*d^3*sin(f*x+e)^4+135*c*d^2*sin(f*x+e)^3+85*d^3*sin(f*x+e)^3+189*c
^2*d*sin(f*x+e)^2+351*c*d^2*sin(f*x+e)^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)*d^2*c+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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.73393, size = 872, normalized size = 3.77 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

Timed out