3.537 \(\int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x))^2 \, dx\)

Optimal. Leaf size=202 \[ -\frac{16 a^2 \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{315 f}-\frac{64 a^3 \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x)}{315 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{105 f}-\frac{4 d (9 c-d) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{63 f}-\frac{2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{9 a f} \]

[Out]

(-64*a^3*(21*c^2 + 30*c*d + 13*d^2)*Cos[e + f*x])/(315*f*Sqrt[a + a*Sin[e + f*x]]) - (16*a^2*(21*c^2 + 30*c*d
+ 13*d^2)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(315*f) - (2*a*(21*c^2 + 30*c*d + 13*d^2)*Cos[e + f*x]*(a + a
*Sin[e + f*x])^(3/2))/(105*f) - (4*(9*c - d)*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(63*f) - (2*d^2*Cos[e
+ f*x]*(a + a*Sin[e + f*x])^(7/2))/(9*a*f)

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Rubi [A]  time = 0.273899, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {2761, 2751, 2647, 2646} \[ -\frac{16 a^2 \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{315 f}-\frac{64 a^3 \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x)}{315 f \sqrt{a \sin (e+f x)+a}}-\frac{2 a \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{105 f}-\frac{4 d (9 c-d) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{63 f}-\frac{2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{9 a f} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^2,x]

[Out]

(-64*a^3*(21*c^2 + 30*c*d + 13*d^2)*Cos[e + f*x])/(315*f*Sqrt[a + a*Sin[e + f*x]]) - (16*a^2*(21*c^2 + 30*c*d
+ 13*d^2)*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(315*f) - (2*a*(21*c^2 + 30*c*d + 13*d^2)*Cos[e + f*x]*(a + a
*Sin[e + f*x])^(3/2))/(105*f) - (4*(9*c - d)*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(63*f) - (2*d^2*Cos[e
+ f*x]*(a + a*Sin[e + f*x])^(7/2))/(9*a*f)

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 2647

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n -
1))/(d*n), x] + Dist[(a*(2*n - 1))/n, Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && Eq
Q[a^2 - b^2, 0] && IGtQ[n - 1/2, 0]

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))^{5/2} (c+d \sin (e+f x))^2 \, dx &=-\frac{2 d^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{9 a f}+\frac{2 \int (a+a \sin (e+f x))^{5/2} \left (\frac{1}{2} a \left (9 c^2+7 d^2\right )+a (9 c-d) d \sin (e+f x)\right ) \, dx}{9 a}\\ &=-\frac{4 (9 c-d) d \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{63 f}-\frac{2 d^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{9 a f}+\frac{1}{21} \left (21 c^2+30 c d+13 d^2\right ) \int (a+a \sin (e+f x))^{5/2} \, dx\\ &=-\frac{2 a \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{105 f}-\frac{4 (9 c-d) d \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{63 f}-\frac{2 d^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{9 a f}+\frac{1}{105} \left (8 a \left (21 c^2+30 c d+13 d^2\right )\right ) \int (a+a \sin (e+f x))^{3/2} \, dx\\ &=-\frac{16 a^2 \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{315 f}-\frac{2 a \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{105 f}-\frac{4 (9 c-d) d \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{63 f}-\frac{2 d^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{9 a f}+\frac{1}{315} \left (32 a^2 \left (21 c^2+30 c d+13 d^2\right )\right ) \int \sqrt{a+a \sin (e+f x)} \, dx\\ &=-\frac{64 a^3 \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x)}{315 f \sqrt{a+a \sin (e+f x)}}-\frac{16 a^2 \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{315 f}-\frac{2 a \left (21 c^2+30 c d+13 d^2\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{105 f}-\frac{4 (9 c-d) d \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{63 f}-\frac{2 d^2 \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{9 a f}\\ \end{align*}

Mathematica [A]  time = 3.35411, size = 180, normalized size = 0.89 \[ -\frac{a^2 \sqrt{a (\sin (e+f x)+1)} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (-4 \left (63 c^2+360 c d+254 d^2\right ) \cos (2 (e+f x))+2352 c^2 \sin (e+f x)+7476 c^2+6060 c d \sin (e+f x)-180 c d \sin (3 (e+f x))+12480 c d+3116 d^2 \sin (e+f x)-260 d^2 \sin (3 (e+f x))+35 d^2 \cos (4 (e+f x))+5653 d^2\right )}{1260 f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^(5/2)*(c + d*Sin[e + f*x])^2,x]

[Out]

-(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]*(7476*c^2 + 12480*c*d + 5653*d^2 - 4*(6
3*c^2 + 360*c*d + 254*d^2)*Cos[2*(e + f*x)] + 35*d^2*Cos[4*(e + f*x)] + 2352*c^2*Sin[e + f*x] + 6060*c*d*Sin[e
+ f*x] + 3116*d^2*Sin[e + f*x] - 180*c*d*Sin[3*(e + f*x)] - 260*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 = 0.638, size = 168, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( fx+e \right ) \right ){a}^{3} \left ( -1+\sin \left ( fx+e \right ) \right ) \left ( 35\,{d}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{4}+90\,cd \left ( \sin \left ( fx+e \right ) \right ) ^{3}+130\,{d}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{3}+63\,{c}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{2}+360\,cd \left ( \sin \left ( fx+e \right ) \right ) ^{2}+219\,{d}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{2}+294\,{c}^{2}\sin \left ( fx+e \right ) +690\,\sin \left ( fx+e \right ) cd+292\,\sin \left ( fx+e \right ){d}^{2}+903\,{c}^{2}+1380\,cd+584\,{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))^(5/2)*(c+d*sin(f*x+e))^2,x)

[Out]

2/315*(1+sin(f*x+e))*a^3*(-1+sin(f*x+e))*(35*d^2*sin(f*x+e)^4+90*c*d*sin(f*x+e)^3+130*d^2*sin(f*x+e)^3+63*c^2*
sin(f*x+e)^2+360*c*d*sin(f*x+e)^2+219*d^2*sin(f*x+e)^2+294*c^2*sin(f*x+e)+690*sin(f*x+e)*c*d+292*sin(f*x+e)*d^
2+903*c^2+1380*c*d+584*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 (a \sin \left (f x + e\right ) + a\right )}^{\frac{5}{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))^(5/2)*(c+d*sin(f*x+e))^2,x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^(5/2)*(d*sin(f*x + e) + c)^2, x)

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Fricas [A]  time = 1.67319, size = 822, normalized size = 4.07 \begin{align*} -\frac{2 \,{\left (35 \, a^{2} d^{2} \cos \left (f x + e\right )^{5} - 5 \,{\left (18 \, a^{2} c d + 19 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{4} + 672 \, a^{2} c^{2} + 960 \, a^{2} c d + 416 \, a^{2} d^{2} -{\left (63 \, a^{2} c^{2} + 360 \, a^{2} c d + 289 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{3} +{\left (231 \, a^{2} c^{2} + 510 \, a^{2} c d + 263 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \,{\left (483 \, a^{2} c^{2} + 870 \, a^{2} c d + 419 \, a^{2} d^{2}\right )} \cos \left (f x + e\right ) -{\left (35 \, a^{2} d^{2} \cos \left (f x + e\right )^{4} + 672 \, a^{2} c^{2} + 960 \, a^{2} c d + 416 \, a^{2} d^{2} + 10 \,{\left (9 \, a^{2} c d + 13 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{3} - 3 \,{\left (21 \, a^{2} c^{2} + 90 \, a^{2} c d + 53 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \,{\left (147 \, a^{2} c^{2} + 390 \, a^{2} c d + 211 \, a^{2} 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))^(5/2)*(c+d*sin(f*x+e))^2,x, algorithm="fricas")

[Out]

-2/315*(35*a^2*d^2*cos(f*x + e)^5 - 5*(18*a^2*c*d + 19*a^2*d^2)*cos(f*x + e)^4 + 672*a^2*c^2 + 960*a^2*c*d + 4
16*a^2*d^2 - (63*a^2*c^2 + 360*a^2*c*d + 289*a^2*d^2)*cos(f*x + e)^3 + (231*a^2*c^2 + 510*a^2*c*d + 263*a^2*d^
2)*cos(f*x + e)^2 + 2*(483*a^2*c^2 + 870*a^2*c*d + 419*a^2*d^2)*cos(f*x + e) - (35*a^2*d^2*cos(f*x + e)^4 + 67
2*a^2*c^2 + 960*a^2*c*d + 416*a^2*d^2 + 10*(9*a^2*c*d + 13*a^2*d^2)*cos(f*x + e)^3 - 3*(21*a^2*c^2 + 90*a^2*c*
d + 53*a^2*d^2)*cos(f*x + e)^2 - 2*(147*a^2*c^2 + 390*a^2*c*d + 211*a^2*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)

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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))**(5/2)*(c+d*sin(f*x+e))**2,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))^(5/2)*(c+d*sin(f*x+e))^2,x, algorithm="giac")

[Out]

Timed out