3.131 \(\int \sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx\)

Optimal. Leaf size=94 \[ \frac{a B \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{5 c f \sqrt{a \sin (e+f x)+a}}-\frac{a (A+B) \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 f \sqrt{a \sin (e+f x)+a}} \]

[Out]

-(a*(A + B)*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(4*f*Sqrt[a + a*Sin[e + f*x]]) + (a*B*Cos[e + f*x]*(c - c
*Sin[e + f*x])^(9/2))/(5*c*f*Sqrt[a + a*Sin[e + f*x]])

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Rubi [A]  time = 0.339329, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {2971, 2738} \[ \frac{a B \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{5 c f \sqrt{a \sin (e+f x)+a}}-\frac{a (A+B) \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 f \sqrt{a \sin (e+f x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2),x]

[Out]

-(a*(A + B)*Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(4*f*Sqrt[a + a*Sin[e + f*x]]) + (a*B*Cos[e + f*x]*(c - c
*Sin[e + f*x])^(9/2))/(5*c*f*Sqrt[a + a*Sin[e + f*x]])

Rule 2971

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

Rule 2738

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] /; FreeQ[{a, b, c, d, e,
f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[n, -2^(-1)]

Rubi steps

\begin{align*} \int \sqrt{a+a \sin (e+f x)} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx &=(A+B) \int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx-\frac{B \int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2} \, dx}{c}\\ &=-\frac{a (A+B) \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 f \sqrt{a+a \sin (e+f x)}}+\frac{a B \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{5 c f \sqrt{a+a \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 0.995429, size = 118, normalized size = 1.26 \[ -\frac{c^3 \sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (4 (23 B-60 A) \sin (e+f x)+4 \cos (2 (e+f x)) (4 (5 A-6 B) \sin (e+f x)-35 A+25 B)+\cos (4 (e+f x)) (5 A+4 B \sin (e+f x)-15 B))}{160 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + a*Sin[e + f*x]]*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2),x]

[Out]

-(c^3*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(4*(-60*A + 23*B)*Sin[e + f*x] + 4*Cos[
2*(e + f*x)]*(-35*A + 25*B + 4*(5*A - 6*B)*Sin[e + f*x]) + Cos[4*(e + f*x)]*(5*A - 15*B + 4*B*Sin[e + f*x])))/
(160*f)

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Maple [B]  time = 0.398, size = 174, normalized size = 1.9 \begin{align*}{\frac{ \left ( -4\,B \left ( \cos \left ( fx+e \right ) \right ) ^{4}+5\,A \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) -15\,B \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) -20\,A \left ( \cos \left ( fx+e \right ) \right ) ^{2}+28\,B \left ( \cos \left ( fx+e \right ) \right ) ^{2}-35\,A\sin \left ( fx+e \right ) +25\,B\sin \left ( fx+e \right ) +40\,A-24\,B \right ) \sin \left ( fx+e \right ) }{20\,f \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) -3\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-4\,\sin \left ( fx+e \right ) +4 \right ) \cos \left ( fx+e \right ) } \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{7}{2}}}\sqrt{a \left ( 1+\sin \left ( fx+e \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2)*(a+a*sin(f*x+e))^(1/2),x)

[Out]

1/20/f*(-4*B*cos(f*x+e)^4+5*A*cos(f*x+e)^2*sin(f*x+e)-15*B*cos(f*x+e)^2*sin(f*x+e)-20*A*cos(f*x+e)^2+28*B*cos(
f*x+e)^2-35*A*sin(f*x+e)+25*B*sin(f*x+e)+40*A-24*B)*(-c*(-1+sin(f*x+e)))^(7/2)*sin(f*x+e)*(a*(1+sin(f*x+e)))^(
1/2)/(cos(f*x+e)^2*sin(f*x+e)-3*cos(f*x+e)^2-4*sin(f*x+e)+4)/cos(f*x+e)

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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 )} \sqrt{a \sin \left (f x + e\right ) + a}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2)*(a+a*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((B*sin(f*x + e) + A)*sqrt(a*sin(f*x + e) + a)*(-c*sin(f*x + e) + c)^(7/2), x)

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Fricas [A]  time = 1.81031, size = 342, normalized size = 3.64 \begin{align*} -\frac{{\left (5 \,{\left (A - 3 \, B\right )} c^{3} \cos \left (f x + e\right )^{4} - 40 \,{\left (A - B\right )} c^{3} \cos \left (f x + e\right )^{2} + 5 \,{\left (7 \, A - 5 \, B\right )} c^{3} + 4 \,{\left (B c^{3} \cos \left (f x + e\right )^{4} +{\left (5 \, A - 7 \, B\right )} c^{3} \cos \left (f x + e\right )^{2} - 2 \,{\left (5 \, A - 3 \, B\right )} c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}}{20 \, f \cos \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2)*(a+a*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

-1/20*(5*(A - 3*B)*c^3*cos(f*x + e)^4 - 40*(A - B)*c^3*cos(f*x + e)^2 + 5*(7*A - 5*B)*c^3 + 4*(B*c^3*cos(f*x +
e)^4 + (5*A - 7*B)*c^3*cos(f*x + e)^2 - 2*(5*A - 3*B)*c^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin
(f*x + e) + c)/(f*cos(f*x + e))

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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+B*sin(f*x+e))*(c-c*sin(f*x+e))**(7/2)*(a+a*sin(f*x+e))**(1/2),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+B*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2)*(a+a*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Timed out