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

Optimal. Leaf size=94 \[ \frac{a (A+B) \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac{a B \cos (e+f x)}{2 c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}} \]

[Out]

(a*(A + B)*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) - (a*B*Cos[e + f*x])/(2*c*f
*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2))

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Rubi [A]  time = 0.33662, 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} \[ \frac{a (A+B) \cos (e+f x)}{3 f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac{a B \cos (e+f x)}{2 c f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}} \]

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])/(3*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) - (a*B*Cos[e + f*x])/(2*c*f
*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2))

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 \frac{\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 \frac{\sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx-\frac{B \int \frac{\sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx}{c}\\ &=\frac{a (A+B) \cos (e+f x)}{3 f \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac{a B \cos (e+f x)}{2 c f \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.627621, size = 103, normalized size = 1.1 \[ \frac{\sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (2 A+3 B \sin (e+f x)-B)}{6 c^4 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a*(1 + Sin[e + f*x])]*(2*A - B + 3*B*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(6*c^4*f*(Cos[(e + f*x)/2]
- Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))

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Maple [B]  time = 0.364, size = 205, normalized size = 2.2 \begin{align*} -{\frac{ \left ( 2\,A \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +2\,A \left ( \cos \left ( fx+e \right ) \right ) ^{3}-B \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) -B \left ( \cos \left ( fx+e \right ) \right ) ^{3}+6\,A\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) -8\,A \left ( \cos \left ( fx+e \right ) \right ) ^{2}-3\,B\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +4\,B \left ( \cos \left ( fx+e \right ) \right ) ^{2}-14\,A\sin \left ( fx+e \right ) -8\,A\cos \left ( fx+e \right ) +4\,B\sin \left ( fx+e \right ) +B\cos \left ( fx+e \right ) +14\,A-4\,B \right ) \sin \left ( fx+e \right ) }{6\,f \left ( 1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) \right ) }\sqrt{a \left ( 1+\sin \left ( fx+e \right ) \right ) } \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6/f*(2*A*cos(f*x+e)^2*sin(f*x+e)+2*A*cos(f*x+e)^3-B*cos(f*x+e)^2*sin(f*x+e)-B*cos(f*x+e)^3+6*A*sin(f*x+e)*c
os(f*x+e)-8*A*cos(f*x+e)^2-3*B*sin(f*x+e)*cos(f*x+e)+4*B*cos(f*x+e)^2-14*A*sin(f*x+e)-8*A*cos(f*x+e)+4*B*sin(f
*x+e)+B*cos(f*x+e)+14*A-4*B)*sin(f*x+e)*(a*(1+sin(f*x+e)))^(1/2)/(1-cos(f*x+e)+sin(f*x+e))/(-c*(-1+sin(f*x+e))
)^(7/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\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))*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(7/2),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 = 1.75888, size = 263, normalized size = 2.8 \begin{align*} -\frac{{\left (3 \, B \sin \left (f x + e\right ) + 2 \, A - B\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}}{6 \,{\left (3 \, c^{4} f \cos \left (f x + e\right )^{3} - 4 \, c^{4} f \cos \left (f x + e\right ) -{\left (c^{4} f \cos \left (f x + e\right )^{3} - 4 \, c^{4} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*B*sin(f*x + e) + 2*A - B)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(3*c^4*f*cos(f*x + e)^3 -
4*c^4*f*cos(f*x + e) - (c^4*f*cos(f*x + e)^3 - 4*c^4*f*cos(f*x + e))*sin(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))*(a+a*sin(f*x+e))**(1/2)/(c-c*sin(f*x+e))**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\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))*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(7/2),x, algorithm="giac")

[Out]

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