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

Optimal. Leaf size=45 \[ \frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{5 a f \sqrt{c-c \sin (e+f x)}} \]

[Out]

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

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Rubi [A]  time = 0.307475, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2841, 2738} \[ \frac{\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{5 a f \sqrt{c-c \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2841

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*
(x_)])^(n_.), x_Symbol] :> Dist[1/(a^(p/2)*c^(p/2)), Int[(a + b*Sin[e + f*x])^(m + p/2)*(c + d*Sin[e + f*x])^(
n + p/2), x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[p
/2]

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

Mathematica [B]  time = 1.44084, size = 142, normalized size = 3.16 \[ \frac{a^3 (\sin (e+f x)+1)^3 \sqrt{a (\sin (e+f x)+1)} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) (210 \sin (e+f x)-45 \sin (3 (e+f x))+\sin (5 (e+f x))-120 \cos (2 (e+f x))+10 \cos (4 (e+f x)))}{80 f \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*Sqrt[a*(1 + Sin[e + f*x])]*(-120*Cos[2*(e + f*
x)] + 10*Cos[4*(e + f*x)] + 210*Sin[e + f*x] - 45*Sin[3*(e + f*x)] + Sin[5*(e + f*x)]))/(80*f*(Cos[(e + f*x)/2
] + Sin[(e + f*x)/2])^7*Sqrt[c - c*Sin[e + f*x]])

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Maple [B]  time = 0.231, size = 245, normalized size = 5.4 \begin{align*}{\frac{ \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{5}+\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}+4\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}-5\,\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{3}-12\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}-7\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) -8\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+15\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +16\,\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) -1 \right ) \sin \left ( fx+e \right ) }{5\,f \left ( \sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{3}+ \left ( \cos \left ( fx+e \right ) \right ) ^{4}-4\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +3\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}-4\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) -8\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+8\,\sin \left ( fx+e \right ) -4\,\cos \left ( fx+e \right ) +8 \right ) } \left ( a \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{7}{2}}}{\frac{1}{\sqrt{-c \left ( -1+\sin \left ( fx+e \right ) \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5/f*(cos(f*x+e)^5+sin(f*x+e)*cos(f*x+e)^4+4*cos(f*x+e)^4-5*sin(f*x+e)*cos(f*x+e)^3-12*cos(f*x+e)^3-7*cos(f*x
+e)^2*sin(f*x+e)-8*cos(f*x+e)^2+15*sin(f*x+e)*cos(f*x+e)+16*cos(f*x+e)+sin(f*x+e)-1)*sin(f*x+e)*(a*(1+sin(f*x+
e)))^(7/2)/(sin(f*x+e)*cos(f*x+e)^3+cos(f*x+e)^4-4*cos(f*x+e)^2*sin(f*x+e)+3*cos(f*x+e)^3-4*sin(f*x+e)*cos(f*x
+e)-8*cos(f*x+e)^2+8*sin(f*x+e)-4*cos(f*x+e)+8)/(-c*(-1+sin(f*x+e)))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{7}{2}} \cos \left (f x + e\right )^{2}}{\sqrt{-c \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.7402, size = 270, normalized size = 6. \begin{align*} \frac{{\left (5 \, a^{3} \cos \left (f x + e\right )^{4} - 20 \, a^{3} \cos \left (f x + e\right )^{2} + 15 \, a^{3} +{\left (a^{3} \cos \left (f x + e\right )^{4} - 12 \, a^{3} \cos \left (f x + e\right )^{2} + 16 \, a^{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}}{5 \, c f \cos \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(5*a^3*cos(f*x + e)^4 - 20*a^3*cos(f*x + e)^2 + 15*a^3 + (a^3*cos(f*x + e)^4 - 12*a^3*cos(f*x + e)^2 + 16*
a^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(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(cos(f*x+e)**2*(a+a*sin(f*x+e))**(7/2)/(c-c*sin(f*x+e))**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{7}{2}} \cos \left (f x + e\right )^{2}}{\sqrt{-c \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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