3.4.41 \(\int \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx\) [341]

Optimal. Leaf size=43 \[ -\frac {a \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt {a+a \sin (e+f x)}} \]

[Out]

-1/3*a*cos(f*x+e)*(c-c*sin(f*x+e))^(5/2)/f/(a+a*sin(f*x+e))^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2817} \begin {gather*} -\frac {a \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt {a \sin (e+f x)+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(a*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(f*Sqrt[a + a*Sin[e + f*x]])

Rule 2817

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

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Mathematica [A]
time = 0.18, size = 74, normalized size = 1.72 \begin {gather*} \frac {c^2 \sec (e+f x) \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)} (6 \cos (2 (e+f x))+15 \sin (e+f x)-\sin (3 (e+f x)))}{12 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(6*Cos[2*(e + f*x)] + 15*Sin[e + f*x] -
Sin[3*(e + f*x)]))/(12*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(77\) vs. \(2(37)=74\).
time = 16.48, size = 78, normalized size = 1.81

method	result	size
default	\(\frac {\left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {5}{2}} \sin \left (f x +e \right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \left (\cos ^{4}\left (f x +e \right )+\sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+2+2 \sin \left (f x +e \right )\right )}{3 f \cos \left (f x +e \right )^{5}}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/3/f*(-c*(sin(f*x+e)-1))^(5/2)*sin(f*x+e)*(a*(1+sin(f*x+e)))^(1/2)*(cos(f*x+e)^4+sin(f*x+e)*cos(f*x+e)^2+2+2*
sin(f*x+e))/cos(f*x+e)^5

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (40) = 80\).
time = 0.34, size = 89, normalized size = 2.07 \begin {gather*} \frac {{\left (3 \, c^{2} \cos \left (f x + e\right )^{2} - 3 \, c^{2} - {\left (c^{2} \cos \left (f x + e\right )^{2} - 4 \, c^{2}\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}}{3 \, f \cos \left (f x + e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

1/3*(3*c^2*cos(f*x + e)^2 - 3*c^2 - (c^2*cos(f*x + e)^2 - 4*c^2)*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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**(1/2)*(c-c*sin(f*x+e))**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4368 deep

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Giac [A]
time = 0.49, size = 54, normalized size = 1.26 \begin {gather*} \frac {8 \, \sqrt {a} c^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6}}{3 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(5/2),x, algorithm="giac")

[Out]

8/3*sqrt(a)*c^(5/2)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*
f*x + 1/2*e)^6/f

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Mupad [B]
time = 7.71, size = 88, normalized size = 2.05 \begin {gather*} \frac {c^2\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (6\,\cos \left (e+f\,x\right )+6\,\cos \left (3\,e+3\,f\,x\right )+14\,\sin \left (2\,e+2\,f\,x\right )-\sin \left (4\,e+4\,f\,x\right )\right )}{12\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(e + f*x))^(1/2)*(c - c*sin(e + f*x))^(5/2),x)

[Out]

(c^2*(a*(sin(e + f*x) + 1))^(1/2)*(-c*(sin(e + f*x) - 1))^(1/2)*(6*cos(e + f*x) + 6*cos(3*e + 3*f*x) + 14*sin(
2*e + 2*f*x) - sin(4*e + 4*f*x)))/(12*f*(cos(2*e + 2*f*x) + 1))

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