3.532 \(\int (a+a \sin (e+f x))^{3/2} \, dx\)

Optimal. Leaf size=59 \[ -\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f} \]

[Out]

-8/3*a^2*cos(f*x+e)/f/(a+a*sin(f*x+e))^(1/2)-2/3*a*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/f

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Rubi [A]  time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2647, 2646} \[ -\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^(3/2),x]

[Out]

(-8*a^2*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]) - (2*a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(3*f)

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]

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]

Rubi steps

\begin {align*} \int (a+a \sin (e+f x))^{3/2} \, dx &=-\frac {2 a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {1}{3} (4 a) \int \sqrt {a+a \sin (e+f x)} \, dx\\ &=-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 89, normalized size = 1.51 \[ -\frac {(a (\sin (e+f x)+1))^{3/2} \left (-9 \sin \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {3}{2} (e+f x)\right )+9 \cos \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {3}{2} (e+f x)\right )\right )}{3 f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^(3/2),x]

[Out]

-1/3*((a*(1 + Sin[e + f*x]))^(3/2)*(9*Cos[(e + f*x)/2] + Cos[(3*(e + f*x))/2] - 9*Sin[(e + f*x)/2] + Sin[(3*(e
+ f*x))/2]))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3)

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fricas [A]  time = 0.42, size = 76, normalized size = 1.29 \[ -\frac {2 \, {\left (a \cos \left (f x + e\right )^{2} + 5 \, a \cos \left (f x + e\right ) + {\left (a \cos \left (f x + e\right ) - 4 \, a\right )} \sin \left (f x + e\right ) + 4 \, a\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{3 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(3/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)sqrt(2*a)*(2*a*sig
n(cos(1/2*(f*x+exp(1))-1/4*pi))*sin(1/4*(2*f*x-pi)+1/2*exp(1))/f-4*a*f*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*cos(
1/4*(2*f*x+2*exp(1)+pi))/(2*f)^2-12*a*f*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*cos(1/4*(6*f*x+6*exp(1)-pi))/(6*f)^
2)

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maple [A]  time = 0.66, size = 53, normalized size = 0.90 \[ \frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) \left (\sin \left (f x +e \right )+5\right )}{3 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^(3/2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^(3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(e + f*x))^(3/2),x)

[Out]

int((a + a*sin(e + f*x))^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sin {\left (e + f x \right )} + a\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**(3/2),x)

[Out]

Integral((a*sin(e + f*x) + a)**(3/2), x)

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