3.486 \(\int \frac{a+a \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=169 \[ -\frac{2 a \cos (e+f x)}{f (c+d) \sqrt{c+d \sin (e+f x)}}+\frac{2 a \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{d f \sqrt{c+d \sin (e+f x)}}-\frac{2 a \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{d f (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}} \]

[Out]

(-2*a*Cos[e + f*x])/((c + d)*f*Sqrt[c + d*Sin[e + f*x]]) - (2*a*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*S
qrt[c + d*Sin[e + f*x]])/(d*(c + d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*a*EllipticF[(e - Pi/2 + f*x)/2,
(2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]])

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Rubi [A]  time = 0.205837, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 a \cos (e+f x)}{f (c+d) \sqrt{c+d \sin (e+f x)}}+\frac{2 a \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{d f \sqrt{c+d \sin (e+f x)}}-\frac{2 a \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{d f (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*Cos[e + f*x])/((c + d)*f*Sqrt[c + d*Sin[e + f*x]]) - (2*a*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*S
qrt[c + d*Sin[e + f*x]])/(d*(c + d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*a*EllipticF[(e - Pi/2 + f*x)/2,
(2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]])

Rule 2754

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

Rule 2752

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]

Rule 2663

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2655

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rubi steps

\begin{align*} \int \frac{a+a \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx &=-\frac{2 a \cos (e+f x)}{(c+d) f \sqrt{c+d \sin (e+f x)}}-\frac{2 \int \frac{-\frac{1}{2} a (c-d)+\frac{1}{2} a (c-d) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{c^2-d^2}\\ &=-\frac{2 a \cos (e+f x)}{(c+d) f \sqrt{c+d \sin (e+f x)}}+\frac{a \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{d}-\frac{a \int \sqrt{c+d \sin (e+f x)} \, dx}{d (c+d)}\\ &=-\frac{2 a \cos (e+f x)}{(c+d) f \sqrt{c+d \sin (e+f x)}}-\frac{\left (a \sqrt{c+d \sin (e+f x)}\right ) \int \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}} \, dx}{d (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left (a \sqrt{\frac{c+d \sin (e+f x)}{c+d}}\right ) \int \frac{1}{\sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{d \sqrt{c+d \sin (e+f x)}}\\ &=-\frac{2 a \cos (e+f x)}{(c+d) f \sqrt{c+d \sin (e+f x)}}-\frac{2 a E\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{c+d \sin (e+f x)}}{d (c+d) f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 a F\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{d f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 6.39223, size = 938, normalized size = 5.55 \[ a \left (\frac{\sqrt{c+d \sin (e+f x)} \left (\frac{2 \csc (e) (c \cos (e)+d \sin (f x))}{d (c+d) f (c+d \sin (e+f x))}-\frac{2 \csc (e) \sec (e)}{d (c+d) f}\right ) (\sin (e+f x)+1)}{\left (\cos \left (\frac{e}{2}+\frac{f x}{2}\right )+\sin \left (\frac{e}{2}+\frac{f x}{2}\right )\right )^2}-\frac{\sec (e) \left (-\frac{F_1\left (-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{\csc (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} \sin (e)\right )}{d \sqrt{\cot ^2(e)+1} \left (1-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}\right )},-\frac{\csc (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} \sin (e)\right )}{d \sqrt{\cot ^2(e)+1} \left (-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}-1\right )}\right ) \cot (e) \sin \left (f x-\tan ^{-1}(\cot (e))\right )}{\sqrt{\cot ^2(e)+1} \sqrt{\frac{\cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} d+\sqrt{\cot ^2(e)+1} d}{d \sqrt{\cot ^2(e)+1}-c \csc (e)}} \sqrt{\frac{d \sqrt{\cot ^2(e)+1}-d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1}}{\sqrt{\cot ^2(e)+1} d+c \csc (e)}} \sqrt{c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} \sin (e)}}-\frac{\frac{2 d \sin (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} \sin (e)\right )}{d^2 \cos ^2(e)+d^2 \sin ^2(e)}-\frac{\cot (e) \sin \left (f x-\tan ^{-1}(\cot (e))\right )}{\sqrt{\cot ^2(e)+1}}}{\sqrt{c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} \sin (e)}}\right ) (\sin (e+f x)+1)}{(c+d) f \left (\cos \left (\frac{e}{2}+\frac{f x}{2}\right )+\sin \left (\frac{e}{2}+\frac{f x}{2}\right )\right )^2}+\frac{2 F_1\left (\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};-\frac{\sec (e) \left (c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1}\right )}{d \sqrt{\tan ^2(e)+1} \left (1-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}\right )},-\frac{\sec (e) \left (c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1}\right )}{d \sqrt{\tan ^2(e)+1} \left (-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}-1\right )}\right ) \sec (e) \sec \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\frac{d \sqrt{\tan ^2(e)+1}-d \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1}}{\sqrt{\tan ^2(e)+1} d+c \sec (e)}} \sqrt{\frac{\sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1} d+\sqrt{\tan ^2(e)+1} d}{d \sqrt{\tan ^2(e)+1}-c \sec (e)}} \sqrt{c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1}} (\sin (e+f x)+1)}{d (c+d) f \left (\cos \left (\frac{e}{2}+\frac{f x}{2}\right )+\sin \left (\frac{e}{2}+\frac{f x}{2}\right )\right )^2 \sqrt{\tan ^2(e)+1}}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

a*(((1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]*((-2*Csc[e]*Sec[e])/(d*(c + d)*f) + (2*Csc[e]*(c*Cos[e] + d*Si
n[f*x]))/(d*(c + d)*f*(c + d*Sin[e + f*x]))))/(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2 - (Sec[e]*(1 + Sin[e
+ f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin
[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e
]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f
*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Co
t[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 +
Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]])
) - ((2*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) -
(Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]
*Sin[e]]))/((c + d)*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (2*AppellF1[1/2, 1/2, 1/2, 3/2, -((Sec[e]
*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1
+ Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*
(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1
+ Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt
[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*Sqrt[c
+ d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]])/(d*(c + d)*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2
])^2*Sqrt[1 + Tan[e]^2]))

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Maple [A]  time = 0.955, size = 246, normalized size = 1.5 \begin{align*} 2\,{\frac{a}{{d}^{2} \left ( c+d \right ) \cos \left ( fx+e \right ) \sqrt{c+d\sin \left ( fx+e \right ) }f} \left ( \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}}\sqrt{-{\frac{ \left ( -1+\sin \left ( fx+e \right ) \right ) d}{c+d}}}\sqrt{-{\frac{d \left ( 1+\sin \left ( fx+e \right ) \right ) }{c-d}}}{\it EllipticE} \left ( \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}},\sqrt{{\frac{c-d}{c+d}}} \right ){c}^{2}-\sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}}\sqrt{-{\frac{ \left ( -1+\sin \left ( fx+e \right ) \right ) d}{c+d}}}\sqrt{-{\frac{d \left ( 1+\sin \left ( fx+e \right ) \right ) }{c-d}}}{\it EllipticE} \left ( \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}},\sqrt{{\frac{c-d}{c+d}}} \right ){d}^{2}+{d}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{2}-{d}^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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