3.513 \(\int \frac{1}{(a+a \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}} \, dx\)

Optimal. Leaf size=257 \[ -\frac{(c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 f (c-d)^2 (\sin (e+f x)+1)}+\frac{(c-2 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 a^2 f (c-d) \sqrt{c+d \sin (e+f x)}}-\frac{(c-3 d) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 a^2 f (c-d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2} \]

[Out]

-((c - 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*(c - d)^2*f*(1 + Sin[e + f*x])) - (Cos[e + f*x]*Sqrt
[c + d*Sin[e + f*x]])/(3*(c - d)*f*(a + a*Sin[e + f*x])^2) - ((c - 3*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c
+ d)]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*(c - d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((c - 2*d)*EllipticF
[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*a^2*(c - d)*f*Sqrt[c + d*Sin[e + f*
x]])

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Rubi [A]  time = 0.441346, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {2766, 2978, 2752, 2663, 2661, 2655, 2653} \[ -\frac{(c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 f (c-d)^2 (\sin (e+f x)+1)}+\frac{(c-2 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 a^2 f (c-d) \sqrt{c+d \sin (e+f x)}}-\frac{(c-3 d) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 a^2 f (c-d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 f (c-d) (a \sin (e+f x)+a)^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + a*Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]]),x]

[Out]

-((c - 3*d)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*(c - d)^2*f*(1 + Sin[e + f*x])) - (Cos[e + f*x]*Sqrt
[c + d*Sin[e + f*x]])/(3*(c - d)*f*(a + a*Sin[e + f*x])^2) - ((c - 3*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c
+ d)]*Sqrt[c + d*Sin[e + f*x]])/(3*a^2*(c - d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((c - 2*d)*EllipticF
[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*a^2*(c - d)*f*Sqrt[c + d*Sin[e + f*
x]])

Rule 2766

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(a*f*(2*m + 1)*(b*c - a*d)), x] + Dis
t[1/(a*(2*m + 1)*(b*c - a*d)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[b*c*(m + 1) - a*d*
(2*m + n + 2) + b*d*(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d,
0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] &&  !GtQ[n, 0] && (IntegersQ[2*m, 2*n] || (IntegerQ
[m] && EqQ[c, 0]))

Rule 2978

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

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{1}{(a+a \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}} \, dx &=-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 (c-d) f (a+a \sin (e+f x))^2}-\frac{\int \frac{-\frac{1}{2} a (2 c-5 d)-\frac{1}{2} a d \sin (e+f x)}{(a+a \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{3 a^2 (c-d)}\\ &=-\frac{(c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 (c-d)^2 f (1+\sin (e+f x))}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 (c-d) f (a+a \sin (e+f x))^2}+\frac{\int \frac{a^2 d^2-\frac{1}{2} a^2 (c-3 d) d \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{3 a^4 (c-d)^2}\\ &=-\frac{(c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 (c-d)^2 f (1+\sin (e+f x))}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 (c-d) f (a+a \sin (e+f x))^2}-\frac{(c-3 d) \int \sqrt{c+d \sin (e+f x)} \, dx}{6 a^2 (c-d)^2}+\frac{(c-2 d) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{6 a^2 (c-d)}\\ &=-\frac{(c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 (c-d)^2 f (1+\sin (e+f x))}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 (c-d) f (a+a \sin (e+f x))^2}-\frac{\left ((c-3 d) \sqrt{c+d \sin (e+f x)}\right ) \int \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}} \, dx}{6 a^2 (c-d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left ((c-2 d) \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}{6 a^2 (c-d) \sqrt{c+d \sin (e+f x)}}\\ &=-\frac{(c-3 d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 a^2 (c-d)^2 f (1+\sin (e+f x))}-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3 (c-d) f (a+a \sin (e+f x))^2}-\frac{(c-3 d) 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)}}{3 a^2 (c-d)^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(c-2 d) 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}}}{3 a^2 (c-d) f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 2.46164, size = 290, normalized size = 1.13 \[ \frac{\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4 \left (-2 d^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-(c-3 d) (c+d \sin (e+f x))-\frac{(c+d \sin (e+f x)) \left ((7 d-3 c) \sin \left (\frac{1}{2} (e+f x)\right )+(c-3 d) \cos \left (\frac{3}{2} (e+f x)\right )+2 d \cos \left (\frac{1}{2} (e+f x)\right )\right )}{\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3}+(c-3 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left ((c+d) E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-c F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )\right )\right )}{3 a^2 f (c-d)^2 (\sin (e+f x)+1)^2 \sqrt{c+d \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + a*Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]]),x]

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(-((c - 3*d)*(c + d*Sin[e + f*x])) - ((2*d*Cos[(e + f*x)/2] + (c - 3*
d)*Cos[(3*(e + f*x))/2] + (-3*c + 7*d)*Sin[(e + f*x)/2])*(c + d*Sin[e + f*x]))/(Cos[(e + f*x)/2] + Sin[(e + f*
x)/2])^3 - 2*d^2*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + (c - 3*d
)*((c + d)*EllipticE[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)] - c*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]
)*Sqrt[(c + d*Sin[e + f*x])/(c + d)]))/(3*a^2*(c - d)^2*f*(1 + Sin[e + f*x])^2*Sqrt[c + d*Sin[e + f*x]])

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Maple [A]  time = 3.622, size = 507, normalized size = 2. \begin{align*}{\frac{1}{{a}^{2}\cos \left ( fx+e \right ) f}\sqrt{- \left ( -d\sin \left ( fx+e \right ) -c \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}} \left ( -{\frac{1}{ \left ( 3\,c-3\,d \right ) \left ( 1+\sin \left ( fx+e \right ) \right ) ^{2}}\sqrt{- \left ( -d\sin \left ( fx+e \right ) -c \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}}}-{\frac{ \left ( - \left ( \sin \left ( fx+e \right ) \right ) ^{2}d-c\sin \left ( fx+e \right ) +d\sin \left ( fx+e \right ) +c \right ) \left ( c-3\,d \right ) }{3\, \left ( c-d \right ) ^{2}}{\frac{1}{\sqrt{ \left ( -d\sin \left ( fx+e \right ) -c \right ) \left ( -1+\sin \left ( fx+e \right ) \right ) \left ( 1+\sin \left ( fx+e \right ) \right ) }}}}+2\,{\frac{{d}^{2}}{ \left ( 3\,{c}^{2}-6\,cd+3\,{d}^{2} \right ) \sqrt{- \left ( -d\sin \left ( fx+e \right ) -c \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}}} \left ({\frac{c}{d}}-1 \right ) \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}}\sqrt{{\frac{d \left ( 1-\sin \left ( fx+e \right ) \right ) }{c+d}}}\sqrt{{\frac{ \left ( -\sin \left ( fx+e \right ) -1 \right ) d}{c-d}}}{\it EllipticF} \left ( \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}},\sqrt{{\frac{c-d}{c+d}}} \right ) }-{\frac{d \left ( c-3\,d \right ) }{3\, \left ( c-d \right ) ^{2}} \left ({\frac{c}{d}}-1 \right ) \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}}\sqrt{{\frac{d \left ( 1-\sin \left ( fx+e \right ) \right ) }{c+d}}}\sqrt{{\frac{ \left ( -\sin \left ( fx+e \right ) -1 \right ) d}{c-d}}} \left ( \left ( -{\frac{c}{d}}-1 \right ){\it EllipticE} \left ( \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}},\sqrt{{\frac{c-d}{c+d}}} \right ) +{\it EllipticF} \left ( \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}},\sqrt{{\frac{c-d}{c+d}}} \right ) \right ){\frac{1}{\sqrt{- \left ( -d\sin \left ( fx+e \right ) -c \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}}}}} \right ){\frac{1}{\sqrt{c+d\sin \left ( fx+e \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/a^2*(-1/3/(c-d)*(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(1+sin(f*x+e))
^2-1/3*(-sin(f*x+e)^2*d-c*sin(f*x+e)+d*sin(f*x+e)+c)/(c-d)^2*(c-3*d)/((-d*sin(f*x+e)-c)*(-1+sin(f*x+e))*(1+sin
(f*x+e)))^(1/2)+2*d^2/(3*c^2-6*c*d+3*d^2)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d))^(1/2
)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(
1/2),((c-d)/(c+d))^(1/2))-1/3*d*(c-3*d)/(c-d)^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d)
)^(1/2)*((-sin(f*x+e)-1)*d/(c-d))^(1/2)/(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(
f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))/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{1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2} \sqrt{d \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((a*sin(f*x + e) + a)^2*sqrt(d*sin(f*x + e) + c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{\sqrt{c + d \sin{\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )} + 2 \sqrt{c + d \sin{\left (e + f x \right )}} \sin{\left (e + f x \right )} + \sqrt{c + d \sin{\left (e + f x \right )}}}\, dx}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(c + d*sin(e + f*x))*sin(e + f*x)**2 + 2*sqrt(c + d*sin(e + f*x))*sin(e + f*x) + sqrt(c + d*si
n(e + f*x))), x)/a**2

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2} \sqrt{d \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((a*sin(f*x + e) + a)^2*sqrt(d*sin(f*x + e) + c)), x)