3.509 \(\int \frac{1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx\)

Optimal. Leaf size=333 \[ -\frac{d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a f (c-d)^3 (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left (3 c^2+20 c d+9 d^2\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 a f (c-d)^3 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{d (3 c+5 d) \cos (e+f x)}{3 a f (c-d)^2 (c+d) (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}+\frac{(3 c+5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 a f (c-d)^2 (c+d) \sqrt{c+d \sin (e+f x)}} \]

[Out]

-(d*(3*c + 5*d)*Cos[e + f*x])/(3*a*(c - d)^2*(c + d)*f*(c + d*Sin[e + f*x])^(3/2)) - Cos[e + f*x]/((c - d)*f*(
a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)) - (d*(3*c^2 + 20*c*d + 9*d^2)*Cos[e + f*x])/(3*a*(c - d)^3*(c
+ d)^2*f*Sqrt[c + d*Sin[e + f*x]]) - ((3*c^2 + 20*c*d + 9*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sq
rt[c + d*Sin[e + f*x]])/(3*a*(c - d)^3*(c + d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((3*c + 5*d)*Elliptic
F[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*a*(c - d)^2*(c + d)*f*Sqrt[c + d*S
in[e + f*x]])

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Rubi [A]  time = 0.488061, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {2768, 2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac{d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a f (c-d)^3 (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{\left (3 c^2+20 c d+9 d^2\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 a f (c-d)^3 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{d (3 c+5 d) \cos (e+f x)}{3 a f (c-d)^2 (c+d) (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^{3/2}}+\frac{(3 c+5 d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 a f (c-d)^2 (c+d) \sqrt{c+d \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2)),x]

[Out]

-(d*(3*c + 5*d)*Cos[e + f*x])/(3*a*(c - d)^2*(c + d)*f*(c + d*Sin[e + f*x])^(3/2)) - Cos[e + f*x]/((c - d)*f*(
a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)) - (d*(3*c^2 + 20*c*d + 9*d^2)*Cos[e + f*x])/(3*a*(c - d)^3*(c
+ d)^2*f*Sqrt[c + d*Sin[e + f*x]]) - ((3*c^2 + 20*c*d + 9*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sq
rt[c + d*Sin[e + f*x]])/(3*a*(c - d)^3*(c + d)^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((3*c + 5*d)*Elliptic
F[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*a*(c - d)^2*(c + d)*f*Sqrt[c + d*S
in[e + f*x]])

Rule 2768

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

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{1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx &=-\frac{\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}+\frac{d \int \frac{-\frac{5 a}{2}+\frac{3}{2} a \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx}{a^2 (c-d)}\\ &=-\frac{d (3 c+5 d) \cos (e+f x)}{3 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac{(2 d) \int \frac{\frac{3}{4} a (5 c+3 d)-\frac{1}{4} a (3 c+5 d) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx}{3 a^2 (c-d)^2 (c+d)}\\ &=-\frac{d (3 c+5 d) \cos (e+f x)}{3 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac{d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a (c-d)^3 (c+d)^2 f \sqrt{c+d \sin (e+f x)}}+\frac{(4 d) \int \frac{-\frac{1}{8} a \left (15 c^2+12 c d+5 d^2\right )-\frac{1}{8} a \left (3 c^2+20 c d+9 d^2\right ) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{3 a^2 (c-d)^3 (c+d)^2}\\ &=-\frac{d (3 c+5 d) \cos (e+f x)}{3 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac{d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a (c-d)^3 (c+d)^2 f \sqrt{c+d \sin (e+f x)}}+\frac{(3 c+5 d) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{6 a (c-d)^2 (c+d)}-\frac{\left (3 c^2+20 c d+9 d^2\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{6 a (c-d)^3 (c+d)^2}\\ &=-\frac{d (3 c+5 d) \cos (e+f x)}{3 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac{d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a (c-d)^3 (c+d)^2 f \sqrt{c+d \sin (e+f x)}}-\frac{\left (\left (3 c^2+20 c d+9 d^2\right ) \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 (c-d)^3 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left ((3 c+5 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 (c-d)^2 (c+d) \sqrt{c+d \sin (e+f x)}}\\ &=-\frac{d (3 c+5 d) \cos (e+f x)}{3 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac{d \left (3 c^2+20 c d+9 d^2\right ) \cos (e+f x)}{3 a (c-d)^3 (c+d)^2 f \sqrt{c+d \sin (e+f x)}}-\frac{\left (3 c^2+20 c d+9 d^2\right ) 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 (c-d)^3 (c+d)^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(3 c+5 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 (c-d)^2 (c+d) f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 4.22967, size = 367, normalized size = 1.1 \[ \frac{\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2 \left (2 (c+d \sin (e+f x)) \left (\frac{3 \sin \left (\frac{1}{2} (e+f x)\right )}{\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )}-\frac{\frac{d^2 \cos (e+f x) \left (8 c^2+d (7 c+3 d) \sin (e+f x)+3 c d-d^2\right )}{(c+d \sin (e+f x))^2}+3 c^2+13 c d+6 d^2}{(c+d)^2}\right )+\frac{\left (3 c^2+20 c d+9 d^2\right ) (c+d \sin (e+f x))+d \left (15 c^2+12 c d+5 d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )+\left (3 c^2+20 c d+9 d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left ((c+d) E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-c F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )\right )}{(c+d)^2}\right )}{3 a f (c-d)^3 (\sin (e+f x)+1) \sqrt{c+d \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(5/2)),x]

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*(((3*c^2 + 20*c*d + 9*d^2)*(c + d*Sin[e + f*x]) + d*(15*c^2 + 12*c*d
+ 5*d^2)*EllipticF[(-2*e + Pi - 2*f*x)/4, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)] + (3*c^2 + 20*c*d
+ 9*d^2)*((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)])/(c + d)^2 + 2*(c + d*Sin[e + f*x])*((3*Sin[(e + f*x)/2])/(Cos[(e
+ f*x)/2] + Sin[(e + f*x)/2]) - (3*c^2 + 13*c*d + 6*d^2 + (d^2*Cos[e + f*x]*(8*c^2 + 3*c*d - d^2 + d*(7*c + 3*
d)*Sin[e + f*x]))/(c + d*Sin[e + f*x])^2)/(c + d)^2)))/(3*a*(c - d)^3*f*(1 + Sin[e + f*x])*Sqrt[c + d*Sin[e +
f*x]])

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Maple [B]  time = 5.365, size = 1291, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-(-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/a*(-d/(c-d)^2*(2*d*cos(f*x+e)^2/(c^2-d^2)/(-(-d*sin(f*x+e)-c)*cos(f*x+
e)^2)^(1/2)+2*c/(c^2-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))+2/(c^2-d^2)*d*(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))))-d/(c-d)*(2/3/(c^2-d^2)/d*(-(
-d*sin(f*x+e)-c)*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+c/d)^2+8/3*d*cos(f*x+e)^2/(c^2-d^2)^2*c/(-(-d*sin(f*x+e)-c)*c
os(f*x+e)^2)^(1/2)+2*(3*c^2+d^2)/(3*c^4-6*c^2*d^2+3*d^4)*(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))+8/3*c*d/(c^2-d^2)^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)*Ellipti
cE(((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))))+1/(c-d)^2*(-(-sin(f*x+e)^2*d-c*sin(f*x+e)+d*sin(f*x+e)+c)/(c-d)/((-d*sin(f*x+e)-c)*(-1+sin(f*x+e))*(1
+sin(f*x+e)))^(1/2)-2*d/(2*c-2*d)*(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))-d/(c-d)*(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(-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(1/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

Timed out

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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}}{a d^{3} \cos \left (f x + e\right )^{4} + a c^{3} + 3 \, a c^{2} d + 3 \, a c d^{2} + a d^{3} -{\left (3 \, a c^{2} d + 3 \, a c d^{2} + 2 \, a d^{3}\right )} \cos \left (f x + e\right )^{2} +{\left (a c^{3} + 3 \, a c^{2} d + 3 \, a c d^{2} + a d^{3} -{\left (3 \, a c d^{2} + a d^{3}\right )} \cos \left (f x + e\right )^{2}\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))/(c+d*sin(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*sin(f*x + e) + c)/(a*d^3*cos(f*x + e)^4 + a*c^3 + 3*a*c^2*d + 3*a*c*d^2 + a*d^3 - (3*a*c^2*d +
3*a*c*d^2 + 2*a*d^3)*cos(f*x + e)^2 + (a*c^3 + 3*a*c^2*d + 3*a*c*d^2 + a*d^3 - (3*a*c*d^2 + a*d^3)*cos(f*x +
e)^2)*sin(f*x + e)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{c^{2} \sqrt{c + d \sin{\left (e + f x \right )}} \sin{\left (e + f x \right )} + c^{2} \sqrt{c + d \sin{\left (e + f x \right )}} + 2 c d \sqrt{c + d \sin{\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )} + 2 c d \sqrt{c + d \sin{\left (e + f x \right )}} \sin{\left (e + f x \right )} + d^{2} \sqrt{c + d \sin{\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )} + d^{2} \sqrt{c + d \sin{\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(c**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x) + c**2*sqrt(c + d*sin(e + f*x)) + 2*c*d*sqrt(c + d*sin(
e + f*x))*sin(e + f*x)**2 + 2*c*d*sqrt(c + d*sin(e + f*x))*sin(e + f*x) + d**2*sqrt(c + d*sin(e + f*x))*sin(e
+ f*x)**3 + d**2*sqrt(c + d*sin(e + f*x))*sin(e + f*x)**2), x)/a

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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 )}{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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