3.6.8 \(\int \frac {1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx\) [508]

Optimal. Leaf size=244 \[ -\frac {d (c+3 d) \cos (e+f x)}{a (c-d)^2 (c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}-\frac {(c+3 d) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{a (c-d)^2 (c+d) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{a (c-d) f \sqrt {c+d \sin (e+f x)}} \]

[Out]

-d*(c+3*d)*cos(f*x+e)/a/(c-d)^2/(c+d)/f/(c+d*sin(f*x+e))^(1/2)-cos(f*x+e)/(c-d)/f/(a+a*sin(f*x+e))/(c+d*sin(f*
x+e))^(1/2)+(c+3*d)*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticE(cos(1/2*e+1/4*Pi+1
/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f*x+e))^(1/2)/a/(c-d)^2/(c+d)/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-(sin(
1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticF(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))
^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/a/(c-d)/f/(c+d*sin(f*x+e))^(1/2)

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Rubi [A]
time = 0.22, antiderivative size = 244, normalized size of antiderivative = 1.00, 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 = {2847, 2833, 2831, 2742, 2740, 2734, 2732} \begin {gather*} -\frac {d (c+3 d) \cos (e+f x)}{a f (c-d)^2 (c+d) \sqrt {c+d \sin (e+f x)}}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) \sqrt {c+d \sin (e+f x)}}+\frac {\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 )}{a 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 )}{a f (c-d)^2 (c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)),x]

[Out]

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

Rule 2732

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

Rule 2734

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/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2740

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

Rule 2742

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/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2831

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 2833

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 2847

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] && Ne
Q[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])

Rubi steps

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

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Mathematica [A]
time = 1.40, size = 264, normalized size = 1.08 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (-\frac {2 \left ((c+d)^2 \cos \left (\frac {1}{2} (e+f x)\right )+d (2 (c+d)+(c+3 d) \cos (e+f x)) \sin \left (\frac {1}{2} (e+f x)\right )\right )}{\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )}+(c+3 d) (c+d \sin (e+f x))+\left (c^2+4 c d+3 d^2\right ) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-\left (c^2-d^2\right ) F\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right )}{a (c-d)^2 (c+d) f (1+\sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(3/2)),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(925\) vs. \(2(298)=596\).
time = 6.78, size = 926, normalized size = 3.80

method	result	size
default	\(-\frac {\sqrt {\sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right ) d +\left (\cos ^{2}\left (f x +e \right )\right ) c}\, \left (4 \sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, \EllipticF \left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2} d -4 \sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, \EllipticF \left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{3}-\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, \EllipticE \left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{3}-3 \sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, \EllipticE \left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2} d +\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, \EllipticE \left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c \,d^{2}+3 \sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c +d}+\frac {d}{c +d}}\, \sqrt {-\frac {d \sin \left (f x +e \right )}{c -d}-\frac {d}{c -d}}\, \EllipticE \left (\sqrt {\frac {d \sin \left (f x +e \right )}{c -d}+\frac {c}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{3}+\left (\cos ^{2}\left (f x +e \right )\right ) c \,d^{2}+3 \left (\cos ^{2}\left (f x +e \right )\right ) d^{3}-c^{2} d \sin \left (f x +e \right )+d^{3} \sin \left (f x +e \right )+c^{2} d -d^{3}\right )}{d \left (c^{2}-d^{2}\right ) \sqrt {-\left (c +d \sin \left (f x +e \right )\right ) \left (\sin \left (f x +e \right )-1\right ) \left (1+\sin \left (f x +e \right )\right )}\, \left (c -d \right ) a \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\)	\(926\)

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))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-(sin(f*x+e)*cos(f*x+e)^2*d+cos(f*x+e)^2*c)^(1/2)*(4*(d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2)*(-d/(c+d)*sin(f*x+e)
+d/(c+d))^(1/2)*(-d/(c-d)*sin(f*x+e)-d/(c-d))^(1/2)*EllipticF((d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/(c+d
))^(1/2))*c^2*d-4*(d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2)*(-d/(c+d)*sin(f*x+e)+d/(c+d))^(1/2)*(-d/(c-d)*sin(f*x+e
)-d/(c-d))^(1/2)*EllipticF((d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/(c+d))^(1/2))*d^3-(d/(c-d)*sin(f*x+e)+1
/(c-d)*c)^(1/2)*(-d/(c+d)*sin(f*x+e)+d/(c+d))^(1/2)*(-d/(c-d)*sin(f*x+e)-d/(c-d))^(1/2)*EllipticE((d/(c-d)*sin
(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/(c+d))^(1/2))*c^3-3*(d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2)*(-d/(c+d)*sin(f*x+e)+
d/(c+d))^(1/2)*(-d/(c-d)*sin(f*x+e)-d/(c-d))^(1/2)*EllipticE((d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/(c+d)
)^(1/2))*c^2*d+(d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2)*(-d/(c+d)*sin(f*x+e)+d/(c+d))^(1/2)*(-d/(c-d)*sin(f*x+e)-d
/(c-d))^(1/2)*EllipticE((d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/(c+d))^(1/2))*c*d^2+3*(d/(c-d)*sin(f*x+e)+
1/(c-d)*c)^(1/2)*(-d/(c+d)*sin(f*x+e)+d/(c+d))^(1/2)*(-d/(c-d)*sin(f*x+e)-d/(c-d))^(1/2)*EllipticE((d/(c-d)*si
n(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/(c+d))^(1/2))*d^3+cos(f*x+e)^2*c*d^2+3*cos(f*x+e)^2*d^3-c^2*d*sin(f*x+e)+d^3*
sin(f*x+e)+c^2*d-d^3)/d/(c^2-d^2)/(-(c+d*sin(f*x+e))*(sin(f*x+e)-1)*(1+sin(f*x+e)))^(1/2)/(c-d)/a/cos(f*x+e)/(
c+d*sin(f*x+e))^(1/2)/f

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.18, size = 1205, normalized size = 4.94 \begin {gather*} \frac {{\left (\sqrt {2} {\left (2 \, c^{2} d - 3 \, c d^{2} - 3 \, d^{3}\right )} \cos \left (f x + e\right )^{2} - \sqrt {2} {\left (2 \, c^{3} - 3 \, c^{2} d - 3 \, c d^{2}\right )} \cos \left (f x + e\right ) - {\left (\sqrt {2} {\left (2 \, c^{2} d - 3 \, c d^{2} - 3 \, d^{3}\right )} \cos \left (f x + e\right ) + \sqrt {2} {\left (2 \, c^{3} - c^{2} d - 6 \, c d^{2} - 3 \, d^{3}\right )}\right )} \sin \left (f x + e\right ) - \sqrt {2} {\left (2 \, c^{3} - c^{2} d - 6 \, c d^{2} - 3 \, d^{3}\right )}\right )} \sqrt {i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) + {\left (\sqrt {2} {\left (2 \, c^{2} d - 3 \, c d^{2} - 3 \, d^{3}\right )} \cos \left (f x + e\right )^{2} - \sqrt {2} {\left (2 \, c^{3} - 3 \, c^{2} d - 3 \, c d^{2}\right )} \cos \left (f x + e\right ) - {\left (\sqrt {2} {\left (2 \, c^{2} d - 3 \, c d^{2} - 3 \, d^{3}\right )} \cos \left (f x + e\right ) + \sqrt {2} {\left (2 \, c^{3} - c^{2} d - 6 \, c d^{2} - 3 \, d^{3}\right )}\right )} \sin \left (f x + e\right ) - \sqrt {2} {\left (2 \, c^{3} - c^{2} d - 6 \, c d^{2} - 3 \, d^{3}\right )}\right )} \sqrt {-i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right ) + 3 \, {\left (\sqrt {2} {\left (i \, c d^{2} + 3 i \, d^{3}\right )} \cos \left (f x + e\right )^{2} + \sqrt {2} {\left (-i \, c^{2} d - 3 i \, c d^{2}\right )} \cos \left (f x + e\right ) + {\left (\sqrt {2} {\left (-i \, c d^{2} - 3 i \, d^{3}\right )} \cos \left (f x + e\right ) + \sqrt {2} {\left (-i \, c^{2} d - 4 i \, c d^{2} - 3 i \, d^{3}\right )}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (-i \, c^{2} d - 4 i \, c d^{2} - 3 i \, d^{3}\right )}\right )} \sqrt {i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) + 3 \, {\left (\sqrt {2} {\left (-i \, c d^{2} - 3 i \, d^{3}\right )} \cos \left (f x + e\right )^{2} + \sqrt {2} {\left (i \, c^{2} d + 3 i \, c d^{2}\right )} \cos \left (f x + e\right ) + {\left (\sqrt {2} {\left (i \, c d^{2} + 3 i \, d^{3}\right )} \cos \left (f x + e\right ) + \sqrt {2} {\left (i \, c^{2} d + 4 i \, c d^{2} + 3 i \, d^{3}\right )}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (i \, c^{2} d + 4 i \, c d^{2} + 3 i \, d^{3}\right )}\right )} \sqrt {-i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right ) + 6 \, {\left (c^{2} d - d^{3} + {\left (c d^{2} + 3 \, d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (c^{2} d + c d^{2} + 2 \, d^{3}\right )} \cos \left (f x + e\right ) - {\left (c^{2} d - d^{3} - {\left (c d^{2} + 3 \, d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{6 \, {\left ({\left (a c^{3} d^{2} - a c^{2} d^{3} - a c d^{4} + a d^{5}\right )} f \cos \left (f x + e\right )^{2} - {\left (a c^{4} d - a c^{3} d^{2} - a c^{2} d^{3} + a c d^{4}\right )} f \cos \left (f x + e\right ) - {\left (a c^{4} d - 2 \, a c^{2} d^{3} + a d^{5}\right )} f - {\left ({\left (a c^{3} d^{2} - a c^{2} d^{3} - a c d^{4} + a d^{5}\right )} f \cos \left (f x + e\right ) + {\left (a c^{4} d - 2 \, a c^{2} d^{3} + a d^{5}\right )} f\right )} \sin \left (f x + e\right )\right )}} \end {gather*}

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))^(3/2),x, algorithm="fricas")

[Out]

1/6*((sqrt(2)*(2*c^2*d - 3*c*d^2 - 3*d^3)*cos(f*x + e)^2 - sqrt(2)*(2*c^3 - 3*c^2*d - 3*c*d^2)*cos(f*x + e) -
(sqrt(2)*(2*c^2*d - 3*c*d^2 - 3*d^3)*cos(f*x + e) + sqrt(2)*(2*c^3 - c^2*d - 6*c*d^2 - 3*d^3))*sin(f*x + e) -
sqrt(2)*(2*c^3 - c^2*d - 6*c*d^2 - 3*d^3))*sqrt(I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*
c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d) + (sqrt(2)*(2*c^2*d - 3*c*d^2 - 3
*d^3)*cos(f*x + e)^2 - sqrt(2)*(2*c^3 - 3*c^2*d - 3*c*d^2)*cos(f*x + e) - (sqrt(2)*(2*c^2*d - 3*c*d^2 - 3*d^3)
*cos(f*x + e) + sqrt(2)*(2*c^3 - c^2*d - 6*c*d^2 - 3*d^3))*sin(f*x + e) - sqrt(2)*(2*c^3 - c^2*d - 6*c*d^2 - 3
*d^3))*sqrt(-I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos
(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d) + 3*(sqrt(2)*(I*c*d^2 + 3*I*d^3)*cos(f*x + e)^2 + sqrt(2)*(-I*c^2*d
- 3*I*c*d^2)*cos(f*x + e) + (sqrt(2)*(-I*c*d^2 - 3*I*d^3)*cos(f*x + e) + sqrt(2)*(-I*c^2*d - 4*I*c*d^2 - 3*I*
d^3))*sin(f*x + e) + sqrt(2)*(-I*c^2*d - 4*I*c*d^2 - 3*I*d^3))*sqrt(I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/
d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2
)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d)) + 3*(sqrt(2)*(-I*c*d^2 - 3*I*d^3)*cos(f*x + e)^
2 + sqrt(2)*(I*c^2*d + 3*I*c*d^2)*cos(f*x + e) + (sqrt(2)*(I*c*d^2 + 3*I*d^3)*cos(f*x + e) + sqrt(2)*(I*c^2*d
+ 4*I*c*d^2 + 3*I*d^3))*sin(f*x + e) + sqrt(2)*(I*c^2*d + 4*I*c*d^2 + 3*I*d^3))*sqrt(-I*d)*weierstrassZeta(-4/
3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(
-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d)) + 6*(c^2*d - d^3 + (c*d^2 +
3*d^3)*cos(f*x + e)^2 + (c^2*d + c*d^2 + 2*d^3)*cos(f*x + e) - (c^2*d - d^3 - (c*d^2 + 3*d^3)*cos(f*x + e))*s
in(f*x + e))*sqrt(d*sin(f*x + e) + c))/((a*c^3*d^2 - a*c^2*d^3 - a*c*d^4 + a*d^5)*f*cos(f*x + e)^2 - (a*c^4*d
- a*c^3*d^2 - a*c^2*d^3 + a*c*d^4)*f*cos(f*x + e) - (a*c^4*d - 2*a*c^2*d^3 + a*d^5)*f - ((a*c^3*d^2 - a*c^2*d^
3 - a*c*d^4 + a*d^5)*f*cos(f*x + e) + (a*c^4*d - 2*a*c^2*d^3 + a*d^5)*f)*sin(f*x + e))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{c \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )} + c \sqrt {c + d \sin {\left (e + f x \right )}} + d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )} + d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}}\, dx}{a} \end {gather*}

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))**(3/2),x)

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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))^(3/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{\left (a+a\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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