3.6.4 \(\int \frac {(c+d \sin (e+f x))^{5/2}}{a+a \sin (e+f x)} \, dx\) [504]

Optimal. Leaf size=246 \[ \frac {(3 c-5 d) d \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 a f}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{f (a+a \sin (e+f x))}-\frac {\left (3 c^2-20 c d+9 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 a f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(3 c-5 d) \left (c^2-d^2\right ) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 a f \sqrt {c+d \sin (e+f x)}} \]

[Out]

-(c-d)*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/f/(a+a*sin(f*x+e))+1/3*(3*c-5*d)*d*cos(f*x+e)*(c+d*sin(f*x+e))^(1/2)/
a/f+1/3*(3*c^2-20*c*d+9*d^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)*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/f/((c+d*sin(f*x+e))/(c+d))^(1/2)-1/3*(3*c-5
*d)*(c^2-d^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/f/(c+d*sin(f*x+e))^(1/2)

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Rubi [A]
time = 0.25, antiderivative size = 246, 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 = {2846, 2832, 2831, 2742, 2740, 2734, 2732} \begin {gather*} \frac {(3 c-5 d) \left (c^2-d^2\right ) \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 \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 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{f (a \sin (e+f x)+a)}+\frac {d (3 c-5 d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 a f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((3*c - 5*d)*d*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*a*f) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/
2))/(f*(a + a*Sin[e + f*x])) - ((3*c^2 - 20*c*d + 9*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[c +
d*Sin[e + f*x]])/(3*a*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + ((3*c - 5*d)*(c^2 - d^2)*EllipticF[(e - Pi/2 +
f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*a*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 2832

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

Rule 2846

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

Rubi steps

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

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Mathematica [A]
time = 0.95, size = 298, normalized size = 1.21 \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 (-3 (c-d)^2 (c+d \sin (e+f x))-2 d^2 \cos (e+f x) (c+d \sin (e+f x))+\frac {6 (c-d)^2 \sin \left (\frac {1}{2} (e+f x)\right ) (c+d \sin (e+f x))}{\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )}-d \left (15 c^2-12 c d+5 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}}+\left (3 c^2-20 c d+9 d^2\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c F\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right )}{3 a f (1+\sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*(-3*(c - d)^2*(c + d*Sin[e + f*x]) - 2*d^2*Cos[e + f*x]*(c + d*Sin[e
+ f*x]) + (6*(c - d)^2*Sin[(e + f*x)/2]*(c + d*Sin[e + f*x]))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) - 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)]))/(3*a*f*(1 + Sin[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. \(1371\) vs. \(2(294)=588\).
time = 6.17, size = 1372, normalized size = 5.58

method	result	size
default	\(\text {Expression too large to display}\)	\(1372\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(sin(f*x+e)*cos(f*x+e)^2*d+cos(f*x+e)^2*c)^(1/2)*(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)*sin(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/(
c+d))^(1/2))*c^4-20*(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*d+6*(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^2+20*(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^3-9*(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))*d^4+12*(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^3*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))*c^2*d^2-12*(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*d^3+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)*E
llipticF((d/(c-d)*sin(f*x+e)+1/(c-d)*c)^(1/2),((c-d)/(c+d))^(1/2))*d^4-2*d^4*sin(f*x+e)*cos(f*x+e)^2-3*cos(f*x
+e)^2*c^2*d^2+4*cos(f*x+e)^2*c*d^3-3*cos(f*x+e)^2*d^4+3*c^3*d*sin(f*x+e)-9*c^2*d^2*sin(f*x+e)+9*c*d^3*sin(f*x+
e)-3*d^4*sin(f*x+e)-3*c^3*d+9*c^2*d^2-9*d^3*c+3*d^4)/d/(-(c+d*sin(f*x+e))*(sin(f*x+e)-1)*(1+sin(f*x+e)))^(1/2)
/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((c+d*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e)),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.18, size = 843, normalized size = 3.43 \begin {gather*} \frac {{\left (\sqrt {2} {\left (6 \, c^{3} + 5 \, c^{2} d - 18 \, c d^{2} + 15 \, d^{3}\right )} \cos \left (f x + e\right ) + \sqrt {2} {\left (6 \, c^{3} + 5 \, c^{2} d - 18 \, c d^{2} + 15 \, d^{3}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (6 \, c^{3} + 5 \, c^{2} d - 18 \, c d^{2} + 15 \, 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 (6 \, c^{3} + 5 \, c^{2} d - 18 \, c d^{2} + 15 \, d^{3}\right )} \cos \left (f x + e\right ) + \sqrt {2} {\left (6 \, c^{3} + 5 \, c^{2} d - 18 \, c d^{2} + 15 \, d^{3}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (6 \, c^{3} + 5 \, c^{2} d - 18 \, c d^{2} + 15 \, 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 (-3 i \, c^{2} d + 20 i \, c d^{2} - 9 i \, d^{3}\right )} \cos \left (f x + e\right ) + \sqrt {2} {\left (-3 i \, c^{2} d + 20 i \, c d^{2} - 9 i \, d^{3}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (-3 i \, c^{2} d + 20 i \, c d^{2} - 9 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 (3 i \, c^{2} d - 20 i \, c d^{2} + 9 i \, d^{3}\right )} \cos \left (f x + e\right ) + \sqrt {2} {\left (3 i \, c^{2} d - 20 i \, c d^{2} + 9 i \, d^{3}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (3 i \, c^{2} d - 20 i \, c d^{2} + 9 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 (2 \, d^{3} \cos \left (f x + e\right )^{2} + 3 \, c^{2} d - 6 \, c d^{2} + 3 \, d^{3} + {\left (3 \, c^{2} d - 6 \, c d^{2} + 5 \, d^{3}\right )} \cos \left (f x + e\right ) + {\left (2 \, d^{3} \cos \left (f x + e\right ) - 3 \, c^{2} d + 6 \, c d^{2} - 3 \, d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}}{18 \, {\left (a d f \cos \left (f x + e\right ) + a d f \sin \left (f x + e\right ) + a d f\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*((sqrt(2)*(6*c^3 + 5*c^2*d - 18*c*d^2 + 15*d^3)*cos(f*x + e) + sqrt(2)*(6*c^3 + 5*c^2*d - 18*c*d^2 + 15*d
^3)*sin(f*x + e) + sqrt(2)*(6*c^3 + 5*c^2*d - 18*c*d^2 + 15*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)
*(6*c^3 + 5*c^2*d - 18*c*d^2 + 15*d^3)*cos(f*x + e) + sqrt(2)*(6*c^3 + 5*c^2*d - 18*c*d^2 + 15*d^3)*sin(f*x +
e) + sqrt(2)*(6*c^3 + 5*c^2*d - 18*c*d^2 + 15*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)*(-3*I*c^2
*d + 20*I*c*d^2 - 9*I*d^3)*cos(f*x + e) + sqrt(2)*(-3*I*c^2*d + 20*I*c*d^2 - 9*I*d^3)*sin(f*x + e) + sqrt(2)*(
-3*I*c^2*d + 20*I*c*d^2 - 9*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)*(3*I*c^2*d - 20*I*c*d^2 + 9*I*d^3)*cos(f*x + e) + sqrt(2)*(3*
I*c^2*d - 20*I*c*d^2 + 9*I*d^3)*sin(f*x + e) + sqrt(2)*(3*I*c^2*d - 20*I*c*d^2 + 9*I*d^3))*sqrt(-I*d)*weierstr
assZeta(-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*(2*d^3*cos(f
*x + e)^2 + 3*c^2*d - 6*c*d^2 + 3*d^3 + (3*c^2*d - 6*c*d^2 + 5*d^3)*cos(f*x + e) + (2*d^3*cos(f*x + e) - 3*c^2
*d + 6*c*d^2 - 3*d^3)*sin(f*x + e))*sqrt(d*sin(f*x + e) + c))/(a*d*f*cos(f*x + e) + a*d*f*sin(f*x + e) + a*d*f
)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}}{a+a\,\sin \left (e+f\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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