\(\int \frac {(3+3 \sin (e+f x))^2}{\sqrt {c+d \sin (e+f x)}} \, dx\) [492]

Optimal result

Integrand size = 27, antiderivative size = 174 \[ \int \frac {(3+3 \sin (e+f x))^2}{\sqrt {c+d \sin (e+f x)}} \, dx=-\frac {6 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f}-\frac {12 (c-3 d) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {12 (c-2 d) (c-d) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{d^2 f \sqrt {c+d \sin (e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2842, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {(3+3 \sin (e+f x))^2}{\sqrt {c+d \sin (e+f x)}} \, dx=\frac {4 a^2 (c-2 d) (c-d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{3 d^2 f \sqrt {c+d \sin (e+f x)}}-\frac {4 a^2 (c-3 d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 a^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f} \]

[In]

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

[Out]

(-2*a^2*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(3*d*f) - (4*a^2*(c - 3*d)*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/
(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(3*d^2*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (4*a^2*(c - 2*d)*(c - d)*Ell
ipticF[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(3*d^2*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 2842

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 - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Dist[1/(
d*(m + n)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c*(m - 2) + b^2*d*(n + 1) + a^2*d
*(m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*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] && GtQ[m, 1] &&  !LtQ[n, -1] && (IntegersQ[2*m,
2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))

Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}+\frac {2 \int \frac {2 a^2 d-a^2 (c-3 d) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 d} \\ & = -\frac {2 a^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}-\frac {\left (2 a^2 (c-3 d)\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{3 d^2}+\frac {\left (2 a^2 (c-2 d) (c-d)\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 d^2} \\ & = -\frac {2 a^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}-\frac {\left (2 a^2 (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}{3 d^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (2 a^2 (c-2 d) (c-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}{3 d^2 \sqrt {c+d \sin (e+f x)}} \\ & = -\frac {2 a^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 d f}-\frac {4 a^2 (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 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {4 a^2 (c-2 d) (c-d) \operatorname {EllipticF}\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 d^2 f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.90 \[ \int \frac {(3+3 \sin (e+f x))^2}{\sqrt {c+d \sin (e+f x)}} \, dx=\frac {3 \left (-2 d \cos (e+f x) (c+d \sin (e+f x))+4 \left (c^2-2 c d-3 d^2\right ) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}-4 \left (c^2-3 c d+2 d^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 e+\pi -2 f x),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right )}{d^2 f \sqrt {c+d \sin (e+f x)}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(720\) vs. \(2(239)=478\).

Time = 3.31 (sec) , antiderivative size = 721, normalized size of antiderivative = 4.14

method	result	size
parts	\(\frac {2 a^{2} \left (c -d \right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{d \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}+\frac {a^{2} \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (-\frac {2 \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}{3 d}+\frac {2 \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )}{3 \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}-\frac {4 c \left (\frac {c}{d}-1\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {\frac {d \left (1-\sin \left (f x +e \right )\right )}{c +d}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) d}{c -d}}\, \left (\left (-\frac {c}{d}-1\right ) E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )+F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )\right )}{3 d \sqrt {-\left (-d \sin \left (f x +e \right )-c \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}\right )}{\cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}-\frac {4 a^{2} \left (c -d \right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, \left (E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c +E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d -F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d \right )}{d^{2} \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\)	\(721\)
default	\(-\frac {2 a^{2} \left (2 \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2} d -12 \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{2} c +10 \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{3}-2 \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{3}+6 \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2} d +2 \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c \,d^{2}-6 \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{3}-\left (\sin ^{3}\left (f x +e \right )\right ) d^{3}-\left (\sin ^{2}\left (f x +e \right )\right ) c \,d^{2}+d^{3} \sin \left (f x +e \right )+c \,d^{2}\right )}{3 d^{3} \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\)	\(758\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 465, normalized size of antiderivative = 2.67 \[ \int \frac {(3+3 \sin (e+f x))^2}{\sqrt {c+d \sin (e+f x)}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {d \sin \left (f x + e\right ) + c} a^{2} d^{2} \cos \left (f x + e\right ) - 2 \, \sqrt {2} {\left (a^{2} c^{2} - 3 \, a^{2} c d + 3 \, a^{2} d^{2}\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 ) - 2 \, \sqrt {2} {\left (a^{2} c^{2} - 3 \, a^{2} c d + 3 \, a^{2} d^{2}\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 \, \sqrt {2} {\left (-i \, a^{2} c d + 3 i \, a^{2} d^{2}\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 \, \sqrt {2} {\left (i \, a^{2} c d - 3 i \, a^{2} d^{2}\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 )\right )}}{9 \, d^{3} f} \]

[In]

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

[Out]

-2/9*(3*sqrt(d*sin(f*x + e) + c)*a^2*d^2*cos(f*x + e) - 2*sqrt(2)*(a^2*c^2 - 3*a^2*c*d + 3*a^2*d^2)*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*s
in(f*x + e) - 2*I*c)/d) - 2*sqrt(2)*(a^2*c^2 - 3*a^2*c*d + 3*a^2*d^2)*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*a^2*c*d + 3*I*a^2*d^2)*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*a^2*c*d - 3*I*a^2*d^2)*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)))/(d^3*f)

Sympy [F]

\[ \int \frac {(3+3 \sin (e+f x))^2}{\sqrt {c+d \sin (e+f x)}} \, dx=a^{2} \left (\int \frac {2 \sin {\left (e + f x \right )}}{\sqrt {c + d \sin {\left (e + f x \right )}}}\, dx + \int \frac {\sin ^{2}{\left (e + f x \right )}}{\sqrt {c + d \sin {\left (e + f x \right )}}}\, dx + \int \frac {1}{\sqrt {c + d \sin {\left (e + f x \right )}}}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {(3+3 \sin (e+f x))^2}{\sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}{\sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {(3+3 \sin (e+f x))^2}{\sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}{\sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(3+3 \sin (e+f x))^2}{\sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2}{\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \]

[In]

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

[Out]

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