\(\int \frac {(3+3 \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{3/2}} \, dx\) [575]

Optimal result

Integrand size = 29, antiderivative size = 114 \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {6 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {d} \cos (e+f x)}{\sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{d^{3/2} f}+\frac {18 (c-d) \cos (e+f x)}{d (c+d) f \sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2841, 21, 2854, 211} \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 a^2 (c-d) \cos (e+f x)}{d f (c+d) \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}-\frac {2 a^{3/2} \arctan \left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{d^{3/2} f} \]

[In]

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

[Out]

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2841

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(-b^2)*(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c
+ a*d))), x] + Dist[b^2/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1
)*Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b*c*(m - 1) - a*d*(m + 2*n + 1))*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[m, 1] && LtQ[n, -1
] && (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))

Rule 2854

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[
-2*(b/f), Subst[Int[1/(b + d*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*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]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(380\) vs. \(2(114)=228\).

Time = 5.37 (sec) , antiderivative size = 380, normalized size of antiderivative = 3.33 \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {3 \sqrt {3} (1+\sin (e+f x))^{3/2} \left (-2 c \sqrt {d} \cos \left (\frac {1}{2} (e+f x)\right )+2 d^{3/2} \cos \left (\frac {1}{2} (e+f x)\right )+2 c \sqrt {d} \sin \left (\frac {1}{2} (e+f x)\right )-2 d^{3/2} \sin \left (\frac {1}{2} (e+f x)\right )-2 (c+d) \arctan \left (\frac {\sqrt {2} \sqrt {d} \sin \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{\sqrt {c+d \sin (e+f x)}}\right ) \sqrt {c+d \sin (e+f x)}-(c+d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{\sqrt {c+d \sin (e+f x)}}\right ) \sqrt {c+d \sin (e+f x)}+c \log \left (\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )+\sqrt {c+d \sin (e+f x)}\right ) \sqrt {c+d \sin (e+f x)}+d \log \left (\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )+\sqrt {c+d \sin (e+f x)}\right ) \sqrt {c+d \sin (e+f x)}\right )}{d^{3/2} (c+d) f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \sqrt {c+d \sin (e+f x)}} \]

[In]

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

[Out]

(-3*Sqrt[3]*(1 + Sin[e + f*x])^(3/2)*(-2*c*Sqrt[d]*Cos[(e + f*x)/2] + 2*d^(3/2)*Cos[(e + f*x)/2] + 2*c*Sqrt[d]
*Sin[(e + f*x)/2] - 2*d^(3/2)*Sin[(e + f*x)/2] - 2*(c + d)*ArcTan[(Sqrt[2]*Sqrt[d]*Sin[(2*e - Pi + 2*f*x)/4])/
Sqrt[c + d*Sin[e + f*x]]]*Sqrt[c + d*Sin[e + f*x]] - (c + d)*ArcTanh[(Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4
])/Sqrt[c + d*Sin[e + f*x]]]*Sqrt[c + d*Sin[e + f*x]] + c*Log[Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4] + Sqrt
[c + d*Sin[e + f*x]]]*Sqrt[c + d*Sin[e + f*x]] + d*Log[Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4] + Sqrt[c + d*
Sin[e + f*x]]]*Sqrt[c + d*Sin[e + f*x]]))/(d^(3/2)*(c + d)*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sqrt[c +
d*Sin[e + f*x]])

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(4781\) vs. \(2(101)=202\).

Time = 1.09 (sec) , antiderivative size = 4782, normalized size of antiderivative = 41.95

\[\text {output too large to display}\]

[In]

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

[Out]

-1/f*sec(f*x+e)*a*(-arctan(((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1
/2))*sin(f*x+e)^2*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*d^6+2*cos(f*x+e)^2*(-(d^2/c^2)^(
1/2)*c)^(1/2)*c^3*d^3-6*cos(f*x+e)^2*(-(d^2/c^2)^(1/2)*c)^(1/2)*c^2*d^4+6*cos(f*x+e)^2*(-(d^2/c^2)^(1/2)*c)^(1
/2)*c*d^5-4*c^3*(-(d^2/c^2)^(1/2)*c)^(1/2)*sin(f*x+e)*d^3+4*c*(-(d^2/c^2)^(1/2)*c)^(1/2)*sin(f*x+e)*d^5-c^4*ar
ctan(((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*((c+d*sin(f*x+e))
*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*d^2+2*c^2*arctan(((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+
d))^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*d^4-2*(-(d^2
/c^2)^(1/2)*c)^(1/2)*sin(f*x+e)*d^6+arctan(((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/
c^2)^(1/2)*c)^(1/2))*sin(f*x+e)*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*d^6+2*c^4*(-(d^2/c
^2)^(1/2)*c)^(1/2)*sin(f*x+e)*d^2+cos(f*x+e)*c*(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*c^2*d^2+d^4*(d^2/c^2)^(
1/2)-4*c^2*d^2-4*d^4)*c)^(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*arctan(((d^2/c^2)^(1/2)*c*sin(f*x+e)+d*cos(f*x+e)-d)
/((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c*sin(f*x+e)+d*cos(f*x+e)-d)*((
d^2/c^2)^(1/2)*c^2-d^2)*c*((d^2/c^2)^(1/2)-1)/(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*c^2*d^2+d^4*(d^2/c^2)^(1
/2)-4*c^2*d^2-4*d^4)*c)^(1/2))*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*d^2+cos(f*x+e)*c^3*
(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*c^2*d^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2)*(-(d^2/c^2)^(1/2
)*c)^(1/2)*arctan(((d^2/c^2)^(1/2)*c*sin(f*x+e)+d*cos(f*x+e)-d)/((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x
+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c*sin(f*x+e)+d*cos(f*x+e)-d)*((d^2/c^2)^(1/2)*c^2-d^2)*c*((d^2/c^2)^(1/2)-1)/(
((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*c^2*d^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2))*((c+d*sin(f*x+e)
)*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*(d^2/c^2)^(1/2)-cos(f*x+e)*(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)
*c^2*d^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*arctan(((d^2/c^2)^(1/2)*c*si
n(f*x+e)+d*cos(f*x+e)-d)/((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c*sin(f
*x+e)+d*cos(f*x+e)-d)*((d^2/c^2)^(1/2)*c^2-d^2)*c*((d^2/c^2)^(1/2)-1)/(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*
c^2*d^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2))*sin(f*x+e)*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f
*x+e)+d))^(1/2)*d^3+cos(f*x+e)*c^2*(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*c^2*d^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d
^2-4*d^4)*c)^(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*arctan(((d^2/c^2)^(1/2)*c*sin(f*x+e)+d*cos(f*x+e)-d)/((c+d*sin(f
*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c*sin(f*x+e)+d*cos(f*x+e)-d)*((d^2/c^2)^(1/
2)*c^2-d^2)*c*((d^2/c^2)^(1/2)-1)/(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*c^2*d^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^
2-4*d^4)*c)^(1/2))*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*d+2*(-(d^2/c^2)^(1/2)*c)^(1/2)*
sin(f*x+e)^2*d^6+cos(f*x+e)*c^2*(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*c^2*d^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-
4*d^4)*c)^(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*arctan(((d^2/c^2)^(1/2)*c*sin(f*x+e)+d*cos(f*x+e)-d)/((c+d*sin(f*x+
e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c*sin(f*x+e)+d*cos(f*x+e)-d)*((d^2/c^2)^(1/2)*
c^2-d^2)*c*((d^2/c^2)^(1/2)-1)/(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*c^2*d^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4
*d^4)*c)^(1/2))*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*(d^2/c^2)^(1/2)*d-cos(f*x+e)*(((d^
2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*c^2*d^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2)*(-(d^2/c^2)^(1/2)*c)^
(1/2)*arctan(((d^2/c^2)^(1/2)*c*sin(f*x+e)+d*cos(f*x+e)-d)/((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d
))^(1/2)/(-(d^2/c^2)^(1/2)*c*sin(f*x+e)+d*cos(f*x+e)-d)*((d^2/c^2)^(1/2)*c^2-d^2)*c*((d^2/c^2)^(1/2)-1)/(((d^2
/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*c^2*d^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2))*sin(f*x+e)*((c+d*sin(
f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*c*d^2-4*(-(d^2/c^2)^(1/2)*c)^(1/2)*c*d^5+4*(-(d^2/c^2)^(1/2)
*c)^(1/2)*c^3*d^3-2*c^4*(-(d^2/c^2)^(1/2)*c)^(1/2)*d^2-cos(f*x+e)*(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*c^2*
d^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*arctan(((d^2/c^2)^(1/2)*c*sin(f*x
+e)+d*cos(f*x+e)-d)/((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c*sin(f*x+e)
+d*cos(f*x+e)-d)*((d^2/c^2)^(1/2)*c^2-d^2)*c*((d^2/c^2)^(1/2)-1)/(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*c^2*d
^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2))*sin(f*x+e)*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)
+d))^(1/2)*(d^2/c^2)^(1/2)*c^2*d-cos(f*x+e)*(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*c^2*d^2+d^4*(d^2/c^2)^(1/2
)-4*c^2*d^2-4*d^4)*c)^(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*arctan(((d^2/c^2)^(1/2)*c*sin(f*x+e)+d*cos(f*x+e)-d)/((
c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c*sin(f*x+e)+d*cos(f*x+e)-d)*((d^2
/c^2)^(1/2)*c^2-d^2)*c*((d^2/c^2)^(1/2)-1)/(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*c^2*d^2+d^4*(d^2/c^2)^(1/2)
-4*c^2*d^2-4*d^4)*c)^(1/2))*sin(f*x+e)*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*(d^2/c^2)^(
1/2)*c*d^2+arctan(((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*((c+
d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*(d^2/c^2)^(1/2)*c^3*cos(f*x+e)^2*d^3+cos(f*x+e)^2*arct
an(((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*((c+d*sin(f*x+e))*d
/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*(d^2/c^2)^(1/2)*c^2*d^4-c^5*arctan(((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/
2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*sin(f*x+e)*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*
x+e)+d))^(1/2)*(d^2/c^2)^(1/2)*d+2*c^3*arctan(((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d
^2/c^2)^(1/2)*c)^(1/2))*sin(f*x+e)*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*(d^2/c^2)^(1/2)
*d^3-arctan(((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*sin(f*x+e)
*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*(d^2/c^2)^(1/2)*d^5*c-cos(f*x+e)^2*arctan(((c+d*s
in(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*((c+d*sin(f*x+e))*d/((d^2/c^2
)^(1/2)*c*sin(f*x+e)+d))^(1/2)*(d^2/c^2)^(1/2)*c^4*d^2+arctan(((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e
)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*sin(f*x+e)^2*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/
2)*(d^2/c^2)^(1/2)*c*d^5+c^5*arctan(((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1
/2)*c)^(1/2))*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*(d^2/c^2)^(1/2)*d-arctan(((c+d*sin(f
*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1
/2)*c*sin(f*x+e)+d))^(1/2)*c^2*cos(f*x+e)^2*d^4+cos(f*x+e)^2*arctan(((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin
(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*c^3*
d^3-cos(f*x+e)^2*arctan(((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2)
)*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*c*d^5+c^4*arctan(((c+d*sin(f*x+e))*d/((d^2/c^2)^
(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*sin(f*x+e)*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin
(f*x+e)+d))^(1/2)*d^2-2*c^2*arctan(((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/
2)*c)^(1/2))*sin(f*x+e)*((c+d*sin(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)*d^4-2*c^3*arctan(((c+d*sin
(f*x+e))*d/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d))^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*((c+d*sin(f*x+e))*d/((d^2/c^2)^
(1/2)*c*sin(f*x+e)+d))^(1/2)*(d^2/c^2)^(1/2)*d^3)*(c+d*sin(f*x+e))^(1/2)*(a*(sin(f*x+e)+1))^(1/2)/(cos(f*x+e)^
2*d^2+c^2-d^2)/d^3/(c^2-2*c*d+d^2)/(-(d^2/c^2)^(1/2)*c)^(1/2)/(c+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (101) = 202\).

Time = 0.61 (sec) , antiderivative size = 1297, normalized size of antiderivative = 11.38 \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/4*((a*c^2 + 2*a*c*d + a*d^2 - (a*c*d + a*d^2)*cos(f*x + e)^2 + (a*c^2 + a*c*d)*cos(f*x + e) + (a*c^2 + 2*a
*c*d + a*d^2 + (a*c*d + a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(-a/d)*log((128*a*d^4*cos(f*x + e)^5 + a*c^4 +
4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + 128*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^4 - 32*(5*a*c^2*d^2 - 14*a*
c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)*cos(f*x + e)^2 - 8*(16*d^4
*cos(f*x + e)^4 - c^3*d + 17*c^2*d^2 - 59*c*d^3 + 51*d^4 + 24*(c*d^3 - d^4)*cos(f*x + e)^3 - 2*(5*c^2*d^2 - 26
*c*d^3 + 33*d^4)*cos(f*x + e)^2 - (c^3*d - 7*c^2*d^2 + 31*c*d^3 - 25*d^4)*cos(f*x + e) + (16*d^4*cos(f*x + e)^
3 + c^3*d - 17*c^2*d^2 + 59*c*d^3 - 51*d^4 - 8*(3*c*d^3 - 5*d^4)*cos(f*x + e)^2 - 2*(5*c^2*d^2 - 14*c*d^3 + 13
*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-a/d) + (a*c^4 - 28*a
*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*cos(f*x + e)^4 + a*c^4 + 4*a*c^3*d
+ 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a*c^2*d^2 - 6*a*c*d^3 + 5*a*
d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e))*sin(f*x + e))/(cos(f*x +
e) + sin(f*x + e) + 1)) + 8*(a*c - a*d + (a*c - a*d)*cos(f*x + e) - (a*c - a*d)*sin(f*x + e))*sqrt(a*sin(f*x
+ e) + a)*sqrt(d*sin(f*x + e) + c))/((c*d^2 + d^3)*f*cos(f*x + e)^2 - (c^2*d + c*d^2)*f*cos(f*x + e) - (c^2*d
+ 2*c*d^2 + d^3)*f - ((c*d^2 + d^3)*f*cos(f*x + e) + (c^2*d + 2*c*d^2 + d^3)*f)*sin(f*x + e)), -1/2*((a*c^2 +
2*a*c*d + a*d^2 - (a*c*d + a*d^2)*cos(f*x + e)^2 + (a*c^2 + a*c*d)*cos(f*x + e) + (a*c^2 + 2*a*c*d + a*d^2 + (
a*c*d + a*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a/d)*arctan(1/4*(8*d^2*cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 -
8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(a/d)/(2*a*d^2*cos(f*x + e)^
3 - (3*a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) - (a*c^2 - a*c*d + 2*a*d^2)*cos(f*x + e))) + 4*(a*c - a*d + (a
*c - a*d)*cos(f*x + e) - (a*c - a*d)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/((c*d^2
+ d^3)*f*cos(f*x + e)^2 - (c^2*d + c*d^2)*f*cos(f*x + e) - (c^2*d + 2*c*d^2 + d^3)*f - ((c*d^2 + d^3)*f*cos(f*
x + e) + (c^2*d + 2*c*d^2 + d^3)*f)*sin(f*x + e))]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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