3.1.28 \(\int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx\) [28]

Optimal. Leaf size=168 \[ -\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {g} \cos (e+f x)}{\sqrt {2} \sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\right )}{\sqrt {a} (c-d) f \sqrt {g}}+\frac {2 d \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \cos (e+f x)}{\sqrt {c+d} \sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\right )}{\sqrt {a} \sqrt {c} (c-d) \sqrt {c+d} f \sqrt {g}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.35, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3017, 2861, 211, 3009} \begin {gather*} \frac {2 d \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a} \sqrt {g \sin (e+f x)}}\right )}{\sqrt {a} \sqrt {c} f \sqrt {g} (c-d) \sqrt {c+d}}-\frac {\sqrt {2} \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {g} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a} \sqrt {g \sin (e+f x)}}\right )}{\sqrt {a} f \sqrt {g} (c-d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]

[Out]

-((Sqrt[2]*ArcTan[(Sqrt[a]*Sqrt[g]*Cos[e + f*x])/(Sqrt[2]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])])/(Sq
rt[a]*(c - d)*f*Sqrt[g])) + (2*d*ArcTan[(Sqrt[a]*Sqrt[c]*Sqrt[g]*Cos[e + f*x])/(Sqrt[c + d]*Sqrt[g*Sin[e + f*x
]]*Sqrt[a + a*Sin[e + f*x]])])/(Sqrt[a]*Sqrt[c]*(c - d)*Sqrt[c + d]*f*Sqrt[g])

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 2861

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> D
ist[-2*(a/f), Subst[Int[1/(2*b^2 - (a*c - 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]

Rule 3009

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) +
(f_.)*(x_)])), x_Symbol] :> Dist[-2*(b/f), Subst[Int[1/(b*c + a*d + c*g*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[g*S
in[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]))], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && EqQ[a^
2 - b^2, 0]

Rule 3017

Int[1/(Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]), x], x
] - Dist[d/(b*c - a*d), Int[Sqrt[a + b*Sin[e + f*x]]/(Sqrt[g*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fr
eeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])

Rubi steps

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

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(3062\) vs. \(2(168)=336\).
time = 24.89, size = 3062, normalized size = 18.23 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]

[Out]

-(((-(d*(c - d + Sqrt[-c^2 + d^2])*Sqrt[d + Sqrt[-c^2 + d^2]]*ArcTan[(Sqrt[2]*Sqrt[-d + Sqrt[-c^2 + d^2]]*Sqrt
[Tan[(e + f*x)/4]])/(Sqrt[c]*Sqrt[-1 + Tan[(e + f*x)/4]^2])]) + 2*Sqrt[c]*Sqrt[-c^2 + d^2]*Sqrt[-d + Sqrt[-c^2
+ d^2]]*Sqrt[d + Sqrt[-c^2 + d^2]]*ArcTanh[(Sqrt[2]*Sqrt[Tan[(e + f*x)/4]])/Sqrt[-1 + Tan[(e + f*x)/4]^2]] -
d*Sqrt[-d + Sqrt[-c^2 + d^2]]*(-c + d + Sqrt[-c^2 + d^2])*ArcTanh[(Sqrt[2]*Sqrt[d + Sqrt[-c^2 + d^2]]*Sqrt[Tan
[(e + f*x)/4]])/(Sqrt[c]*Sqrt[-1 + Tan[(e + f*x)/4]^2])])*Sec[(e + f*x)/4]^2*Sqrt[Sin[e + f*x]])/(Sqrt[2]*Sqrt
[c]*(c - d)*Sqrt[-c^2 + d^2]*Sqrt[-d + Sqrt[-c^2 + d^2]]*Sqrt[d + Sqrt[-c^2 + d^2]]*f*Sqrt[g*Sin[e + f*x]]*Sqr
t[a*(1 + Sin[e + f*x])]*(c + d*Sin[e + f*x])*Sqrt[Tan[(e + f*x)/4]]*Sqrt[-1 + Tan[(e + f*x)/4]^2]*(((-(d*(c -
d + Sqrt[-c^2 + d^2])*Sqrt[d + Sqrt[-c^2 + d^2]]*ArcTan[(Sqrt[2]*Sqrt[-d + Sqrt[-c^2 + d^2]]*Sqrt[Tan[(e + f*x
)/4]])/(Sqrt[c]*Sqrt[-1 + Tan[(e + f*x)/4]^2])]) + 2*Sqrt[c]*Sqrt[-c^2 + d^2]*Sqrt[-d + Sqrt[-c^2 + d^2]]*Sqrt
[d + Sqrt[-c^2 + d^2]]*ArcTanh[(Sqrt[2]*Sqrt[Tan[(e + f*x)/4]])/Sqrt[-1 + Tan[(e + f*x)/4]^2]] - d*Sqrt[-d + S
qrt[-c^2 + d^2]]*(-c + d + Sqrt[-c^2 + d^2])*ArcTanh[(Sqrt[2]*Sqrt[d + Sqrt[-c^2 + d^2]]*Sqrt[Tan[(e + f*x)/4]
])/(Sqrt[c]*Sqrt[-1 + Tan[(e + f*x)/4]^2])])*Sec[(e + f*x)/4]^4*Sqrt[Sin[e + f*x]]*Sqrt[Tan[(e + f*x)/4]])/(4*
Sqrt[2]*Sqrt[c]*(c - d)*Sqrt[-c^2 + d^2]*Sqrt[-d + Sqrt[-c^2 + d^2]]*Sqrt[d + Sqrt[-c^2 + d^2]]*(-1 + Tan[(e +
f*x)/4]^2)^(3/2)) + ((-(d*(c - d + Sqrt[-c^2 + d^2])*Sqrt[d + Sqrt[-c^2 + d^2]]*ArcTan[(Sqrt[2]*Sqrt[-d + Sqr
t[-c^2 + d^2]]*Sqrt[Tan[(e + f*x)/4]])/(Sqrt[c]*Sqrt[-1 + Tan[(e + f*x)/4]^2])]) + 2*Sqrt[c]*Sqrt[-c^2 + d^2]*
Sqrt[-d + Sqrt[-c^2 + d^2]]*Sqrt[d + Sqrt[-c^2 + d^2]]*ArcTanh[(Sqrt[2]*Sqrt[Tan[(e + f*x)/4]])/Sqrt[-1 + Tan[
(e + f*x)/4]^2]] - d*Sqrt[-d + Sqrt[-c^2 + d^2]]*(-c + d + Sqrt[-c^2 + d^2])*ArcTanh[(Sqrt[2]*Sqrt[d + Sqrt[-c
^2 + d^2]]*Sqrt[Tan[(e + f*x)/4]])/(Sqrt[c]*Sqrt[-1 + Tan[(e + f*x)/4]^2])])*Sec[(e + f*x)/4]^4*Sqrt[Sin[e + f
*x]])/(8*Sqrt[2]*Sqrt[c]*(c - d)*Sqrt[-c^2 + d^2]*Sqrt[-d + Sqrt[-c^2 + d^2]]*Sqrt[d + Sqrt[-c^2 + d^2]]*Tan[(
e + f*x)/4]^(3/2)*Sqrt[-1 + Tan[(e + f*x)/4]^2]) - ((-(d*(c - d + Sqrt[-c^2 + d^2])*Sqrt[d + Sqrt[-c^2 + d^2]]
*ArcTan[(Sqrt[2]*Sqrt[-d + Sqrt[-c^2 + d^2]]*Sqrt[Tan[(e + f*x)/4]])/(Sqrt[c]*Sqrt[-1 + Tan[(e + f*x)/4]^2])])
+ 2*Sqrt[c]*Sqrt[-c^2 + d^2]*Sqrt[-d + Sqrt[-c^2 + d^2]]*Sqrt[d + Sqrt[-c^2 + d^2]]*ArcTanh[(Sqrt[2]*Sqrt[Tan
[(e + f*x)/4]])/Sqrt[-1 + Tan[(e + f*x)/4]^2]] - d*Sqrt[-d + Sqrt[-c^2 + d^2]]*(-c + d + Sqrt[-c^2 + d^2])*Arc
Tanh[(Sqrt[2]*Sqrt[d + Sqrt[-c^2 + d^2]]*Sqrt[Tan[(e + f*x)/4]])/(Sqrt[c]*Sqrt[-1 + Tan[(e + f*x)/4]^2])])*Cos
[e + f*x]*Sec[(e + f*x)/4]^2)/(2*Sqrt[2]*Sqrt[c]*(c - d)*Sqrt[-c^2 + d^2]*Sqrt[-d + Sqrt[-c^2 + d^2]]*Sqrt[d +
Sqrt[-c^2 + d^2]]*Sqrt[Sin[e + f*x]]*Sqrt[Tan[(e + f*x)/4]]*Sqrt[-1 + Tan[(e + f*x)/4]^2]) - ((-(d*(c - d + S
qrt[-c^2 + d^2])*Sqrt[d + Sqrt[-c^2 + d^2]]*ArcTan[(Sqrt[2]*Sqrt[-d + Sqrt[-c^2 + d^2]]*Sqrt[Tan[(e + f*x)/4]]
)/(Sqrt[c]*Sqrt[-1 + Tan[(e + f*x)/4]^2])]) + 2*Sqrt[c]*Sqrt[-c^2 + d^2]*Sqrt[-d + Sqrt[-c^2 + d^2]]*Sqrt[d +
Sqrt[-c^2 + d^2]]*ArcTanh[(Sqrt[2]*Sqrt[Tan[(e + f*x)/4]])/Sqrt[-1 + Tan[(e + f*x)/4]^2]] - d*Sqrt[-d + Sqrt[-
c^2 + d^2]]*(-c + d + Sqrt[-c^2 + d^2])*ArcTanh[(Sqrt[2]*Sqrt[d + Sqrt[-c^2 + d^2]]*Sqrt[Tan[(e + f*x)/4]])/(S
qrt[c]*Sqrt[-1 + Tan[(e + f*x)/4]^2])])*Sec[(e + f*x)/4]^2*Sqrt[Sin[e + f*x]]*Sqrt[Tan[(e + f*x)/4]])/(2*Sqrt[
2]*Sqrt[c]*(c - d)*Sqrt[-c^2 + d^2]*Sqrt[-d + Sqrt[-c^2 + d^2]]*Sqrt[d + Sqrt[-c^2 + d^2]]*Sqrt[-1 + Tan[(e +
f*x)/4]^2]) - (Sec[(e + f*x)/4]^2*Sqrt[Sin[e + f*x]]*((2*Sqrt[c]*Sqrt[-c^2 + d^2]*Sqrt[-d + Sqrt[-c^2 + d^2]]*
Sqrt[d + Sqrt[-c^2 + d^2]]*(-1/2*(Sec[(e + f*x)/4]^2*Tan[(e + f*x)/4]^(3/2))/(Sqrt[2]*(-1 + Tan[(e + f*x)/4]^2
)^(3/2)) + Sec[(e + f*x)/4]^2/(4*Sqrt[2]*Sqrt[Tan[(e + f*x)/4]]*Sqrt[-1 + Tan[(e + f*x)/4]^2])))/(1 - (2*Tan[(
e + f*x)/4])/(-1 + Tan[(e + f*x)/4]^2)) - (d*(c - d + Sqrt[-c^2 + d^2])*Sqrt[d + Sqrt[-c^2 + d^2]]*(-1/2*(Sqrt
[-d + Sqrt[-c^2 + d^2]]*Sec[(e + f*x)/4]^2*Tan[(e + f*x)/4]^(3/2))/(Sqrt[2]*Sqrt[c]*(-1 + Tan[(e + f*x)/4]^2)^
(3/2)) + (Sqrt[-d + Sqrt[-c^2 + d^2]]*Sec[(e + f*x)/4]^2)/(4*Sqrt[2]*Sqrt[c]*Sqrt[Tan[(e + f*x)/4]]*Sqrt[-1 +
Tan[(e + f*x)/4]^2])))/(1 + (2*(-d + Sqrt[-c^2 + d^2])*Tan[(e + f*x)/4])/(c*(-1 + Tan[(e + f*x)/4]^2))) - (d*S
qrt[-d + Sqrt[-c^2 + d^2]]*(-c + d + Sqrt[-c^2 + d^2])*(-1/2*(Sqrt[d + Sqrt[-c^2 + d^2]]*Sec[(e + f*x)/4]^2*Ta
n[(e + f*x)/4]^(3/2))/(Sqrt[2]*Sqrt[c]*(-1 + Tan[(e + f*x)/4]^2)^(3/2)) + (Sqrt[d + Sqrt[-c^2 + d^2]]*Sec[(e +
f*x)/4]^2)/(4*Sqrt[2]*Sqrt[c]*Sqrt[Tan[(e + f*x)/4]]*Sqrt[-1 + Tan[(e + f*x)/4]^2])))/(1 - (2*(d + Sqrt[-c^2
+ d^2])*Tan[(e + f*x)/4])/(c*(-1 + Tan[(e + f*x)/4]^2)))))/(Sqrt[2]*Sqrt[c]*(c - d)*Sqrt[-c^2 + d^2]*Sqrt[-d +
Sqrt[-c^2 + d^2]]*Sqrt[d + Sqrt[-c^2 + d^2]]*Sqrt[Tan[(e + f*x)/4]]*Sqrt[-1 + Tan[(e + f*x)/4]^2]))))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(620\) vs. \(2(133)=266\).
time = 0.26, size = 621, normalized size = 3.70

method	result	size
default	\(\frac {\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \left (-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )\right ) \sin \left (f x +e \right ) \left (2 \arctan \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\right ) \sqrt {-\left (c -d \right ) \left (c +d \right )}\, \sqrt {\left (\sqrt {-\left (c -d \right ) \left (c +d \right )}-d \right ) c}\, \sqrt {\left (\sqrt {-\left (c -d \right ) \left (c +d \right )}+d \right ) c}-\sqrt {\left (\sqrt {-\left (c -d \right ) \left (c +d \right )}-d \right ) c}\, \arctan \left (\frac {c \sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}}{\sqrt {\left (\sqrt {-\left (c -d \right ) \left (c +d \right )}+d \right ) c}}\right ) \sqrt {-\left (c -d \right ) \left (c +d \right )}\, d +\arctan \left (\frac {c \sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}}{\sqrt {\left (\sqrt {-\left (c -d \right ) \left (c +d \right )}+d \right ) c}}\right ) \sqrt {\left (\sqrt {-\left (c -d \right ) \left (c +d \right )}-d \right ) c}\, c d -\sqrt {\left (\sqrt {-\left (c -d \right ) \left (c +d \right )}-d \right ) c}\, \arctan \left (\frac {c \sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}}{\sqrt {\left (\sqrt {-\left (c -d \right ) \left (c +d \right )}+d \right ) c}}\right ) d^{2}+\arctanh \left (\frac {c \sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}}{\sqrt {\left (\sqrt {-\left (c -d \right ) \left (c +d \right )}-d \right ) c}}\right ) \sqrt {\left (\sqrt {-\left (c -d \right ) \left (c +d \right )}+d \right ) c}\, \sqrt {-\left (c -d \right ) \left (c +d \right )}\, d +\sqrt {\left (\sqrt {-\left (c -d \right ) \left (c +d \right )}+d \right ) c}\, \arctanh \left (\frac {c \sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}}{\sqrt {\left (\sqrt {-\left (c -d \right ) \left (c +d \right )}-d \right ) c}}\right ) c d -\arctanh \left (\frac {c \sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}}{\sqrt {\left (\sqrt {-\left (c -d \right ) \left (c +d \right )}-d \right ) c}}\right ) \sqrt {\left (\sqrt {-\left (c -d \right ) \left (c +d \right )}+d \right ) c}\, d^{2}\right )}{f \sqrt {g \sin \left (f x +e \right )}\, \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \left (-1+\cos \left (f x +e \right )\right ) \left (c -d \right ) \sqrt {-\left (c -d \right ) \left (c +d \right )}\, \sqrt {\left (\sqrt {-\left (c -d \right ) \left (c +d \right )}-d \right ) c}\, \sqrt {\left (\sqrt {-\left (c -d \right ) \left (c +d \right )}+d \right ) c}}\)	\(621\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (139) = 278\).
time = 1.88, size = 3297, normalized size = 19.62 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(sqrt(2)*(a*c^2 + a*c*d)*g*sqrt(-1/(a*g))*log((4*sqrt(2)*(3*cos(f*x + e)^2 + (3*cos(f*x + e) + 4)*sin(f*
x + e) - cos(f*x + e) - 4)*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))*sqrt(-1/(a*g)) + 17*cos(f*x + e)^3 +
3*cos(f*x + e)^2 + (17*cos(f*x + e)^2 + 14*cos(f*x + e) - 4)*sin(f*x + e) - 18*cos(f*x + e) - 4)/(cos(f*x + e)
^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 4)*sin(f*x + e) - 2*cos(f*x + e) - 4)) - sqrt(-(a*c
^2 + a*c*d)*g)*d*log(((128*a*c^4 + 256*a*c^3*d + 160*a*c^2*d^2 + 32*a*c*d^3 + a*d^4)*g*cos(f*x + e)^5 - (128*a
*c^4 + 192*a*c^3*d + 64*a*c^2*d^2 - 4*a*c*d^3 - a*d^4)*g*cos(f*x + e)^4 - 2*(208*a*c^4 + 368*a*c^3*d + 195*a*c
^2*d^2 + 32*a*c*d^3 + a*d^4)*g*cos(f*x + e)^3 + 2*(64*a*c^4 + 94*a*c^3*d + 29*a*c^2*d^2 - 4*a*c*d^3 - a*d^4)*g
*cos(f*x + e)^2 + (289*a*c^4 + 480*a*c^3*d + 230*a*c^2*d^2 + 32*a*c*d^3 + a*d^4)*g*cos(f*x + e) + 8*((16*c^3 +
24*c^2*d + 10*c*d^2 + d^3)*cos(f*x + e)^4 - (24*c^3 + 28*c^2*d + 7*c*d^2)*cos(f*x + e)^3 + 51*c^3 + 59*c^2*d
+ 17*c*d^2 + d^3 - (66*c^3 + 83*c^2*d + 27*c*d^2 + 2*d^3)*cos(f*x + e)^2 + (25*c^3 + 28*c^2*d + 7*c*d^2)*cos(f
*x + e) + ((16*c^3 + 24*c^2*d + 10*c*d^2 + d^3)*cos(f*x + e)^3 - 51*c^3 - 59*c^2*d - 17*c*d^2 - d^3 + (40*c^3
+ 52*c^2*d + 17*c*d^2 + d^3)*cos(f*x + e)^2 - (26*c^3 + 31*c^2*d + 10*c*d^2 + d^3)*cos(f*x + e))*sin(f*x + e))
*sqrt(-(a*c^2 + a*c*d)*g)*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e)) + (a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4
*a*c*d^3 + a*d^4)*g + ((128*a*c^4 + 256*a*c^3*d + 160*a*c^2*d^2 + 32*a*c*d^3 + a*d^4)*g*cos(f*x + e)^4 + 4*(64
*a*c^4 + 112*a*c^3*d + 56*a*c^2*d^2 + 7*a*c*d^3)*g*cos(f*x + e)^3 - 2*(80*a*c^4 + 144*a*c^3*d + 83*a*c^2*d^2 +
18*a*c*d^3 + a*d^4)*g*cos(f*x + e)^2 - 4*(72*a*c^4 + 119*a*c^3*d + 56*a*c^2*d^2 + 7*a*c*d^3)*g*cos(f*x + e) +
(a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4)*g)*sin(f*x + e))/(d^4*cos(f*x + e)^5 + (4*c*d^3 + d^4)*
cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 2*(3*c^2*d^2 + d^4)*cos(f*x + e)^3 - 2*(2*c^3*d +
3*c^2*d^2 + 4*c*d^3 + d^4)*cos(f*x + e)^2 + (c^4 + 6*c^2*d^2 + d^4)*cos(f*x + e) + (d^4*cos(f*x + e)^4 - 4*c*
d^3*cos(f*x + e)^3 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 2*(3*c^2*d^2 + 2*c*d^3 + d^4)*cos(f*x + e)^2
+ 4*(c^3*d + c*d^3)*cos(f*x + e))*sin(f*x + e))))/((a*c^3 - a*c*d^2)*f*g), 1/4*(2*sqrt(2)*(a*c^2 + a*c*d)*g*sq
rt(1/(a*g))*arctan(1/4*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))*sqrt(1/(a*g))*(3*sin(f*x + e) - 1
)/(cos(f*x + e)*sin(f*x + e))) + sqrt(-(a*c^2 + a*c*d)*g)*d*log(((128*a*c^4 + 256*a*c^3*d + 160*a*c^2*d^2 + 32
*a*c*d^3 + a*d^4)*g*cos(f*x + e)^5 - (128*a*c^4 + 192*a*c^3*d + 64*a*c^2*d^2 - 4*a*c*d^3 - a*d^4)*g*cos(f*x +
e)^4 - 2*(208*a*c^4 + 368*a*c^3*d + 195*a*c^2*d^2 + 32*a*c*d^3 + a*d^4)*g*cos(f*x + e)^3 + 2*(64*a*c^4 + 94*a*
c^3*d + 29*a*c^2*d^2 - 4*a*c*d^3 - a*d^4)*g*cos(f*x + e)^2 + (289*a*c^4 + 480*a*c^3*d + 230*a*c^2*d^2 + 32*a*c
*d^3 + a*d^4)*g*cos(f*x + e) + 8*((16*c^3 + 24*c^2*d + 10*c*d^2 + d^3)*cos(f*x + e)^4 - (24*c^3 + 28*c^2*d + 7
*c*d^2)*cos(f*x + e)^3 + 51*c^3 + 59*c^2*d + 17*c*d^2 + d^3 - (66*c^3 + 83*c^2*d + 27*c*d^2 + 2*d^3)*cos(f*x +
e)^2 + (25*c^3 + 28*c^2*d + 7*c*d^2)*cos(f*x + e) + ((16*c^3 + 24*c^2*d + 10*c*d^2 + d^3)*cos(f*x + e)^3 - 51
*c^3 - 59*c^2*d - 17*c*d^2 - d^3 + (40*c^3 + 52*c^2*d + 17*c*d^2 + d^3)*cos(f*x + e)^2 - (26*c^3 + 31*c^2*d +
10*c*d^2 + d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(-(a*c^2 + a*c*d)*g)*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x +
e)) + (a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4)*g + ((128*a*c^4 + 256*a*c^3*d + 160*a*c^2*d^2 + 3
2*a*c*d^3 + a*d^4)*g*cos(f*x + e)^4 + 4*(64*a*c^4 + 112*a*c^3*d + 56*a*c^2*d^2 + 7*a*c*d^3)*g*cos(f*x + e)^3 -
2*(80*a*c^4 + 144*a*c^3*d + 83*a*c^2*d^2 + 18*a*c*d^3 + a*d^4)*g*cos(f*x + e)^2 - 4*(72*a*c^4 + 119*a*c^3*d +
56*a*c^2*d^2 + 7*a*c*d^3)*g*cos(f*x + e) + (a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4)*g)*sin(f*x +
e))/(d^4*cos(f*x + e)^5 + (4*c*d^3 + d^4)*cos(f*x + e)^4 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 2*(3*c
^2*d^2 + d^4)*cos(f*x + e)^3 - 2*(2*c^3*d + 3*c^2*d^2 + 4*c*d^3 + d^4)*cos(f*x + e)^2 + (c^4 + 6*c^2*d^2 + d^4
)*cos(f*x + e) + (d^4*cos(f*x + e)^4 - 4*c*d^3*cos(f*x + e)^3 + c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4 - 2*
(3*c^2*d^2 + 2*c*d^3 + d^4)*cos(f*x + e)^2 + 4*(c^3*d + c*d^3)*cos(f*x + e))*sin(f*x + e))))/((a*c^3 - a*c*d^2
)*f*g), -1/4*(sqrt(2)*(a*c^2 + a*c*d)*g*sqrt(-1/(a*g))*log((4*sqrt(2)*(3*cos(f*x + e)^2 + (3*cos(f*x + e) + 4)
*sin(f*x + e) - cos(f*x + e) - 4)*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))*sqrt(-1/(a*g)) + 17*cos(f*x +
e)^3 + 3*cos(f*x + e)^2 + (17*cos(f*x + e)^2 + 14*cos(f*x + e) - 4)*sin(f*x + e) - 18*cos(f*x + e) - 4)/(cos(f
*x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 4)*sin(f*x + e) - 2*cos(f*x + e) - 4)) + 2*s
qrt((a*c^2 + a*c*d)*g)*d*arctan(1/4*((8*c^2 + 8*c*d + d^2)*cos(f*x + e)^2 - 9*c^2 - 8*c*d - d^2 + 2*(4*c^2 + 3
*c*d)*sin(f*x + e))*sqrt((a*c^2 + a*c*d)*g)*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))/((2*a*c^3 + 3*a*c^2*
d + a*c*d^2)*g*cos(f*x + e)^3 + (a*c^3 + a*c^2*...

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {g\,\sin \left (e+f\,x\right )}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c+d\,\sin \left (e+f\,x\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________