3.1.27 \(\int \frac {\sqrt {g \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx\) [27]
Optimal. Leaf size=166 \[ \frac {\sqrt {2} \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {g} \cos (e+f x)}{\sqrt {2} \sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\right )}{\sqrt {a} (c-d) f}-\frac {2 \sqrt {c} \sqrt {g} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \cos (e+f x)}{\sqrt {c+d} \sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\right )}{\sqrt {a} (c-d) \sqrt {c+d} f} \]
[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)*g^(1/2)/(c-
d)/f/a^(1/2)-2*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^(1/2)*g^(1/2)/(c-d)/f/a^(1/2)/(c+d)^(1/2)
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Rubi [A]
time = 0.35, antiderivative size = 166, 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 = {3015, 2861,
211, 3009} \begin {gather*} \frac {\sqrt {2} \sqrt {g} \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 (c-d)}-\frac {2 \sqrt {c} \sqrt {g} \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} f (c-d) \sqrt {c+d}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sqrt[g*Sin[e + f*x]]/(Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]
[Out]
(Sqrt[2]*Sqrt[g]*ArcTan[(Sqrt[a]*Sqrt[g]*Cos[e + f*x])/(Sqrt[2]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])
])/(Sqrt[a]*(c - d)*f) - (2*Sqrt[c]*Sqrt[g]*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]*(c - d)*Sqrt[c + d]*f)
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 3015
Int[Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) +
(f_.)*(x_)])), x_Symbol] :> Dist[(-a)*(g/(b*c - a*d)), Int[1/(Sqrt[g*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]),
x], x] + Dist[c*(g/(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] /; FreeQ[{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 {\sqrt {g \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx &=-\frac {g \int \frac {1}{\sqrt {g \sin (e+f x)} \sqrt {a+a \sin (e+f x)}} \, dx}{c-d}+\frac {(c g) \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 g) \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 c g) \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} \sqrt {g} \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}-\frac {2 \sqrt {c} \sqrt {g} \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} (c-d) \sqrt {c+d} f}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(2568\) vs. \(2(166)=332\).
time = 24.66, size = 2568, normalized size = 15.47 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[Sqrt[g*Sin[e + f*x]]/(Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])),x]
[Out]
((-((Sqrt[c]*(c - d + Sqrt[-c^2 + d^2])*ArcTan[(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])])/(Sqrt[-c^2 + d^2]*Sqrt[-d + Sqrt[-c^2 + d^2]])) + 2*ArcTanh[(Sqrt[2]*S
qrt[Tan[(e + f*x)/4]])/Sqrt[-1 + Tan[(e + f*x)/4]^2]] - (Sqrt[c]*(-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])])/(Sqrt[-c^2 + d^2]
*Sqrt[d + Sqrt[-c^2 + d^2]]))*Sec[(e + f*x)/4]^2*Sqrt[Sin[e + f*x]]*Sqrt[g*Sin[e + f*x]])/(Sqrt[2]*(c - d)*f*S
qrt[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]*(-1/4*((-(
(Sqrt[c]*(c - 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])])/(Sqrt[-c^2 + d^2]*Sqrt[-d + Sqrt[-c^2 + d^2]])) + 2*ArcTanh[(Sqrt[2]*Sqrt[
Tan[(e + f*x)/4]])/Sqrt[-1 + Tan[(e + f*x)/4]^2]] - (Sqrt[c]*(-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])])/(Sqrt[-c^2 + d^2]*Sqr
t[d + Sqrt[-c^2 + d^2]]))*Sec[(e + f*x)/4]^4*Sqrt[Sin[e + f*x]]*Sqrt[Tan[(e + f*x)/4]])/(Sqrt[2]*(c - d)*(-1 +
Tan[(e + f*x)/4]^2)^(3/2)) - ((-((Sqrt[c]*(c - 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])])/(Sqrt[-c^2 + d^2]*Sqrt[-d + Sqrt[-c^2 +
d^2]])) + 2*ArcTanh[(Sqrt[2]*Sqrt[Tan[(e + f*x)/4]])/Sqrt[-1 + Tan[(e + f*x)/4]^2]] - (Sqrt[c]*(-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])])/(Sqrt[-c^2 + d^2]*Sqrt[d + Sqrt[-c^2 + d^2]]))*Sec[(e + f*x)/4]^4*Sqrt[Sin[e + f*x]])/(8*Sqrt[2]*
(c - d)*Tan[(e + f*x)/4]^(3/2)*Sqrt[-1 + Tan[(e + f*x)/4]^2]) + ((-((Sqrt[c]*(c - 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])])/(Sqrt[
-c^2 + d^2]*Sqrt[-d + Sqrt[-c^2 + d^2]])) + 2*ArcTanh[(Sqrt[2]*Sqrt[Tan[(e + f*x)/4]])/Sqrt[-1 + Tan[(e + f*x)
/4]^2]] - (Sqrt[c]*(-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])])/(Sqrt[-c^2 + d^2]*Sqrt[d + Sqrt[-c^2 + d^2]]))*Cos[e + f*x]*Sec
[(e + f*x)/4]^2)/(2*Sqrt[2]*(c - d)*Sqrt[Sin[e + f*x]]*Sqrt[Tan[(e + f*x)/4]]*Sqrt[-1 + Tan[(e + f*x)/4]^2]) +
((-((Sqrt[c]*(c - 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])])/(Sqrt[-c^2 + d^2]*Sqrt[-d + Sqrt[-c^2 + d^2]])) + 2*ArcTanh[(Sqrt[2]*
Sqrt[Tan[(e + f*x)/4]])/Sqrt[-1 + Tan[(e + f*x)/4]^2]] - (Sqrt[c]*(-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])])/(Sqrt[-c^2 + d^2
]*Sqrt[d + Sqrt[-c^2 + d^2]]))*Sec[(e + f*x)/4]^2*Sqrt[Sin[e + f*x]]*Sqrt[Tan[(e + f*x)/4]])/(2*Sqrt[2]*(c - d
)*Sqrt[-1 + Tan[(e + f*x)/4]^2]) + (Sec[(e + f*x)/4]^2*Sqrt[Sin[e + f*x]]*((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)) - (Sqrt[c]*(c - 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]*Sqr
t[Tan[(e + f*x)/4]]*Sqrt[-1 + Tan[(e + f*x)/4]^2])))/(Sqrt[-c^2 + d^2]*Sqrt[-d + Sqrt[-c^2 + d^2]]*(1 + (2*(-d
+ Sqrt[-c^2 + d^2])*Tan[(e + f*x)/4])/(c*(-1 + Tan[(e + f*x)/4]^2)))) - (Sqrt[c]*(-c + 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]]*S
qrt[-1 + Tan[(e + f*x)/4]^2])))/(Sqrt[-c^2 + d^2]*Sqrt[d + Sqrt[-c^2 + d^2]]*(1 - (2*(d + Sqrt[-c^2 + d^2])*Ta
n[(e + f*x)/4])/(c*(-1 + Tan[(e + f*x)/4]^2))))))/(Sqrt[2]*(c - d)*Sqrt[Tan[(e + f*x)/4]]*Sqrt[-1 + Tan[(e + f
*x)/4]^2])))
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(612\) vs.
\(2(131)=262\).
time = 0.37, size = 613, normalized size = 3.69
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method |
result |
size |
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default |
\(\frac {\sqrt {g \sin \left (f x +e \right )}\, \left (-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )\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}-\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}\, \sqrt {-\left (c -d \right ) \left (c +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 (\sqrt {-\left (c -d \right ) \left (c +d \right )}-d \right ) c}\, c^{2}-\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}\, \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 (c -d \right ) \left (c +d \right )}\, c +\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^{2}-\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 \right )}{f \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sin \left (f x +e \right ) \sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \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}}\) |
\(613\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((g*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))/(a+a*sin(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
[Out]
1/f*(g*sin(f*x+e))^(1/2)*(-1+cos(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)-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))^(1/2)*c+a
rctan(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^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)*c)^(1/2)*c*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)*(c+d))^(1/2)*c+(((-(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^2-(((-(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)/(a*(1+sin(f*x+e)))^(1/2)/sin(f*x+e)/(-
(-1+cos(f*x+e))/sin(f*x+e))^(1/2)/(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)
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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((g*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))/(a+a*sin(f*x+e))^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(g*sin(f*x + e))/(sqrt(a*sin(f*x + e) + a)*(d*sin(f*x + e) + c)), x)
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Fricas [A]
time = 1.81, size = 3170, normalized size = 19.10 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((g*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))/(a+a*sin(f*x+e))^(1/2),x, algorithm="fricas")
[Out]
[-1/4*(sqrt(2)*sqrt(-g/a)*log((17*g*cos(f*x + e)^3 - 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(-g/a) + 3*g*cos(f*x + e)^2 - 18*
g*cos(f*x + e) + (17*g*cos(f*x + e)^2 + 14*g*cos(f*x + e) - 4*g)*sin(f*x + e) - 4*g)/(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(-c*g/(a*c + a*d))
*log(((128*c^4 + 256*c^3*d + 160*c^2*d^2 + 32*c*d^3 + d^4)*g*cos(f*x + e)^5 - (128*c^4 + 192*c^3*d + 64*c^2*d^
2 - 4*c*d^3 - d^4)*g*cos(f*x + e)^4 - 2*(208*c^4 + 368*c^3*d + 195*c^2*d^2 + 32*c*d^3 + d^4)*g*cos(f*x + e)^3
+ 2*(64*c^4 + 94*c^3*d + 29*c^2*d^2 - 4*c*d^3 - d^4)*g*cos(f*x + e)^2 + (289*c^4 + 480*c^3*d + 230*c^2*d^2 + 3
2*c*d^3 + d^4)*g*cos(f*x + e) + 8*((16*c^4 + 40*c^3*d + 34*c^2*d^2 + 11*c*d^3 + d^4)*cos(f*x + e)^4 + 51*c^4 +
110*c^3*d + 76*c^2*d^2 + 18*c*d^3 + d^4 - (24*c^4 + 52*c^3*d + 35*c^2*d^2 + 7*c*d^3)*cos(f*x + e)^3 - (66*c^4
+ 149*c^3*d + 110*c^2*d^2 + 29*c*d^3 + 2*d^4)*cos(f*x + e)^2 + (25*c^4 + 53*c^3*d + 35*c^2*d^2 + 7*c*d^3)*cos
(f*x + e) - (51*c^4 + 110*c^3*d + 76*c^2*d^2 + 18*c*d^3 + d^4 - (16*c^4 + 40*c^3*d + 34*c^2*d^2 + 11*c*d^3 + d
^4)*cos(f*x + e)^3 - (40*c^4 + 92*c^3*d + 69*c^2*d^2 + 18*c*d^3 + d^4)*cos(f*x + e)^2 + (26*c^4 + 57*c^3*d + 4
1*c^2*d^2 + 11*c*d^3 + d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))*sqrt(-c*
g/(a*c + a*d)) + (c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4)*g + ((128*c^4 + 256*c^3*d + 160*c^2*d^2 + 32*c*d^
3 + d^4)*g*cos(f*x + e)^4 + 4*(64*c^4 + 112*c^3*d + 56*c^2*d^2 + 7*c*d^3)*g*cos(f*x + e)^3 - 2*(80*c^4 + 144*c
^3*d + 83*c^2*d^2 + 18*c*d^3 + d^4)*g*cos(f*x + e)^2 - 4*(72*c^4 + 119*c^3*d + 56*c^2*d^2 + 7*c*d^3)*g*cos(f*x
+ e) + (c^4 + 4*c^3*d + 6*c^2*d^2 + 4*c*d^3 + 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))))/((c - d)*f), -1/4*(sqrt(2)*sqrt(-g/a)*log((17*g*cos(f*x + e)^3
- 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(-g/a) + 3*g*cos(f*x + e)^2 - 18*g*cos(f*x + e) + (17*g*cos(f*x + e)^2 + 14*g*cos(f*
x + e) - 4*g)*sin(f*x + e) - 4*g)/(cos(f*x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 4)*s
in(f*x + e) - 2*cos(f*x + e) - 4)) - 2*sqrt(c*g/(a*c + a*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*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))*sqrt(c*g
/(a*c + a*d))/((2*c^2 + c*d)*g*cos(f*x + e)^3 + c^2*g*cos(f*x + e)*sin(f*x + e) - (2*c^2 + c*d)*g*cos(f*x + e)
)))/((c - d)*f), -1/4*(2*sqrt(2)*sqrt(g/a)*arctan(1/4*sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))*sq
rt(g/a)*(3*sin(f*x + e) - 1)/(g*cos(f*x + e)*sin(f*x + e))) + sqrt(-c*g/(a*c + a*d))*log(((128*c^4 + 256*c^3*d
+ 160*c^2*d^2 + 32*c*d^3 + d^4)*g*cos(f*x + e)^5 - (128*c^4 + 192*c^3*d + 64*c^2*d^2 - 4*c*d^3 - d^4)*g*cos(f
*x + e)^4 - 2*(208*c^4 + 368*c^3*d + 195*c^2*d^2 + 32*c*d^3 + d^4)*g*cos(f*x + e)^3 + 2*(64*c^4 + 94*c^3*d + 2
9*c^2*d^2 - 4*c*d^3 - d^4)*g*cos(f*x + e)^2 + (289*c^4 + 480*c^3*d + 230*c^2*d^2 + 32*c*d^3 + d^4)*g*cos(f*x +
e) + 8*((16*c^4 + 40*c^3*d + 34*c^2*d^2 + 11*c*d^3 + d^4)*cos(f*x + e)^4 + 51*c^4 + 110*c^3*d + 76*c^2*d^2 +
18*c*d^3 + d^4 - (24*c^4 + 52*c^3*d + 35*c^2*d^2 + 7*c*d^3)*cos(f*x + e)^3 - (66*c^4 + 149*c^3*d + 110*c^2*d^2
+ 29*c*d^3 + 2*d^4)*cos(f*x + e)^2 + (25*c^4 + 53*c^3*d + 35*c^2*d^2 + 7*c*d^3)*cos(f*x + e) - (51*c^4 + 110*
c^3*d + 76*c^2*d^2 + 18*c*d^3 + d^4 - (16*c^4 + 40*c^3*d + 34*c^2*d^2 + 11*c*d^3 + d^4)*cos(f*x + e)^3 - (40*c
^4 + 92*c^3*d + 69*c^2*d^2 + 18*c*d^3 + d^4)*cos(f*x + e)^2 + (26*c^4 + 57*c^3*d + 41*c^2*d^2 + 11*c*d^3 + d^4
)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))*sqrt(-c*g/(a*c + a*d)) + (c^4 + 4*
c^3*d + 6*c^2*d^2 + 4*c*d^3 + d^4)*g + ((128*c^4 + 256*c^3*d + 160*c^2*d^2 + 32*c*d^3 + d^4)*g*cos(f*x + e)^4
+ 4*(64*c^4 + 112*c^3*d + 56*c^2*d^2 + 7*c*d^3)*g*cos(f*x + e)^3 - 2*(80*c^4 + 144*c^3*d + 83*c^2*d^2 + 18*c*d
^3 + d^4)*g*cos(f*x + e)^2 - 4*(72*c^4 + 119*c^3*d + 56*c^2*d^2 + 7*c*d^3)*g*cos(f*x + e) + (c^4 + 4*c^3*d + 6
*c^2*d^2 + 4*c*d^3 + 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)*c
os(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))))/((c - d)*f), -1/2*(sqrt(2)...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {g \sin {\left (e + f x \right )}}}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\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((g*sin(f*x+e))**(1/2)/(c+d*sin(f*x+e))/(a+a*sin(f*x+e))**(1/2),x)
[Out]
Integral(sqrt(g*sin(e + f*x))/(sqrt(a*(sin(e + f*x) + 1))*(c + d*sin(e + f*x))), x)
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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((g*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e))/(a+a*sin(f*x+e))^(1/2),x, algorithm="giac")
[Out]
Timed out
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\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((g*sin(e + f*x))^(1/2)/((a + a*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))),x)
[Out]
int((g*sin(e + f*x))^(1/2)/((a + a*sin(e + f*x))^(1/2)*(c + d*sin(e + f*x))), x)
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