3.10.37 \(\int \frac {\cos ^2(e+f x)}{(a+a \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx\) [937]

Optimal. Leaf size=141 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{a^{3/2} \sqrt {d} f}-\frac {2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{a^{3/2} \sqrt {c-d} f} \]

[Out]

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

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Rubi [A]
time = 0.43, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {2995, 3061, 2861, 214, 2854, 211} \begin {gather*} \frac {2 \text {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 )}{a^{3/2} \sqrt {d} f}-\frac {2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{a^{3/2} f \sqrt {c-d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])])/(a^(3/2)*Sqrt[d
]*f) - (2*Sqrt[2]*ArcTanh[(Sqrt[a]*Sqrt[c - d]*Cos[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c + d*Sin[
e + f*x]])])/(a^(3/2)*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 214

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

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]

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 2995

Int[cos[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_), x_Symbol] :> Dist[1/b^2, Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*(a - b*Sin[e + f*x])
, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n]

Rule 3061

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

Rubi steps

\begin {align*} \int \frac {\cos ^2(e+f x)}{(a+a \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx &=\frac {\int \frac {a-a \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{a^2}\\ &=-\frac {\int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{a^2}+\frac {2 \int \frac {1}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{a}\\ &=-\frac {4 \text {Subst}\left (\int \frac {1}{2 a^2-(a c-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 )}{f}+\frac {2 \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 )}{a f}\\ &=\frac {2 \tan ^{-1}\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 )}{a^{3/2} \sqrt {d} f}-\frac {2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{a^{3/2} \sqrt {c-d} f}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 42.27, size = 208404, normalized size = 1478.04 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(4460\) vs. \(2(114)=228\).
time = 0.33, size = 4461, normalized size = 31.64 \[\text {output too large to display}\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/f*(arctan(((c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*(2*c-2*
d)^(1/2)*((c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)*d^4*cos(f*x+e)+2*arctan(((c+d*sin(f*x+e))
/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*(2*c-2*d)^(1/2)*((c+d*sin(f*x+e))/((d^2
/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)*c*d^3-arctan(((c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)/
(-(d^2/c^2)^(1/2)*c)^(1/2))*(2*c-2*d)^(1/2)*((c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)*d^4+(d
^2/c^2)^(1/2)*arctan(((c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*(
2*c-2*d)^(1/2)*((c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)*c^3*d+(d^2/c^2)^(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^2/c^2)^(1/2)*c*sin(f*x+e)+d)*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)*d^2*c^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2))*(2*c-2*d)^(1/2)*((c+d*sin(f*x+e))/((d^2/c
^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*d^2*c^2+
d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2)*c+4*2^(1/2)*((c+d*sin(f*x+e))/(1+cos(f*x+e)))^(1/2)*ln(2*((2*c-2
*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)+c*sin(f*x+e)-d*sin(f*x+e)+cos(f*x+e)*c-d*
cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*(-(d^2/c^2)^(1/2)*c)^(1/2)*c^2*d^2*sin(f*x+e)-8*2^(1/2)*((c+d*sin(f
*x+e))/(1+cos(f*x+e)))^(1/2)*ln(2*((2*c-2*d)^(1/2)*2^(1/2)*((c+d*sin(f*x+e))/(1+cos(f*x+e)))^(1/2)*sin(f*x+e)+
c*sin(f*x+e)-d*sin(f*x+e)+cos(f*x+e)*c-d*cos(f*x+e)-c+d)/(1-cos(f*x+e)+sin(f*x+e)))*(-(d^2/c^2)^(1/2)*c)^(1/2)
*c*d^3*sin(f*x+e)-(d^2/c^2)^(1/2)*arctan(((c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)/(-(d^2/c^
2)^(1/2)*c)^(1/2))*(2*c-2*d)^(1/2)*((c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)*c^3*d*sin(f*x+e
)+2*(d^2/c^2)^(1/2)*arctan(((c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1
/2))*(2*c-2*d)^(1/2)*((c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)*c^2*d^2*sin(f*x+e)-(d^2/c^2)^
(1/2)*arctan(((c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*(2*c-2*d)
^(1/2)*((c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)*c*d^3*sin(f*x+e)-(d^2/c^2)^(1/2)*arctan(((c
+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*(2*c-2*d)^(1/2)*((c+d*sin
(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)*c^3*d*cos(f*x+e)+2*(d^2/c^2)^(1/2)*arctan(((c+d*sin(f*x+e))
/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*(2*c-2*d)^(1/2)*((c+d*sin(f*x+e))/((d^2
/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)*c^2*d^2*cos(f*x+e)-(d^2/c^2)^(1/2)*arctan(((c+d*sin(f*x+e))/((d^2/c^2)^(1
/2)*c*sin(f*x+e)+d)*d)^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*(2*c-2*d)^(1/2)*((c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*
sin(f*x+e)+d)*d)^(1/2)*c*d^3*cos(f*x+e)+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^2/c^2)^(1/2)*c*sin(f*x+e)+d)*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)*d^2*c^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4
)*c)^(1/2))*(2*c-2*d)^(1/2)*((c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)*(-(d^2/c^2)^(1/2)*c)^(
1/2)*(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*d^2*c^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2)*d*sin(f*x+e
)-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^2/c^2)^(1/2)*c*sin(f*x+e)+d)*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)*d^2*c^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2))*(2*c-2*d)^(1/2)*((c+d*sin(
f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)
^(1/2)*d^2*c^2+d^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2)*d*cos(f*x+e)+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^2/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)/((d^2/c^2)^(1/2)*c*sin(f*x+e)-d*c
os(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)*d^2*c^2+d
^4*(d^2/c^2)^(1/2)-4*c^2*d^2-4*d^4)*c)^(1/2))*(2*c-2*d)^(1/2)*((c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+
d)*d)^(1/2)*(-(d^2/c^2)^(1/2)*c)^(1/2)*(((d^2/c^2)^(1/2)*c^4+6*(d^2/c^2)^(1/2)*d^2*c^2+d^4*(d^2/c^2)^(1/2)-4*c
^2*d^2-4*d^4)*c)^(1/2)*d+arctan(((c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)/(-(d^2/c^2)^(1/2)*
c)^(1/2))*(2*c-2*d)^(1/2)*((c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)*d^4*sin(f*x+e)-arctan(((
c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*(2*c-2*d)^(1/2)*((c+d*si
n(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*x+e)+d)*d)^(1/2)*c^2*d^2+arctan(((c+d*sin(f*x+e))/((d^2/c^2)^(1/2)*c*sin(f*
x+e)+d)*d)^(1/2)/(-(d^2/c^2)^(1/2)*c)^(1/2))*(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(cos(f*x+e)^2/(a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (120) = 240\).
time = 0.80, size = 2161, normalized size = 15.33 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^2/(a+a*sin(f*x+e))^(3/2)/(c+d*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

[1/4*(4*sqrt(2)*a*d*log(-((c - 3*d)*cos(f*x + e)^2 - 2*sqrt(2)*((c - d)*cos(f*x + e) - (c - d)*sin(f*x + e) +
c - d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/sqrt(a*c - a*d) + (3*c - d)*cos(f*x + e) - ((c - 3*d)
*cos(f*x + e) - 2*c - 2*d)*sin(f*x + e) + 2*c + 2*d)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f
*x + e) - 2))/sqrt(a*c - a*d) - 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^3*cos(f*x + e)^4 + 24*(c*d^2
- d^3)*cos(f*x + e)^3 - c^3 + 17*c^2*d - 59*c*d^2 + 51*d^3 - 2*(5*c^2*d - 26*c*d^2 + 33*d^3)*cos(f*x + e)^2 -
(c^3 - 7*c^2*d + 31*c*d^2 - 25*d^3)*cos(f*x + e) + (16*d^3*cos(f*x + e)^3 + c^3 - 17*c^2*d + 59*c*d^2 - 51*d^
3 - 8*(3*c*d^2 - 5*d^3)*cos(f*x + e)^2 - 2*(5*c^2*d - 14*c*d^2 + 13*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(-a*d
)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c) + (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)))/(a^2*d*f), 1/2*(2
*sqrt(2)*a*d*log(-((c - 3*d)*cos(f*x + e)^2 - 2*sqrt(2)*((c - d)*cos(f*x + e) - (c - d)*sin(f*x + e) + c - d)*
sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/sqrt(a*c - a*d) + (3*c - d)*cos(f*x + e) - ((c - 3*d)*cos(f*
x + e) - 2*c - 2*d)*sin(f*x + e) + 2*c + 2*d)/(cos(f*x + e)^2 - (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e)
- 2))/sqrt(a*c - a*d) - 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*d)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(2*a*d^3*cos(f*x + e)^3 - (3*a*c*d^2 - a
*d^3)*cos(f*x + e)*sin(f*x + e) - (a*c^2*d - a*c*d^2 + 2*a*d^3)*cos(f*x + e))))/(a^2*d*f), 1/4*(8*sqrt(2)*a*d*
sqrt(-1/(a*c - a*d))*arctan(sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-1/(a*c - a*d))/cos
(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^3*cos(f*x + e)^4 + 24*(c*d^2 - d^3)*cos(f*x + e
)^3 - c^3 + 17*c^2*d - 59*c*d^2 + 51*d^3 - 2*(5*c^2*d - 26*c*d^2 + 33*d^3)*cos(f*x + e)^2 - (c^3 - 7*c^2*d + 3
1*c*d^2 - 25*d^3)*cos(f*x + e) + (16*d^3*cos(f*x + e)^3 + c^3 - 17*c^2*d + 59*c*d^2 - 51*d^3 - 8*(3*c*d^2 - 5*
d^3)*cos(f*x + e)^2 - 2*(5*c^2*d - 14*c*d^2 + 13*d^3)*cos(f*x + e))*sin(f*x + e))*sqrt(-a*d)*sqrt(a*sin(f*x +
e) + a)*sqrt(d*sin(f*x + e) + c) + (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)))/(a^2*d*f), 1/2*(4*sqrt(2)*a*d*sqrt(-
1/(a*c - a*d))*arctan(sqrt(2)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*sqrt(-1/(a*c - a*d))/cos(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*d
)*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(2*a*d^3*cos(f*x + e)^3 - (3*a*c*d^2 - a*d^3)*cos(f*x + e)
*sin(f*x + e) - (a*c^2*d - a*c*d^2 + 2*a*d^3)*cos(f*x + e))))/(a^2*d*f)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos ^{2}{\left (e + f x \right )}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \sqrt {c + d \sin {\left (e + f x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(e + f*x)**2/((a*(sin(e + f*x) + 1))**(3/2)*sqrt(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(cos(f*x+e)^2/(a+a*sin(f*x+e))^(3/2)/(c+d*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 {{\cos \left (e+f\,x\right )}^2}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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