3.3.99 \(\int \frac {(e \cos (c+d x))^{3/2}}{\sqrt {a+a \sin (c+d x)}} \, dx\) [299]

Optimal. Leaf size=200 \[ \frac {e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{a d}-\frac {e^{3/2} \sinh ^{-1}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{a d (1+\cos (c+d x)+\sin (c+d x))}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{a d (1+\cos (c+d x)+\sin (c+d x))} \]

[Out]

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

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Rubi [A]
time = 0.19, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2764, 2756, 2854, 209, 2912, 65, 221} \begin {gather*} \frac {e^{3/2} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \text {ArcTan}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{a d (\sin (c+d x)+\cos (c+d x)+1)}-\frac {e^{3/2} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \sinh ^{-1}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{a d (\sin (c+d x)+\cos (c+d x)+1)}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a*d) - (e^(3/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[
1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a*d*(1 + Cos[c + d*x] + Sin[c + d*x])) + (e^(3/2)*ArcTan[(Sqrt[e]
*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])
/(a*d*(1 + Cos[c + d*x] + Sin[c + d*x]))

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 2756

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

Rule 2764

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(3/2)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[g*Sqrt
[g*Cos[e + f*x]]*(Sqrt[a + b*Sin[e + f*x]]/(b*f)), x] + Dist[g^2/(2*a), Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[g*Co
s[e + f*x]], x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 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]

Rule 2912

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

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.12, size = 77, normalized size = 0.38 \begin {gather*} -\frac {2\ 2^{3/4} (e \cos (c+d x))^{5/2} \, _2F_1\left (\frac {1}{4},\frac {5}{4};\frac {9}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{5 d e (1+\sin (c+d x))^{3/4} \sqrt {a (1+\sin (c+d x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*2^(3/4)*(e*Cos[c + d*x])^(5/2)*Hypergeometric2F1[1/4, 5/4, 9/4, (1 - Sin[c + d*x])/2])/(5*d*e*(1 + Sin[c +
d*x])^(3/4)*Sqrt[a*(1 + Sin[c + d*x])])

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Maple [A]
time = 0.16, size = 212, normalized size = 1.06

method	result	size
default	\(\frac {\left (\sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \sin \left (d x +c \right )-\sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \sin \left (d x +c \right )+2 \cos \left (d x +c \right ) \sin \left (d x +c \right )-2 \left (\cos ^{2}\left (d x +c \right )\right )+2 \cos \left (d x +c \right )\right ) \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{2 d \left (\cos \left (d x +c \right )-1+\sin \left (d x +c \right )\right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, \cos \left (d x +c \right )}\)	\(212\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3207 vs. \(2 (160) = 320\).
time = 186.73, size = 3207, normalized size = 16.04 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*sqrt(2)*a*d*(1/(a^2*d^4))^(1/4)*arctan(-1/4*(2*sqrt(1/2)*((sqrt(2)*a^2*d^3*cos(d*x + c)^6 + 5*sqrt(2)*a
^2*d^3*cos(d*x + c)^5 - 8*sqrt(2)*a^2*d^3*cos(d*x + c)^4 - 20*sqrt(2)*a^2*d^3*cos(d*x + c)^3 + 8*sqrt(2)*a^2*d
^3*cos(d*x + c)^2 + 16*sqrt(2)*a^2*d^3*cos(d*x + c) + (sqrt(2)*a^2*d^3*cos(d*x + c)^5 - 4*sqrt(2)*a^2*d^3*cos(
d*x + c)^4 - 12*sqrt(2)*a^2*d^3*cos(d*x + c)^3 + 8*sqrt(2)*a^2*d^3*cos(d*x + c)^2 + 16*sqrt(2)*a^2*d^3*cos(d*x
+ c))*sin(d*x + c))*(1/(a^2*d^4))^(3/4)*e^(9/2) + (sqrt(2)*a*d*cos(d*x + c)^6*e^3 - 3*sqrt(2)*a*d*cos(d*x + c
)^5*e^3 - 8*sqrt(2)*a*d*cos(d*x + c)^4*e^3 + 4*sqrt(2)*a*d*cos(d*x + c)^3*e^3 + 8*sqrt(2)*a*d*cos(d*x + c)^2*e
^3 - (sqrt(2)*a*d*cos(d*x + c)^5*e^3 + 4*sqrt(2)*a*d*cos(d*x + c)^4*e^3 - 4*sqrt(2)*a*d*cos(d*x + c)^3*e^3 - 8
*sqrt(2)*a*d*cos(d*x + c)^2*e^3)*sin(d*x + c))*(1/(a^2*d^4))^(1/4)*e^(3/2) - (cos(d*x + c)^4*e^(9/2) - 3*cos(d
*x + c)^3*e^(9/2) - 8*cos(d*x + c)^2*e^(9/2) + (2*a*d^2*cos(d*x + c)^5*e^(3/2) - 5*a*d^2*cos(d*x + c)^4*e^(3/2
) - 19*a*d^2*cos(d*x + c)^3*e^(3/2) + 20*a*d^2*cos(d*x + c)*e^(3/2) + 8*a*d^2*e^(3/2) - (2*a*d^2*cos(d*x + c)^
4*e^(3/2) + 9*a*d^2*cos(d*x + c)^3*e^(3/2) - 4*a*d^2*cos(d*x + c)^2*e^(3/2) - 20*a*d^2*cos(d*x + c)*e^(3/2) -
8*a*d^2*e^(3/2))*sin(d*x + c))*sqrt(1/(a^2*d^4))*e^3 + 4*cos(d*x + c)*e^(9/2) - (cos(d*x + c)^3*e^(9/2) + 4*co
s(d*x + c)^2*e^(9/2) - 4*cos(d*x + c)*e^(9/2) - 8*e^(9/2))*sin(d*x + c) + 8*e^(9/2))*sqrt(a*sin(d*x + c) + a)*
sqrt(cos(d*x + c)))*sqrt((2*a*cos(d*x + c)*e^9*sin(d*x + c) + 2*a*cos(d*x + c)*e^9 + (a^2*d^2*e^6*sin(d*x + c)
+ a^2*d^2*e^6)*sqrt(1/(a^2*d^4))*e^3 + (sqrt(2)*a^2*d^3*(1/(a^2*d^4))^(3/4)*cos(d*x + c)*e^9 + (sqrt(2)*a*d*e
^(15/2)*sin(d*x + c) + sqrt(2)*a*d*e^(15/2))*(1/(a^2*d^4))^(1/4)*e^(3/2))*sqrt(a*sin(d*x + c) + a)*sqrt(cos(d*
x + c)))/(sin(d*x + c) + 1)) - ((7*sqrt(2)*a^2*d^3*cos(d*x + c)^4*e^(9/2) + 3*sqrt(2)*a^2*d^3*cos(d*x + c)^3*e
^(9/2) - 16*sqrt(2)*a^2*d^3*cos(d*x + c)^2*e^(9/2) - 4*sqrt(2)*a^2*d^3*cos(d*x + c)*e^(9/2) + 8*sqrt(2)*a^2*d^
3*e^(9/2) + (2*sqrt(2)*a^2*d^3*cos(d*x + c)^4*e^(9/2) + sqrt(2)*a^2*d^3*cos(d*x + c)^3*e^(9/2) - 12*sqrt(2)*a^
2*d^3*cos(d*x + c)^2*e^(9/2) - 4*sqrt(2)*a^2*d^3*cos(d*x + c)*e^(9/2) + 8*sqrt(2)*a^2*d^3*e^(9/2))*sin(d*x + c
))*(1/(a^2*d^4))^(3/4)*e^(9/2) + (2*sqrt(2)*a*d*cos(d*x + c)^5*e^(15/2) + sqrt(2)*a*d*cos(d*x + c)^4*e^(15/2)
- 13*sqrt(2)*a*d*cos(d*x + c)^3*e^(15/2) - 8*sqrt(2)*a*d*cos(d*x + c)^2*e^(15/2) + 12*sqrt(2)*a*d*cos(d*x + c)
*e^(15/2) + 8*sqrt(2)*a*d*e^(15/2) - (7*sqrt(2)*a*d*cos(d*x + c)^3*e^(15/2) + 4*sqrt(2)*a*d*cos(d*x + c)^2*e^(
15/2) - 12*sqrt(2)*a*d*cos(d*x + c)*e^(15/2) - 8*sqrt(2)*a*d*e^(15/2))*sin(d*x + c))*(1/(a^2*d^4))^(1/4)*e^(3/
2))*sqrt(a*sin(d*x + c) + a)*sqrt(cos(d*x + c)))/(a*cos(d*x + c)^6*e^9 + a*cos(d*x + c)^5*e^9 - 8*a*cos(d*x +
c)^4*e^9 - 8*a*cos(d*x + c)^3*e^9 + 8*a*cos(d*x + c)^2*e^9 + 8*a*cos(d*x + c)*e^9 - 4*(a*cos(d*x + c)^4*e^9 +
a*cos(d*x + c)^3*e^9 - 2*a*cos(d*x + c)^2*e^9 - 2*a*cos(d*x + c)*e^9)*sin(d*x + c)))*e^(3/2) - 4*sqrt(2)*a*d*(
1/(a^2*d^4))^(1/4)*arctan(1/4*(2*sqrt(1/2)*((sqrt(2)*a^2*d^3*cos(d*x + c)^6 + 5*sqrt(2)*a^2*d^3*cos(d*x + c)^5
- 8*sqrt(2)*a^2*d^3*cos(d*x + c)^4 - 20*sqrt(2)*a^2*d^3*cos(d*x + c)^3 + 8*sqrt(2)*a^2*d^3*cos(d*x + c)^2 + 1
6*sqrt(2)*a^2*d^3*cos(d*x + c) + (sqrt(2)*a^2*d^3*cos(d*x + c)^5 - 4*sqrt(2)*a^2*d^3*cos(d*x + c)^4 - 12*sqrt(
2)*a^2*d^3*cos(d*x + c)^3 + 8*sqrt(2)*a^2*d^3*cos(d*x + c)^2 + 16*sqrt(2)*a^2*d^3*cos(d*x + c))*sin(d*x + c))*
(1/(a^2*d^4))^(3/4)*e^(9/2) + (sqrt(2)*a*d*cos(d*x + c)^6*e^3 - 3*sqrt(2)*a*d*cos(d*x + c)^5*e^3 - 8*sqrt(2)*a
*d*cos(d*x + c)^4*e^3 + 4*sqrt(2)*a*d*cos(d*x + c)^3*e^3 + 8*sqrt(2)*a*d*cos(d*x + c)^2*e^3 - (sqrt(2)*a*d*cos
(d*x + c)^5*e^3 + 4*sqrt(2)*a*d*cos(d*x + c)^4*e^3 - 4*sqrt(2)*a*d*cos(d*x + c)^3*e^3 - 8*sqrt(2)*a*d*cos(d*x
+ c)^2*e^3)*sin(d*x + c))*(1/(a^2*d^4))^(1/4)*e^(3/2) + (cos(d*x + c)^4*e^(9/2) - 3*cos(d*x + c)^3*e^(9/2) - 8
*cos(d*x + c)^2*e^(9/2) + (2*a*d^2*cos(d*x + c)^5*e^(3/2) - 5*a*d^2*cos(d*x + c)^4*e^(3/2) - 19*a*d^2*cos(d*x
+ c)^3*e^(3/2) + 20*a*d^2*cos(d*x + c)*e^(3/2) + 8*a*d^2*e^(3/2) - (2*a*d^2*cos(d*x + c)^4*e^(3/2) + 9*a*d^2*c
os(d*x + c)^3*e^(3/2) - 4*a*d^2*cos(d*x + c)^2*e^(3/2) - 20*a*d^2*cos(d*x + c)*e^(3/2) - 8*a*d^2*e^(3/2))*sin(
d*x + c))*sqrt(1/(a^2*d^4))*e^3 + 4*cos(d*x + c)*e^(9/2) - (cos(d*x + c)^3*e^(9/2) + 4*cos(d*x + c)^2*e^(9/2)
- 4*cos(d*x + c)*e^(9/2) - 8*e^(9/2))*sin(d*x + c) + 8*e^(9/2))*sqrt(a*sin(d*x + c) + a)*sqrt(cos(d*x + c)))*s
qrt((2*a*cos(d*x + c)*e^9*sin(d*x + c) + 2*a*cos(d*x + c)*e^9 + (a^2*d^2*e^6*sin(d*x + c) + a^2*d^2*e^6)*sqrt(
1/(a^2*d^4))*e^3 - (sqrt(2)*a^2*d^3*(1/(a^2*d^4))^(3/4)*cos(d*x + c)*e^9 + (sqrt(2)*a*d*e^(15/2)*sin(d*x + c)
+ sqrt(2)*a*d*e^(15/2))*(1/(a^2*d^4))^(1/4)*e^(3/2))*sqrt(a*sin(d*x + c) + a)*sqrt(cos(d*x + c)))/(sin(d*x + c
) + 1)) - ((7*sqrt(2)*a^2*d^3*cos(d*x + c)^4*e^(9/2) + 3*sqrt(2)*a^2*d^3*cos(d*x + c)^3*e^(9/2) - 16*sqrt(2)*a
^2*d^3*cos(d*x + c)^2*e^(9/2) - 4*sqrt(2)*a^2*d^3*cos(d*x + c)*e^(9/2) + 8*sqrt(2)*a^2*d^3*e^(9/2) + (2*sqrt(2
)*a^2*d^3*cos(d*x + c)^4*e^(9/2) + sqrt(2)*a^2*...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((e*cos(c + d*x))**(3/2)/sqrt(a*(sin(c + d*x) + 1)), 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((e*cos(d*x+c))^(3/2)/(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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