3.3.100 \(\int \frac {\sqrt {e \cos (c+d x)}}{\sqrt {a+a \sin (c+d x)}} \, dx\) [300]

Optimal. Leaf size=169 \[ \frac {2 \sqrt {e} \sinh ^{-1}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {2 \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d (a+a \cos (c+d x)+a \sin (c+d x))} \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2763, 2854, 209, 2912, 65, 221} \begin {gather*} \frac {2 \sqrt {e} \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 )}{d (a \sin (c+d x)+a \cos (c+d x)+a)}+\frac {2 \sqrt {e} \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 )}{d (a \sin (c+d x)+a \cos (c+d x)+a)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[e*Cos[c + d*x]]/Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(2*Sqrt[e]*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*(a + a*Co
s[c + d*x] + a*Sin[c + d*x])) + (2*Sqrt[e]*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]])/(d*(a + a*Cos[c + d*x] + a*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 2763

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[g*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[g*Sqrt[1 + Cos[e + f*x]]*(Sqrt[a + b*Sin[e + f*x]]/(b + b*Cos[e + f*x] + a
*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 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 {\sqrt {e \cos (c+d x)}}{\sqrt {a+a \sin (c+d x)}} \, dx &=\frac {\left (e \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}{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 ) \int \frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}} \, dx}{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 {e x} \sqrt {1+x}} \, dx,x,\cos (c+d x)\right )}{d (a+a \cos (c+d x)+a \sin (c+d x))}-\frac {\left (2 e \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 {2 \sqrt {e} \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 (2 \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 {2 \sqrt {e} \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 {2 \sqrt {e} \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 = 0.08, size = 77, normalized size = 0.46 \begin {gather*} -\frac {2 \sqrt [4]{2} (e \cos (c+d x))^{3/2} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{3 d e \sqrt [4]{1+\sin (c+d x)} \sqrt {a (1+\sin (c+d x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[e*Cos[c + d*x]]/Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(-2*2^(1/4)*(e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[3/4, 3/4, 7/4, (1 - Sin[c + d*x])/2])/(3*d*e*(1 + Sin[c +
d*x])^(1/4)*Sqrt[a*(1 + Sin[c + d*x])])

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Maple [A]
time = 0.16, size = 142, normalized size = 0.84

method	result	size
default	\(-\frac {\sqrt {e \cos \left (d x +c \right )}\, \left (\cos \left (d x +c \right )-1-\sin \left (d x +c \right )\right ) \left (\arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right )+\arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right )\right ) \sqrt {2}}{d \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, \sin \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\)	\(142\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

e^(1/2)*integrate(sqrt(cos(d*x + c))/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. 3171 vs. \(2 (134) = 268\).
time = 183.84, size = 3171, normalized size = 18.76 \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))^(1/2)/(a+a*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {e\,\cos \left (c+d\,x\right )}}{\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))^(1/2)/(a + a*sin(c + d*x))^(1/2),x)

[Out]

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

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