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

Optimal result

Integrand size = 27, antiderivative size = 200 \[ \int \frac {(e \cos (c+d x))^{3/2}}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{a d}-\frac {e^{3/2} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{a d (1+\cos (c+d x)+\sin (c+d x))}+\frac {e^{3/2} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{a d (1+\cos (c+d x)+\sin (c+d x))} \]

[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))

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2764, 2756, 2854, 209, 2912, 65, 221} \[ \int \frac {(e \cos (c+d x))^{3/2}}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {e^{3/2} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{a d (\sin (c+d x)+\cos (c+d x)+1)}+\frac {e^{3/2} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{a d (\sin (c+d x)+\cos (c+d x)+1)}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d} \]

[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*} \text {integral}& = \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} \arctan \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} \text {arcsinh}\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} \arctan \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*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.07 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.38 \[ \int \frac {(e \cos (c+d x))^{3/2}}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {2\ 2^{3/4} (e \cos (c+d x))^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {5}{4},\frac {9}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{5 d e (1+\sin (c+d x))^{3/4} \sqrt {a (1+\sin (c+d x))}} \]

[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])])

Maple [A] (verified)

Time = 6.79 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.44

method	result	size
default	\(\frac {\left (\operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )+\arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )+\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )+\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )+\cos ^{2}\left (d x +c \right )+\cos \left (d x +c \right ) \sin \left (d x +c \right )+\cos \left (d x +c \right )\right ) \sqrt {e \cos \left (d x +c \right )}\, e}{d \left (\cos \left (d x +c \right )-\sin \left (d x +c \right )+1\right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}}\)	\(288\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.48 (sec) , antiderivative size = 971, normalized size of antiderivative = 4.86 \[ \int \frac {(e \cos (c+d x))^{3/2}}{\sqrt {a+a \sin (c+d x)}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/4*(I*a*d*(-e^6/(a^2*d^4))^(1/4)*log(-1/2*(2*(e^4*sin(d*x + c) + (a*d^2*e*cos(d*x + c) + a*d^2*e)*sqrt(-e^6/
(a^2*d^4)))*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a) + (I*a^2*d^3*cos(d*x + c) + I*a^2*d^3 + (2*I*a^2*d^3
*cos(d*x + c) + I*a^2*d^3)*sin(d*x + c))*(-e^6/(a^2*d^4))^(3/4) - (2*I*a*d*e^3*cos(d*x + c)^2 + I*a*d*e^3*cos(
d*x + c) - I*a*d*e^3*sin(d*x + c) - I*a*d*e^3)*(-e^6/(a^2*d^4))^(1/4))/(cos(d*x + c) + sin(d*x + c) + 1)) - I*
a*d*(-e^6/(a^2*d^4))^(1/4)*log(-1/2*(2*(e^4*sin(d*x + c) + (a*d^2*e*cos(d*x + c) + a*d^2*e)*sqrt(-e^6/(a^2*d^4
)))*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a) + (-I*a^2*d^3*cos(d*x + c) - I*a^2*d^3 + (-2*I*a^2*d^3*cos(d
*x + c) - I*a^2*d^3)*sin(d*x + c))*(-e^6/(a^2*d^4))^(3/4) - (-2*I*a*d*e^3*cos(d*x + c)^2 - I*a*d*e^3*cos(d*x +
c) + I*a*d*e^3*sin(d*x + c) + I*a*d*e^3)*(-e^6/(a^2*d^4))^(1/4))/(cos(d*x + c) + sin(d*x + c) + 1)) - a*d*(-e
^6/(a^2*d^4))^(1/4)*log(-1/2*(2*(e^4*sin(d*x + c) - (a*d^2*e*cos(d*x + c) + a*d^2*e)*sqrt(-e^6/(a^2*d^4)))*sqr
t(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a) + (a^2*d^3*cos(d*x + c) + a^2*d^3 + (2*a^2*d^3*cos(d*x + c) + a^2*d
^3)*sin(d*x + c))*(-e^6/(a^2*d^4))^(3/4) + (2*a*d*e^3*cos(d*x + c)^2 + a*d*e^3*cos(d*x + c) - a*d*e^3*sin(d*x
+ c) - a*d*e^3)*(-e^6/(a^2*d^4))^(1/4))/(cos(d*x + c) + sin(d*x + c) + 1)) + a*d*(-e^6/(a^2*d^4))^(1/4)*log(-1
/2*(2*(e^4*sin(d*x + c) - (a*d^2*e*cos(d*x + c) + a*d^2*e)*sqrt(-e^6/(a^2*d^4)))*sqrt(e*cos(d*x + c))*sqrt(a*s
in(d*x + c) + a) - (a^2*d^3*cos(d*x + c) + a^2*d^3 + (2*a^2*d^3*cos(d*x + c) + a^2*d^3)*sin(d*x + c))*(-e^6/(a
^2*d^4))^(3/4) - (2*a*d*e^3*cos(d*x + c)^2 + a*d*e^3*cos(d*x + c) - a*d*e^3*sin(d*x + c) - a*d*e^3)*(-e^6/(a^2
*d^4))^(1/4))/(cos(d*x + c) + sin(d*x + c) + 1)) - 4*sqrt(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a)*e)/(a*d)

Sympy [F]

\[ \int \frac {(e \cos (c+d x))^{3/2}}{\sqrt {a+a \sin (c+d x)}} \, dx=\int \frac {\left (e \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}\, dx \]

[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)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {(e \cos (c+d x))^{3/2}}{\sqrt {a+a \sin (c+d x)}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(e \cos (c+d x))^{3/2}}{\sqrt {a+a \sin (c+d x)}} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}}{\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \]

[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)