\(\int \frac {1}{(a+a \sec (c+d x)) (e \sin (c+d x))^{3/2}} \, dx\) [125]

Optimal result

Integrand size = 25, antiderivative size = 135 \[ \int \frac {1}{(a+a \sec (c+d x)) (e \sin (c+d x))^{3/2}} \, dx=-\frac {2 e}{5 a d (e \sin (c+d x))^{5/2}}+\frac {2 e \cos (c+d x)}{5 a d (e \sin (c+d x))^{5/2}}-\frac {4 \cos (c+d x)}{5 a d e \sqrt {e \sin (c+d x)}}-\frac {4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a d e^2 \sqrt {\sin (c+d x)}} \]

[Out]

-2/5*e/a/d/(e*sin(d*x+c))^(5/2)+2/5*e*cos(d*x+c)/a/d/(e*sin(d*x+c))^(5/2)-4/5*cos(d*x+c)/a/d/e/(e*sin(d*x+c))^
(1/2)+4/5*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^
(1/2))*(e*sin(d*x+c))^(1/2)/a/d/e^2/sin(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3957, 2918, 2644, 30, 2647, 2716, 2721, 2719} \[ \int \frac {1}{(a+a \sec (c+d x)) (e \sin (c+d x))^{3/2}} \, dx=-\frac {4 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a d e^2 \sqrt {\sin (c+d x)}}-\frac {2 e}{5 a d (e \sin (c+d x))^{5/2}}+\frac {2 e \cos (c+d x)}{5 a d (e \sin (c+d x))^{5/2}}-\frac {4 \cos (c+d x)}{5 a d e \sqrt {e \sin (c+d x)}} \]

[In]

Int[1/((a + a*Sec[c + d*x])*(e*Sin[c + d*x])^(3/2)),x]

[Out]

(-2*e)/(5*a*d*(e*Sin[c + d*x])^(5/2)) + (2*e*Cos[c + d*x])/(5*a*d*(e*Sin[c + d*x])^(5/2)) - (4*Cos[c + d*x])/(
5*a*d*e*Sqrt[e*Sin[c + d*x]]) - (4*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*a*d*e^2*Sqrt[Sin[
c + d*x]])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 2647

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(a*Cos[e +
f*x])^(m - 1)*((b*Sin[e + f*x])^(n + 1)/(b*f*(n + 1))), x] + Dist[a^2*((m - 1)/(b^2*(n + 1))), Int[(a*Cos[e +
f*x])^(m - 2)*(b*Sin[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && (Intege
rsQ[2*m, 2*n] || EqQ[m + n, 0])

Rule 2716

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1
))), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*n]

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

Rule 2918

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

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cos (c+d x)}{(-a-a \cos (c+d x)) (e \sin (c+d x))^{3/2}} \, dx \\ & = \frac {e^2 \int \frac {\cos (c+d x)}{(e \sin (c+d x))^{7/2}} \, dx}{a}-\frac {e^2 \int \frac {\cos ^2(c+d x)}{(e \sin (c+d x))^{7/2}} \, dx}{a} \\ & = \frac {2 e \cos (c+d x)}{5 a d (e \sin (c+d x))^{5/2}}+\frac {2 \int \frac {1}{(e \sin (c+d x))^{3/2}} \, dx}{5 a}+\frac {e \text {Subst}\left (\int \frac {1}{x^{7/2}} \, dx,x,e \sin (c+d x)\right )}{a d} \\ & = -\frac {2 e}{5 a d (e \sin (c+d x))^{5/2}}+\frac {2 e \cos (c+d x)}{5 a d (e \sin (c+d x))^{5/2}}-\frac {4 \cos (c+d x)}{5 a d e \sqrt {e \sin (c+d x)}}-\frac {2 \int \sqrt {e \sin (c+d x)} \, dx}{5 a e^2} \\ & = -\frac {2 e}{5 a d (e \sin (c+d x))^{5/2}}+\frac {2 e \cos (c+d x)}{5 a d (e \sin (c+d x))^{5/2}}-\frac {4 \cos (c+d x)}{5 a d e \sqrt {e \sin (c+d x)}}-\frac {\left (2 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{5 a e^2 \sqrt {\sin (c+d x)}} \\ & = -\frac {2 e}{5 a d (e \sin (c+d x))^{5/2}}+\frac {2 e \cos (c+d x)}{5 a d (e \sin (c+d x))^{5/2}}-\frac {4 \cos (c+d x)}{5 a d e \sqrt {e \sin (c+d x)}}-\frac {4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a d e^2 \sqrt {\sin (c+d x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 1.52 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(a+a \sec (c+d x)) (e \sin (c+d x))^{3/2}} \, dx=\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) (\cos (c+d x)+i \sin (c+d x)) \left (-6-9 \cos (c+d x)+2 \sqrt {1-e^{2 i (c+d x)}} (1+\cos (c+d x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},e^{2 i (c+d x)}\right )+3 i \sin (c+d x)\right )}{15 a d e \sqrt {e \sin (c+d x)}} \]

[In]

Integrate[1/((a + a*Sec[c + d*x])*(e*Sin[c + d*x])^(3/2)),x]

[Out]

(Sec[(c + d*x)/2]^2*(Cos[c + d*x] + I*Sin[c + d*x])*(-6 - 9*Cos[c + d*x] + 2*Sqrt[1 - E^((2*I)*(c + d*x))]*(1
+ Cos[c + d*x])*Hypergeometric2F1[1/2, 3/4, 7/4, E^((2*I)*(c + d*x))] + (3*I)*Sin[c + d*x]))/(15*a*d*e*Sqrt[e*
Sin[c + d*x]])

Maple [A] (verified)

Time = 5.84 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.39

method	result	size
default	\(\frac {-\frac {2 e}{5 a \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {\frac {4 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sin \left (d x +c \right )^{\frac {7}{2}} \operatorname {EllipticE}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )}{5}-\frac {2 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sin \left (d x +c \right )^{\frac {7}{2}} \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {4 \sin \left (d x +c \right )^{5}}{5}-\frac {6 \sin \left (d x +c \right )^{3}}{5}+\frac {2 \sin \left (d x +c \right )}{5}}{e a \sin \left (d x +c \right )^{3} \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\)	\(187\)

[In]

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

[Out]

(-2/5/a*e/(e*sin(d*x+c))^(5/2)+2/5/e*(2*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(7/2)*Elliptic
E((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))-(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(7/2)*EllipticF((
-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))+2*sin(d*x+c)^5-3*sin(d*x+c)^3+sin(d*x+c))/a/sin(d*x+c)^3/cos(d*x+c)/(e*sin(d
*x+c))^(1/2))/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.21 \[ \int \frac {1}{(a+a \sec (c+d x)) (e \sin (c+d x))^{3/2}} \, dx=-\frac {2 \, {\left ({\left (i \, \sqrt {2} \cos \left (d x + c\right ) + i \, \sqrt {2}\right )} \sqrt {-i \, e} \sin \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + {\left (-i \, \sqrt {2} \cos \left (d x + c\right ) - i \, \sqrt {2}\right )} \sqrt {i \, e} \sin \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + {\left (2 \, \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {e \sin \left (d x + c\right )}\right )}}{5 \, {\left (a d e^{2} \cos \left (d x + c\right ) + a d e^{2}\right )} \sin \left (d x + c\right )} \]

[In]

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

[Out]

-2/5*((I*sqrt(2)*cos(d*x + c) + I*sqrt(2))*sqrt(-I*e)*sin(d*x + c)*weierstrassZeta(4, 0, weierstrassPInverse(4
, 0, cos(d*x + c) + I*sin(d*x + c))) + (-I*sqrt(2)*cos(d*x + c) - I*sqrt(2))*sqrt(I*e)*sin(d*x + c)*weierstras
sZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c))) + (2*cos(d*x + c)^2 + 2*cos(d*x + c) + 1
)*sqrt(e*sin(d*x + c)))/((a*d*e^2*cos(d*x + c) + a*d*e^2)*sin(d*x + c))

Sympy [F]

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

[In]

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

[Out]

Integral(1/((e*sin(c + d*x))**(3/2)*sec(c + d*x) + (e*sin(c + d*x))**(3/2)), x)/a

Maxima [F(-1)]

Timed out. \[ \int \frac {1}{(a+a \sec (c+d x)) (e \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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