\(\int \frac {1}{\cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx\) [821]

Optimal result

Integrand size = 23, antiderivative size = 29 \[ \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx=\frac {2 \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{(a+b) d} \]

[Out]

2*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2*a/(a+b),2^(1/2))/(a+b)/d

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4349, 3934, 2884} \[ \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx=\frac {2 \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{d (a+b)} \]

[In]

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

[Out]

(2*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/((a + b)*d)

Rule 2884

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]

Rule 3934

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[d*Sqrt[d*S
in[e + f*x]]*Sqrt[d*Csc[e + f*x]], Int[1/(Sqrt[d*Sin[e + f*x]]*(b + a*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d
, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4349

Int[(u_)*((c_.)*sin[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Dist[(c*Csc[a + b*x])^m*(c*Sin[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Csc[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSecantIntegrandQ[
u, x]

Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx \\ & = \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))} \, dx \\ & = \frac {2 \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{(a+b) d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx=\frac {2 \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{(a+b) d} \]

[In]

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

[Out]

(2*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/((a + b)*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(149\) vs. \(2(55)=110\).

Time = 3.99 (sec) , antiderivative size = 150, normalized size of antiderivative = 5.17

method	result	size
default	\(\frac {2 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticPi}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 a}{a -b}, \sqrt {2}\right )}{\left (a -b \right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\)	\(150\)

[In]

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

[Out]

2*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^
2+1)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2*a/(a-b),2^(1/2))/(a-b)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^
2)^(1/2)/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

integrate(1/cos(d*x+c)**(3/2)/(a+b*sec(d*x+c)),x)

[Out]

Integral(1/((a + b*sec(c + d*x))*cos(c + d*x)**(3/2)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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