\(\int \frac {\sec ^{\frac {3}{2}}(d+e x)}{(a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \, dx\) [451]

Optimal result

Integrand size = 33, antiderivative size = 240 \[ \int \frac {\sec ^{\frac {3}{2}}(d+e x)}{(a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \, dx=-\frac {2 \sec ^{\frac {3}{2}}(d+e x) (c \cos (d+e x)-a \sin (d+e x)) (b+a \cos (d+e x)+c \sin (d+e x))}{\left (a^2-b^2+c^2\right ) e (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}-\frac {2 E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(a,c)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) \sec ^{\frac {3}{2}}(d+e x) (b+a \cos (d+e x)+c \sin (d+e x))^2}{\left (a^2-b^2+c^2\right ) e \sqrt {\frac {b+a \cos (d+e x)+c \sin (d+e x)}{b+\sqrt {a^2+c^2}}} (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \]

[Out]

-2*sec(e*x+d)^(3/2)*(c*cos(e*x+d)-a*sin(e*x+d))*(b+a*cos(e*x+d)+c*sin(e*x+d))/(a^2-b^2+c^2)/e/(a+b*sec(e*x+d)+
c*tan(e*x+d))^(3/2)-2*(cos(1/2*d+1/2*e*x-1/2*arctan(a,c))^2)^(1/2)/cos(1/2*d+1/2*e*x-1/2*arctan(a,c))*Elliptic
E(sin(1/2*d+1/2*e*x-1/2*arctan(a,c)),2^(1/2)*((a^2+c^2)^(1/2)/(b+(a^2+c^2)^(1/2)))^(1/2))*sec(e*x+d)^(3/2)*(b+
a*cos(e*x+d)+c*sin(e*x+d))^2/(a^2-b^2+c^2)/e/((b+a*cos(e*x+d)+c*sin(e*x+d))/(b+(a^2+c^2)^(1/2)))^(1/2)/(a+b*se
c(e*x+d)+c*tan(e*x+d))^(3/2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3246, 3207, 3198, 2732} \[ \int \frac {\sec ^{\frac {3}{2}}(d+e x)}{(a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \, dx=-\frac {2 \sec ^{\frac {3}{2}}(d+e x) (a \cos (d+e x)+b+c \sin (d+e x))^2 E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(a,c)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \left (a^2-b^2+c^2\right ) \sqrt {\frac {a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt {a^2+c^2}+b}} (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}-\frac {2 \sec ^{\frac {3}{2}}(d+e x) (c \cos (d+e x)-a \sin (d+e x)) (a \cos (d+e x)+b+c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \]

[In]

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

[Out]

(-2*Sec[d + e*x]^(3/2)*(c*Cos[d + e*x] - a*Sin[d + e*x])*(b + a*Cos[d + e*x] + c*Sin[d + e*x]))/((a^2 - b^2 +
c^2)*e*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2)) - (2*EllipticE[(d + e*x - ArcTan[a, c])/2, (2*Sqrt[a^2 + c
^2])/(b + Sqrt[a^2 + c^2])]*Sec[d + e*x]^(3/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^2)/((a^2 - b^2 + c^2)*e*S
qrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))

Rule 2732

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

Rule 3198

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*C
os[d + e*x] + c*Sin[d + e*x]]/Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])], Int[Sqrt[a/(a
+ Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - ArcTan[b, c]]], x], x] /; FreeQ[{a
, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] &&  !GtQ[a + Sqrt[b^2 + c^2], 0]

Rule 3207

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(-3/2), x_Symbol] :> Simp[2*((c*Cos
[d + e*x] - b*Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])), x] + Dist[1/(a^2
- b^2 - c^2), Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 -
b^2 - c^2, 0]

Rule 3246

Int[sec[(d_.) + (e_.)*(x_)]^(n_.)*((a_.) + (b_.)*sec[(d_.) + (e_.)*(x_)] + (c_.)*tan[(d_.) + (e_.)*(x_)])^(m_)
, x_Symbol] :> Dist[Sec[d + e*x]^n*((b + a*Cos[d + e*x] + c*Sin[d + e*x])^n/(a + b*Sec[d + e*x] + c*Tan[d + e*
x])^n), Int[1/(b + a*Cos[d + e*x] + c*Sin[d + e*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + n, 0] &&
!IntegerQ[n]

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

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 6.94 (sec) , antiderivative size = 1732, normalized size of antiderivative = 7.22 \[ \int \frac {\sec ^{\frac {3}{2}}(d+e x)}{(a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \, dx=\frac {\sec ^{\frac {3}{2}}(d+e x) (b+a \cos (d+e x)+c \sin (d+e x))^2 \left (-\frac {2 \left (a^2+c^2\right )}{a c \left (a^2-b^2+c^2\right )}+\frac {2 \left (b c+a^2 \sin (d+e x)+c^2 \sin (d+e x)\right )}{a \left (a^2-b^2+c^2\right ) (b+a \cos (d+e x)+c \sin (d+e x))}\right )}{e (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}-\frac {2 b \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},-\frac {b+\sqrt {1+\frac {a^2}{c^2}} c \sin \left (d+e x+\arctan \left (\frac {a}{c}\right )\right )}{\sqrt {1+\frac {a^2}{c^2}} \left (1-\frac {b}{\sqrt {1+\frac {a^2}{c^2}} c}\right ) c},-\frac {b+\sqrt {1+\frac {a^2}{c^2}} c \sin \left (d+e x+\arctan \left (\frac {a}{c}\right )\right )}{\sqrt {1+\frac {a^2}{c^2}} \left (-1-\frac {b}{\sqrt {1+\frac {a^2}{c^2}} c}\right ) c}\right ) \sec ^{\frac {3}{2}}(d+e x) \sec \left (d+e x+\arctan \left (\frac {a}{c}\right )\right ) (b+a \cos (d+e x)+c \sin (d+e x))^{3/2} \sqrt {\frac {c \sqrt {\frac {a^2+c^2}{c^2}}-c \sqrt {\frac {a^2+c^2}{c^2}} \sin \left (d+e x+\arctan \left (\frac {a}{c}\right )\right )}{b+c \sqrt {\frac {a^2+c^2}{c^2}}}} \sqrt {b+c \sqrt {\frac {a^2+c^2}{c^2}} \sin \left (d+e x+\arctan \left (\frac {a}{c}\right )\right )} \sqrt {\frac {c \sqrt {\frac {a^2+c^2}{c^2}}+c \sqrt {\frac {a^2+c^2}{c^2}} \sin \left (d+e x+\arctan \left (\frac {a}{c}\right )\right )}{-b+c \sqrt {\frac {a^2+c^2}{c^2}}}}}{\sqrt {1+\frac {a^2}{c^2}} c \left (a^2-b^2+c^2\right ) e (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}-\frac {a^2 \sec ^{\frac {3}{2}}(d+e x) (b+a \cos (d+e x)+c \sin (d+e x))^{3/2} \left (-\frac {c \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},-\frac {b+a \sqrt {1+\frac {c^2}{a^2}} \cos \left (d+e x-\arctan \left (\frac {c}{a}\right )\right )}{a \sqrt {1+\frac {c^2}{a^2}} \left (1-\frac {b}{a \sqrt {1+\frac {c^2}{a^2}}}\right )},-\frac {b+a \sqrt {1+\frac {c^2}{a^2}} \cos \left (d+e x-\arctan \left (\frac {c}{a}\right )\right )}{a \sqrt {1+\frac {c^2}{a^2}} \left (-1-\frac {b}{a \sqrt {1+\frac {c^2}{a^2}}}\right )}\right ) \sin \left (d+e x-\arctan \left (\frac {c}{a}\right )\right )}{a \sqrt {1+\frac {c^2}{a^2}} \sqrt {\frac {a \sqrt {\frac {a^2+c^2}{a^2}}-a \sqrt {\frac {a^2+c^2}{a^2}} \cos \left (d+e x-\arctan \left (\frac {c}{a}\right )\right )}{b+a \sqrt {\frac {a^2+c^2}{a^2}}}} \sqrt {b+a \sqrt {\frac {a^2+c^2}{a^2}} \cos \left (d+e x-\arctan \left (\frac {c}{a}\right )\right )} \sqrt {\frac {a \sqrt {\frac {a^2+c^2}{a^2}}+a \sqrt {\frac {a^2+c^2}{a^2}} \cos \left (d+e x-\arctan \left (\frac {c}{a}\right )\right )}{-b+a \sqrt {\frac {a^2+c^2}{a^2}}}}}-\frac {\frac {2 a \left (b+a \sqrt {1+\frac {c^2}{a^2}} \cos \left (d+e x-\arctan \left (\frac {c}{a}\right )\right )\right )}{a^2+c^2}-\frac {c \sin \left (d+e x-\arctan \left (\frac {c}{a}\right )\right )}{a \sqrt {1+\frac {c^2}{a^2}}}}{\sqrt {b+a \sqrt {1+\frac {c^2}{a^2}} \cos \left (d+e x-\arctan \left (\frac {c}{a}\right )\right )}}\right )}{c \left (a^2-b^2+c^2\right ) e (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}-\frac {c \sec ^{\frac {3}{2}}(d+e x) (b+a \cos (d+e x)+c \sin (d+e x))^{3/2} \left (-\frac {c \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},-\frac {b+a \sqrt {1+\frac {c^2}{a^2}} \cos \left (d+e x-\arctan \left (\frac {c}{a}\right )\right )}{a \sqrt {1+\frac {c^2}{a^2}} \left (1-\frac {b}{a \sqrt {1+\frac {c^2}{a^2}}}\right )},-\frac {b+a \sqrt {1+\frac {c^2}{a^2}} \cos \left (d+e x-\arctan \left (\frac {c}{a}\right )\right )}{a \sqrt {1+\frac {c^2}{a^2}} \left (-1-\frac {b}{a \sqrt {1+\frac {c^2}{a^2}}}\right )}\right ) \sin \left (d+e x-\arctan \left (\frac {c}{a}\right )\right )}{a \sqrt {1+\frac {c^2}{a^2}} \sqrt {\frac {a \sqrt {\frac {a^2+c^2}{a^2}}-a \sqrt {\frac {a^2+c^2}{a^2}} \cos \left (d+e x-\arctan \left (\frac {c}{a}\right )\right )}{b+a \sqrt {\frac {a^2+c^2}{a^2}}}} \sqrt {b+a \sqrt {\frac {a^2+c^2}{a^2}} \cos \left (d+e x-\arctan \left (\frac {c}{a}\right )\right )} \sqrt {\frac {a \sqrt {\frac {a^2+c^2}{a^2}}+a \sqrt {\frac {a^2+c^2}{a^2}} \cos \left (d+e x-\arctan \left (\frac {c}{a}\right )\right )}{-b+a \sqrt {\frac {a^2+c^2}{a^2}}}}}-\frac {\frac {2 a \left (b+a \sqrt {1+\frac {c^2}{a^2}} \cos \left (d+e x-\arctan \left (\frac {c}{a}\right )\right )\right )}{a^2+c^2}-\frac {c \sin \left (d+e x-\arctan \left (\frac {c}{a}\right )\right )}{a \sqrt {1+\frac {c^2}{a^2}}}}{\sqrt {b+a \sqrt {1+\frac {c^2}{a^2}} \cos \left (d+e x-\arctan \left (\frac {c}{a}\right )\right )}}\right )}{\left (a^2-b^2+c^2\right ) e (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \]

[In]

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

[Out]

(Sec[d + e*x]^(3/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^2*((-2*(a^2 + c^2))/(a*c*(a^2 - b^2 + c^2)) + (2*(b*
c + a^2*Sin[d + e*x] + c^2*Sin[d + e*x]))/(a*(a^2 - b^2 + c^2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x]))))/(e*(a
+ b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2)) - (2*b*AppellF1[1/2, 1/2, 1/2, 3/2, -((b + Sqrt[1 + a^2/c^2]*c*Sin[d
+ e*x + ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(1 - b/(Sqrt[1 + a^2/c^2]*c))*c)), -((b + Sqrt[1 + a^2/c^2]*c*Sin[d
+ e*x + ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(-1 - b/(Sqrt[1 + a^2/c^2]*c))*c))]*Sec[d + e*x]^(3/2)*Sec[d + e*x +
ArcTan[a/c]]*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^(3/2)*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] - c*Sqrt[(a^2 + c^2)/c^
2]*Sin[d + e*x + ArcTan[a/c]])/(b + c*Sqrt[(a^2 + c^2)/c^2])]*Sqrt[b + c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + A
rcTan[a/c]]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] + c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]])/(-b + c*Sqrt[
(a^2 + c^2)/c^2])])/(Sqrt[1 + a^2/c^2]*c*(a^2 - b^2 + c^2)*e*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2)) - (a
^2*Sec[d + e*x]^(3/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^(3/2)*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((b +
a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(1 - b/(a*Sqrt[1 + c^2/a^2])))), -((b +
a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(-1 - b/(a*Sqrt[1 + c^2/a^2]))))]*Sin[d +
e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] - a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x
- ArcTan[c/a]])/(b + a*Sqrt[(a^2 + c^2)/a^2])]*Sqrt[b + a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]]]*Sq
rt[(a*Sqrt[(a^2 + c^2)/a^2] + a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]])/(-b + a*Sqrt[(a^2 + c^2)/a^2
])])) - ((2*a*(b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]]))/(a^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/a]
])/(a*Sqrt[1 + c^2/a^2]))/Sqrt[b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]]]))/(c*(a^2 - b^2 + c^2)*e*(a
+ b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2)) - (c*Sec[d + e*x]^(3/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^(3/2)
*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/
a^2]*(1 - b/(a*Sqrt[1 + c^2/a^2])))), -((b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a
^2]*(-1 - b/(a*Sqrt[1 + c^2/a^2]))))]*Sin[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*Sqrt[(a*Sqrt[(a^2 + c^2
)/a^2] - a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]])/(b + a*Sqrt[(a^2 + c^2)/a^2])]*Sqrt[b + a*Sqrt[(a
^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]]]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] + a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x
- ArcTan[c/a]])/(-b + a*Sqrt[(a^2 + c^2)/a^2])])) - ((2*a*(b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]]
))/(a^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]))/Sqrt[b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*
x - ArcTan[c/a]]]))/((a^2 - b^2 + c^2)*e*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 20.45 (sec) , antiderivative size = 40064, normalized size of antiderivative = 166.93

method	result	size
default	\(\text {Expression too large to display}\)	\(40064\)

[In]

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

[Out]

result too large to display

Fricas [C] (verification not implemented)

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

Time = 0.16 (sec) , antiderivative size = 1724, normalized size of antiderivative = 7.18 \[ \int \frac {\sec ^{\frac {3}{2}}(d+e x)}{(a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/3*((I*a*b^2 - b^2*c + (I*a^2*b - a*b*c)*cos(e*x + d) + (I*a*b*c - b*c^2)*sin(e*x + d))*sqrt(2*a - 2*I*c)*wei
erstrassPInverse(-4/3*(3*a^4 - 4*a^2*b^2 + 4*b^2*c^2 + 6*I*a*c^3 - 3*c^4 + 2*I*(3*a^3 - 4*a*b^2)*c)/(a^4 + 2*a
^2*c^2 + c^4), 8/27*(9*a^5*b - 8*a^3*b^3 - 27*a*b*c^4 - 9*I*b*c^5 + 2*I*(9*a^2*b + 4*b^3)*c^3 - 6*(3*a^3*b - 4
*a*b^3)*c^2 + 3*I*(9*a^4*b - 8*a^2*b^3)*c)/(a^6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6), 1/3*(2*a*b + 2*I*b*c + 3*(a^2
+ c^2)*cos(e*x + d) - 3*(-I*a^2 - I*c^2)*sin(e*x + d))/(a^2 + c^2)) + (-I*a*b^2 - b^2*c + (-I*a^2*b - a*b*c)*c
os(e*x + d) + (-I*a*b*c - b*c^2)*sin(e*x + d))*sqrt(2*a + 2*I*c)*weierstrassPInverse(-4/3*(3*a^4 - 4*a^2*b^2 +
4*b^2*c^2 - 6*I*a*c^3 - 3*c^4 - 2*I*(3*a^3 - 4*a*b^2)*c)/(a^4 + 2*a^2*c^2 + c^4), 8/27*(9*a^5*b - 8*a^3*b^3 -
27*a*b*c^4 + 9*I*b*c^5 - 2*I*(9*a^2*b + 4*b^3)*c^3 - 6*(3*a^3*b - 4*a*b^3)*c^2 - 3*I*(9*a^4*b - 8*a^2*b^3)*c)
/(a^6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6), 1/3*(2*a*b - 2*I*b*c + 3*(a^2 + c^2)*cos(e*x + d) - 3*(I*a^2 + I*c^2)*si
n(e*x + d))/(a^2 + c^2)) - 3*(I*a^2*b + I*b*c^2 + (I*a^3 + I*a*c^2)*cos(e*x + d) + (I*a^2*c + I*c^3)*sin(e*x +
d))*sqrt(2*a - 2*I*c)*weierstrassZeta(-4/3*(3*a^4 - 4*a^2*b^2 + 4*b^2*c^2 + 6*I*a*c^3 - 3*c^4 + 2*I*(3*a^3 -
4*a*b^2)*c)/(a^4 + 2*a^2*c^2 + c^4), 8/27*(9*a^5*b - 8*a^3*b^3 - 27*a*b*c^4 - 9*I*b*c^5 + 2*I*(9*a^2*b + 4*b^3
)*c^3 - 6*(3*a^3*b - 4*a*b^3)*c^2 + 3*I*(9*a^4*b - 8*a^2*b^3)*c)/(a^6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6), weierstr
assPInverse(-4/3*(3*a^4 - 4*a^2*b^2 + 4*b^2*c^2 + 6*I*a*c^3 - 3*c^4 + 2*I*(3*a^3 - 4*a*b^2)*c)/(a^4 + 2*a^2*c^
2 + c^4), 8/27*(9*a^5*b - 8*a^3*b^3 - 27*a*b*c^4 - 9*I*b*c^5 + 2*I*(9*a^2*b + 4*b^3)*c^3 - 6*(3*a^3*b - 4*a*b^
3)*c^2 + 3*I*(9*a^4*b - 8*a^2*b^3)*c)/(a^6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6), 1/3*(2*a*b + 2*I*b*c + 3*(a^2 + c^2
)*cos(e*x + d) - 3*(-I*a^2 - I*c^2)*sin(e*x + d))/(a^2 + c^2))) - 3*(-I*a^2*b - I*b*c^2 + (-I*a^3 - I*a*c^2)*c
os(e*x + d) + (-I*a^2*c - I*c^3)*sin(e*x + d))*sqrt(2*a + 2*I*c)*weierstrassZeta(-4/3*(3*a^4 - 4*a^2*b^2 + 4*b
^2*c^2 - 6*I*a*c^3 - 3*c^4 - 2*I*(3*a^3 - 4*a*b^2)*c)/(a^4 + 2*a^2*c^2 + c^4), 8/27*(9*a^5*b - 8*a^3*b^3 - 27*
a*b*c^4 + 9*I*b*c^5 - 2*I*(9*a^2*b + 4*b^3)*c^3 - 6*(3*a^3*b - 4*a*b^3)*c^2 - 3*I*(9*a^4*b - 8*a^2*b^3)*c)/(a^
6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6), weierstrassPInverse(-4/3*(3*a^4 - 4*a^2*b^2 + 4*b^2*c^2 - 6*I*a*c^3 - 3*c^4
- 2*I*(3*a^3 - 4*a*b^2)*c)/(a^4 + 2*a^2*c^2 + c^4), 8/27*(9*a^5*b - 8*a^3*b^3 - 27*a*b*c^4 + 9*I*b*c^5 - 2*I*(
9*a^2*b + 4*b^3)*c^3 - 6*(3*a^3*b - 4*a*b^3)*c^2 - 3*I*(9*a^4*b - 8*a^2*b^3)*c)/(a^6 + 3*a^4*c^2 + 3*a^2*c^4 +
c^6), 1/3*(2*a*b - 2*I*b*c + 3*(a^2 + c^2)*cos(e*x + d) - 3*(I*a^2 + I*c^2)*sin(e*x + d))/(a^2 + c^2))) - 6*(
(a^2*c + c^3)*cos(e*x + d)^2 - (a^3 + a*c^2)*cos(e*x + d)*sin(e*x + d))*sqrt((a*cos(e*x + d) + c*sin(e*x + d)
+ b)/cos(e*x + d))/sqrt(cos(e*x + d)))/((a^5 - a^3*b^2 + a*c^4 + (2*a^3 - a*b^2)*c^2)*e*cos(e*x + d) + (c^5 +
(2*a^2 - b^2)*c^3 + (a^4 - a^2*b^2)*c)*e*sin(e*x + d) + (a^4*b - a^2*b^3 + b*c^4 + (2*a^2*b - b^3)*c^2)*e)

Sympy [F]

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

[In]

integrate(sec(e*x+d)**(3/2)/(a+b*sec(e*x+d)+c*tan(e*x+d))**(3/2),x)

[Out]

Integral(sec(d + e*x)**(3/2)/(a + b*sec(d + e*x) + c*tan(d + e*x))**(3/2), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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