∫ ∘ __________ a + b sec(e + fx) (c+d sec(e+fx))3∕2 dx [209]

3.3.9 \(\int \frac {\sqrt {a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{3/2}} \, dx\) [209]

Optimal. Leaf size=598 \[ -\frac {2 \sqrt {c+d} \cot (e+f x) \Pi \left (\frac {a (c+d)}{(a+b) c};\text {ArcSin}\left (\frac {\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sqrt {-\frac {(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sec (e+f x))}{(c-d) (a+b \sec (e+f x))}} (a+b \sec (e+f x))}{\sqrt {a+b} c^2 f}-\frac {2 \sqrt {a+b} d \cot (e+f x) E\left (\text {ArcSin}\left (\frac {\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) (1+\sec (e+f x)) \sqrt {\frac {(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}}}{c (c-d) \sqrt {c+d} f \sqrt {-\frac {(b c-a d) (1+\sec (e+f x))}{(a-b) (c+d \sec (e+f x))}}}-\frac {2 (a-b) \sqrt {a+b} d \cot (e+f x) F\left (\text {ArcSin}\left (\frac {\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {\frac {(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sec (e+f x))}{(a-b) (c+d \sec (e+f x))}} (c+d \sec (e+f x))}{c (c-d) \sqrt {c+d} (b c-a d) f} \]

[Out]

-2*cot(f*x+e)*EllipticPi((a+b)^(1/2)*(c+d*sec(f*x+e))^(1/2)/(c+d)^(1/2)/(a+b*sec(f*x+e))^(1/2),a*(c+d)/(a+b)/c

,((a-b)*(c+d)/(a+b)/(c-d))^(1/2))*(a+b*sec(f*x+e))*(c+d)^(1/2)*(-(-a*d+b*c)*(1-sec(f*x+e))/(c+d)/(a+b*sec(f*x+

e)))^(1/2)*((-a*d+b*c)*(1+sec(f*x+e))/(c-d)/(a+b*sec(f*x+e)))^(1/2)/c^2/f/(a+b)^(1/2)-2*d*cot(f*x+e)*EllipticE

((c+d)^(1/2)*(a+b*sec(f*x+e))^(1/2)/(a+b)^(1/2)/(c+d*sec(f*x+e))^(1/2),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(1+sec

(f*x+e))*(a+b)^(1/2)*((-a*d+b*c)*(1-sec(f*x+e))/(a+b)/(c+d*sec(f*x+e)))^(1/2)/c/(c-d)/f/(c+d)^(1/2)/(-(-a*d+b*

c)*(1+sec(f*x+e))/(a-b)/(c+d*sec(f*x+e)))^(1/2)-2*(a-b)*d*cot(f*x+e)*EllipticF((c+d)^(1/2)*(a+b*sec(f*x+e))^(1

/2)/(a+b)^(1/2)/(c+d*sec(f*x+e))^(1/2),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(c+d*sec(f*x+e))*(a+b)^(1/2)*((-a*d+b*

c)*(1-sec(f*x+e))/(a+b)/(c+d*sec(f*x+e)))^(1/2)*(-(-a*d+b*c)*(1+sec(f*x+e))/(a-b)/(c+d*sec(f*x+e)))^(1/2)/c/(c

-d)/(-a*d+b*c)/f/(c+d)^(1/2)

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Rubi [A]
time = 0.59, antiderivative size = 598, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {4024, 4021, 4071, 4069, 4079} \begin {gather*} -\frac {2 \sqrt {c+d} \cot (e+f x) (a+b \sec (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt {\frac {(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \Pi \left (\frac {a (c+d)}{(a+b) c};\text {ArcSin}\left (\frac {\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{c^2 f \sqrt {a+b}}-\frac {2 d (a-b) \sqrt {a+b} \cot (e+f x) (c+d \sec (e+f x)) \sqrt {\frac {(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt {-\frac {(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}} F\left (\text {ArcSin}\left (\frac {\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c f (c-d) \sqrt {c+d} (b c-a d)}-\frac {2 d \sqrt {a+b} \cot (e+f x) (\sec (e+f x)+1) \sqrt {\frac {(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} E\left (\text {ArcSin}\left (\frac {\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c f (c-d) \sqrt {c+d} \sqrt {-\frac {(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*Sqrt[c + d]*Cot[e + f*x]*EllipticPi[(a*(c + d))/((a + b)*c), ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])

/(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])], ((a - b)*(c + d))/((a + b)*(c - d))]*Sqrt[-(((b*c - a*d)*(1 - Sec[e

+ f*x]))/((c + d)*(a + b*Sec[e + f*x])))]*Sqrt[((b*c - a*d)*(1 + Sec[e + f*x]))/((c - d)*(a + b*Sec[e + f*x]))

]*(a + b*Sec[e + f*x]))/(Sqrt[a + b]*c^2*f) - (2*Sqrt[a + b]*d*Cot[e + f*x]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt

[a + b*Sec[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*(1 + Sec[e

+ f*x])*Sqrt[((b*c - a*d)*(1 - Sec[e + f*x]))/((a + b)*(c + d*Sec[e + f*x]))])/(c*(c - d)*Sqrt[c + d]*f*Sqrt[

-(((b*c - a*d)*(1 + Sec[e + f*x]))/((a - b)*(c + d*Sec[e + f*x])))]) - (2*(a - b)*Sqrt[a + b]*d*Cot[e + f*x]*E

llipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sec[e + f*x]])], ((a + b)*(c -

d))/((a - b)*(c + d))]*Sqrt[((b*c - a*d)*(1 - Sec[e + f*x]))/((a + b)*(c + d*Sec[e + f*x]))]*Sqrt[-(((b*c - a*

d)*(1 + Sec[e + f*x]))/((a - b)*(c + d*Sec[e + f*x])))]*(c + d*Sec[e + f*x]))/(c*(c - d)*Sqrt[c + d]*(b*c - a*

d)*f)

Rule 4021

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)], x_Symbol] :> Simp[2

*((a + b*Csc[e + f*x])/(c*f*Rt[(a + b)/(c + d), 2]*Cot[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Csc[e + f*x])/((c - d

)*(a + b*Csc[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Csc[e + f*x])/((c + d)*(a + b*Csc[e + f*x])))]*EllipticPi[

a*((c + d)/(c*(a + b))), ArcSin[Rt[(a + b)/(c + d), 2]*(Sqrt[c + d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]])], (

a - b)*((c + d)/((a + b)*(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]

Rule 4024

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(3/2), x_Symbol] :> Dist

[1/c, Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]], x], x] - Dist[d/c, Int[Csc[e + f*x]*(Sqrt[a + b*C

sc[e + f*x]]/(c + d*Csc[e + f*x])^(3/2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[c

^2 - d^2, 0]

Rule 4069

Int[csc[(e_.) + (f_.)*(x_)]/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.) + (

c_)]), x_Symbol] :> Simp[-2*((c + d*Csc[e + f*x])/(f*(b*c - a*d)*Rt[(c + d)/(a + b), 2]*Cot[e + f*x]))*Sqrt[(b

*c - a*d)*((1 - Csc[e + f*x])/((a + b)*(c + d*Csc[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 + Csc[e + f*x])/((a - b

)*(c + d*Csc[e + f*x])))]*EllipticF[ArcSin[Rt[(c + d)/(a + b), 2]*(Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e +

f*x]])], (a + b)*((c - d)/((a - b)*(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]

Rule 4071

Int[(csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)])/(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)

)^(3/2), x_Symbol] :> Dist[(a - b)/(c - d), Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[e + f*x]

]), x], x] + Dist[(b*c - a*d)/(c - d), Int[Csc[e + f*x]*((1 + Csc[e + f*x])/(Sqrt[a + b*Csc[e + f*x]]*(c + d*C

sc[e + f*x])^(3/2))), x], 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]

Rule 4079

Int[(sec[(e_.) + (f_.)*(x_)]*((A_) + (B_.)*sec[(e_.) + (f_.)*(x_)]))/(Sqrt[(a_) + (b_.)*sec[(e_.) + (f_.)*(x_)

]]*((c_) + (d_.)*sec[(e_.) + (f_.)*(x_)])^(3/2)), x_Symbol] :> Simp[2*A*(1 + Sec[e + f*x])*(Sqrt[(b*c - a*d)*(

(1 - Sec[e + f*x])/((a + b)*(c + d*Sec[e + f*x])))]/(f*(b*c - a*d)*Rt[(c + d)/(a + b), 2]*Tan[e + f*x]*Sqrt[(-

(b*c - a*d))*((1 + Sec[e + f*x])/((a - b)*(c + d*Sec[e + f*x])))]))*EllipticE[ArcSin[Rt[(c + d)/(a + b), 2]*(S

qrt[a + b*Sec[e + f*x]]/Sqrt[c + d*Sec[e + f*x]])], (a + b)*((c - d)/((a - b)*(c + d)))], x] /; FreeQ[{a, b, c

, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B]

Rubi steps

\begin {align*} \int \frac {\sqrt {a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{3/2}} \, dx &=\frac {\int \frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx}{c}-\frac {d \int \frac {\sec (e+f x) \sqrt {a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{3/2}} \, dx}{c}\\ &=-\frac {2 \sqrt {c+d} \cot (e+f x) \Pi \left (\frac {a (c+d)}{(a+b) c};\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sqrt {-\frac {(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sec (e+f x))}{(c-d) (a+b \sec (e+f x))}} (a+b \sec (e+f x))}{\sqrt {a+b} c^2 f}-\frac {((a-b) d) \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx}{c (c-d)}-\frac {(d (b c-a d)) \int \frac {\sec (e+f x) (1+\sec (e+f x))}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))^{3/2}} \, dx}{c (c-d)}\\ &=-\frac {2 \sqrt {c+d} \cot (e+f x) \Pi \left (\frac {a (c+d)}{(a+b) c};\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sqrt {-\frac {(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sec (e+f x))}{(c-d) (a+b \sec (e+f x))}} (a+b \sec (e+f x))}{\sqrt {a+b} c^2 f}-\frac {2 \sqrt {a+b} d \cot (e+f x) E\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) (1+\sec (e+f x)) \sqrt {\frac {(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}}}{c (c-d) \sqrt {c+d} f \sqrt {-\frac {(b c-a d) (1+\sec (e+f x))}{(a-b) (c+d \sec (e+f x))}}}-\frac {2 (a-b) \sqrt {a+b} d \cot (e+f x) F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {\frac {(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sec (e+f x))}{(a-b) (c+d \sec (e+f x))}} (c+d \sec (e+f x))}{c (c-d) \sqrt {c+d} (b c-a d) f}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1708\) vs. \(2(598)=1196\).
time = 9.31, size = 1708, normalized size = 2.86 \begin {gather*} \frac {(d+c \cos (e+f x))^{3/2} \sec (e+f x) \sqrt {a+b \sec (e+f x)} \left (\frac {4 b c (b c-a d) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} (e+f x)\right )}{c-d}} \sqrt {\frac {(c+d) (b+a \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \csc (e+f x) F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}}}{\sqrt {2}}\right )|\frac {2 (b c-a d)}{(a+b) (c-d)}\right ) \sin ^4\left (\frac {1}{2} (e+f x)\right )}{(a+b) (c+d) \sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}}+4 (b c-a d) (a c+b d) \left (\frac {\sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} (e+f x)\right )}{c-d}} \sqrt {\frac {(c+d) (b+a \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \csc (e+f x) F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}}}{\sqrt {2}}\right )|\frac {2 (b c-a d)}{(a+b) (c-d)}\right ) \sin ^4\left (\frac {1}{2} (e+f x)\right )}{(a+b) (c+d) \sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}}-\frac {\sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} (e+f x)\right )}{c-d}} \sqrt {\frac {(c+d) (b+a \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \csc (e+f x) \Pi \left (\frac {b c-a d}{(a+b) c};\text {ArcSin}\left (\frac {\sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}}}{\sqrt {2}}\right )|\frac {2 (b c-a d)}{(a+b) (c-d)}\right ) \sin ^4\left (\frac {1}{2} (e+f x)\right )}{(a+b) c \sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}}\right )+2 a d \left (\frac {\sqrt {\frac {-a+b}{a+b}} (a+b) \cos \left (\frac {1}{2} (e+f x)\right ) \sqrt {d+c \cos (e+f x)} E\left (\text {ArcSin}\left (\frac {\sqrt {\frac {-a+b}{a+b}} \sin \left (\frac {1}{2} (e+f x)\right )}{\sqrt {\frac {b+a \cos (e+f x)}{a+b}}}\right )|\frac {2 (b c-a d)}{(-a+b) (c+d)}\right )}{a c \sqrt {\frac {(a+b) \cos ^2\left (\frac {1}{2} (e+f x)\right )}{b+a \cos (e+f x)}} \sqrt {b+a \cos (e+f x)} \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \sqrt {\frac {(a+b) (d+c \cos (e+f x))}{(c+d) (b+a \cos (e+f x))}}}-\frac {2 (b c-a d) \left (\frac {(b c+(a+b) d) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} (e+f x)\right )}{c-d}} \sqrt {\frac {(c+d) (b+a \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \csc (e+f x) F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}}}{\sqrt {2}}\right )|\frac {2 (b c-a d)}{(a+b) (c-d)}\right ) \sin ^4\left (\frac {1}{2} (e+f x)\right )}{(a+b) (c+d) \sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}}-\frac {(b c+a d) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} (e+f x)\right )}{c-d}} \sqrt {\frac {(c+d) (b+a \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \csc (e+f x) \Pi \left (\frac {b c-a d}{(a+b) c};\text {ArcSin}\left (\frac {\sqrt {\frac {(-a-b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}}}{\sqrt {2}}\right )|\frac {2 (b c-a d)}{(a+b) (c-d)}\right ) \sin ^4\left (\frac {1}{2} (e+f x)\right )}{(a+b) c \sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}}\right )}{a c}+\frac {\sqrt {d+c \cos (e+f x)} \sin (e+f x)}{c \sqrt {b+a \cos (e+f x)}}\right )\right )}{(c-d) (c+d) f \sqrt {b+a \cos (e+f x)} (c+d \sec (e+f x))^{3/2}}+\frac {2 d (d+c \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \tan (e+f x)}{\left (-c^2+d^2\right ) f (c+d \sec (e+f x))^{3/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((d + c*Cos[e + f*x])^(3/2)*Sec[e + f*x]*Sqrt[a + b*Sec[e + f*x]]*((4*b*c*(b*c - a*d)*Sqrt[((c + d)*Cot[(e + f

*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*C

os[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x]

)*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*(

c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) + 4*(b*c - a*d)*(a*c + b*d)*((Sqrt[((c + d)*Cot[(e +

f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c

*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*

x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)

*(c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) - (Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt

[((c + d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f

*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticPi[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f

*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b

)*c*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])) + 2*a*d*((Sqrt[(-a + b)/(a + b)]*(a + b)*Cos[(e + f*x)

/2]*Sqrt[d + c*Cos[e + f*x]]*EllipticE[ArcSin[(Sqrt[(-a + b)/(a + b)]*Sin[(e + f*x)/2])/Sqrt[(b + a*Cos[e + f*

x])/(a + b)]], (2*(b*c - a*d))/((-a + b)*(c + d))])/(a*c*Sqrt[((a + b)*Cos[(e + f*x)/2]^2)/(b + a*Cos[e + f*x]

)]*Sqrt[b + a*Cos[e + f*x]]*Sqrt[(b + a*Cos[e + f*x])/(a + b)]*Sqrt[((a + b)*(d + c*Cos[e + f*x]))/((c + d)*(b

+ a*Cos[e + f*x]))]) - (2*(b*c - a*d)*(((b*c + (a + b)*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c

+ d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/

2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c

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

*x]]*Sqrt[d + c*Cos[e + f*x]]) - ((b*c + a*d)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*(b + a*

Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c -

a*d)]*Csc[e + f*x]*EllipticPi[(b*c - a*d)/((a + b)*c), ArcSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x

)/2]^2)/(b*c - a*d)]/Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)*c*Sqrt[b + a*Co

s[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])))/(a*c) + (Sqrt[d + c*Cos[e + f*x]]*Sin[e + f*x])/(c*Sqrt[b + a*Cos[e +

f*x]]))))/((c - d)*(c + d)*f*Sqrt[b + a*Cos[e + f*x]]*(c + d*Sec[e + f*x])^(3/2)) + (2*d*(d + c*Cos[e + f*x])*

Sqrt[a + b*Sec[e + f*x]]*Tan[e + f*x])/((-c^2 + d^2)*f*(c + d*Sec[e + f*x])^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2846\) vs. \(2(553)=1106\).
time = 2.74, size = 2847, normalized size = 4.76


method	result	size

default	\(\text {Expression too large to display}\)	\(2847\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/f*(-((a-b)/(a+b))^(1/2)*b*c*d-sin(f*x+e)*EllipticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-

d)/(a-b)/(c+d))^(1/2))*((a*cos(f*x+e)+b)/(cos(f*x+e)+1)/(a+b))^(1/2)*((d+c*cos(f*x+e))/(cos(f*x+e)+1)/(c+d))^(

1/2)*b*c^2-((a*cos(f*x+e)+b)/(cos(f*x+e)+1)/(a+b))^(1/2)*((d+c*cos(f*x+e))/(cos(f*x+e)+1)/(c+d))^(1/2)*Ellipti

cE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a*d^2*sin(f*x+e)+((a*cos(f*

x+e)+b)/(cos(f*x+e)+1)/(a+b))^(1/2)*((d+c*cos(f*x+e))/(cos(f*x+e)+1)/(c+d))^(1/2)*EllipticE((-1+cos(f*x+e))*((

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

+1)/(a+b))^(1/2)*((d+c*cos(f*x+e))/(cos(f*x+e)+1)/(c+d))^(1/2)*EllipticPi((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/

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

+e)+1)/(a+b))^(1/2)*((d+c*cos(f*x+e))/(cos(f*x+e)+1)/(c+d))^(1/2)*EllipticPi((-1+cos(f*x+e))*((a-b)/(a+b))^(1/

2)/sin(f*x+e),-(a+b)/(a-b),((c-d)/(c+d))^(1/2)/((a-b)/(a+b))^(1/2))*a*d^2*sin(f*x+e)+cos(f*x+e)*sin(f*x+e)*Ell

ipticE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*((a*cos(f*x+e)+b)/(cos(

f*x+e)+1)/(a+b))^(1/2)*((d+c*cos(f*x+e))/(cos(f*x+e)+1)/(c+d))^(1/2)*b*c*d+EllipticF((-1+cos(f*x+e))*((a-b)/(a

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

(f*x+e))/(cos(f*x+e)+1)/(c+d))^(1/2)*a*c^2*sin(f*x+e)+((a-b)/(a+b))^(1/2)*b*d^2+cos(f*x+e)*sin(f*x+e)*Elliptic

F((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*((a*cos(f*x+e)+b)/(cos(f*x+e

)+1)/(a+b))^(1/2)*((d+c*cos(f*x+e))/(cos(f*x+e)+1)/(c+d))^(1/2)*a*c^2-cos(f*x+e)*sin(f*x+e)*EllipticF((-1+cos(

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

)^(1/2)*((d+c*cos(f*x+e))/(cos(f*x+e)+1)/(c+d))^(1/2)*b*c^2-cos(f*x+e)*sin(f*x+e)*EllipticE((-1+cos(f*x+e))*((

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

d+c*cos(f*x+e))/(cos(f*x+e)+1)/(c+d))^(1/2)*a*d^2+cos(f*x+e)*sin(f*x+e)*EllipticE((-1+cos(f*x+e))*((a-b)/(a+b)

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

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

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

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

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

1)/(a+b))^(1/2)*((d+c*cos(f*x+e))/(cos(f*x+e)+1)/(c+d))^(1/2)*a*d^2+sin(f*x+e)*EllipticF((-1+cos(f*x+e))*((a-b

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

*cos(f*x+e))/(cos(f*x+e)+1)/(c+d))^(1/2)*a*c*d-sin(f*x+e)*EllipticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*

x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*((a*cos(f*x+e)+b)/(cos(f*x+e)+1)/(a+b))^(1/2)*((d+c*cos(f*x+e))/(cos(f*x

+e)+1)/(c+d))^(1/2)*b*c*d-sin(f*x+e)*EllipticE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-

b)/(c+d))^(1/2))*((a*cos(f*x+e)+b)/(cos(f*x+e)+1)/(a+b))^(1/2)*((d+c*cos(f*x+e))/(cos(f*x+e)+1)/(c+d))^(1/2)*a

*c*d+sin(f*x+e)*EllipticE((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*((a*

cos(f*x+e)+b)/(cos(f*x+e)+1)/(a+b))^(1/2)*((d+c*cos(f*x+e))/(cos(f*x+e)+1)/(c+d))^(1/2)*b*c*d+cos(f*x+e)*sin(f

*x+e)*EllipticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*((a*cos(f*x+e)

+b)/(cos(f*x+e)+1)/(a+b))^(1/2)*((d+c*cos(f*x+e))/(cos(f*x+e)+1)/(c+d))^(1/2)*a*c*d-cos(f*x+e)*sin(f*x+e)*Elli

pticF((-1+cos(f*x+e))*((a-b)/(a+b))^(1/2)/sin(f*x+e),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*((a*cos(f*x+e)+b)/(cos(f

*x+e)+1)/(a+b))^(1/2)*((d+c*cos(f*x+e))/(cos(f*x+e)+1)/(c+d))^(1/2)*b*c*d-cos(f*x+e)*sin(f*x+e)*EllipticE((-1+

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

a+b))^(1/2)*((d+c*cos(f*x+e))/(cos(f*x+e)+1)/(c+d))^(1/2)*a*c*d+((a-b)/(a+b))^(1/2)*cos(f*x+e)^2*a*c*d-((a-b)/

(a+b))^(1/2)*cos(f*x+e)*a*c*d+((a-b)/(a+b))^(1/2)*cos(f*x+e)*b*c*d+((a-b)/(a+b))^(1/2)*cos(f*x+e)*a*d^2-((a-b)

/(a+b))^(1/2)*cos(f*x+e)*b*d^2-((a-b)/(a+b))^(1/2)*cos(f*x+e)^2*a*d^2)*cos(f*x+e)*((a*cos(f*x+e)+b)/cos(f*x+e)

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

c-d)/(c+d)/c

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

[Out]

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

________________________________________________________________________________________

Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + b \sec {\left (e + f x \right )}}}{\left (c + d \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}}{{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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