\(\int \frac {c+d \sec (e+f x)}{(a+b \sec (e+f x))^{5/2}} \, dx\) [206]

Optimal result

Integrand size = 25, antiderivative size = 495 \[ \int \frac {c+d \sec (e+f x)}{(a+b \sec (e+f x))^{5/2}} \, dx=\frac {2 \left (7 a^2 b c-3 b^3 c-4 a^3 d\right ) \cot (e+f x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{3 a^2 (a-b) b (a+b)^{3/2} f}-\frac {2 \left (6 a^2 b c-a b^2 c-3 b^3 c-3 a^3 d+a^2 b d\right ) \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{3 a^2 (a-b) b (a+b)^{3/2} f}-\frac {2 \sqrt {a+b} c \cot (e+f x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{a^3 f}+\frac {2 b (b c-a d) \tan (e+f x)}{3 a \left (a^2-b^2\right ) f (a+b \sec (e+f x))^{3/2}}+\frac {2 b \left (7 a^2 b c-3 b^3 c-4 a^3 d\right ) \tan (e+f x)}{3 a^2 \left (a^2-b^2\right )^2 f \sqrt {a+b \sec (e+f x)}} \]

[Out]

2/3*(-4*a^3*d+7*a^2*b*c-3*b^3*c)*cot(f*x+e)*EllipticE((a+b*sec(f*x+e))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*
(b*(1-sec(f*x+e))/(a+b))^(1/2)*(-b*(1+sec(f*x+e))/(a-b))^(1/2)/a^2/(a-b)/b/(a+b)^(3/2)/f-2/3*(-3*a^3*d+6*a^2*b
*c+a^2*b*d-a*b^2*c-3*b^3*c)*cot(f*x+e)*EllipticF((a+b*sec(f*x+e))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1
-sec(f*x+e))/(a+b))^(1/2)*(-b*(1+sec(f*x+e))/(a-b))^(1/2)/a^2/(a-b)/b/(a+b)^(3/2)/f-2*c*cot(f*x+e)*EllipticPi(
(a+b*sec(f*x+e))^(1/2)/(a+b)^(1/2),(a+b)/a,((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(f*x+e))/(a+b))^(1/2)*(-b
*(1+sec(f*x+e))/(a-b))^(1/2)/a^3/f+2/3*b*(-a*d+b*c)*tan(f*x+e)/a/(a^2-b^2)/f/(a+b*sec(f*x+e))^(3/2)+2/3*b*(-4*
a^3*d+7*a^2*b*c-3*b^3*c)*tan(f*x+e)/a^2/(a^2-b^2)^2/f/(a+b*sec(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {4008, 4145, 4143, 4006, 3869, 3917, 4089} \[ \int \frac {c+d \sec (e+f x)}{(a+b \sec (e+f x))^{5/2}} \, dx=-\frac {2 c \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a^3 f}+\frac {2 b (b c-a d) \tan (e+f x)}{3 a f \left (a^2-b^2\right ) (a+b \sec (e+f x))^{3/2}}+\frac {2 \left (-4 a^3 d+7 a^2 b c-3 b^3 c\right ) \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{3 a^2 b f (a-b) (a+b)^{3/2}}-\frac {2 \left (-3 a^3 d+6 a^2 b c+a^2 b d-a b^2 c-3 b^3 c\right ) \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{3 a^2 b f (a-b) (a+b)^{3/2}}+\frac {2 b \left (-4 a^3 d+7 a^2 b c-3 b^3 c\right ) \tan (e+f x)}{3 a^2 f \left (a^2-b^2\right )^2 \sqrt {a+b \sec (e+f x)}} \]

[In]

Int[(c + d*Sec[e + f*x])/(a + b*Sec[e + f*x])^(5/2),x]

[Out]

(2*(7*a^2*b*c - 3*b^3*c - 4*a^3*d)*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b
)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(3*a^2*(a - b)*b*(a +
b)^(3/2)*f) - (2*(6*a^2*b*c - a*b^2*c - 3*b^3*c - 3*a^3*d + a^2*b*d)*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[a + b
*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]
))/(a - b))])/(3*a^2*(a - b)*b*(a + b)^(3/2)*f) - (2*Sqrt[a + b]*c*Cot[e + f*x]*EllipticPi[(a + b)/a, ArcSin[S
qrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec
[e + f*x]))/(a - b))])/(a^3*f) + (2*b*(b*c - a*d)*Tan[e + f*x])/(3*a*(a^2 - b^2)*f*(a + b*Sec[e + f*x])^(3/2))
+ (2*b*(7*a^2*b*c - 3*b^3*c - 4*a^3*d)*Tan[e + f*x])/(3*a^2*(a^2 - b^2)^2*f*Sqrt[a + b*Sec[e + f*x]])

Rule 3869

Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*(Rt[a + b, 2]/(a*d*Cot[c + d*x]))*Sqrt[b
*((1 - Csc[c + d*x])/(a + b))]*Sqrt[(-b)*((1 + Csc[c + d*x])/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b
*Csc[c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]

Rule 3917

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(Rt[a + b, 2]/(b*
f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin
[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4006

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

Rule 4008

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Simp[b*(b
*c - a*d)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(a*f*(m + 1)*(a^2 - b^2))), x] + Dist[1/(a*(m + 1)*(a^2 -
b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[c*(a^2 - b^2)*(m + 1) - (a*(b*c - a*d)*(m + 1))*Csc[e + f*x] + b
*(b*c - a*d)*(m + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m,
-1] && NeQ[a^2 - b^2, 0] && IntegerQ[2*m]

Rule 4089

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)
], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + C
sc[e + f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b*(B/A), 2]], (a
*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]

Rule 4143

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_
.) + (a_)], x_Symbol] :> Int[(A + (B - C)*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + Dist[C, Int[Csc[e + f*x
]*((1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0
]

Rule 4145

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) +
(a_))^(m_), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(a*f*(m + 1)
*(a^2 - b^2))), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[A*(a^2 - b^2)*(m +
1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {2 b (b c-a d) \tan (e+f x)}{3 a \left (a^2-b^2\right ) f (a+b \sec (e+f x))^{3/2}}-\frac {2 \int \frac {-\frac {3}{2} \left (a^2-b^2\right ) c+\frac {3}{2} a (b c-a d) \sec (e+f x)-\frac {1}{2} b (b c-a d) \sec ^2(e+f x)}{(a+b \sec (e+f x))^{3/2}} \, dx}{3 a \left (a^2-b^2\right )} \\ & = \frac {2 b (b c-a d) \tan (e+f x)}{3 a \left (a^2-b^2\right ) f (a+b \sec (e+f x))^{3/2}}+\frac {2 b \left (7 a^2 b c-3 b^3 c-4 a^3 d\right ) \tan (e+f x)}{3 a^2 \left (a^2-b^2\right )^2 f \sqrt {a+b \sec (e+f x)}}+\frac {4 \int \frac {\frac {3}{4} \left (a^2-b^2\right )^2 c-\frac {1}{4} a \left (6 a^2 b c-2 b^3 c-3 a^3 d-a b^2 d\right ) \sec (e+f x)-\frac {1}{4} b \left (7 a^2 b c-3 b^3 c-4 a^3 d\right ) \sec ^2(e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx}{3 a^2 \left (a^2-b^2\right )^2} \\ & = \frac {2 b (b c-a d) \tan (e+f x)}{3 a \left (a^2-b^2\right ) f (a+b \sec (e+f x))^{3/2}}+\frac {2 b \left (7 a^2 b c-3 b^3 c-4 a^3 d\right ) \tan (e+f x)}{3 a^2 \left (a^2-b^2\right )^2 f \sqrt {a+b \sec (e+f x)}}+\frac {4 \int \frac {\frac {3}{4} \left (a^2-b^2\right )^2 c+\left (\frac {1}{4} b \left (7 a^2 b c-3 b^3 c-4 a^3 d\right )-\frac {1}{4} a \left (6 a^2 b c-2 b^3 c-3 a^3 d-a b^2 d\right )\right ) \sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx}{3 a^2 \left (a^2-b^2\right )^2}-\frac {\left (b \left (7 a^2 b c-3 b^3 c-4 a^3 d\right )\right ) \int \frac {\sec (e+f x) (1+\sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx}{3 a^2 \left (a^2-b^2\right )^2} \\ & = \frac {2 \left (7 a^2 b c-3 b^3 c-4 a^3 d\right ) \cot (e+f x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{3 a^2 (a-b) b (a+b)^{3/2} f}+\frac {2 b (b c-a d) \tan (e+f x)}{3 a \left (a^2-b^2\right ) f (a+b \sec (e+f x))^{3/2}}+\frac {2 b \left (7 a^2 b c-3 b^3 c-4 a^3 d\right ) \tan (e+f x)}{3 a^2 \left (a^2-b^2\right )^2 f \sqrt {a+b \sec (e+f x)}}+\frac {c \int \frac {1}{\sqrt {a+b \sec (e+f x)}} \, dx}{a^2}+\frac {\left (a b^2 c+3 b^3 c+3 a^3 d-a^2 b (6 c+d)\right ) \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx}{3 a^2 (a-b) (a+b)^2} \\ & = \frac {2 \left (7 a^2 b c-3 b^3 c-4 a^3 d\right ) \cot (e+f x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{3 a^2 (a-b) b (a+b)^{3/2} f}+\frac {2 \left (a b^2 c+3 b^3 c+3 a^3 d-a^2 b (6 c+d)\right ) \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{3 a^2 (a-b) b (a+b)^{3/2} f}-\frac {2 \sqrt {a+b} c \cot (e+f x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{a^3 f}+\frac {2 b (b c-a d) \tan (e+f x)}{3 a \left (a^2-b^2\right ) f (a+b \sec (e+f x))^{3/2}}+\frac {2 b \left (7 a^2 b c-3 b^3 c-4 a^3 d\right ) \tan (e+f x)}{3 a^2 \left (a^2-b^2\right )^2 f \sqrt {a+b \sec (e+f x)}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1589\) vs. \(2(495)=990\).

Time = 14.93 (sec) , antiderivative size = 1589, normalized size of antiderivative = 3.21 \[ \int \frac {c+d \sec (e+f x)}{(a+b \sec (e+f x))^{5/2}} \, dx=\frac {(b+a \cos (e+f x))^3 \sec ^2(e+f x) (c+d \sec (e+f x)) \left (\frac {2 \left (-7 a^2 b c+3 b^3 c+4 a^3 d\right ) \sin (e+f x)}{3 a^2 \left (a^2-b^2\right )^2}-\frac {2 \left (b^3 c \sin (e+f x)-a b^2 d \sin (e+f x)\right )}{3 a^2 \left (a^2-b^2\right ) (b+a \cos (e+f x))^2}-\frac {2 \left (-8 a^2 b^2 c \sin (e+f x)+4 b^4 c \sin (e+f x)+5 a^3 b d \sin (e+f x)-a b^3 d \sin (e+f x)\right )}{3 a^2 \left (a^2-b^2\right )^2 (b+a \cos (e+f x))}\right )}{f (d+c \cos (e+f x)) (a+b \sec (e+f x))^{5/2}}-\frac {2 (b+a \cos (e+f x))^{5/2} \sec ^{\frac {3}{2}}(e+f x) (c+d \sec (e+f x)) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{1+\tan ^2\left (\frac {1}{2} (e+f x)\right )}} \left (-7 a^3 b c \tan \left (\frac {1}{2} (e+f x)\right )-7 a^2 b^2 c \tan \left (\frac {1}{2} (e+f x)\right )+3 a b^3 c \tan \left (\frac {1}{2} (e+f x)\right )+3 b^4 c \tan \left (\frac {1}{2} (e+f x)\right )+4 a^4 d \tan \left (\frac {1}{2} (e+f x)\right )+4 a^3 b d \tan \left (\frac {1}{2} (e+f x)\right )+14 a^3 b c \tan ^3\left (\frac {1}{2} (e+f x)\right )-6 a b^3 c \tan ^3\left (\frac {1}{2} (e+f x)\right )-8 a^4 d \tan ^3\left (\frac {1}{2} (e+f x)\right )-7 a^3 b c \tan ^5\left (\frac {1}{2} (e+f x)\right )+7 a^2 b^2 c \tan ^5\left (\frac {1}{2} (e+f x)\right )+3 a b^3 c \tan ^5\left (\frac {1}{2} (e+f x)\right )-3 b^4 c \tan ^5\left (\frac {1}{2} (e+f x)\right )+4 a^4 d \tan ^5\left (\frac {1}{2} (e+f x)\right )-4 a^3 b d \tan ^5\left (\frac {1}{2} (e+f x)\right )-6 a^4 c \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}+12 a^2 b^2 c \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}-6 b^4 c \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}-6 a^4 c \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}+12 a^2 b^2 c \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}-6 b^4 c \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}+(a+b) \left (-7 a^2 b c+3 b^3 c+4 a^3 d\right ) E\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}+a (a+b) \left (-2 b^2 c+3 a^2 (c-d)+a b (3 c-d)\right ) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}\right )}{3 a^2 \left (a^2-b^2\right )^2 f (d+c \cos (e+f x)) (a+b \sec (e+f x))^{5/2} \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {\frac {1+\tan ^2\left (\frac {1}{2} (e+f x)\right )}{1-\tan ^2\left (\frac {1}{2} (e+f x)\right )}} \left (a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-b \left (1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )} \]

[In]

Integrate[(c + d*Sec[e + f*x])/(a + b*Sec[e + f*x])^(5/2),x]

[Out]

((b + a*Cos[e + f*x])^3*Sec[e + f*x]^2*(c + d*Sec[e + f*x])*((2*(-7*a^2*b*c + 3*b^3*c + 4*a^3*d)*Sin[e + f*x])
/(3*a^2*(a^2 - b^2)^2) - (2*(b^3*c*Sin[e + f*x] - a*b^2*d*Sin[e + f*x]))/(3*a^2*(a^2 - b^2)*(b + a*Cos[e + f*x
])^2) - (2*(-8*a^2*b^2*c*Sin[e + f*x] + 4*b^4*c*Sin[e + f*x] + 5*a^3*b*d*Sin[e + f*x] - a*b^3*d*Sin[e + f*x]))
/(3*a^2*(a^2 - b^2)^2*(b + a*Cos[e + f*x]))))/(f*(d + c*Cos[e + f*x])*(a + b*Sec[e + f*x])^(5/2)) - (2*(b + a*
Cos[e + f*x])^(5/2)*Sec[e + f*x]^(3/2)*(c + d*Sec[e + f*x])*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*
x)/2]^2)/(1 + Tan[(e + f*x)/2]^2)]*(-7*a^3*b*c*Tan[(e + f*x)/2] - 7*a^2*b^2*c*Tan[(e + f*x)/2] + 3*a*b^3*c*Tan
[(e + f*x)/2] + 3*b^4*c*Tan[(e + f*x)/2] + 4*a^4*d*Tan[(e + f*x)/2] + 4*a^3*b*d*Tan[(e + f*x)/2] + 14*a^3*b*c*
Tan[(e + f*x)/2]^3 - 6*a*b^3*c*Tan[(e + f*x)/2]^3 - 8*a^4*d*Tan[(e + f*x)/2]^3 - 7*a^3*b*c*Tan[(e + f*x)/2]^5
+ 7*a^2*b^2*c*Tan[(e + f*x)/2]^5 + 3*a*b^3*c*Tan[(e + f*x)/2]^5 - 3*b^4*c*Tan[(e + f*x)/2]^5 + 4*a^4*d*Tan[(e
+ f*x)/2]^5 - 4*a^3*b*d*Tan[(e + f*x)/2]^5 - 6*a^4*c*EllipticPi[-1, ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]
*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] + 12*a^2*b^2
*c*EllipticPi[-1, ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[
(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] - 6*b^4*c*EllipticPi[-1, ArcSin[Tan[(e + f*x)/2]], (a - b)/(a
+ b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] - 6*a^4
*c*EllipticPi[-1, ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]*Tan[(e + f*x)/2]^2*Sqrt[1 - Tan[(e + f*x)/2]^2]*S
qrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] + 12*a^2*b^2*c*EllipticPi[-1, ArcSin[Tan[(e
+ f*x)/2]], (a - b)/(a + b)]*Tan[(e + f*x)/2]^2*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]
^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] - 6*b^4*c*EllipticPi[-1, ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]*Tan[(e
+ f*x)/2]^2*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)]
+ (a + b)*(-7*a^2*b*c + 3*b^3*c + 4*a^3*d)*EllipticE[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(
e + f*x)/2]^2]*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] +
a*(a + b)*(-2*b^2*c + 3*a^2*(c - d) + a*b*(3*c - d))*EllipticF[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]*Sqrt
[1 - Tan[(e + f*x)/2]^2]*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(
a + b)]))/(3*a^2*(a^2 - b^2)^2*f*(d + c*Cos[e + f*x])*(a + b*Sec[e + f*x])^(5/2)*(-1 + Tan[(e + f*x)/2]^2)*Sqr
t[(1 + Tan[(e + f*x)/2]^2)/(1 - Tan[(e + f*x)/2]^2)]*(a*(-1 + Tan[(e + f*x)/2]^2) - b*(1 + Tan[(e + f*x)/2]^2)
))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(7172\) vs. \(2(456)=912\).

Time = 17.34 (sec) , antiderivative size = 7173, normalized size of antiderivative = 14.49

method	result	size
parts	\(\text {Expression too large to display}\)	\(7173\)
default	\(\text {Expression too large to display}\)	\(7241\)

[In]

int((c+d*sec(f*x+e))/(a+b*sec(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F]

\[ \int \frac {c+d \sec (e+f x)}{(a+b \sec (e+f x))^{5/2}} \, dx=\int { \frac {d \sec \left (f x + e\right ) + c}{{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((c+d*sec(f*x+e))/(a+b*sec(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {c+d \sec (e+f x)}{(a+b \sec (e+f x))^{5/2}} \, dx=\int \frac {c + d \sec {\left (e + f x \right )}}{\left (a + b \sec {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate((c+d*sec(f*x+e))/(a+b*sec(f*x+e))**(5/2),x)

[Out]

Integral((c + d*sec(e + f*x))/(a + b*sec(e + f*x))**(5/2), x)

Maxima [F]

\[ \int \frac {c+d \sec (e+f x)}{(a+b \sec (e+f x))^{5/2}} \, dx=\int { \frac {d \sec \left (f x + e\right ) + c}{{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((c+d*sec(f*x+e))/(a+b*sec(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

integrate((d*sec(f*x + e) + c)/(b*sec(f*x + e) + a)^(5/2), x)

Giac [F]

\[ \int \frac {c+d \sec (e+f x)}{(a+b \sec (e+f x))^{5/2}} \, dx=\int { \frac {d \sec \left (f x + e\right ) + c}{{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((c+d*sec(f*x+e))/(a+b*sec(f*x+e))^(5/2),x, algorithm="giac")

[Out]

integrate((d*sec(f*x + e) + c)/(b*sec(f*x + e) + a)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {c+d \sec (e+f x)}{(a+b \sec (e+f x))^{5/2}} \, dx=\int \frac {c+\frac {d}{\cos \left (e+f\,x\right )}}{{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]

[In]

int((c + d/cos(e + f*x))/(a + b/cos(e + f*x))^(5/2),x)

[Out]

int((c + d/cos(e + f*x))/(a + b/cos(e + f*x))^(5/2), x)