3.7.16 \(\int \frac {1}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))^2} \, dx\) [616]

Optimal. Leaf size=700 \[ \frac {9 a b^{7/2} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt [4]{\sec ^2(e+f x)}}{2 \left (a^2+b^2\right )^{13/4} d^2 f \sqrt {d \sec (e+f x)}}-\frac {9 a b^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt [4]{\sec ^2(e+f x)}}{2 \left (a^2+b^2\right )^{13/4} d^2 f \sqrt {d \sec (e+f x)}}+\frac {3 \left (2 a^4+10 a^2 b^2-7 b^4\right ) E\left (\left .\frac {1}{2} \text {ArcTan}(\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{5 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}-\frac {3 \left (2 a^4+10 a^2 b^2-7 b^4\right ) \tan (e+f x)}{5 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}-\frac {9 a^2 b^3 \cot (e+f x) \Pi \left (-\frac {b}{\sqrt {a^2+b^2}};\left .\text {ArcSin}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right ) \sqrt [4]{\sec ^2(e+f x)} \sqrt {-\tan ^2(e+f x)}}{2 \left (a^2+b^2\right )^{7/2} d^2 f \sqrt {d \sec (e+f x)}}+\frac {9 a^2 b^3 \cot (e+f x) \Pi \left (\frac {b}{\sqrt {a^2+b^2}};\left .\text {ArcSin}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right ) \sqrt [4]{\sec ^2(e+f x)} \sqrt {-\tan ^2(e+f x)}}{2 \left (a^2+b^2\right )^{7/2} d^2 f \sqrt {d \sec (e+f x)}}+\frac {3 b \left (2 a^4+10 a^2 b^2-7 b^4\right ) \sec ^2(e+f x)}{5 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}+\frac {2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}-\frac {2 \left (b \left (2 a^2-7 b^2\right )-3 a \left (a^2+4 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))} \]

[Out]

9/2*a*b^(7/2)*arctan((sec(f*x+e)^2)^(1/4)*b^(1/2)/(a^2+b^2)^(1/4))*(sec(f*x+e)^2)^(1/4)/(a^2+b^2)^(13/4)/d^2/f
/(d*sec(f*x+e))^(1/2)-9/2*a*b^(7/2)*arctanh((sec(f*x+e)^2)^(1/4)*b^(1/2)/(a^2+b^2)^(1/4))*(sec(f*x+e)^2)^(1/4)
/(a^2+b^2)^(13/4)/d^2/f/(d*sec(f*x+e))^(1/2)+3/5*(2*a^4+10*a^2*b^2-7*b^4)*(cos(1/2*arctan(tan(f*x+e)))^2)^(1/2
)/cos(1/2*arctan(tan(f*x+e)))*EllipticE(sin(1/2*arctan(tan(f*x+e))),2^(1/2))*(sec(f*x+e)^2)^(1/4)/(a^2+b^2)^3/
d^2/f/(d*sec(f*x+e))^(1/2)-9/2*a^2*b^3*cot(f*x+e)*EllipticPi((sec(f*x+e)^2)^(1/4),-b/(a^2+b^2)^(1/2),I)*(sec(f
*x+e)^2)^(1/4)*(-tan(f*x+e)^2)^(1/2)/(a^2+b^2)^(7/2)/d^2/f/(d*sec(f*x+e))^(1/2)+9/2*a^2*b^3*cot(f*x+e)*Ellipti
cPi((sec(f*x+e)^2)^(1/4),b/(a^2+b^2)^(1/2),I)*(sec(f*x+e)^2)^(1/4)*(-tan(f*x+e)^2)^(1/2)/(a^2+b^2)^(7/2)/d^2/f
/(d*sec(f*x+e))^(1/2)-3/5*(2*a^4+10*a^2*b^2-7*b^4)*tan(f*x+e)/(a^2+b^2)^3/d^2/f/(d*sec(f*x+e))^(1/2)+3/5*b*(2*
a^4+10*a^2*b^2-7*b^4)*sec(f*x+e)^2/(a^2+b^2)^3/d^2/f/(d*sec(f*x+e))^(1/2)/(a+b*tan(f*x+e))+2/5*cos(f*x+e)^2*(b
+a*tan(f*x+e))/(a^2+b^2)/d^2/f/(d*sec(f*x+e))^(1/2)/(a+b*tan(f*x+e))-2/5*(b*(2*a^2-7*b^2)-3*a*(a^2+4*b^2)*tan(
f*x+e))/(a^2+b^2)^2/d^2/f/(d*sec(f*x+e))^(1/2)/(a+b*tan(f*x+e))

________________________________________________________________________________________

Rubi [A]
time = 0.55, antiderivative size = 700, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 17, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {3593, 755, 837, 849, 858, 233, 202, 760, 408, 504, 1227, 551, 455, 65, 304, 211, 214} \begin {gather*} -\frac {9 a^2 b^3 \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \Pi \left (-\frac {b}{\sqrt {a^2+b^2}};\left .\text {ArcSin}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{2 d^2 f \left (a^2+b^2\right )^{7/2} \sqrt {d \sec (e+f x)}}+\frac {9 a^2 b^3 \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \Pi \left (\frac {b}{\sqrt {a^2+b^2}};\left .\text {ArcSin}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{2 d^2 f \left (a^2+b^2\right )^{7/2} \sqrt {d \sec (e+f x)}}+\frac {9 a b^{7/2} \sqrt [4]{\sec ^2(e+f x)} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{2 d^2 f \left (a^2+b^2\right )^{13/4} \sqrt {d \sec (e+f x)}}-\frac {2 \left (b \left (2 a^2-7 b^2\right )-3 a \left (a^2+4 b^2\right ) \tan (e+f x)\right )}{5 d^2 f \left (a^2+b^2\right )^2 \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}+\frac {2 \cos ^2(e+f x) (a \tan (e+f x)+b)}{5 d^2 f \left (a^2+b^2\right ) \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}-\frac {9 a b^{7/2} \sqrt [4]{\sec ^2(e+f x)} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{2 d^2 f \left (a^2+b^2\right )^{13/4} \sqrt {d \sec (e+f x)}}+\frac {3 \left (2 a^4+10 a^2 b^2-7 b^4\right ) \sqrt [4]{\sec ^2(e+f x)} E\left (\left .\frac {1}{2} \text {ArcTan}(\tan (e+f x))\right |2\right )}{5 d^2 f \left (a^2+b^2\right )^3 \sqrt {d \sec (e+f x)}}+\frac {3 b \left (2 a^4+10 a^2 b^2-7 b^4\right ) \sec ^2(e+f x)}{5 d^2 f \left (a^2+b^2\right )^3 \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}-\frac {3 \left (2 a^4+10 a^2 b^2-7 b^4\right ) \tan (e+f x)}{5 d^2 f \left (a^2+b^2\right )^3 \sqrt {d \sec (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d*Sec[e + f*x])^(5/2)*(a + b*Tan[e + f*x])^2),x]

[Out]

(9*a*b^(7/2)*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(Sec[e + f*x]^2)^(1/4))/(2*(a^2 + b^2)
^(13/4)*d^2*f*Sqrt[d*Sec[e + f*x]]) - (9*a*b^(7/2)*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]
*(Sec[e + f*x]^2)^(1/4))/(2*(a^2 + b^2)^(13/4)*d^2*f*Sqrt[d*Sec[e + f*x]]) + (3*(2*a^4 + 10*a^2*b^2 - 7*b^4)*E
llipticE[ArcTan[Tan[e + f*x]]/2, 2]*(Sec[e + f*x]^2)^(1/4))/(5*(a^2 + b^2)^3*d^2*f*Sqrt[d*Sec[e + f*x]]) - (3*
(2*a^4 + 10*a^2*b^2 - 7*b^4)*Tan[e + f*x])/(5*(a^2 + b^2)^3*d^2*f*Sqrt[d*Sec[e + f*x]]) - (9*a^2*b^3*Cot[e + f
*x]*EllipticPi[-(b/Sqrt[a^2 + b^2]), ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(Sec[e + f*x]^2)^(1/4)*Sqrt[-Tan[e +
f*x]^2])/(2*(a^2 + b^2)^(7/2)*d^2*f*Sqrt[d*Sec[e + f*x]]) + (9*a^2*b^3*Cot[e + f*x]*EllipticPi[b/Sqrt[a^2 + b^
2], ArcSin[(Sec[e + f*x]^2)^(1/4)], -1]*(Sec[e + f*x]^2)^(1/4)*Sqrt[-Tan[e + f*x]^2])/(2*(a^2 + b^2)^(7/2)*d^2
*f*Sqrt[d*Sec[e + f*x]]) + (3*b*(2*a^4 + 10*a^2*b^2 - 7*b^4)*Sec[e + f*x]^2)/(5*(a^2 + b^2)^3*d^2*f*Sqrt[d*Sec
[e + f*x]]*(a + b*Tan[e + f*x])) + (2*Cos[e + f*x]^2*(b + a*Tan[e + f*x]))/(5*(a^2 + b^2)*d^2*f*Sqrt[d*Sec[e +
f*x]]*(a + b*Tan[e + f*x])) - (2*(b*(2*a^2 - 7*b^2) - 3*a*(a^2 + 4*b^2)*Tan[e + f*x]))/(5*(a^2 + b^2)^2*d^2*f
*Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x]))

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 202

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]))*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]
*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 233

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[2*(x/(a + b*x^2)^(1/4)), x] - Dist[a, Int[1/(a + b*x^2)^(5
/4), x], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 408

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Dist[2*(Sqrt[(-b)*(x^2/a)]/x), Subst[I
nt[x^2/(Sqrt[1 - x^4/a]*(b*c - a*d + d*x^4)), x], x, (a + b*x^2)^(1/4)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0]

Rule 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 504

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s
= Denominator[Rt[-a/b, 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), Int[1/
((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 551

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

Rule 755

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(a*e + c*d*x)*
((a + c*x^2)^(p + 1)/(2*a*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^
m*Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[
{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 760

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(1/4)), x_Symbol] :> Dist[d, Int[1/((d^2 - e^2*x^2)*(a + c*x^
2)^(1/4)), x], x] - Dist[e, Int[x/((d^2 - e^2*x^2)*(a + c*x^2)^(1/4)), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ
[c*d^2 + a*e^2, 0]

Rule 837

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(
m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] +
Dist[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^
2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g},
x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 849

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*
(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 1227

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[Sqrt[-c],
Int[1/((d + e*x^2)*Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[c,
0]

Rule 3593

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[d^(2*
IntPart[m/2])*((d*Sec[e + f*x])^(2*FracPart[m/2])/(b*f*(Sec[e + f*x]^2)^FracPart[m/2])), Subst[Int[(a + x)^n*(
1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 + b^2, 0] &&
!IntegerQ[m/2]

Rubi steps

\begin {align*} \int \frac {1}{(d \sec (e+f x))^{5/2} (a+b \tan (e+f x))^2} \, dx &=\frac {\sqrt [4]{\sec ^2(e+f x)} \text {Subst}\left (\int \frac {1}{(a+x)^2 \left (1+\frac {x^2}{b^2}\right )^{9/4}} \, dx,x,b \tan (e+f x)\right )}{b d^2 f \sqrt {d \sec (e+f x)}}\\ &=\frac {2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}-\frac {\left (2 b \sqrt [4]{\sec ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {\frac {1}{2} \left (-7-\frac {3 a^2}{b^2}\right )-\frac {5 a x}{2 b^2}}{(a+x)^2 \left (1+\frac {x^2}{b^2}\right )^{5/4}} \, dx,x,b \tan (e+f x)\right )}{5 \left (a^2+b^2\right ) d^2 f \sqrt {d \sec (e+f x)}}\\ &=\frac {2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}-\frac {2 \left (b \left (2 a^2-7 b^2\right )-3 a \left (a^2+4 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}+\frac {\left (4 b^5 \sqrt [4]{\sec ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {-\frac {3 \left (a^4+6 a^2 b^2-7 b^4\right )}{4 b^6}+\frac {3 a \left (a^2+4 b^2\right ) x}{4 b^6}}{(a+x)^2 \sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)}}\\ &=\frac {3 b \left (2 a^4+10 a^2 b^2-7 b^4\right ) \sec ^2(e+f x)}{5 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}+\frac {2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}-\frac {2 \left (b \left (2 a^2-7 b^2\right )-3 a \left (a^2+4 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}-\frac {\left (4 b^7 \sqrt [4]{\sec ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {\frac {3 a \left (a^4+5 a^2 b^2-11 b^4\right )}{4 b^8}+\frac {3 \left (2 a^4+10 a^2 b^2-7 b^4\right ) x}{8 b^8}}{(a+x) \sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}\\ &=\frac {3 b \left (2 a^4+10 a^2 b^2-7 b^4\right ) \sec ^2(e+f x)}{5 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}+\frac {2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}-\frac {2 \left (b \left (2 a^2-7 b^2\right )-3 a \left (a^2+4 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}+\frac {\left (9 a b^3 \sqrt [4]{\sec ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{(a+x) \sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{2 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}-\frac {\left (3 \left (2 a^4+10 a^2 b^2-7 b^4\right ) \sqrt [4]{\sec ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{10 b \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}\\ &=-\frac {3 \left (2 a^4+10 a^2 b^2-7 b^4\right ) \tan (e+f x)}{5 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}+\frac {3 b \left (2 a^4+10 a^2 b^2-7 b^4\right ) \sec ^2(e+f x)}{5 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}+\frac {2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}-\frac {2 \left (b \left (2 a^2-7 b^2\right )-3 a \left (a^2+4 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}-\frac {\left (9 a b^3 \sqrt [4]{\sec ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {x}{\left (a^2-x^2\right ) \sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{2 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}+\frac {\left (9 a^2 b^3 \sqrt [4]{\sec ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-x^2\right ) \sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{2 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}+\frac {\left (3 \left (2 a^4+10 a^2 b^2-7 b^4\right ) \sqrt [4]{\sec ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {x^2}{b^2}\right )^{5/4}} \, dx,x,b \tan (e+f x)\right )}{10 b \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}\\ &=\frac {3 \left (2 a^4+10 a^2 b^2-7 b^4\right ) E\left (\left .\frac {1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{5 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}-\frac {3 \left (2 a^4+10 a^2 b^2-7 b^4\right ) \tan (e+f x)}{5 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}+\frac {3 b \left (2 a^4+10 a^2 b^2-7 b^4\right ) \sec ^2(e+f x)}{5 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}+\frac {2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}-\frac {2 \left (b \left (2 a^2-7 b^2\right )-3 a \left (a^2+4 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}-\frac {\left (9 a b^3 \sqrt [4]{\sec ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-x\right ) \sqrt [4]{1+\frac {x}{b^2}}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{4 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}+\frac {\left (9 a^2 b^2 \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^4} \left (1+\frac {a^2}{b^2}-x^4\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{\left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}\\ &=\frac {3 \left (2 a^4+10 a^2 b^2-7 b^4\right ) E\left (\left .\frac {1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{5 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}-\frac {3 \left (2 a^4+10 a^2 b^2-7 b^4\right ) \tan (e+f x)}{5 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}+\frac {3 b \left (2 a^4+10 a^2 b^2-7 b^4\right ) \sec ^2(e+f x)}{5 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}+\frac {2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}-\frac {2 \left (b \left (2 a^2-7 b^2\right )-3 a \left (a^2+4 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}-\frac {\left (9 a b^5 \sqrt [4]{\sec ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {x^2}{a^2+b^2-b^2 x^4} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{\left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}+\frac {\left (9 a^2 b^3 \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a^2+b^2}-b x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}-\frac {\left (9 a^2 b^3 \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a^2+b^2}+b x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}\\ &=\frac {3 \left (2 a^4+10 a^2 b^2-7 b^4\right ) E\left (\left .\frac {1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{5 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}-\frac {3 \left (2 a^4+10 a^2 b^2-7 b^4\right ) \tan (e+f x)}{5 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}+\frac {3 b \left (2 a^4+10 a^2 b^2-7 b^4\right ) \sec ^2(e+f x)}{5 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}+\frac {2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}-\frac {2 \left (b \left (2 a^2-7 b^2\right )-3 a \left (a^2+4 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}-\frac {\left (9 a b^4 \sqrt [4]{\sec ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2+b^2}-b x^2} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}+\frac {\left (9 a b^4 \sqrt [4]{\sec ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2+b^2}+b x^2} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}+\frac {\left (9 a^2 b^3 \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (\sqrt {a^2+b^2}-b x^2\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}-\frac {\left (9 a^2 b^3 \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (\sqrt {a^2+b^2}+b x^2\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{2 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}\\ &=\frac {9 a b^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt [4]{\sec ^2(e+f x)}}{2 \left (a^2+b^2\right )^{13/4} d^2 f \sqrt {d \sec (e+f x)}}-\frac {9 a b^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sqrt [4]{\sec ^2(e+f x)}}{2 \left (a^2+b^2\right )^{13/4} d^2 f \sqrt {d \sec (e+f x)}}+\frac {3 \left (2 a^4+10 a^2 b^2-7 b^4\right ) E\left (\left .\frac {1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{5 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}-\frac {3 \left (2 a^4+10 a^2 b^2-7 b^4\right ) \tan (e+f x)}{5 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)}}-\frac {9 a^2 b^3 \cot (e+f x) \Pi \left (-\frac {b}{\sqrt {a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right ) \sqrt [4]{\sec ^2(e+f x)} \sqrt {-\tan ^2(e+f x)}}{2 \left (a^2+b^2\right )^{7/2} d^2 f \sqrt {d \sec (e+f x)}}+\frac {9 a^2 b^3 \cot (e+f x) \Pi \left (\frac {b}{\sqrt {a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right ) \sqrt [4]{\sec ^2(e+f x)} \sqrt {-\tan ^2(e+f x)}}{2 \left (a^2+b^2\right )^{7/2} d^2 f \sqrt {d \sec (e+f x)}}+\frac {3 b \left (2 a^4+10 a^2 b^2-7 b^4\right ) \sec ^2(e+f x)}{5 \left (a^2+b^2\right )^3 d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}+\frac {2 \cos ^2(e+f x) (b+a \tan (e+f x))}{5 \left (a^2+b^2\right ) d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}-\frac {2 \left (b \left (2 a^2-7 b^2\right )-3 a \left (a^2+4 b^2\right ) \tan (e+f x)\right )}{5 \left (a^2+b^2\right )^2 d^2 f \sqrt {d \sec (e+f x)} (a+b \tan (e+f x))}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 91.44, size = 5832, normalized size = 8.33 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((d*Sec[e + f*x])^(5/2)*(a + b*Tan[e + f*x])^2),x]

[Out]

Result too large to show

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 44328 vs. \(2 (645 ) = 1290\).
time = 2.03, size = 44329, normalized size = 63.33

method	result	size
default	\(\text {Expression too large to display}\)	\(44329\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*sec(f*x+e))^(5/2)/(a+b*tan(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

result too large to display

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

[Out]

integrate(1/((d*sec(f*x + e))^(5/2)*(b*tan(f*x + e) + a)^2), x)

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{2}} \left (a + b \tan {\left (e + f x \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*sec(f*x+e))**(5/2)/(a+b*tan(f*x+e))**2,x)

[Out]

Integral(1/((d*sec(e + f*x))**(5/2)*(a + b*tan(e + f*x))**2), x)

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

[Out]

integrate(1/((d*sec(f*x + e))^(5/2)*(b*tan(f*x + e) + a)^2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{5/2}\,{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d/cos(e + f*x))^(5/2)*(a + b*tan(e + f*x))^2),x)

[Out]

int(1/((d/cos(e + f*x))^(5/2)*(a + b*tan(e + f*x))^2), x)

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