3.7.22 \(\int \frac {1}{(d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^3} \, dx\) [622]

Optimal. Leaf size=620 \[ -\frac {7 b^{5/2} \left (9 a^2-2 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}}{8 \left (a^2+b^2\right )^{15/4} f (d \sec (e+f x))^{3/2}}-\frac {7 b^{5/2} \left (9 a^2-2 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}}{8 \left (a^2+b^2\right )^{15/4} f (d \sec (e+f x))^{3/2}}+\frac {a \left (8 a^2-69 b^2\right ) F\left (\left .\frac {1}{2} \text {ArcTan}(\tan (e+f x))\right |2\right ) \sec ^2(e+f x)^{3/4}}{12 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}+\frac {7 a b^2 \left (9 a^2-2 b^2\right ) \cot (e+f x) \Pi \left (-\frac {b}{\sqrt {a^2+b^2}};\left .\text {ArcSin}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right ) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}}{8 \left (a^2+b^2\right )^4 f (d \sec (e+f x))^{3/2}}+\frac {7 a b^2 \left (9 a^2-2 b^2\right ) \cot (e+f x) \Pi \left (\frac {b}{\sqrt {a^2+b^2}};\left .\text {ArcSin}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right ) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}}{8 \left (a^2+b^2\right )^4 f (d \sec (e+f x))^{3/2}}+\frac {b \left (4 a^2-7 b^2\right ) \sec ^2(e+f x)}{6 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac {2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac {a b \left (8 a^2-69 b^2\right ) \sec ^2(e+f x)}{12 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))} \]

[Out]

-7/8*b^(5/2)*(9*a^2-2*b^2)*arctan((sec(f*x+e)^2)^(1/4)*b^(1/2)/(a^2+b^2)^(1/4))*(sec(f*x+e)^2)^(3/4)/(a^2+b^2)
^(15/4)/f/(d*sec(f*x+e))^(3/2)-7/8*b^(5/2)*(9*a^2-2*b^2)*arctanh((sec(f*x+e)^2)^(1/4)*b^(1/2)/(a^2+b^2)^(1/4))
*(sec(f*x+e)^2)^(3/4)/(a^2+b^2)^(15/4)/f/(d*sec(f*x+e))^(3/2)+1/12*a*(8*a^2-69*b^2)*(cos(1/2*arctan(tan(f*x+e)
))^2)^(1/2)/cos(1/2*arctan(tan(f*x+e)))*EllipticF(sin(1/2*arctan(tan(f*x+e))),2^(1/2))*(sec(f*x+e)^2)^(3/4)/(a
^2+b^2)^3/f/(d*sec(f*x+e))^(3/2)+7/8*a*b^2*(9*a^2-2*b^2)*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)^(3/4)*(-tan(f*x+e)^2)^(1/2)/(a^2+b^2)^4/f/(d*sec(f*x+e))^(3/2)+7/8*a*b^2*(9*a^2-2*b
^2)*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)^(3/4)*(-tan(f*x+e)^2)^(1/2)
/(a^2+b^2)^4/f/(d*sec(f*x+e))^(3/2)+1/6*b*(4*a^2-7*b^2)*sec(f*x+e)^2/(a^2+b^2)^2/f/(d*sec(f*x+e))^(3/2)/(a+b*t
an(f*x+e))^2+2/3*(b+a*tan(f*x+e))/(a^2+b^2)/f/(d*sec(f*x+e))^(3/2)/(a+b*tan(f*x+e))^2+1/12*a*b*(8*a^2-69*b^2)*
sec(f*x+e)^2/(a^2+b^2)^3/f/(d*sec(f*x+e))^(3/2)/(a+b*tan(f*x+e))

________________________________________________________________________________________

Rubi [A]
time = 0.56, antiderivative size = 620, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 16, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3593, 755, 849, 858, 237, 761, 410, 109, 418, 1227, 551, 455, 65, 218, 214, 211} \begin {gather*} \frac {7 a b^2 \left (9 a^2-2 b^2\right ) \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \sec ^2(e+f x)^{3/4} \Pi \left (-\frac {b}{\sqrt {a^2+b^2}};\left .\text {ArcSin}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{8 f \left (a^2+b^2\right )^4 (d \sec (e+f x))^{3/2}}+\frac {7 a b^2 \left (9 a^2-2 b^2\right ) \sqrt {-\tan ^2(e+f x)} \cot (e+f x) \sec ^2(e+f x)^{3/4} \Pi \left (\frac {b}{\sqrt {a^2+b^2}};\left .\text {ArcSin}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{8 f \left (a^2+b^2\right )^4 (d \sec (e+f x))^{3/2}}+\frac {a \left (8 a^2-69 b^2\right ) \sec ^2(e+f x)^{3/4} F\left (\left .\frac {1}{2} \text {ArcTan}(\tan (e+f x))\right |2\right )}{12 f \left (a^2+b^2\right )^3 (d \sec (e+f x))^{3/2}}-\frac {7 b^{5/2} \left (9 a^2-2 b^2\right ) \sec ^2(e+f x)^{3/4} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{8 f \left (a^2+b^2\right )^{15/4} (d \sec (e+f x))^{3/2}}+\frac {a b \left (8 a^2-69 b^2\right ) \sec ^2(e+f x)}{12 f \left (a^2+b^2\right )^3 (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac {b \left (4 a^2-7 b^2\right ) \sec ^2(e+f x)}{6 f \left (a^2+b^2\right )^2 (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac {2 (a \tan (e+f x)+b)}{3 f \left (a^2+b^2\right ) (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}-\frac {7 b^{5/2} \left (9 a^2-2 b^2\right ) \sec ^2(e+f x)^{3/4} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{8 f \left (a^2+b^2\right )^{15/4} (d \sec (e+f x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x])^3),x]

[Out]

(-7*b^(5/2)*(9*a^2 - 2*b^2)*ArcTan[(Sqrt[b]*(Sec[e + f*x]^2)^(1/4))/(a^2 + b^2)^(1/4)]*(Sec[e + f*x]^2)^(3/4))
/(8*(a^2 + b^2)^(15/4)*f*(d*Sec[e + f*x])^(3/2)) - (7*b^(5/2)*(9*a^2 - 2*b^2)*ArcTanh[(Sqrt[b]*(Sec[e + f*x]^2
)^(1/4))/(a^2 + b^2)^(1/4)]*(Sec[e + f*x]^2)^(3/4))/(8*(a^2 + b^2)^(15/4)*f*(d*Sec[e + f*x])^(3/2)) + (a*(8*a^
2 - 69*b^2)*EllipticF[ArcTan[Tan[e + f*x]]/2, 2]*(Sec[e + f*x]^2)^(3/4))/(12*(a^2 + b^2)^3*f*(d*Sec[e + f*x])^
(3/2)) + (7*a*b^2*(9*a^2 - 2*b^2)*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)^(3/4)*Sqrt[-Tan[e + f*x]^2])/(8*(a^2 + b^2)^4*f*(d*Sec[e + f*x])^(3/2)) + (7*a*b^2*(9*a
^2 - 2*b^2)*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)^(3
/4)*Sqrt[-Tan[e + f*x]^2])/(8*(a^2 + b^2)^4*f*(d*Sec[e + f*x])^(3/2)) + (b*(4*a^2 - 7*b^2)*Sec[e + f*x]^2)/(6*
(a^2 + b^2)^2*f*(d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x])^2) + (2*(b + a*Tan[e + f*x]))/(3*(a^2 + b^2)*f*(d*
Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x])^2) + (a*b*(8*a^2 - 69*b^2)*Sec[e + f*x]^2)/(12*(a^2 + b^2)^3*f*(d*Sec
[e + f*x])^(3/2)*(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 109

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(3/4)), x_Symbol] :> Dist[-4, Subst[
Int[1/((b*e - a*f - b*x^4)*Sqrt[c - d*(e/f) + d*(x^4/f)]), x], x, (e + f*x)^(1/4)], x] /; FreeQ[{a, b, c, d, e
, f}, x] && GtQ[-f/(d*e - c*f), 0]

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 218

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

Rule 237

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]))*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]
*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rule 410

Int[1/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Dist[Sqrt[(-b)*(x^2/a)]/(2*x), Subst[I
nt[1/(Sqrt[(-b)*(x/a)]*(a + b*x)^(3/4)*(c + d*x)), x], x, x^2], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0]

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1
- Rt[-d/c, 2]*x^2)), x], x] + Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], 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 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 761

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

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))^{3/2} (a+b \tan (e+f x))^3} \, dx &=\frac {\sec ^2(e+f x)^{3/4} \text {Subst}\left (\int \frac {1}{(a+x)^3 \left (1+\frac {x^2}{b^2}\right )^{7/4}} \, dx,x,b \tan (e+f x)\right )}{b f (d \sec (e+f x))^{3/2}}\\ &=\frac {2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}-\frac {\left (2 b \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {\frac {1}{2} \left (-7-\frac {a^2}{b^2}\right )-\frac {5 a x}{2 b^2}}{(a+x)^3 \left (1+\frac {x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2}}\\ &=\frac {b \left (4 a^2-7 b^2\right ) \sec ^2(e+f x)}{6 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac {2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac {\left (b^3 \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {\frac {a \left (a^2+12 b^2\right )}{b^4}+\frac {3 \left (4 a^2-7 b^2\right ) x}{4 b^4}}{(a+x)^2 \left (1+\frac {x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{3 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2}}\\ &=\frac {b \left (4 a^2-7 b^2\right ) \sec ^2(e+f x)}{6 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac {2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac {a b \left (8 a^2-69 b^2\right ) \sec ^2(e+f x)}{12 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}-\frac {\left (b^5 \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {-\frac {4 a^4+60 a^2 b^2-21 b^4}{4 b^6}-\frac {a \left (8 a^2-69 b^2\right ) x}{8 b^6}}{(a+x) \left (1+\frac {x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{3 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}\\ &=\frac {b \left (4 a^2-7 b^2\right ) \sec ^2(e+f x)}{6 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac {2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac {a b \left (8 a^2-69 b^2\right ) \sec ^2(e+f x)}{12 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac {\left (a \left (8 a^2-69 b^2\right ) \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{24 b \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}+\frac {\left (7 b \left (9 a^2-2 b^2\right ) \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{(a+x) \left (1+\frac {x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{8 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}\\ &=\frac {a \left (8 a^2-69 b^2\right ) F\left (\left .\frac {1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sec ^2(e+f x)^{3/4}}{12 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}+\frac {b \left (4 a^2-7 b^2\right ) \sec ^2(e+f x)}{6 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac {2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac {a b \left (8 a^2-69 b^2\right ) \sec ^2(e+f x)}{12 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}-\frac {\left (7 b \left (9 a^2-2 b^2\right ) \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {x}{\left (a^2-x^2\right ) \left (1+\frac {x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{8 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}+\frac {\left (7 a b \left (9 a^2-2 b^2\right ) \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-x^2\right ) \left (1+\frac {x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{8 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}\\ &=\frac {a \left (8 a^2-69 b^2\right ) F\left (\left .\frac {1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sec ^2(e+f x)^{3/4}}{12 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}+\frac {b \left (4 a^2-7 b^2\right ) \sec ^2(e+f x)}{6 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac {2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac {a b \left (8 a^2-69 b^2\right ) \sec ^2(e+f x)}{12 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}-\frac {\left (7 b \left (9 a^2-2 b^2\right ) \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-x\right ) \left (1+\frac {x}{b^2}\right )^{3/4}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{16 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}+\frac {\left (7 a \left (9 a^2-2 b^2\right ) \cot (e+f x) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-x\right ) \sqrt {-\frac {x}{b^2}} \left (1+\frac {x}{b^2}\right )^{3/4}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{16 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}\\ &=\frac {a \left (8 a^2-69 b^2\right ) F\left (\left .\frac {1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sec ^2(e+f x)^{3/4}}{12 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}+\frac {b \left (4 a^2-7 b^2\right ) \sec ^2(e+f x)}{6 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac {2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac {a b \left (8 a^2-69 b^2\right ) \sec ^2(e+f x)}{12 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}-\frac {\left (7 b^3 \left (9 a^2-2 b^2\right ) \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{a^2+b^2-b^2 x^4} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{4 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}-\frac {\left (7 a \left (9 a^2-2 b^2\right ) \cot (e+f x) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^4} \left (-1-\frac {a^2}{b^2}+x^4\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{4 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}\\ &=\frac {a \left (8 a^2-69 b^2\right ) F\left (\left .\frac {1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sec ^2(e+f x)^{3/4}}{12 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}+\frac {b \left (4 a^2-7 b^2\right ) \sec ^2(e+f x)}{6 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac {2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac {a b \left (8 a^2-69 b^2\right ) \sec ^2(e+f x)}{12 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}-\frac {\left (7 b^3 \left (9 a^2-2 b^2\right ) \sec ^2(e+f x)^{3/4}\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 )}{8 \left (a^2+b^2\right )^{7/2} f (d \sec (e+f x))^{3/2}}-\frac {\left (7 b^3 \left (9 a^2-2 b^2\right ) \sec ^2(e+f x)^{3/4}\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 )}{8 \left (a^2+b^2\right )^{7/2} f (d \sec (e+f x))^{3/2}}+\frac {\left (7 a b^2 \left (9 a^2-2 b^2\right ) \cot (e+f x) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {b x^2}{\sqrt {a^2+b^2}}\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{8 \left (a^2+b^2\right )^4 f (d \sec (e+f x))^{3/2}}+\frac {\left (7 a b^2 \left (9 a^2-2 b^2\right ) \cot (e+f x) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {b x^2}{\sqrt {a^2+b^2}}\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{8 \left (a^2+b^2\right )^4 f (d \sec (e+f x))^{3/2}}\\ &=-\frac {7 b^{5/2} \left (9 a^2-2 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}}{8 \left (a^2+b^2\right )^{15/4} f (d \sec (e+f x))^{3/2}}-\frac {7 b^{5/2} \left (9 a^2-2 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}}{8 \left (a^2+b^2\right )^{15/4} f (d \sec (e+f x))^{3/2}}+\frac {a \left (8 a^2-69 b^2\right ) F\left (\left .\frac {1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sec ^2(e+f x)^{3/4}}{12 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}+\frac {b \left (4 a^2-7 b^2\right ) \sec ^2(e+f x)}{6 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac {2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac {a b \left (8 a^2-69 b^2\right ) \sec ^2(e+f x)}{12 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}+\frac {\left (7 a b^2 \left (9 a^2-2 b^2\right ) \cot (e+f x) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1-\frac {b x^2}{\sqrt {a^2+b^2}}\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{8 \left (a^2+b^2\right )^4 f (d \sec (e+f x))^{3/2}}+\frac {\left (7 a b^2 \left (9 a^2-2 b^2\right ) \cot (e+f x) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1+\frac {b x^2}{\sqrt {a^2+b^2}}\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{8 \left (a^2+b^2\right )^4 f (d \sec (e+f x))^{3/2}}\\ &=-\frac {7 b^{5/2} \left (9 a^2-2 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}}{8 \left (a^2+b^2\right )^{15/4} f (d \sec (e+f x))^{3/2}}-\frac {7 b^{5/2} \left (9 a^2-2 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) \sec ^2(e+f x)^{3/4}}{8 \left (a^2+b^2\right )^{15/4} f (d \sec (e+f x))^{3/2}}+\frac {a \left (8 a^2-69 b^2\right ) F\left (\left .\frac {1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sec ^2(e+f x)^{3/4}}{12 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2}}+\frac {7 a b^2 \left (9 a^2-2 b^2\right ) \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 ) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}}{8 \left (a^2+b^2\right )^4 f (d \sec (e+f x))^{3/2}}+\frac {7 a b^2 \left (9 a^2-2 b^2\right ) \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 ) \sec ^2(e+f x)^{3/4} \sqrt {-\tan ^2(e+f x)}}{8 \left (a^2+b^2\right )^4 f (d \sec (e+f x))^{3/2}}+\frac {b \left (4 a^2-7 b^2\right ) \sec ^2(e+f x)}{6 \left (a^2+b^2\right )^2 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac {2 (b+a \tan (e+f x))}{3 \left (a^2+b^2\right ) f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))^2}+\frac {a b \left (8 a^2-69 b^2\right ) \sec ^2(e+f x)}{12 \left (a^2+b^2\right )^3 f (d \sec (e+f x))^{3/2} (a+b \tan (e+f x))}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 69.79, size = 5131, normalized size = 8.28 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((d*Sec[e + f*x])^(3/2)*(a + b*Tan[e + f*x])^3),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. 82288 vs. \(2 (573 ) = 1146\).
time = 3.24, size = 82289, normalized size = 132.72

method	result	size
default	\(\text {Expression too large to display}\)	\(82289\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((d*sec(f*x + e))^(3/2)*(b*tan(f*x + e) + a)^3), 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))^(3/2)/(a+b*tan(f*x+e))^3,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 {3}{2}} \left (a + b \tan {\left (e + f x \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(1/((d*sec(f*x + e))^(3/2)*(b*tan(f*x + e) + a)^3), 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 )}^{3/2}\,{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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