3.621 \(\int \frac{1}{\sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^3} \, dx\)

Optimal. Leaf size=664 \[ \frac{5 b^{3/2} \left (7 a^2-2 b^2\right ) \sqrt [4]{\sec ^2(e+f x)} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{8 f \left (a^2+b^2\right )^{13/4} \sqrt{d \sec (e+f x)}}+\frac{a b \left (8 a^2-37 b^2\right ) \sec ^2(e+f x)}{4 f \left (a^2+b^2\right )^3 \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}+\frac{b \left (4 a^2-5 b^2\right ) \sec ^2(e+f x)}{2 f \left (a^2+b^2\right )^2 \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}-\frac{5 b^{3/2} \left (7 a^2-2 b^2\right ) \sqrt [4]{\sec ^2(e+f x)} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{8 f \left (a^2+b^2\right )^{13/4} \sqrt{d \sec (e+f x)}}-\frac{a \left (8 a^2-37 b^2\right ) \tan (e+f x)}{4 f \left (a^2+b^2\right )^3 \sqrt{d \sec (e+f x)}}+\frac{2 (a \tan (e+f x)+b)}{f \left (a^2+b^2\right ) \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac{a \left (8 a^2-37 b^2\right ) \sqrt [4]{\sec ^2(e+f x)} E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right )}{4 f \left (a^2+b^2\right )^3 \sqrt{d \sec (e+f x)}}-\frac{5 a b \left (7 a^2-2 b^2\right ) \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \Pi \left (-\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{8 f \left (a^2+b^2\right )^{7/2} \sqrt{d \sec (e+f x)}}+\frac{5 a b \left (7 a^2-2 b^2\right ) \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \Pi \left (\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{8 f \left (a^2+b^2\right )^{7/2} \sqrt{d \sec (e+f x)}} \]

[Out]

(5*b^(3/2)*(7*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)^(1/4))/
(8*(a^2 + b^2)^(13/4)*f*Sqrt[d*Sec[e + f*x]]) - (5*b^(3/2)*(7*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)^(1/4))/(8*(a^2 + b^2)^(13/4)*f*Sqrt[d*Sec[e + f*x]]) + (a*(8*a^2 - 3
7*b^2)*EllipticE[ArcTan[Tan[e + f*x]]/2, 2]*(Sec[e + f*x]^2)^(1/4))/(4*(a^2 + b^2)^3*f*Sqrt[d*Sec[e + f*x]]) -
(a*(8*a^2 - 37*b^2)*Tan[e + f*x])/(4*(a^2 + b^2)^3*f*Sqrt[d*Sec[e + f*x]]) - (5*a*b*(7*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)^(1/4)*Sqrt[-Tan[e +
f*x]^2])/(8*(a^2 + b^2)^(7/2)*f*Sqrt[d*Sec[e + f*x]]) + (5*a*b*(7*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)^(1/4)*Sqrt[-Tan[e + f*x]^2])/(8*(a^2 + b^2)^(
7/2)*f*Sqrt[d*Sec[e + f*x]]) + (b*(4*a^2 - 5*b^2)*Sec[e + f*x]^2)/(2*(a^2 + b^2)^2*f*Sqrt[d*Sec[e + f*x]]*(a +
b*Tan[e + f*x])^2) + (2*(b + a*Tan[e + f*x]))/((a^2 + b^2)*f*Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x])^2) + (
a*b*(8*a^2 - 37*b^2)*Sec[e + f*x]^2)/(4*(a^2 + b^2)^3*f*Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.761036, antiderivative size = 664, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 16, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.64, Rules used = {3512, 741, 835, 844, 227, 196, 746, 399, 490, 1213, 537, 444, 63, 298, 205, 208} \[ \frac{5 b^{3/2} \left (7 a^2-2 b^2\right ) \sqrt [4]{\sec ^2(e+f x)} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{8 f \left (a^2+b^2\right )^{13/4} \sqrt{d \sec (e+f x)}}+\frac{a b \left (8 a^2-37 b^2\right ) \sec ^2(e+f x)}{4 f \left (a^2+b^2\right )^3 \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}+\frac{b \left (4 a^2-5 b^2\right ) \sec ^2(e+f x)}{2 f \left (a^2+b^2\right )^2 \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}-\frac{5 b^{3/2} \left (7 a^2-2 b^2\right ) \sqrt [4]{\sec ^2(e+f x)} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{8 f \left (a^2+b^2\right )^{13/4} \sqrt{d \sec (e+f x)}}-\frac{a \left (8 a^2-37 b^2\right ) \tan (e+f x)}{4 f \left (a^2+b^2\right )^3 \sqrt{d \sec (e+f x)}}+\frac{2 (a \tan (e+f x)+b)}{f \left (a^2+b^2\right ) \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac{a \left (8 a^2-37 b^2\right ) \sqrt [4]{\sec ^2(e+f x)} E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right )}{4 f \left (a^2+b^2\right )^3 \sqrt{d \sec (e+f x)}}-\frac{5 a b \left (7 a^2-2 b^2\right ) \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \Pi \left (-\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{8 f \left (a^2+b^2\right )^{7/2} \sqrt{d \sec (e+f x)}}+\frac{5 a b \left (7 a^2-2 b^2\right ) \sqrt{-\tan ^2(e+f x)} \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \Pi \left (\frac{b}{\sqrt{a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{8 f \left (a^2+b^2\right )^{7/2} \sqrt{d \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*b^(3/2)*(7*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)^(1/4))/
(8*(a^2 + b^2)^(13/4)*f*Sqrt[d*Sec[e + f*x]]) - (5*b^(3/2)*(7*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)^(1/4))/(8*(a^2 + b^2)^(13/4)*f*Sqrt[d*Sec[e + f*x]]) + (a*(8*a^2 - 3
7*b^2)*EllipticE[ArcTan[Tan[e + f*x]]/2, 2]*(Sec[e + f*x]^2)^(1/4))/(4*(a^2 + b^2)^3*f*Sqrt[d*Sec[e + f*x]]) -
(a*(8*a^2 - 37*b^2)*Tan[e + f*x])/(4*(a^2 + b^2)^3*f*Sqrt[d*Sec[e + f*x]]) - (5*a*b*(7*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)^(1/4)*Sqrt[-Tan[e +
f*x]^2])/(8*(a^2 + b^2)^(7/2)*f*Sqrt[d*Sec[e + f*x]]) + (5*a*b*(7*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)^(1/4)*Sqrt[-Tan[e + f*x]^2])/(8*(a^2 + b^2)^(
7/2)*f*Sqrt[d*Sec[e + f*x]]) + (b*(4*a^2 - 5*b^2)*Sec[e + f*x]^2)/(2*(a^2 + b^2)^2*f*Sqrt[d*Sec[e + f*x]]*(a +
b*Tan[e + f*x])^2) + (2*(b + a*Tan[e + f*x]))/((a^2 + b^2)*f*Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x])^2) + (
a*b*(8*a^2 - 37*b^2)*Sec[e + f*x]^2)/(4*(a^2 + b^2)^3*f*Sqrt[d*Sec[e + f*x]]*(a + b*Tan[e + f*x]))

Rule 3512

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]

Rule 741

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 835

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 844

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 227

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 196

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

Rule 746

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 399

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 490

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), In
t[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 1213

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 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), 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 444

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 63

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 298

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 205

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

Rule 208

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^3} \, dx &=\frac{\sqrt [4]{\sec ^2(e+f x)} \operatorname{Subst}\left (\int \frac{1}{(a+x)^3 \left (1+\frac{x^2}{b^2}\right )^{5/4}} \, dx,x,b \tan (e+f x)\right )}{b f \sqrt{d \sec (e+f x)}}\\ &=\frac{2 (b+a \tan (e+f x))}{\left (a^2+b^2\right ) f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}-\frac{\left (2 b \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (-5+\frac{a^2}{b^2}\right )-\frac{3 a x}{2 b^2}}{(a+x)^3 \sqrt [4]{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{\left (a^2+b^2\right ) f \sqrt{d \sec (e+f x)}}\\ &=\frac{b \left (4 a^2-5 b^2\right ) \sec ^2(e+f x)}{2 \left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac{2 (b+a \tan (e+f x))}{\left (a^2+b^2\right ) f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac{\left (b^3 \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{-\frac{a \left (a^2-8 b^2\right )}{b^4}+\frac{\left (4 a^2-5 b^2\right ) x}{4 b^4}}{(a+x)^2 \sqrt [4]{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{\left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)}}\\ &=\frac{b \left (4 a^2-5 b^2\right ) \sec ^2(e+f x)}{2 \left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac{2 (b+a \tan (e+f x))}{\left (a^2+b^2\right ) f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac{a b \left (8 a^2-37 b^2\right ) \sec ^2(e+f x)}{4 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}-\frac{\left (b^5 \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{\frac{1}{4} \left (\frac{4 a^4}{b^6}-\frac{36 a^2}{b^4}+\frac{5}{b^2}\right )+\frac{a \left (8 a^2-37 b^2\right ) x}{8 b^6}}{(a+x) \sqrt [4]{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{\left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)}}\\ &=\frac{b \left (4 a^2-5 b^2\right ) \sec ^2(e+f x)}{2 \left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac{2 (b+a \tan (e+f x))}{\left (a^2+b^2\right ) f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac{a b \left (8 a^2-37 b^2\right ) \sec ^2(e+f x)}{4 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}-\frac{\left (a \left (8 a^2-37 b^2\right ) \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{8 b \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)}}+\frac{\left (5 b \left (7 a^2-2 b^2\right ) \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{(a+x) \sqrt [4]{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{8 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)}}\\ &=-\frac{a \left (8 a^2-37 b^2\right ) \tan (e+f x)}{4 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)}}+\frac{b \left (4 a^2-5 b^2\right ) \sec ^2(e+f x)}{2 \left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac{2 (b+a \tan (e+f x))}{\left (a^2+b^2\right ) f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac{a b \left (8 a^2-37 b^2\right ) \sec ^2(e+f x)}{4 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}+\frac{\left (a \left (8 a^2-37 b^2\right ) \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{x^2}{b^2}\right )^{5/4}} \, dx,x,b \tan (e+f x)\right )}{8 b \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)}}-\frac{\left (5 b \left (7 a^2-2 b^2\right ) \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{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 )}{8 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)}}+\frac{\left (5 a b \left (7 a^2-2 b^2\right ) \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{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 )}{8 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)}}\\ &=\frac{a \left (8 a^2-37 b^2\right ) E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{4 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)}}-\frac{a \left (8 a^2-37 b^2\right ) \tan (e+f x)}{4 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)}}+\frac{b \left (4 a^2-5 b^2\right ) \sec ^2(e+f x)}{2 \left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac{2 (b+a \tan (e+f x))}{\left (a^2+b^2\right ) f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac{a b \left (8 a^2-37 b^2\right ) \sec ^2(e+f x)}{4 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}-\frac{\left (5 b \left (7 a^2-2 b^2\right ) \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{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 )}{16 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)}}+\frac{\left (5 a \left (7 a^2-2 b^2\right ) \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \sqrt{-\tan ^2(e+f x)}\right ) \operatorname{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 )}{4 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)}}\\ &=\frac{a \left (8 a^2-37 b^2\right ) E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{4 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)}}-\frac{a \left (8 a^2-37 b^2\right ) \tan (e+f x)}{4 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)}}+\frac{b \left (4 a^2-5 b^2\right ) \sec ^2(e+f x)}{2 \left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac{2 (b+a \tan (e+f x))}{\left (a^2+b^2\right ) f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac{a b \left (8 a^2-37 b^2\right ) \sec ^2(e+f x)}{4 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}-\frac{\left (5 b^3 \left (7 a^2-2 b^2\right ) \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{Subst}\left (\int \frac{x^2}{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 \sqrt{d \sec (e+f x)}}+\frac{\left (5 a b \left (7 a^2-2 b^2\right ) \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \sqrt{-\tan ^2(e+f x)}\right ) \operatorname{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 )}{8 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)}}-\frac{\left (5 a b \left (7 a^2-2 b^2\right ) \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \sqrt{-\tan ^2(e+f x)}\right ) \operatorname{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 )}{8 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)}}\\ &=\frac{a \left (8 a^2-37 b^2\right ) E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{4 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)}}-\frac{a \left (8 a^2-37 b^2\right ) \tan (e+f x)}{4 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)}}+\frac{b \left (4 a^2-5 b^2\right ) \sec ^2(e+f x)}{2 \left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac{2 (b+a \tan (e+f x))}{\left (a^2+b^2\right ) f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac{a b \left (8 a^2-37 b^2\right ) \sec ^2(e+f x)}{4 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}-\frac{\left (5 b^2 \left (7 a^2-2 b^2\right ) \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{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 )^3 f \sqrt{d \sec (e+f x)}}+\frac{\left (5 b^2 \left (7 a^2-2 b^2\right ) \sqrt [4]{\sec ^2(e+f x)}\right ) \operatorname{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 )^3 f \sqrt{d \sec (e+f x)}}+\frac{\left (5 a b \left (7 a^2-2 b^2\right ) \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \sqrt{-\tan ^2(e+f x)}\right ) \operatorname{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 )}{8 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)}}-\frac{\left (5 a b \left (7 a^2-2 b^2\right ) \cot (e+f x) \sqrt [4]{\sec ^2(e+f x)} \sqrt{-\tan ^2(e+f x)}\right ) \operatorname{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 )}{8 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)}}\\ &=\frac{5 b^{3/2} \left (7 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 ) \sqrt [4]{\sec ^2(e+f x)}}{8 \left (a^2+b^2\right )^{13/4} f \sqrt{d \sec (e+f x)}}-\frac{5 b^{3/2} \left (7 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 ) \sqrt [4]{\sec ^2(e+f x)}}{8 \left (a^2+b^2\right )^{13/4} f \sqrt{d \sec (e+f x)}}+\frac{a \left (8 a^2-37 b^2\right ) E\left (\left .\frac{1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) \sqrt [4]{\sec ^2(e+f x)}}{4 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)}}-\frac{a \left (8 a^2-37 b^2\right ) \tan (e+f x)}{4 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)}}-\frac{5 a b \left (7 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 ) \sqrt [4]{\sec ^2(e+f x)} \sqrt{-\tan ^2(e+f x)}}{8 \left (a^2+b^2\right )^{7/2} f \sqrt{d \sec (e+f x)}}+\frac{5 a b \left (7 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 ) \sqrt [4]{\sec ^2(e+f x)} \sqrt{-\tan ^2(e+f x)}}{8 \left (a^2+b^2\right )^{7/2} f \sqrt{d \sec (e+f x)}}+\frac{b \left (4 a^2-5 b^2\right ) \sec ^2(e+f x)}{2 \left (a^2+b^2\right )^2 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac{2 (b+a \tan (e+f x))}{\left (a^2+b^2\right ) f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))^2}+\frac{a b \left (8 a^2-37 b^2\right ) \sec ^2(e+f x)}{4 \left (a^2+b^2\right )^3 f \sqrt{d \sec (e+f x)} (a+b \tan (e+f x))}\\ \end{align*}

Mathematica [C]  time = 29.2325, size = 14684, normalized size = 22.11 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [B]  time = 11.363, size = 100394, normalized size = 151.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{d \sec{\left (e + f x \right )}} \left (a + b \tan{\left (e + f x \right )}\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{d \sec \left (f x + e\right )}{\left (b \tan \left (f x + e\right ) + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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