\(\int \frac {\tanh ^4(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx\) [147]

Optimal result

Integrand size = 23, antiderivative size = 907 \[ \int \frac {\tanh ^4(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=-\frac {2 \coth (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{a \sqrt {a+b} d}+\frac {4 a \coth (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{b^2 \sqrt {a+b} d}-\frac {2 a \left (8 a^2-5 b^2\right ) \coth (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{3 b^4 \sqrt {a+b} d}+\frac {2 \coth (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{a \sqrt {a+b} d}+\frac {4 \coth (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{b \sqrt {a+b} d}-\frac {2 (2 a+b) (4 a+b) \coth (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{3 b^3 \sqrt {a+b} d}+\frac {2 \sqrt {a+b} \coth (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{a^2 d}-\frac {4 a \tanh (c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 b^2 \tanh (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}-\frac {2 a^2 \text {sech}(c+d x) \tanh (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 \left (4 a^2-b^2\right ) \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x)}{3 b^2 \left (a^2-b^2\right ) d} \]

[Out]

-2*coth(d*x+c)*EllipticE((a+b*sech(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1-sech(d*x+c))/(a+b))^(1
/2)*(-b*(1+sech(d*x+c))/(a-b))^(1/2)/a/d/(a+b)^(1/2)+4*a*coth(d*x+c)*EllipticE((a+b*sech(d*x+c))^(1/2)/(a+b)^(
1/2),((a+b)/(a-b))^(1/2))*(b*(1-sech(d*x+c))/(a+b))^(1/2)*(-b*(1+sech(d*x+c))/(a-b))^(1/2)/b^2/d/(a+b)^(1/2)-2
/3*a*(8*a^2-5*b^2)*coth(d*x+c)*EllipticE((a+b*sech(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1-sech(d
*x+c))/(a+b))^(1/2)*(-b*(1+sech(d*x+c))/(a-b))^(1/2)/b^4/d/(a+b)^(1/2)+2*coth(d*x+c)*EllipticF((a+b*sech(d*x+c
))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1-sech(d*x+c))/(a+b))^(1/2)*(-b*(1+sech(d*x+c))/(a-b))^(1/2)/a/d
/(a+b)^(1/2)+4*coth(d*x+c)*EllipticF((a+b*sech(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1-sech(d*x+c
))/(a+b))^(1/2)*(-b*(1+sech(d*x+c))/(a-b))^(1/2)/b/d/(a+b)^(1/2)-2/3*(2*a+b)*(4*a+b)*coth(d*x+c)*EllipticF((a+
b*sech(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1-sech(d*x+c))/(a+b))^(1/2)*(-b*(1+sech(d*x+c))/(a-b
))^(1/2)/b^3/d/(a+b)^(1/2)+2*coth(d*x+c)*EllipticPi((a+b*sech(d*x+c))^(1/2)/(a+b)^(1/2),(a+b)/a,((a+b)/(a-b))^
(1/2))*(a+b)^(1/2)*(b*(1-sech(d*x+c))/(a+b))^(1/2)*(-b*(1+sech(d*x+c))/(a-b))^(1/2)/a^2/d-4*a*tanh(d*x+c)/(a^2
-b^2)/d/(a+b*sech(d*x+c))^(1/2)+2*b^2*tanh(d*x+c)/a/(a^2-b^2)/d/(a+b*sech(d*x+c))^(1/2)-2*a^2*sech(d*x+c)*tanh
(d*x+c)/b/(a^2-b^2)/d/(a+b*sech(d*x+c))^(1/2)+2/3*(4*a^2-b^2)*(a+b*sech(d*x+c))^(1/2)*tanh(d*x+c)/b^2/(a^2-b^2
)/d

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 907, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3980, 3870, 4143, 4006, 3869, 3917, 4089, 3921, 4090, 3930, 4167} \[ \int \frac {\tanh ^4(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=-\frac {2 \text {sech}(c+d x) \tanh (c+d x) a^2}{b \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}-\frac {4 \tanh (c+d x) a}{\left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}-\frac {2 \left (8 a^2-5 b^2\right ) \coth (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} a}{3 b^4 \sqrt {a+b} d}+\frac {4 \coth (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} a}{b^2 \sqrt {a+b} d}+\frac {2 \left (4 a^2-b^2\right ) \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {2 (2 a+b) (4 a+b) \coth (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}}}{3 b^3 \sqrt {a+b} d}+\frac {4 \coth (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}}}{b \sqrt {a+b} d}+\frac {2 b^2 \tanh (c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)} a}-\frac {2 \coth (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}}}{\sqrt {a+b} d a}+\frac {2 \coth (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}}}{\sqrt {a+b} d a}+\frac {2 \sqrt {a+b} \coth (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}}}{d a^2} \]

[In]

Int[Tanh[c + d*x]^4/(a + b*Sech[c + d*x])^(3/2),x]

[Out]

(-2*Coth[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[
c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) + (4*a*Coth[c + d*x]*EllipticE
[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-(
(b*(1 + Sech[c + d*x]))/(a - b))])/(b^2*Sqrt[a + b]*d) - (2*a*(8*a^2 - 5*b^2)*Coth[c + d*x]*EllipticE[ArcSin[S
qrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + S
ech[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*d) + (2*Coth[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sech[c + d*x]]/
Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])
/(a*Sqrt[a + b]*d) + (4*Coth[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)
]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(b*Sqrt[a + b]*d) - (2*(2*a
+ b)*(4*a + b)*Coth[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b
*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(3*b^3*Sqrt[a + b]*d) + (2*Sqrt[a + b
]*Coth[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*
(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(a^2*d) - (4*a*Tanh[c + d*x])/((a^2 -
b^2)*d*Sqrt[a + b*Sech[c + d*x]]) + (2*b^2*Tanh[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sech[c + d*x]]) - (2*a^2
*Sech[c + d*x]*Tanh[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sech[c + d*x]]) + (2*(4*a^2 - b^2)*Sqrt[a + b*Sech[c
+ d*x]]*Tanh[c + d*x])/(3*b^2*(a^2 - b^2)*d)

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 3870

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

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 3921

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

Rule 3930

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-a^2)
*d^3*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^(n - 3)/(b*f*(m + 1)*(a^2 - b^2))), x] + Dist
[d^3/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 3)*Simp[a^2*(n - 3) + a*b
*(m + 1)*Csc[e + f*x] - (a^2*(n - 2) + b^2*(m + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f}, x] &&
NeQ[a^2 - b^2, 0] && LtQ[m, -1] && (IGtQ[n, 3] || (IntegersQ[n + 1/2, 2*m] && GtQ[n, 2]))

Rule 3980

Int[cot[(c_.) + (d_.)*(x_)]^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Int[ExpandIntegrand
[(a + b*Csc[c + d*x])^n, (-1 + Csc[c + d*x]^2)^(m/2), x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0]
&& IGtQ[m/2, 0] && IntegerQ[n - 1/2]

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 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 4090

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)
], x_Symbol] :> Dist[A - B, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[B, 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}, x] && NeQ[a^2 - b^2, 0] && NeQ[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 4167

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{(a+b \text {sech}(c+d x))^{3/2}}-\frac {2 \text {sech}^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}}+\frac {\text {sech}^4(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}}\right ) \, dx \\ & = -\left (2 \int \frac {\text {sech}^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx\right )+\int \frac {1}{(a+b \text {sech}(c+d x))^{3/2}} \, dx+\int \frac {\text {sech}^4(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx \\ & = -\frac {4 a \tanh (c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 b^2 \tanh (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}-\frac {2 a^2 \text {sech}(c+d x) \tanh (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}-\frac {4 \int \frac {\text {sech}(c+d x) \left (-\frac {b}{2}-\frac {1}{2} a \text {sech}(c+d x)\right )}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{a^2-b^2}-\frac {2 \int \frac {\frac {1}{2} \left (-a^2+b^2\right )+\frac {1}{2} a b \text {sech}(c+d x)+\frac {1}{2} b^2 \text {sech}^2(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{a \left (a^2-b^2\right )}-\frac {2 \int \frac {\text {sech}(c+d x) \left (a^2-\frac {1}{2} a b \text {sech}(c+d x)-\frac {1}{2} \left (4 a^2-b^2\right ) \text {sech}^2(c+d x)\right )}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{b \left (a^2-b^2\right )} \\ & = -\frac {4 a \tanh (c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 b^2 \tanh (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}-\frac {2 a^2 \text {sech}(c+d x) \tanh (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 \left (4 a^2-b^2\right ) \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {2 \int \frac {\text {sech}(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{a+b}-\frac {2 \int \frac {\frac {1}{2} \left (-a^2+b^2\right )+\left (\frac {a b}{2}-\frac {b^2}{2}\right ) \text {sech}(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{a \left (a^2-b^2\right )}+\frac {(2 a) \int \frac {\text {sech}(c+d x) (1+\text {sech}(c+d x))}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{a^2-b^2}-\frac {4 \int \frac {\text {sech}(c+d x) \left (\frac {1}{4} b \left (2 a^2+b^2\right )+\frac {1}{4} a \left (8 a^2-5 b^2\right ) \text {sech}(c+d x)\right )}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{3 b^2 \left (a^2-b^2\right )}-\frac {b^2 \int \frac {\text {sech}(c+d x) (1+\text {sech}(c+d x))}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{a \left (a^2-b^2\right )} \\ & = -\frac {2 \coth (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{a \sqrt {a+b} d}+\frac {4 a \coth (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{b^2 \sqrt {a+b} d}+\frac {4 \coth (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{b \sqrt {a+b} d}-\frac {4 a \tanh (c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 b^2 \tanh (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}-\frac {2 a^2 \text {sech}(c+d x) \tanh (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 \left (4 a^2-b^2\right ) \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}+\frac {\int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{a}-\frac {b \int \frac {\text {sech}(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{a (a+b)}+\frac {((2 a+b) (4 a+b)) \int \frac {\text {sech}(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{3 b^2 (a+b)}-\frac {\left (a \left (8 a^2-5 b^2\right )\right ) \int \frac {\text {sech}(c+d x) (1+\text {sech}(c+d x))}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{3 b^2 \left (a^2-b^2\right )} \\ & = -\frac {2 \coth (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{a \sqrt {a+b} d}+\frac {4 a \coth (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{b^2 \sqrt {a+b} d}-\frac {2 a \left (8 a^2-5 b^2\right ) \coth (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{3 b^4 \sqrt {a+b} d}+\frac {2 \coth (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{a \sqrt {a+b} d}+\frac {4 \coth (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{b \sqrt {a+b} d}-\frac {2 (2 a+b) (4 a+b) \coth (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{3 b^3 \sqrt {a+b} d}+\frac {2 \sqrt {a+b} \coth (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{a^2 d}-\frac {4 a \tanh (c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 b^2 \tanh (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}-\frac {2 a^2 \text {sech}(c+d x) \tanh (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 \left (4 a^2-b^2\right ) \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x)}{3 b^2 \left (a^2-b^2\right ) d} \\ \end{align*}

Mathematica [F]

\[ \int \frac {\tanh ^4(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int \frac {\tanh ^4(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx \]

[In]

Integrate[Tanh[c + d*x]^4/(a + b*Sech[c + d*x])^(3/2),x]

[Out]

Integrate[Tanh[c + d*x]^4/(a + b*Sech[c + d*x])^(3/2), x]

Maple [F]

\[\int \frac {\tanh \left (d x +c \right )^{4}}{\left (a +b \,\operatorname {sech}\left (d x +c \right )\right )^{\frac {3}{2}}}d x\]

[In]

int(tanh(d*x+c)^4/(a+b*sech(d*x+c))^(3/2),x)

[Out]

int(tanh(d*x+c)^4/(a+b*sech(d*x+c))^(3/2),x)

Fricas [F]

\[ \int \frac {\tanh ^4(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int { \frac {\tanh \left (d x + c\right )^{4}}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(tanh(d*x+c)^4/(a+b*sech(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*sech(d*x + c) + a)*tanh(d*x + c)^4/(b^2*sech(d*x + c)^2 + 2*a*b*sech(d*x + c) + a^2), x)

Sympy [F]

\[ \int \frac {\tanh ^4(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int \frac {\tanh ^{4}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(tanh(d*x+c)**4/(a+b*sech(d*x+c))**(3/2),x)

[Out]

Integral(tanh(c + d*x)**4/(a + b*sech(c + d*x))**(3/2), x)

Maxima [F]

\[ \int \frac {\tanh ^4(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int { \frac {\tanh \left (d x + c\right )^{4}}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(tanh(d*x+c)^4/(a+b*sech(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate(tanh(d*x + c)^4/(b*sech(d*x + c) + a)^(3/2), x)

Giac [F]

\[ \int \frac {\tanh ^4(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int { \frac {\tanh \left (d x + c\right )^{4}}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(tanh(d*x+c)^4/(a+b*sech(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate(tanh(d*x + c)^4/(b*sech(d*x + c) + a)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\tanh ^4(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int \frac {{\mathrm {tanh}\left (c+d\,x\right )}^4}{{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{3/2}} \,d x \]

[In]

int(tanh(c + d*x)^4/(a + b/cosh(c + d*x))^(3/2),x)

[Out]

int(tanh(c + d*x)^4/(a + b/cosh(c + d*x))^(3/2), x)