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

Optimal. Leaf size=907 \[ \text{result too large to display} \]

[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)

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Rubi [A]  time = 1.36637, antiderivative size = 907, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {3895, 3785, 4058, 3921, 3784, 3832, 4004, 3836, 4005, 3845, 4082} \[ -\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 (\sin ^{-1}\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 (\sin ^{-1}\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) F\left (\sin ^{-1}\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) F\left (\sin ^{-1}\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 (\sin ^{-1}\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) F\left (\sin ^{-1}\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) \Pi \left (\frac{a+b}{a};\sin ^{-1}\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} \]

Antiderivative was successfully verified.

[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 3895

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 3785

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 4058

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 3921

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 3784

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

Rule 3832

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

Rule 4004

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 + Cs
c[e + f*x]))/(a - b))]*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + (b*B)/A, 2]], (a*A + b*B)/(a*A - b*B)]
)/(b^2*f*Cot[e + f*x]), x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]

Rule 3836

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 4005

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 3845

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] && N
eQ[a^2 - b^2, 0] && LtQ[m, -1] && (IGtQ[n, 3] || (IntegersQ[n + 1/2, 2*m] && GtQ[n, 2]))

Rule 4082

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*} \int \frac{\tanh ^4(c+d x)}{(a+b \text{sech}(c+d x))^{3/2}} \, dx &=\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 (\sin ^{-1}\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 (\sin ^{-1}\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) F\left (\sin ^{-1}\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 (\sin ^{-1}\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 (\sin ^{-1}\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 (\sin ^{-1}\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) F\left (\sin ^{-1}\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) F\left (\sin ^{-1}\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) F\left (\sin ^{-1}\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) \Pi \left (\frac{a+b}{a};\sin ^{-1}\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]  time = 180.004, size = 0, normalized size = 0. \[ \text{\$Aborted} \]

Verification is Not applicable to the result.

[In]

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

[Out]

$Aborted

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Maple [F]  time = 0.171, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \tanh \left ( dx+c \right ) \right ) ^{4} \left ( a+b{\rm sech} \left (dx+c\right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \operatorname{sech}\left (d x + c\right ) + a} \tanh \left (d x + c\right )^{4}}{b^{2} \operatorname{sech}\left (d x + c\right )^{2} + 2 \, a b \operatorname{sech}\left (d x + c\right ) + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh ^{4}{\left (c + d x \right )}}{\left (a + b \operatorname{sech}{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)