3.212 \(\int \frac{\tanh ^5(x)}{(a+b \text{sech}^2(x))^{5/2}} \, dx\)

Optimal. Leaf size=76 \[ -\frac{\frac{1}{a^2}-\frac{1}{b^2}}{\sqrt{a+b \text{sech}^2(x)}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{(a+b)^2}{3 a b^2 \left (a+b \text{sech}^2(x)\right )^{3/2}} \]

[Out]

ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]]/a^(5/2) - (a + b)^2/(3*a*b^2*(a + b*Sech[x]^2)^(3/2)) - (a^(-2) - b^(-2
))/Sqrt[a + b*Sech[x]^2]

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Rubi [A]  time = 0.157898, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {4139, 446, 87, 63, 208} \[ -\frac{\frac{1}{a^2}-\frac{1}{b^2}}{\sqrt{a+b \text{sech}^2(x)}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{(a+b)^2}{3 a b^2 \left (a+b \text{sech}^2(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Tanh[x]^5/(a + b*Sech[x]^2)^(5/2),x]

[Out]

ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]]/a^(5/2) - (a + b)^2/(3*a*b^2*(a + b*Sech[x]^2)^(3/2)) - (a^(-2) - b^(-2
))/Sqrt[a + b*Sech[x]^2]

Rule 4139

Int[((a_) + (b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With
[{ff = FreeFactors[Sec[e + f*x], x]}, Dist[1/f, Subst[Int[((-1 + ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p)/x
, x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (GtQ[m, 0] || EqQ[
n, 2] || EqQ[n, 4] || IGtQ[p, 0] || IntegersQ[2*n, p])

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(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] && IntegerQ[Simplify[(m + 1)/n]]

Rule 87

Int[(((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_))/((a_.) + (b_.)*(x_)), x_Symbol] :> Int[ExpandIntegr
and[(e + f*x)^FractionalPart[p], ((c + d*x)^n*(e + f*x)^IntegerPart[p])/(a + b*x), x], x] /; FreeQ[{a, b, c, d
, e, f}, x] && IGtQ[n, 0] && LtQ[p, -1] && FractionQ[p]

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 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{\tanh ^5(x)}{\left (a+b \text{sech}^2(x)\right )^{5/2}} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (-1+x^2\right )^2}{x \left (a+b x^2\right )^{5/2}} \, dx,x,\text{sech}(x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{(-1+x)^2}{x (a+b x)^{5/2}} \, dx,x,\text{sech}^2(x)\right )\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{(a+b)^2}{a b (a+b x)^{5/2}}+\frac{a^2-b^2}{a^2 b (a+b x)^{3/2}}+\frac{1}{a^2 x \sqrt{a+b x}}\right ) \, dx,x,\text{sech}^2(x)\right )\right )\\ &=-\frac{(a+b)^2}{3 a b^2 \left (a+b \text{sech}^2(x)\right )^{3/2}}-\frac{\frac{1}{a^2}-\frac{1}{b^2}}{\sqrt{a+b \text{sech}^2(x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\text{sech}^2(x)\right )}{2 a^2}\\ &=-\frac{(a+b)^2}{3 a b^2 \left (a+b \text{sech}^2(x)\right )^{3/2}}-\frac{\frac{1}{a^2}-\frac{1}{b^2}}{\sqrt{a+b \text{sech}^2(x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \text{sech}^2(x)}\right )}{a^2 b}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{(a+b)^2}{3 a b^2 \left (a+b \text{sech}^2(x)\right )^{3/2}}-\frac{\frac{1}{a^2}-\frac{1}{b^2}}{\sqrt{a+b \text{sech}^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.521306, size = 126, normalized size = 1.66 \[ \frac{\text{sech}^5(x) \left (\frac{4 (a+b) \cosh (x) \left (a^2+a (a-2 b) \cosh (2 x)+a b-3 b^2\right ) (a \cosh (2 x)+a+2 b)}{3 a^2 b^2}+\frac{\sqrt{2} (a \cosh (2 x)+a+2 b)^{5/2} \log \left (\sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{a} \cosh (x)\right )}{a^{5/2}}\right )}{8 \left (a+b \text{sech}^2(x)\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[x]^5/(a + b*Sech[x]^2)^(5/2),x]

[Out]

(((4*(a + b)*Cosh[x]*(a + 2*b + a*Cosh[2*x])*(a^2 + a*b - 3*b^2 + a*(a - 2*b)*Cosh[2*x]))/(3*a^2*b^2) + (Sqrt[
2]*(a + 2*b + a*Cosh[2*x])^(5/2)*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[a + 2*b + a*Cosh[2*x]]])/a^(5/2))*Sech[x]^
5)/(8*(a + b*Sech[x]^2)^(5/2))

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Maple [F]  time = 0.108, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \tanh \left ( x \right ) \right ) ^{5} \left ( a+b \left ({\rm sech} \left (x\right ) \right ) ^{2} \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^5/(a+b*sech(x)^2)^(5/2),x)

[Out]

int(tanh(x)^5/(a+b*sech(x)^2)^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh \left (x\right )^{5}}{{\left (b \operatorname{sech}\left (x\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^5/(a+b*sech(x)^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(tanh(x)^5/(b*sech(x)^2 + a)^(5/2), x)

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Fricas [B]  time = 4.85522, size = 12878, normalized size = 169.45 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^5/(a+b*sech(x)^2)^(5/2),x, algorithm="fricas")

[Out]

[1/12*(3*(a^2*b^2*cosh(x)^8 + 8*a^2*b^2*cosh(x)*sinh(x)^7 + a^2*b^2*sinh(x)^8 + 4*(a^2*b^2 + 2*a*b^3)*cosh(x)^
6 + 4*(7*a^2*b^2*cosh(x)^2 + a^2*b^2 + 2*a*b^3)*sinh(x)^6 + 8*(7*a^2*b^2*cosh(x)^3 + 3*(a^2*b^2 + 2*a*b^3)*cos
h(x))*sinh(x)^5 + 2*(3*a^2*b^2 + 8*a*b^3 + 8*b^4)*cosh(x)^4 + 2*(35*a^2*b^2*cosh(x)^4 + 3*a^2*b^2 + 8*a*b^3 +
8*b^4 + 30*(a^2*b^2 + 2*a*b^3)*cosh(x)^2)*sinh(x)^4 + a^2*b^2 + 8*(7*a^2*b^2*cosh(x)^5 + 10*(a^2*b^2 + 2*a*b^3
)*cosh(x)^3 + (3*a^2*b^2 + 8*a*b^3 + 8*b^4)*cosh(x))*sinh(x)^3 + 4*(a^2*b^2 + 2*a*b^3)*cosh(x)^2 + 4*(7*a^2*b^
2*cosh(x)^6 + 15*(a^2*b^2 + 2*a*b^3)*cosh(x)^4 + a^2*b^2 + 2*a*b^3 + 3*(3*a^2*b^2 + 8*a*b^3 + 8*b^4)*cosh(x)^2
)*sinh(x)^2 + 8*(a^2*b^2*cosh(x)^7 + 3*(a^2*b^2 + 2*a*b^3)*cosh(x)^5 + (3*a^2*b^2 + 8*a*b^3 + 8*b^4)*cosh(x)^3
+ (a^2*b^2 + 2*a*b^3)*cosh(x))*sinh(x))*sqrt(a)*log(((a^3 + 2*a^2*b + a*b^2)*cosh(x)^8 + 8*(a^3 + 2*a^2*b + a
*b^2)*cosh(x)*sinh(x)^7 + (a^3 + 2*a^2*b + a*b^2)*sinh(x)^8 + 2*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3)*cosh(x)^6 +
2*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3 + 14*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^2)*sinh(x)^6 + 4*(14*(a^3 + 2*a^2*b +
a*b^2)*cosh(x)^3 + 3*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3)*cosh(x))*sinh(x)^5 + (6*a^3 + 14*a^2*b + 9*a*b^2)*cosh(
x)^4 + (70*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^4 + 6*a^3 + 14*a^2*b + 9*a*b^2 + 30*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^
3)*cosh(x)^2)*sinh(x)^4 + 4*(14*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^5 + 10*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3)*cosh(
x)^3 + (6*a^3 + 14*a^2*b + 9*a*b^2)*cosh(x))*sinh(x)^3 + a^3 + 2*(2*a^3 + 3*a^2*b)*cosh(x)^2 + 2*(14*(a^3 + 2*
a^2*b + a*b^2)*cosh(x)^6 + 15*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3)*cosh(x)^4 + 2*a^3 + 3*a^2*b + 3*(6*a^3 + 14*a^
2*b + 9*a*b^2)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*((a^2 + 2*a*b + b^2)*cosh(x)^6 + 6*(a^2 + 2*a*b + b^2)*cosh(x)*s
inh(x)^5 + (a^2 + 2*a*b + b^2)*sinh(x)^6 + 3*(a^2 + 2*a*b + b^2)*cosh(x)^4 + 3*(5*(a^2 + 2*a*b + b^2)*cosh(x)^
2 + a^2 + 2*a*b + b^2)*sinh(x)^4 + 4*(5*(a^2 + 2*a*b + b^2)*cosh(x)^3 + 3*(a^2 + 2*a*b + b^2)*cosh(x))*sinh(x)
^3 + (3*a^2 + 4*a*b)*cosh(x)^2 + (15*(a^2 + 2*a*b + b^2)*cosh(x)^4 + 18*(a^2 + 2*a*b + b^2)*cosh(x)^2 + 3*a^2
+ 4*a*b)*sinh(x)^2 + a^2 + 2*(3*(a^2 + 2*a*b + b^2)*cosh(x)^5 + 6*(a^2 + 2*a*b + b^2)*cosh(x)^3 + (3*a^2 + 4*a
*b)*cosh(x))*sinh(x))*sqrt(a)*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh
(x)^2)) + 4*(2*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^7 + 3*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3)*cosh(x)^5 + (6*a^3 + 14
*a^2*b + 9*a*b^2)*cosh(x)^3 + (2*a^3 + 3*a^2*b)*cosh(x))*sinh(x))/(cosh(x)^6 + 6*cosh(x)^5*sinh(x) + 15*cosh(x
)^4*sinh(x)^2 + 20*cosh(x)^3*sinh(x)^3 + 15*cosh(x)^2*sinh(x)^4 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6)) + 3*(a^2*b
^2*cosh(x)^8 + 8*a^2*b^2*cosh(x)*sinh(x)^7 + a^2*b^2*sinh(x)^8 + 4*(a^2*b^2 + 2*a*b^3)*cosh(x)^6 + 4*(7*a^2*b^
2*cosh(x)^2 + a^2*b^2 + 2*a*b^3)*sinh(x)^6 + 8*(7*a^2*b^2*cosh(x)^3 + 3*(a^2*b^2 + 2*a*b^3)*cosh(x))*sinh(x)^5
+ 2*(3*a^2*b^2 + 8*a*b^3 + 8*b^4)*cosh(x)^4 + 2*(35*a^2*b^2*cosh(x)^4 + 3*a^2*b^2 + 8*a*b^3 + 8*b^4 + 30*(a^2
*b^2 + 2*a*b^3)*cosh(x)^2)*sinh(x)^4 + a^2*b^2 + 8*(7*a^2*b^2*cosh(x)^5 + 10*(a^2*b^2 + 2*a*b^3)*cosh(x)^3 + (
3*a^2*b^2 + 8*a*b^3 + 8*b^4)*cosh(x))*sinh(x)^3 + 4*(a^2*b^2 + 2*a*b^3)*cosh(x)^2 + 4*(7*a^2*b^2*cosh(x)^6 + 1
5*(a^2*b^2 + 2*a*b^3)*cosh(x)^4 + a^2*b^2 + 2*a*b^3 + 3*(3*a^2*b^2 + 8*a*b^3 + 8*b^4)*cosh(x)^2)*sinh(x)^2 + 8
*(a^2*b^2*cosh(x)^7 + 3*(a^2*b^2 + 2*a*b^3)*cosh(x)^5 + (3*a^2*b^2 + 8*a*b^3 + 8*b^4)*cosh(x)^3 + (a^2*b^2 + 2
*a*b^3)*cosh(x))*sinh(x))*sqrt(a)*log(-(a*cosh(x)^4 + 4*a*cosh(x)*sinh(x)^3 + a*sinh(x)^4 + 2*b*cosh(x)^2 + 2*
(3*a*cosh(x)^2 + b)*sinh(x)^2 + sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*sqrt(a)*sqrt((a*cosh(x
)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 4*(a*cosh(x)^3 + b*cosh(x))*sinh(x
) + a)/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)) + 8*sqrt(2)*((a^4 - a^3*b - 2*a^2*b^2)*cosh(x)^6 + 6*(a^4
- a^3*b - 2*a^2*b^2)*cosh(x)*sinh(x)^5 + (a^4 - a^3*b - 2*a^2*b^2)*sinh(x)^6 + 3*(a^4 + a^3*b - 2*a^2*b^2 - 2*
a*b^3)*cosh(x)^4 + 3*(a^4 + a^3*b - 2*a^2*b^2 - 2*a*b^3 + 5*(a^4 - a^3*b - 2*a^2*b^2)*cosh(x)^2)*sinh(x)^4 + a
^4 - a^3*b - 2*a^2*b^2 + 4*(5*(a^4 - a^3*b - 2*a^2*b^2)*cosh(x)^3 + 3*(a^4 + a^3*b - 2*a^2*b^2 - 2*a*b^3)*cosh
(x))*sinh(x)^3 + 3*(a^4 + a^3*b - 2*a^2*b^2 - 2*a*b^3)*cosh(x)^2 + 3*(5*(a^4 - a^3*b - 2*a^2*b^2)*cosh(x)^4 +
a^4 + a^3*b - 2*a^2*b^2 - 2*a*b^3 + 6*(a^4 + a^3*b - 2*a^2*b^2 - 2*a*b^3)*cosh(x)^2)*sinh(x)^2 + 6*((a^4 - a^3
*b - 2*a^2*b^2)*cosh(x)^5 + 2*(a^4 + a^3*b - 2*a^2*b^2 - 2*a*b^3)*cosh(x)^3 + (a^4 + a^3*b - 2*a^2*b^2 - 2*a*b
^3)*cosh(x))*sinh(x))*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))
/(a^5*b^2*cosh(x)^8 + 8*a^5*b^2*cosh(x)*sinh(x)^7 + a^5*b^2*sinh(x)^8 + a^5*b^2 + 4*(a^5*b^2 + 2*a^4*b^3)*cosh
(x)^6 + 4*(7*a^5*b^2*cosh(x)^2 + a^5*b^2 + 2*a^4*b^3)*sinh(x)^6 + 8*(7*a^5*b^2*cosh(x)^3 + 3*(a^5*b^2 + 2*a^4*
b^3)*cosh(x))*sinh(x)^5 + 2*(3*a^5*b^2 + 8*a^4*b^3 + 8*a^3*b^4)*cosh(x)^4 + 2*(35*a^5*b^2*cosh(x)^4 + 3*a^5*b^
2 + 8*a^4*b^3 + 8*a^3*b^4 + 30*(a^5*b^2 + 2*a^4*b^3)*cosh(x)^2)*sinh(x)^4 + 8*(7*a^5*b^2*cosh(x)^5 + 10*(a^5*b
^2 + 2*a^4*b^3)*cosh(x)^3 + (3*a^5*b^2 + 8*a^4*b^3 + 8*a^3*b^4)*cosh(x))*sinh(x)^3 + 4*(a^5*b^2 + 2*a^4*b^3)*c
osh(x)^2 + 4*(7*a^5*b^2*cosh(x)^6 + a^5*b^2 + 2*a^4*b^3 + 15*(a^5*b^2 + 2*a^4*b^3)*cosh(x)^4 + 3*(3*a^5*b^2 +
8*a^4*b^3 + 8*a^3*b^4)*cosh(x)^2)*sinh(x)^2 + 8*(a^5*b^2*cosh(x)^7 + 3*(a^5*b^2 + 2*a^4*b^3)*cosh(x)^5 + (3*a^
5*b^2 + 8*a^4*b^3 + 8*a^3*b^4)*cosh(x)^3 + (a^5*b^2 + 2*a^4*b^3)*cosh(x))*sinh(x)), -1/6*(3*(a^2*b^2*cosh(x)^8
+ 8*a^2*b^2*cosh(x)*sinh(x)^7 + a^2*b^2*sinh(x)^8 + 4*(a^2*b^2 + 2*a*b^3)*cosh(x)^6 + 4*(7*a^2*b^2*cosh(x)^2
+ a^2*b^2 + 2*a*b^3)*sinh(x)^6 + 8*(7*a^2*b^2*cosh(x)^3 + 3*(a^2*b^2 + 2*a*b^3)*cosh(x))*sinh(x)^5 + 2*(3*a^2*
b^2 + 8*a*b^3 + 8*b^4)*cosh(x)^4 + 2*(35*a^2*b^2*cosh(x)^4 + 3*a^2*b^2 + 8*a*b^3 + 8*b^4 + 30*(a^2*b^2 + 2*a*b
^3)*cosh(x)^2)*sinh(x)^4 + a^2*b^2 + 8*(7*a^2*b^2*cosh(x)^5 + 10*(a^2*b^2 + 2*a*b^3)*cosh(x)^3 + (3*a^2*b^2 +
8*a*b^3 + 8*b^4)*cosh(x))*sinh(x)^3 + 4*(a^2*b^2 + 2*a*b^3)*cosh(x)^2 + 4*(7*a^2*b^2*cosh(x)^6 + 15*(a^2*b^2 +
2*a*b^3)*cosh(x)^4 + a^2*b^2 + 2*a*b^3 + 3*(3*a^2*b^2 + 8*a*b^3 + 8*b^4)*cosh(x)^2)*sinh(x)^2 + 8*(a^2*b^2*co
sh(x)^7 + 3*(a^2*b^2 + 2*a*b^3)*cosh(x)^5 + (3*a^2*b^2 + 8*a*b^3 + 8*b^4)*cosh(x)^3 + (a^2*b^2 + 2*a*b^3)*cosh
(x))*sinh(x))*sqrt(-a)*arctan(sqrt(2)*((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 + a)*
sqrt(-a)*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/((a^2 + a*b)*
cosh(x)^4 + 4*(a^2 + a*b)*cosh(x)*sinh(x)^3 + (a^2 + a*b)*sinh(x)^4 + (2*a^2 + 3*a*b)*cosh(x)^2 + (6*(a^2 + a*
b)*cosh(x)^2 + 2*a^2 + 3*a*b)*sinh(x)^2 + a^2 + 2*(2*(a^2 + a*b)*cosh(x)^3 + (2*a^2 + 3*a*b)*cosh(x))*sinh(x))
) + 3*(a^2*b^2*cosh(x)^8 + 8*a^2*b^2*cosh(x)*sinh(x)^7 + a^2*b^2*sinh(x)^8 + 4*(a^2*b^2 + 2*a*b^3)*cosh(x)^6 +
4*(7*a^2*b^2*cosh(x)^2 + a^2*b^2 + 2*a*b^3)*sinh(x)^6 + 8*(7*a^2*b^2*cosh(x)^3 + 3*(a^2*b^2 + 2*a*b^3)*cosh(x
))*sinh(x)^5 + 2*(3*a^2*b^2 + 8*a*b^3 + 8*b^4)*cosh(x)^4 + 2*(35*a^2*b^2*cosh(x)^4 + 3*a^2*b^2 + 8*a*b^3 + 8*b
^4 + 30*(a^2*b^2 + 2*a*b^3)*cosh(x)^2)*sinh(x)^4 + a^2*b^2 + 8*(7*a^2*b^2*cosh(x)^5 + 10*(a^2*b^2 + 2*a*b^3)*c
osh(x)^3 + (3*a^2*b^2 + 8*a*b^3 + 8*b^4)*cosh(x))*sinh(x)^3 + 4*(a^2*b^2 + 2*a*b^3)*cosh(x)^2 + 4*(7*a^2*b^2*c
osh(x)^6 + 15*(a^2*b^2 + 2*a*b^3)*cosh(x)^4 + a^2*b^2 + 2*a*b^3 + 3*(3*a^2*b^2 + 8*a*b^3 + 8*b^4)*cosh(x)^2)*s
inh(x)^2 + 8*(a^2*b^2*cosh(x)^7 + 3*(a^2*b^2 + 2*a*b^3)*cosh(x)^5 + (3*a^2*b^2 + 8*a*b^3 + 8*b^4)*cosh(x)^3 +
(a^2*b^2 + 2*a*b^3)*cosh(x))*sinh(x))*sqrt(-a)*arctan(sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)*
sqrt(-a)*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/(a*cosh(x)^4
+ 4*a*cosh(x)*sinh(x)^3 + a*sinh(x)^4 + 2*(a + 2*b)*cosh(x)^2 + 2*(3*a*cosh(x)^2 + a + 2*b)*sinh(x)^2 + 4*(a*c
osh(x)^3 + (a + 2*b)*cosh(x))*sinh(x) + a)) - 4*sqrt(2)*((a^4 - a^3*b - 2*a^2*b^2)*cosh(x)^6 + 6*(a^4 - a^3*b
- 2*a^2*b^2)*cosh(x)*sinh(x)^5 + (a^4 - a^3*b - 2*a^2*b^2)*sinh(x)^6 + 3*(a^4 + a^3*b - 2*a^2*b^2 - 2*a*b^3)*c
osh(x)^4 + 3*(a^4 + a^3*b - 2*a^2*b^2 - 2*a*b^3 + 5*(a^4 - a^3*b - 2*a^2*b^2)*cosh(x)^2)*sinh(x)^4 + a^4 - a^3
*b - 2*a^2*b^2 + 4*(5*(a^4 - a^3*b - 2*a^2*b^2)*cosh(x)^3 + 3*(a^4 + a^3*b - 2*a^2*b^2 - 2*a*b^3)*cosh(x))*sin
h(x)^3 + 3*(a^4 + a^3*b - 2*a^2*b^2 - 2*a*b^3)*cosh(x)^2 + 3*(5*(a^4 - a^3*b - 2*a^2*b^2)*cosh(x)^4 + a^4 + a^
3*b - 2*a^2*b^2 - 2*a*b^3 + 6*(a^4 + a^3*b - 2*a^2*b^2 - 2*a*b^3)*cosh(x)^2)*sinh(x)^2 + 6*((a^4 - a^3*b - 2*a
^2*b^2)*cosh(x)^5 + 2*(a^4 + a^3*b - 2*a^2*b^2 - 2*a*b^3)*cosh(x)^3 + (a^4 + a^3*b - 2*a^2*b^2 - 2*a*b^3)*cosh
(x))*sinh(x))*sqrt((a*cosh(x)^2 + a*sinh(x)^2 + a + 2*b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/(a^5*b^
2*cosh(x)^8 + 8*a^5*b^2*cosh(x)*sinh(x)^7 + a^5*b^2*sinh(x)^8 + a^5*b^2 + 4*(a^5*b^2 + 2*a^4*b^3)*cosh(x)^6 +
4*(7*a^5*b^2*cosh(x)^2 + a^5*b^2 + 2*a^4*b^3)*sinh(x)^6 + 8*(7*a^5*b^2*cosh(x)^3 + 3*(a^5*b^2 + 2*a^4*b^3)*cos
h(x))*sinh(x)^5 + 2*(3*a^5*b^2 + 8*a^4*b^3 + 8*a^3*b^4)*cosh(x)^4 + 2*(35*a^5*b^2*cosh(x)^4 + 3*a^5*b^2 + 8*a^
4*b^3 + 8*a^3*b^4 + 30*(a^5*b^2 + 2*a^4*b^3)*cosh(x)^2)*sinh(x)^4 + 8*(7*a^5*b^2*cosh(x)^5 + 10*(a^5*b^2 + 2*a
^4*b^3)*cosh(x)^3 + (3*a^5*b^2 + 8*a^4*b^3 + 8*a^3*b^4)*cosh(x))*sinh(x)^3 + 4*(a^5*b^2 + 2*a^4*b^3)*cosh(x)^2
+ 4*(7*a^5*b^2*cosh(x)^6 + a^5*b^2 + 2*a^4*b^3 + 15*(a^5*b^2 + 2*a^4*b^3)*cosh(x)^4 + 3*(3*a^5*b^2 + 8*a^4*b^
3 + 8*a^3*b^4)*cosh(x)^2)*sinh(x)^2 + 8*(a^5*b^2*cosh(x)^7 + 3*(a^5*b^2 + 2*a^4*b^3)*cosh(x)^5 + (3*a^5*b^2 +
8*a^4*b^3 + 8*a^3*b^4)*cosh(x)^3 + (a^5*b^2 + 2*a^4*b^3)*cosh(x))*sinh(x))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh ^{5}{\left (x \right )}}{\left (a + b \operatorname{sech}^{2}{\left (x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)**5/(a+b*sech(x)**2)**(5/2),x)

[Out]

Integral(tanh(x)**5/(a + b*sech(x)**2)**(5/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh \left (x\right )^{5}}{{\left (b \operatorname{sech}\left (x\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^5/(a+b*sech(x)^2)^(5/2),x, algorithm="giac")

[Out]

integrate(tanh(x)^5/(b*sech(x)^2 + a)^(5/2), x)