3.142 \(\int \frac{\tanh ^5(c+d x)}{(a+b \text{sech}(c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=148 \[ -\frac{2 \left (3 a^2-2 b^2\right ) \sqrt{a+b \text{sech}(c+d x)}}{b^4 d}-\frac{2 \left (a^2-b^2\right )^2}{a b^4 d \sqrt{a+b \text{sech}(c+d x)}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a}}\right )}{a^{3/2} d}-\frac{2 (a+b \text{sech}(c+d x))^{5/2}}{5 b^4 d}+\frac{2 a (a+b \text{sech}(c+d x))^{3/2}}{b^4 d} \]

[Out]

(2*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) - (2*(a^2 - b^2)^2)/(a*b^4*d*Sqrt[a + b*Sech[c + d*
x]]) - (2*(3*a^2 - 2*b^2)*Sqrt[a + b*Sech[c + d*x]])/(b^4*d) + (2*a*(a + b*Sech[c + d*x])^(3/2))/(b^4*d) - (2*
(a + b*Sech[c + d*x])^(5/2))/(5*b^4*d)

________________________________________________________________________________________

Rubi [A]  time = 0.191165, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3885, 898, 1261, 206} \[ -\frac{2 \left (3 a^2-2 b^2\right ) \sqrt{a+b \text{sech}(c+d x)}}{b^4 d}-\frac{2 \left (a^2-b^2\right )^2}{a b^4 d \sqrt{a+b \text{sech}(c+d x)}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a}}\right )}{a^{3/2} d}-\frac{2 (a+b \text{sech}(c+d x))^{5/2}}{5 b^4 d}+\frac{2 a (a+b \text{sech}(c+d x))^{3/2}}{b^4 d} \]

Antiderivative was successfully verified.

[In]

Int[Tanh[c + d*x]^5/(a + b*Sech[c + d*x])^(3/2),x]

[Out]

(2*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) - (2*(a^2 - b^2)^2)/(a*b^4*d*Sqrt[a + b*Sech[c + d*
x]]) - (2*(3*a^2 - 2*b^2)*Sqrt[a + b*Sech[c + d*x]])/(b^4*d) + (2*a*(a + b*Sech[c + d*x])^(3/2))/(b^4*d) - (2*
(a + b*Sech[c + d*x])^(5/2))/(5*b^4*d)

Rule 3885

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

Rule 898

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{q = De
nominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 + a*e^2)/e^2 - (2*c
*d*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*
g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegersQ[n, p] && FractionQ[m]

Rule 1261

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] &&
NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\tanh ^5(c+d x)}{(a+b \text{sech}(c+d x))^{3/2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (b^2-x^2\right )^2}{x (a+x)^{3/2}} \, dx,x,b \text{sech}(c+d x)\right )}{b^4 d}\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{\left (-a^2+b^2+2 a x^2-x^4\right )^2}{x^2 \left (-a+x^2\right )} \, dx,x,\sqrt{a+b \text{sech}(c+d x)}\right )}{b^4 d}\\ &=-\frac{2 \operatorname{Subst}\left (\int \left (3 a^2 \left (1-\frac{2 b^2}{3 a^2}\right )-\frac{\left (a^2-b^2\right )^2}{a x^2}-3 a x^2+x^4-\frac{b^4}{a \left (a-x^2\right )}\right ) \, dx,x,\sqrt{a+b \text{sech}(c+d x)}\right )}{b^4 d}\\ &=-\frac{2 \left (a^2-b^2\right )^2}{a b^4 d \sqrt{a+b \text{sech}(c+d x)}}-\frac{2 \left (3 a^2-2 b^2\right ) \sqrt{a+b \text{sech}(c+d x)}}{b^4 d}+\frac{2 a (a+b \text{sech}(c+d x))^{3/2}}{b^4 d}-\frac{2 (a+b \text{sech}(c+d x))^{5/2}}{5 b^4 d}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{a-x^2} \, dx,x,\sqrt{a+b \text{sech}(c+d x)}\right )}{a d}\\ &=\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a}}\right )}{a^{3/2} d}-\frac{2 \left (a^2-b^2\right )^2}{a b^4 d \sqrt{a+b \text{sech}(c+d x)}}-\frac{2 \left (3 a^2-2 b^2\right ) \sqrt{a+b \text{sech}(c+d x)}}{b^4 d}+\frac{2 a (a+b \text{sech}(c+d x))^{3/2}}{b^4 d}-\frac{2 (a+b \text{sech}(c+d x))^{5/2}}{5 b^4 d}\\ \end{align*}

Mathematica [A]  time = 3.07674, size = 155, normalized size = 1.05 \[ -\frac{2 \left (-2 a^2 b^2 \text{sech}^2(c+d x)+2 a b \left (4 a^2-5 b^2\right ) \text{sech}(c+d x)-20 a^2 b^2+16 a^4+a b^3 \text{sech}^3(c+d x)-\frac{5 b^4 \sqrt{a \cosh (c+d x)+b} \tanh ^{-1}\left (\frac{\sqrt{a \cosh (c+d x)+b}}{\sqrt{a \cosh (c+d x)}}\right )}{\sqrt{a \cosh (c+d x)}}+5 b^4\right )}{5 a b^4 d \sqrt{a+b \text{sech}(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[c + d*x]^5/(a + b*Sech[c + d*x])^(3/2),x]

[Out]

(-2*(16*a^4 - 20*a^2*b^2 + 5*b^4 - (5*b^4*ArcTanh[Sqrt[b + a*Cosh[c + d*x]]/Sqrt[a*Cosh[c + d*x]]]*Sqrt[b + a*
Cosh[c + d*x]])/Sqrt[a*Cosh[c + d*x]] + 2*a*b*(4*a^2 - 5*b^2)*Sech[c + d*x] - 2*a^2*b^2*Sech[c + d*x]^2 + a*b^
3*Sech[c + d*x]^3))/(5*a*b^4*d*Sqrt[a + b*Sech[c + d*x]])

________________________________________________________________________________________

Maple [F]  time = 0.174, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \tanh \left ( dx+c \right ) \right ) ^{5} \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)^5/(a+b*sech(d*x+c))^(3/2),x)

[Out]

int(tanh(d*x+c)^5/(a+b*sech(d*x+c))^(3/2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh \left (d x + c\right )^{5}}{{\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)^5/(a+b*sech(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate(tanh(d*x + c)^5/(b*sech(d*x + c) + a)^(3/2), x)

________________________________________________________________________________________

Fricas [B]  time = 12.5299, size = 9172, normalized size = 61.97 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)^5/(a+b*sech(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

[1/10*(5*(a*b^4*cosh(d*x + c)^6 + a*b^4*sinh(d*x + c)^6 + 2*b^5*cosh(d*x + c)^5 + 3*a*b^4*cosh(d*x + c)^4 + 4*
b^5*cosh(d*x + c)^3 + 3*a*b^4*cosh(d*x + c)^2 + 2*b^5*cosh(d*x + c) + 2*(3*a*b^4*cosh(d*x + c) + b^5)*sinh(d*x
+ c)^5 + a*b^4 + (15*a*b^4*cosh(d*x + c)^2 + 10*b^5*cosh(d*x + c) + 3*a*b^4)*sinh(d*x + c)^4 + 4*(5*a*b^4*cos
h(d*x + c)^3 + 5*b^5*cosh(d*x + c)^2 + 3*a*b^4*cosh(d*x + c) + b^5)*sinh(d*x + c)^3 + (15*a*b^4*cosh(d*x + c)^
4 + 20*b^5*cosh(d*x + c)^3 + 18*a*b^4*cosh(d*x + c)^2 + 12*b^5*cosh(d*x + c) + 3*a*b^4)*sinh(d*x + c)^2 + 2*(3
*a*b^4*cosh(d*x + c)^5 + 5*b^5*cosh(d*x + c)^4 + 6*a*b^4*cosh(d*x + c)^3 + 6*b^5*cosh(d*x + c)^2 + 3*a*b^4*cos
h(d*x + c) + b^5)*sinh(d*x + c))*sqrt(a)*log(-(2*a^2*cosh(d*x + c)^4 + 2*a^2*sinh(d*x + c)^4 + 4*a*b*cosh(d*x
+ c)^3 + 4*(2*a^2*cosh(d*x + c) + a*b)*sinh(d*x + c)^3 + 4*a*b*cosh(d*x + c) + (4*a^2 + b^2)*cosh(d*x + c)^2 +
(12*a^2*cosh(d*x + c)^2 + 12*a*b*cosh(d*x + c) + 4*a^2 + b^2)*sinh(d*x + c)^2 + 2*a^2 + 2*(a*cosh(d*x + c)^4
+ a*sinh(d*x + c)^4 + b*cosh(d*x + c)^3 + (4*a*cosh(d*x + c) + b)*sinh(d*x + c)^3 + 2*a*cosh(d*x + c)^2 + (6*a
*cosh(d*x + c)^2 + 3*b*cosh(d*x + c) + 2*a)*sinh(d*x + c)^2 + b*cosh(d*x + c) + (4*a*cosh(d*x + c)^3 + 3*b*cos
h(d*x + c)^2 + 4*a*cosh(d*x + c) + b)*sinh(d*x + c) + a)*sqrt(a)*sqrt((a*cosh(d*x + c) + b)/cosh(d*x + c)) + 2
*(4*a^2*cosh(d*x + c)^3 + 6*a*b*cosh(d*x + c)^2 + 2*a*b + (4*a^2 + b^2)*cosh(d*x + c))*sinh(d*x + c))/(cosh(d*
x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) - 4*((16*a^5 - 20*a^3*b^2 + 5*a*b^4)*cosh(d*x + c
)^6 + (16*a^5 - 20*a^3*b^2 + 5*a*b^4)*sinh(d*x + c)^6 + 4*(4*a^4*b - 5*a^2*b^3)*cosh(d*x + c)^5 + 2*(8*a^4*b -
10*a^2*b^3 + 3*(16*a^5 - 20*a^3*b^2 + 5*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 16*a^5 - 20*a^3*b^2 + 5*a*b^4
+ (48*a^5 - 68*a^3*b^2 + 15*a*b^4)*cosh(d*x + c)^4 + (48*a^5 - 68*a^3*b^2 + 15*a*b^4 + 15*(16*a^5 - 20*a^3*b^
2 + 5*a*b^4)*cosh(d*x + c)^2 + 20*(4*a^4*b - 5*a^2*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 32*(a^4*b - a^2*b^3)*
cosh(d*x + c)^3 + 4*(8*a^4*b - 8*a^2*b^3 + 5*(16*a^5 - 20*a^3*b^2 + 5*a*b^4)*cosh(d*x + c)^3 + 10*(4*a^4*b - 5
*a^2*b^3)*cosh(d*x + c)^2 + (48*a^5 - 68*a^3*b^2 + 15*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + (48*a^5 - 68*a^3
*b^2 + 15*a*b^4)*cosh(d*x + c)^2 + (48*a^5 - 68*a^3*b^2 + 15*a*b^4 + 15*(16*a^5 - 20*a^3*b^2 + 5*a*b^4)*cosh(d
*x + c)^4 + 40*(4*a^4*b - 5*a^2*b^3)*cosh(d*x + c)^3 + 6*(48*a^5 - 68*a^3*b^2 + 15*a*b^4)*cosh(d*x + c)^2 + 96
*(a^4*b - a^2*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + 4*(4*a^4*b - 5*a^2*b^3)*cosh(d*x + c) + 2*(3*(16*a^5 - 20*
a^3*b^2 + 5*a*b^4)*cosh(d*x + c)^5 + 8*a^4*b - 10*a^2*b^3 + 10*(4*a^4*b - 5*a^2*b^3)*cosh(d*x + c)^4 + 2*(48*a
^5 - 68*a^3*b^2 + 15*a*b^4)*cosh(d*x + c)^3 + 48*(a^4*b - a^2*b^3)*cosh(d*x + c)^2 + (48*a^5 - 68*a^3*b^2 + 15
*a*b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt((a*cosh(d*x + c) + b)/cosh(d*x + c)))/(a^3*b^4*d*cosh(d*x + c)^6 +
a^3*b^4*d*sinh(d*x + c)^6 + 2*a^2*b^5*d*cosh(d*x + c)^5 + 3*a^3*b^4*d*cosh(d*x + c)^4 + 4*a^2*b^5*d*cosh(d*x +
c)^3 + 3*a^3*b^4*d*cosh(d*x + c)^2 + 2*a^2*b^5*d*cosh(d*x + c) + a^3*b^4*d + 2*(3*a^3*b^4*d*cosh(d*x + c) + a
^2*b^5*d)*sinh(d*x + c)^5 + (15*a^3*b^4*d*cosh(d*x + c)^2 + 10*a^2*b^5*d*cosh(d*x + c) + 3*a^3*b^4*d)*sinh(d*x
+ c)^4 + 4*(5*a^3*b^4*d*cosh(d*x + c)^3 + 5*a^2*b^5*d*cosh(d*x + c)^2 + 3*a^3*b^4*d*cosh(d*x + c) + a^2*b^5*d
)*sinh(d*x + c)^3 + (15*a^3*b^4*d*cosh(d*x + c)^4 + 20*a^2*b^5*d*cosh(d*x + c)^3 + 18*a^3*b^4*d*cosh(d*x + c)^
2 + 12*a^2*b^5*d*cosh(d*x + c) + 3*a^3*b^4*d)*sinh(d*x + c)^2 + 2*(3*a^3*b^4*d*cosh(d*x + c)^5 + 5*a^2*b^5*d*c
osh(d*x + c)^4 + 6*a^3*b^4*d*cosh(d*x + c)^3 + 6*a^2*b^5*d*cosh(d*x + c)^2 + 3*a^3*b^4*d*cosh(d*x + c) + a^2*b
^5*d)*sinh(d*x + c)), -1/5*(5*(a*b^4*cosh(d*x + c)^6 + a*b^4*sinh(d*x + c)^6 + 2*b^5*cosh(d*x + c)^5 + 3*a*b^4
*cosh(d*x + c)^4 + 4*b^5*cosh(d*x + c)^3 + 3*a*b^4*cosh(d*x + c)^2 + 2*b^5*cosh(d*x + c) + 2*(3*a*b^4*cosh(d*x
+ c) + b^5)*sinh(d*x + c)^5 + a*b^4 + (15*a*b^4*cosh(d*x + c)^2 + 10*b^5*cosh(d*x + c) + 3*a*b^4)*sinh(d*x +
c)^4 + 4*(5*a*b^4*cosh(d*x + c)^3 + 5*b^5*cosh(d*x + c)^2 + 3*a*b^4*cosh(d*x + c) + b^5)*sinh(d*x + c)^3 + (15
*a*b^4*cosh(d*x + c)^4 + 20*b^5*cosh(d*x + c)^3 + 18*a*b^4*cosh(d*x + c)^2 + 12*b^5*cosh(d*x + c) + 3*a*b^4)*s
inh(d*x + c)^2 + 2*(3*a*b^4*cosh(d*x + c)^5 + 5*b^5*cosh(d*x + c)^4 + 6*a*b^4*cosh(d*x + c)^3 + 6*b^5*cosh(d*x
+ c)^2 + 3*a*b^4*cosh(d*x + c) + b^5)*sinh(d*x + c))*sqrt(-a)*arctan((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 +
b*cosh(d*x + c) + (2*a*cosh(d*x + c) + b)*sinh(d*x + c) + a)*sqrt(-a)*sqrt((a*cosh(d*x + c) + b)/cosh(d*x + c
))/(a^2*cosh(d*x + c)^2 + a^2*sinh(d*x + c)^2 + 2*a*b*cosh(d*x + c) + a^2 + 2*(a^2*cosh(d*x + c) + a*b)*sinh(d
*x + c))) + 2*((16*a^5 - 20*a^3*b^2 + 5*a*b^4)*cosh(d*x + c)^6 + (16*a^5 - 20*a^3*b^2 + 5*a*b^4)*sinh(d*x + c)
^6 + 4*(4*a^4*b - 5*a^2*b^3)*cosh(d*x + c)^5 + 2*(8*a^4*b - 10*a^2*b^3 + 3*(16*a^5 - 20*a^3*b^2 + 5*a*b^4)*cos
h(d*x + c))*sinh(d*x + c)^5 + 16*a^5 - 20*a^3*b^2 + 5*a*b^4 + (48*a^5 - 68*a^3*b^2 + 15*a*b^4)*cosh(d*x + c)^4
+ (48*a^5 - 68*a^3*b^2 + 15*a*b^4 + 15*(16*a^5 - 20*a^3*b^2 + 5*a*b^4)*cosh(d*x + c)^2 + 20*(4*a^4*b - 5*a^2*
b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 32*(a^4*b - a^2*b^3)*cosh(d*x + c)^3 + 4*(8*a^4*b - 8*a^2*b^3 + 5*(16*a^
5 - 20*a^3*b^2 + 5*a*b^4)*cosh(d*x + c)^3 + 10*(4*a^4*b - 5*a^2*b^3)*cosh(d*x + c)^2 + (48*a^5 - 68*a^3*b^2 +
15*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + (48*a^5 - 68*a^3*b^2 + 15*a*b^4)*cosh(d*x + c)^2 + (48*a^5 - 68*a^3
*b^2 + 15*a*b^4 + 15*(16*a^5 - 20*a^3*b^2 + 5*a*b^4)*cosh(d*x + c)^4 + 40*(4*a^4*b - 5*a^2*b^3)*cosh(d*x + c)^
3 + 6*(48*a^5 - 68*a^3*b^2 + 15*a*b^4)*cosh(d*x + c)^2 + 96*(a^4*b - a^2*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 +
4*(4*a^4*b - 5*a^2*b^3)*cosh(d*x + c) + 2*(3*(16*a^5 - 20*a^3*b^2 + 5*a*b^4)*cosh(d*x + c)^5 + 8*a^4*b - 10*a
^2*b^3 + 10*(4*a^4*b - 5*a^2*b^3)*cosh(d*x + c)^4 + 2*(48*a^5 - 68*a^3*b^2 + 15*a*b^4)*cosh(d*x + c)^3 + 48*(a
^4*b - a^2*b^3)*cosh(d*x + c)^2 + (48*a^5 - 68*a^3*b^2 + 15*a*b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt((a*cosh(
d*x + c) + b)/cosh(d*x + c)))/(a^3*b^4*d*cosh(d*x + c)^6 + a^3*b^4*d*sinh(d*x + c)^6 + 2*a^2*b^5*d*cosh(d*x +
c)^5 + 3*a^3*b^4*d*cosh(d*x + c)^4 + 4*a^2*b^5*d*cosh(d*x + c)^3 + 3*a^3*b^4*d*cosh(d*x + c)^2 + 2*a^2*b^5*d*c
osh(d*x + c) + a^3*b^4*d + 2*(3*a^3*b^4*d*cosh(d*x + c) + a^2*b^5*d)*sinh(d*x + c)^5 + (15*a^3*b^4*d*cosh(d*x
+ c)^2 + 10*a^2*b^5*d*cosh(d*x + c) + 3*a^3*b^4*d)*sinh(d*x + c)^4 + 4*(5*a^3*b^4*d*cosh(d*x + c)^3 + 5*a^2*b^
5*d*cosh(d*x + c)^2 + 3*a^3*b^4*d*cosh(d*x + c) + a^2*b^5*d)*sinh(d*x + c)^3 + (15*a^3*b^4*d*cosh(d*x + c)^4 +
20*a^2*b^5*d*cosh(d*x + c)^3 + 18*a^3*b^4*d*cosh(d*x + c)^2 + 12*a^2*b^5*d*cosh(d*x + c) + 3*a^3*b^4*d)*sinh(
d*x + c)^2 + 2*(3*a^3*b^4*d*cosh(d*x + c)^5 + 5*a^2*b^5*d*cosh(d*x + c)^4 + 6*a^3*b^4*d*cosh(d*x + c)^3 + 6*a^
2*b^5*d*cosh(d*x + c)^2 + 3*a^3*b^4*d*cosh(d*x + c) + a^2*b^5*d)*sinh(d*x + c))]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh ^{5}{\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)**5/(a+b*sech(d*x+c))**(3/2),x)

[Out]

Integral(tanh(c + d*x)**5/(a + b*sech(c + d*x))**(3/2), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh \left (d x + c\right )^{5}}{{\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)^5/(a+b*sech(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate(tanh(d*x + c)^5/(b*sech(d*x + c) + a)^(3/2), x)