3.21 \(\int \text {csch}(c+d x) (a+b \tanh ^2(c+d x))^3 \, dx\)

Optimal. Leaf size=84 \[ -\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {b \left (3 a^2+3 a b+b^2\right ) \text {sech}(c+d x)}{d}+\frac {b^2 (3 a+2 b) \text {sech}^3(c+d x)}{3 d}-\frac {b^3 \text {sech}^5(c+d x)}{5 d} \]

[Out]

-a^3*arctanh(cosh(d*x+c))/d-b*(3*a^2+3*a*b+b^2)*sech(d*x+c)/d+1/3*b^2*(3*a+2*b)*sech(d*x+c)^3/d-1/5*b^3*sech(d
*x+c)^5/d

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Rubi [A]  time = 0.08, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3664, 390, 207} \[ -\frac {b \left (3 a^2+3 a b+b^2\right ) \text {sech}(c+d x)}{d}-\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {b^2 (3 a+2 b) \text {sech}^3(c+d x)}{3 d}-\frac {b^3 \text {sech}^5(c+d x)}{5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 207

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

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 3664

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sec[e + f*x], x]}, Dist[1/(f*ff^m), Subst[Int[((-1 + ff^2*x^2)^((m - 1)/2)*(a - b + b*ff^2*x^2)^p)/x^(
m + 1), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

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

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Mathematica [A]  time = 0.33, size = 79, normalized size = 0.94 \[ \frac {15 a^3 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )-15 b \left (3 a^2+3 a b+b^2\right ) \text {sech}(c+d x)+5 b^2 (3 a+2 b) \text {sech}^3(c+d x)-3 b^3 \text {sech}^5(c+d x)}{15 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(15*a^3*Log[Tanh[(c + d*x)/2]] - 15*b*(3*a^2 + 3*a*b + b^2)*Sech[c + d*x] + 5*b^2*(3*a + 2*b)*Sech[c + d*x]^3
- 3*b^3*Sech[c + d*x]^5)/(15*d)

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fricas [B]  time = 0.79, size = 2277, normalized size = 27.11 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(30*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^9 + 270*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)*sinh(d*x + c
)^8 + 30*(3*a^2*b + 3*a*b^2 + b^3)*sinh(d*x + c)^9 + 40*(9*a^2*b + 6*a*b^2 + b^3)*cosh(d*x + c)^7 + 40*(9*a^2*
b + 6*a*b^2 + b^3 + 27*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 280*(9*(3*a^2*b + 3*a*b^2
+ b^3)*cosh(d*x + c)^3 + (9*a^2*b + 6*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^6 + 4*(135*a^2*b + 75*a*b^2 +
29*b^3)*cosh(d*x + c)^5 + 4*(945*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 + 135*a^2*b + 75*a*b^2 + 29*b^3 + 2
10*(9*a^2*b + 6*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 20*(189*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c
)^5 + 70*(9*a^2*b + 6*a*b^2 + b^3)*cosh(d*x + c)^3 + (135*a^2*b + 75*a*b^2 + 29*b^3)*cosh(d*x + c))*sinh(d*x +
c)^4 + 40*(9*a^2*b + 6*a*b^2 + b^3)*cosh(d*x + c)^3 + 40*(63*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^6 + 35*(
9*a^2*b + 6*a*b^2 + b^3)*cosh(d*x + c)^4 + 9*a^2*b + 6*a*b^2 + b^3 + (135*a^2*b + 75*a*b^2 + 29*b^3)*cosh(d*x
+ c)^2)*sinh(d*x + c)^3 + 40*(27*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^7 + 21*(9*a^2*b + 6*a*b^2 + b^3)*cosh
(d*x + c)^5 + (135*a^2*b + 75*a*b^2 + 29*b^3)*cosh(d*x + c)^3 + 3*(9*a^2*b + 6*a*b^2 + b^3)*cosh(d*x + c))*sin
h(d*x + c)^2 + 30*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c) + 15*(a^3*cosh(d*x + c)^10 + 10*a^3*cosh(d*x + c)*si
nh(d*x + c)^9 + a^3*sinh(d*x + c)^10 + 5*a^3*cosh(d*x + c)^8 + 10*a^3*cosh(d*x + c)^6 + 5*(9*a^3*cosh(d*x + c)
^2 + a^3)*sinh(d*x + c)^8 + 40*(3*a^3*cosh(d*x + c)^3 + a^3*cosh(d*x + c))*sinh(d*x + c)^7 + 10*a^3*cosh(d*x +
c)^4 + 10*(21*a^3*cosh(d*x + c)^4 + 14*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^6 + 4*(63*a^3*cosh(d*x + c)^5
+ 70*a^3*cosh(d*x + c)^3 + 15*a^3*cosh(d*x + c))*sinh(d*x + c)^5 + 5*a^3*cosh(d*x + c)^2 + 10*(21*a^3*cosh(d*
x + c)^6 + 35*a^3*cosh(d*x + c)^4 + 15*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^4 + 40*(3*a^3*cosh(d*x + c)^7
+ 7*a^3*cosh(d*x + c)^5 + 5*a^3*cosh(d*x + c)^3 + a^3*cosh(d*x + c))*sinh(d*x + c)^3 + a^3 + 5*(9*a^3*cosh(d*x
+ c)^8 + 28*a^3*cosh(d*x + c)^6 + 30*a^3*cosh(d*x + c)^4 + 12*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^2 + 10
*(a^3*cosh(d*x + c)^9 + 4*a^3*cosh(d*x + c)^7 + 6*a^3*cosh(d*x + c)^5 + 4*a^3*cosh(d*x + c)^3 + a^3*cosh(d*x +
c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) - 15*(a^3*cosh(d*x + c)^10 + 10*a^3*cosh(d*x + c)*s
inh(d*x + c)^9 + a^3*sinh(d*x + c)^10 + 5*a^3*cosh(d*x + c)^8 + 10*a^3*cosh(d*x + c)^6 + 5*(9*a^3*cosh(d*x + c
)^2 + a^3)*sinh(d*x + c)^8 + 40*(3*a^3*cosh(d*x + c)^3 + a^3*cosh(d*x + c))*sinh(d*x + c)^7 + 10*a^3*cosh(d*x
+ c)^4 + 10*(21*a^3*cosh(d*x + c)^4 + 14*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^6 + 4*(63*a^3*cosh(d*x + c)^
5 + 70*a^3*cosh(d*x + c)^3 + 15*a^3*cosh(d*x + c))*sinh(d*x + c)^5 + 5*a^3*cosh(d*x + c)^2 + 10*(21*a^3*cosh(d
*x + c)^6 + 35*a^3*cosh(d*x + c)^4 + 15*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^4 + 40*(3*a^3*cosh(d*x + c)^7
+ 7*a^3*cosh(d*x + c)^5 + 5*a^3*cosh(d*x + c)^3 + a^3*cosh(d*x + c))*sinh(d*x + c)^3 + a^3 + 5*(9*a^3*cosh(d*
x + c)^8 + 28*a^3*cosh(d*x + c)^6 + 30*a^3*cosh(d*x + c)^4 + 12*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^2 + 1
0*(a^3*cosh(d*x + c)^9 + 4*a^3*cosh(d*x + c)^7 + 6*a^3*cosh(d*x + c)^5 + 4*a^3*cosh(d*x + c)^3 + a^3*cosh(d*x
+ c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 10*(27*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^8
+ 28*(9*a^2*b + 6*a*b^2 + b^3)*cosh(d*x + c)^6 + 2*(135*a^2*b + 75*a*b^2 + 29*b^3)*cosh(d*x + c)^4 + 9*a^2*b
+ 9*a*b^2 + 3*b^3 + 12*(9*a^2*b + 6*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/(d*cosh(d*x + c)^10 + 10*d*co
sh(d*x + c)*sinh(d*x + c)^9 + d*sinh(d*x + c)^10 + 5*d*cosh(d*x + c)^8 + 5*(9*d*cosh(d*x + c)^2 + d)*sinh(d*x
+ c)^8 + 40*(3*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c)^7 + 10*d*cosh(d*x + c)^6 + 10*(21*d*cosh(d*x
+ c)^4 + 14*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^6 + 4*(63*d*cosh(d*x + c)^5 + 70*d*cosh(d*x + c)^3 + 15*d*co
sh(d*x + c))*sinh(d*x + c)^5 + 10*d*cosh(d*x + c)^4 + 10*(21*d*cosh(d*x + c)^6 + 35*d*cosh(d*x + c)^4 + 15*d*c
osh(d*x + c)^2 + d)*sinh(d*x + c)^4 + 40*(3*d*cosh(d*x + c)^7 + 7*d*cosh(d*x + c)^5 + 5*d*cosh(d*x + c)^3 + d*
cosh(d*x + c))*sinh(d*x + c)^3 + 5*d*cosh(d*x + c)^2 + 5*(9*d*cosh(d*x + c)^8 + 28*d*cosh(d*x + c)^6 + 30*d*co
sh(d*x + c)^4 + 12*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^2 + 10*(d*cosh(d*x + c)^9 + 4*d*cosh(d*x + c)^7 + 6*d*
cosh(d*x + c)^5 + 4*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c) + d)

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giac [B]  time = 0.37, size = 262, normalized size = 3.12 \[ -\frac {15 \, a^{3} \log \left (e^{\left (d x + c\right )} + 1\right ) - 15 \, a^{3} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) + \frac {2 \, {\left (45 \, a^{2} b e^{\left (9 \, d x + 9 \, c\right )} + 45 \, a b^{2} e^{\left (9 \, d x + 9 \, c\right )} + 15 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} + 180 \, a^{2} b e^{\left (7 \, d x + 7 \, c\right )} + 120 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} + 20 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} + 270 \, a^{2} b e^{\left (5 \, d x + 5 \, c\right )} + 150 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 58 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 180 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} + 120 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 20 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 45 \, a^{2} b e^{\left (d x + c\right )} + 45 \, a b^{2} e^{\left (d x + c\right )} + 15 \, b^{3} e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}}{15 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(15*a^3*log(e^(d*x + c) + 1) - 15*a^3*log(abs(e^(d*x + c) - 1)) + 2*(45*a^2*b*e^(9*d*x + 9*c) + 45*a*b^2
*e^(9*d*x + 9*c) + 15*b^3*e^(9*d*x + 9*c) + 180*a^2*b*e^(7*d*x + 7*c) + 120*a*b^2*e^(7*d*x + 7*c) + 20*b^3*e^(
7*d*x + 7*c) + 270*a^2*b*e^(5*d*x + 5*c) + 150*a*b^2*e^(5*d*x + 5*c) + 58*b^3*e^(5*d*x + 5*c) + 180*a^2*b*e^(3
*d*x + 3*c) + 120*a*b^2*e^(3*d*x + 3*c) + 20*b^3*e^(3*d*x + 3*c) + 45*a^2*b*e^(d*x + c) + 45*a*b^2*e^(d*x + c)
+ 15*b^3*e^(d*x + c))/(e^(2*d*x + 2*c) + 1)^5)/d

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maple [A]  time = 0.24, size = 118, normalized size = 1.40 \[ \frac {-2 a^{3} \arctanh \left ({\mathrm e}^{d x +c}\right )-\frac {3 a^{2} b}{\cosh \left (d x +c \right )}+3 a \,b^{2} \left (-\frac {\sinh ^{2}\left (d x +c \right )}{\cosh \left (d x +c \right )^{3}}-\frac {2}{3 \cosh \left (d x +c \right )^{3}}\right )+b^{3} \left (-\frac {\sinh ^{4}\left (d x +c \right )}{\cosh \left (d x +c \right )^{5}}-\frac {4 \left (\sinh ^{2}\left (d x +c \right )\right )}{3 \cosh \left (d x +c \right )^{5}}-\frac {8}{15 \cosh \left (d x +c \right )^{5}}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)*(a+b*tanh(d*x+c)^2)^3,x)

[Out]

1/d*(-2*a^3*arctanh(exp(d*x+c))-3*a^2*b/cosh(d*x+c)+3*a*b^2*(-sinh(d*x+c)^2/cosh(d*x+c)^3-2/3/cosh(d*x+c)^3)+b
^3*(-sinh(d*x+c)^4/cosh(d*x+c)^5-4/3*sinh(d*x+c)^2/cosh(d*x+c)^5-8/15/cosh(d*x+c)^5))

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maxima [B]  time = 0.33, size = 560, normalized size = 6.67 \[ -\frac {2}{15} \, b^{3} {\left (\frac {15 \, e^{\left (-d x - c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {20 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {58 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {20 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {15 \, e^{\left (-9 \, d x - 9 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} - 2 \, a b^{2} {\left (\frac {3 \, e^{\left (-d x - c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac {2 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac {3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + \frac {a^{3} \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} - \frac {6 \, a^{2} b}{d {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*b^3*(15*e^(-d*x - c)/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x -
8*c) + e^(-10*d*x - 10*c) + 1)) + 20*e^(-3*d*x - 3*c)/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*
d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1)) + 58*e^(-5*d*x - 5*c)/(d*(5*e^(-2*d*x - 2*c) + 10*e
^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1)) + 20*e^(-7*d*x - 7*c)/(d
*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1
)) + 15*e^(-9*d*x - 9*c)/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*
c) + e^(-10*d*x - 10*c) + 1))) - 2*a*b^2*(3*e^(-d*x - c)/(d*(3*e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + e^(-6*d
*x - 6*c) + 1)) + 2*e^(-3*d*x - 3*c)/(d*(3*e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) + 1)) + 3*
e^(-5*d*x - 5*c)/(d*(3*e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) + 1))) + a^3*log(tanh(1/2*d*x
+ 1/2*c))/d - 6*a^2*b/(d*(e^(d*x + c) + e^(-d*x - c)))

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mupad [B]  time = 0.25, size = 317, normalized size = 3.77 \[ \frac {8\,{\mathrm {e}}^{c+d\,x}\,\left (2\,b^3+3\,a\,b^2\right )}{3\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {2\,\mathrm {atan}\left (\frac {a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^6}}\right )\,\sqrt {a^6}}{\sqrt {-d^2}}-\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (3\,a^2\,b+3\,a\,b^2+b^3\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {64\,b^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {8\,{\mathrm {e}}^{c+d\,x}\,\left (22\,b^3+15\,a\,b^2\right )}{15\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {32\,b^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8*exp(c + d*x)*(3*a*b^2 + 2*b^3))/(3*d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) - (2*atan((a^3*exp(d*x)*e
xp(c)*(-d^2)^(1/2))/(d*(a^6)^(1/2)))*(a^6)^(1/2))/(-d^2)^(1/2) - (2*exp(c + d*x)*(3*a*b^2 + 3*a^2*b + b^3))/(d
*(exp(2*c + 2*d*x) + 1)) + (64*b^3*exp(c + d*x))/(5*d*(4*exp(2*c + 2*d*x) + 6*exp(4*c + 4*d*x) + 4*exp(6*c + 6
*d*x) + exp(8*c + 8*d*x) + 1)) - (8*exp(c + d*x)*(15*a*b^2 + 22*b^3))/(15*d*(3*exp(2*c + 2*d*x) + 3*exp(4*c +
4*d*x) + exp(6*c + 6*d*x) + 1)) - (32*b^3*exp(c + d*x))/(5*d*(5*exp(2*c + 2*d*x) + 10*exp(4*c + 4*d*x) + 10*ex
p(6*c + 6*d*x) + 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) + 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3} \operatorname {csch}{\left (c + d x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)*(a+b*tanh(d*x+c)**2)**3,x)

[Out]

Integral((a + b*tanh(c + d*x)**2)**3*csch(c + d*x), x)

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