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

Optimal. Leaf size=98 \[ -\frac {a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {a b \tan ^{-1}(\sinh (c+d x))}{d}-\frac {a b \tanh (c+d x) \text {sech}(c+d x)}{d}-\frac {b^2 \text {sech}^5(c+d x)}{5 d}+\frac {2 b^2 \text {sech}^3(c+d x)}{3 d}-\frac {b^2 \text {sech}(c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.14, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3666, 3770, 2611, 2606, 194} \[ -\frac {a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {a b \tan ^{-1}(\sinh (c+d x))}{d}-\frac {a b \tanh (c+d x) \text {sech}(c+d x)}{d}-\frac {b^2 \text {sech}^5(c+d x)}{5 d}+\frac {2 b^2 \text {sech}^3(c+d x)}{3 d}-\frac {b^2 \text {sech}(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 3666

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> Int[ExpandTrig[(d*sin[e + f*x])^m*(a + b*(c*tan[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n},
x] && IGtQ[p, 0]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

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Mathematica [A]  time = 0.17, size = 106, normalized size = 1.08 \[ \frac {a^2 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {2 a b \tan ^{-1}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{d}-\frac {a b \tanh (c+d x) \text {sech}(c+d x)}{d}-\frac {b^2 \text {sech}^5(c+d x)}{5 d}+\frac {2 b^2 \text {sech}^3(c+d x)}{3 d}-\frac {b^2 \text {sech}(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.77, size = 2498, normalized size = 25.49 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(30*(a*b + b^2)*cosh(d*x + c)^9 + 270*(a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^8 + 30*(a*b + b^2)*sinh(d*
x + c)^9 + 20*(3*a*b + 2*b^2)*cosh(d*x + c)^7 + 20*(54*(a*b + b^2)*cosh(d*x + c)^2 + 3*a*b + 2*b^2)*sinh(d*x +
c)^7 + 116*b^2*cosh(d*x + c)^5 + 140*(18*(a*b + b^2)*cosh(d*x + c)^3 + (3*a*b + 2*b^2)*cosh(d*x + c))*sinh(d*
x + c)^6 + 4*(945*(a*b + b^2)*cosh(d*x + c)^4 + 105*(3*a*b + 2*b^2)*cosh(d*x + c)^2 + 29*b^2)*sinh(d*x + c)^5
+ 20*(189*(a*b + b^2)*cosh(d*x + c)^5 + 35*(3*a*b + 2*b^2)*cosh(d*x + c)^3 + 29*b^2*cosh(d*x + c))*sinh(d*x +
c)^4 - 20*(3*a*b - 2*b^2)*cosh(d*x + c)^3 + 20*(126*(a*b + b^2)*cosh(d*x + c)^6 + 35*(3*a*b + 2*b^2)*cosh(d*x
+ c)^4 + 58*b^2*cosh(d*x + c)^2 - 3*a*b + 2*b^2)*sinh(d*x + c)^3 + 20*(54*(a*b + b^2)*cosh(d*x + c)^7 + 21*(3*
a*b + 2*b^2)*cosh(d*x + c)^5 + 58*b^2*cosh(d*x + c)^3 - 3*(3*a*b - 2*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 - 30*
(a*b*cosh(d*x + c)^10 + 10*a*b*cosh(d*x + c)*sinh(d*x + c)^9 + a*b*sinh(d*x + c)^10 + 5*a*b*cosh(d*x + c)^8 +
5*(9*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c)^8 + 10*a*b*cosh(d*x + c)^6 + 40*(3*a*b*cosh(d*x + c)^3 + a*b*cos
h(d*x + c))*sinh(d*x + c)^7 + 10*(21*a*b*cosh(d*x + c)^4 + 14*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c)^6 + 10*
a*b*cosh(d*x + c)^4 + 4*(63*a*b*cosh(d*x + c)^5 + 70*a*b*cosh(d*x + c)^3 + 15*a*b*cosh(d*x + c))*sinh(d*x + c)
^5 + 10*(21*a*b*cosh(d*x + c)^6 + 35*a*b*cosh(d*x + c)^4 + 15*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c)^4 + 5*a
*b*cosh(d*x + c)^2 + 40*(3*a*b*cosh(d*x + c)^7 + 7*a*b*cosh(d*x + c)^5 + 5*a*b*cosh(d*x + c)^3 + a*b*cosh(d*x
+ c))*sinh(d*x + c)^3 + 5*(9*a*b*cosh(d*x + c)^8 + 28*a*b*cosh(d*x + c)^6 + 30*a*b*cosh(d*x + c)^4 + 12*a*b*co
sh(d*x + c)^2 + a*b)*sinh(d*x + c)^2 + a*b + 10*(a*b*cosh(d*x + c)^9 + 4*a*b*cosh(d*x + c)^7 + 6*a*b*cosh(d*x
+ c)^5 + 4*a*b*cosh(d*x + c)^3 + a*b*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) - 30*
(a*b - b^2)*cosh(d*x + c) + 15*(a^2*cosh(d*x + c)^10 + 10*a^2*cosh(d*x + c)*sinh(d*x + c)^9 + a^2*sinh(d*x + c
)^10 + 5*a^2*cosh(d*x + c)^8 + 5*(9*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*x + c)^8 + 10*a^2*cosh(d*x + c)^6 + 40*(
3*a^2*cosh(d*x + c)^3 + a^2*cosh(d*x + c))*sinh(d*x + c)^7 + 10*(21*a^2*cosh(d*x + c)^4 + 14*a^2*cosh(d*x + c)
^2 + a^2)*sinh(d*x + c)^6 + 10*a^2*cosh(d*x + c)^4 + 4*(63*a^2*cosh(d*x + c)^5 + 70*a^2*cosh(d*x + c)^3 + 15*a
^2*cosh(d*x + c))*sinh(d*x + c)^5 + 10*(21*a^2*cosh(d*x + c)^6 + 35*a^2*cosh(d*x + c)^4 + 15*a^2*cosh(d*x + c)
^2 + a^2)*sinh(d*x + c)^4 + 5*a^2*cosh(d*x + c)^2 + 40*(3*a^2*cosh(d*x + c)^7 + 7*a^2*cosh(d*x + c)^5 + 5*a^2*
cosh(d*x + c)^3 + a^2*cosh(d*x + c))*sinh(d*x + c)^3 + 5*(9*a^2*cosh(d*x + c)^8 + 28*a^2*cosh(d*x + c)^6 + 30*
a^2*cosh(d*x + c)^4 + 12*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*x + c)^2 + a^2 + 10*(a^2*cosh(d*x + c)^9 + 4*a^2*co
sh(d*x + c)^7 + 6*a^2*cosh(d*x + c)^5 + 4*a^2*cosh(d*x + c)^3 + a^2*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x
+ c) + sinh(d*x + c) + 1) - 15*(a^2*cosh(d*x + c)^10 + 10*a^2*cosh(d*x + c)*sinh(d*x + c)^9 + a^2*sinh(d*x +
c)^10 + 5*a^2*cosh(d*x + c)^8 + 5*(9*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*x + c)^8 + 10*a^2*cosh(d*x + c)^6 + 40*
(3*a^2*cosh(d*x + c)^3 + a^2*cosh(d*x + c))*sinh(d*x + c)^7 + 10*(21*a^2*cosh(d*x + c)^4 + 14*a^2*cosh(d*x + c
)^2 + a^2)*sinh(d*x + c)^6 + 10*a^2*cosh(d*x + c)^4 + 4*(63*a^2*cosh(d*x + c)^5 + 70*a^2*cosh(d*x + c)^3 + 15*
a^2*cosh(d*x + c))*sinh(d*x + c)^5 + 10*(21*a^2*cosh(d*x + c)^6 + 35*a^2*cosh(d*x + c)^4 + 15*a^2*cosh(d*x + c
)^2 + a^2)*sinh(d*x + c)^4 + 5*a^2*cosh(d*x + c)^2 + 40*(3*a^2*cosh(d*x + c)^7 + 7*a^2*cosh(d*x + c)^5 + 5*a^2
*cosh(d*x + c)^3 + a^2*cosh(d*x + c))*sinh(d*x + c)^3 + 5*(9*a^2*cosh(d*x + c)^8 + 28*a^2*cosh(d*x + c)^6 + 30
*a^2*cosh(d*x + c)^4 + 12*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*x + c)^2 + a^2 + 10*(a^2*cosh(d*x + c)^9 + 4*a^2*c
osh(d*x + c)^7 + 6*a^2*cosh(d*x + c)^5 + 4*a^2*cosh(d*x + c)^3 + a^2*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*
x + c) + sinh(d*x + c) - 1) + 10*(27*(a*b + b^2)*cosh(d*x + c)^8 + 14*(3*a*b + 2*b^2)*cosh(d*x + c)^6 + 58*b^2
*cosh(d*x + c)^4 - 6*(3*a*b - 2*b^2)*cosh(d*x + c)^2 - 3*a*b + 3*b^2)*sinh(d*x + c))/(d*cosh(d*x + c)^10 + 10*
d*cosh(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*cosh(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*cosh(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*cosh(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 [A]  time = 0.29, size = 178, normalized size = 1.82 \[ \frac {30 \, a b \arctan \left (e^{\left (d x + c\right )}\right ) - 15 \, a^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) + 15 \, a^{2} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac {2 \, {\left (15 \, a b e^{\left (9 \, d x + 9 \, c\right )} + 15 \, b^{2} e^{\left (9 \, d x + 9 \, c\right )} + 30 \, a b e^{\left (7 \, d x + 7 \, c\right )} + 20 \, b^{2} e^{\left (7 \, d x + 7 \, c\right )} + 58 \, b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 30 \, a b e^{\left (3 \, d x + 3 \, c\right )} + 20 \, b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 15 \, a b e^{\left (d x + c\right )} + 15 \, b^{2} e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}}{15 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(30*a*b*arctan(e^(d*x + c)) - 15*a^2*log(e^(d*x + c) + 1) + 15*a^2*log(abs(e^(d*x + c) - 1)) - 2*(15*a*b*
e^(9*d*x + 9*c) + 15*b^2*e^(9*d*x + 9*c) + 30*a*b*e^(7*d*x + 7*c) + 20*b^2*e^(7*d*x + 7*c) + 58*b^2*e^(5*d*x +
5*c) - 30*a*b*e^(3*d*x + 3*c) + 20*b^2*e^(3*d*x + 3*c) - 15*a*b*e^(d*x + c) + 15*b^2*e^(d*x + c))/(e^(2*d*x +
2*c) + 1)^5)/d

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maple [A]  time = 0.39, size = 134, normalized size = 1.37 \[ -\frac {2 a^{2} \arctanh \left ({\mathrm e}^{d x +c}\right )}{d}-\frac {2 a b \sinh \left (d x +c \right )}{d \cosh \left (d x +c \right )^{2}}+\frac {a b \,\mathrm {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{d}+\frac {2 a b \arctan \left ({\mathrm e}^{d x +c}\right )}{d}-\frac {b^{2} \left (\sinh ^{4}\left (d x +c \right )\right )}{d \cosh \left (d x +c \right )^{5}}-\frac {4 b^{2} \left (\sinh ^{2}\left (d x +c \right )\right )}{3 d \cosh \left (d x +c \right )^{5}}-\frac {8 b^{2}}{15 d \cosh \left (d x +c \right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.41, size = 447, normalized size = 4.56 \[ -2 \, a b {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} - \frac {2}{15} \, b^{2} {\left (\frac {15 \, e^{\left (-d x - c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {20 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {58 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {20 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {15 \, e^{\left (-9 \, d x - 9 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} + \frac {a^{2} \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*b*(arctan(e^(-d*x - c))/d + (e^(-d*x - c) - e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c) +
1))) - 2/15*b^2*(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))) + a^2*log(tanh(1/2*d*x + 1/2*c))/d

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mupad [B]  time = 3.28, size = 522, normalized size = 5.33 \[ \frac {a^2\,\ln \left (32\,a^6+32\,a^4\,b^2-32\,a^6\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-32\,a^4\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{d}-\frac {176\,b^2\,{\mathrm {e}}^{c+d\,x}}{15\,\left (d+3\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,d\,{\mathrm {e}}^{4\,c+4\,d\,x}+d\,{\mathrm {e}}^{6\,c+6\,d\,x}\right )}-\frac {32\,b^2\,{\mathrm {e}}^{c+d\,x}}{5\,\left (d+5\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,d\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,d\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,d\,{\mathrm {e}}^{8\,c+8\,d\,x}+d\,{\mathrm {e}}^{10\,c+10\,d\,x}\right )}-\frac {a^2\,\ln \left (-32\,a^6-32\,a^4\,b^2-32\,a^6\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-32\,a^4\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{d}-\frac {2\,b^2\,{\mathrm {e}}^{c+d\,x}}{d+d\,{\mathrm {e}}^{2\,c+2\,d\,x}}+\frac {16\,b^2\,{\mathrm {e}}^{c+d\,x}}{3\,\left (d+2\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}+d\,{\mathrm {e}}^{4\,c+4\,d\,x}\right )}+\frac {64\,b^2\,{\mathrm {e}}^{c+d\,x}}{5\,\left (d+4\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,d\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,d\,{\mathrm {e}}^{6\,c+6\,d\,x}+d\,{\mathrm {e}}^{8\,c+8\,d\,x}\right )}-\frac {2\,a\,b\,{\mathrm {e}}^{c+d\,x}}{d+d\,{\mathrm {e}}^{2\,c+2\,d\,x}}+\frac {4\,a\,b\,{\mathrm {e}}^{c+d\,x}}{d+2\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}+d\,{\mathrm {e}}^{4\,c+4\,d\,x}}-\frac {a\,b\,\left (\ln \left (32\,a^3\,b^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+32\,a^5\,b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-a^5\,b\,32{}\mathrm {i}-a^3\,b^3\,32{}\mathrm {i}\right )\,1{}\mathrm {i}-\ln \left (32\,a^3\,b^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+32\,a^5\,b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+a^5\,b\,32{}\mathrm {i}+a^3\,b^3\,32{}\mathrm {i}\right )\,1{}\mathrm {i}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^2*log(32*a^6 + 32*a^4*b^2 - 32*a^6*exp(d*x)*exp(c) - 32*a^4*b^2*exp(d*x)*exp(c)))/d - (176*b^2*exp(c + d*x)
)/(15*(d + 3*d*exp(2*c + 2*d*x) + 3*d*exp(4*c + 4*d*x) + d*exp(6*c + 6*d*x))) - (32*b^2*exp(c + d*x))/(5*(d +
5*d*exp(2*c + 2*d*x) + 10*d*exp(4*c + 4*d*x) + 10*d*exp(6*c + 6*d*x) + 5*d*exp(8*c + 8*d*x) + d*exp(10*c + 10*
d*x))) - (a^2*log(- 32*a^6 - 32*a^4*b^2 - 32*a^6*exp(d*x)*exp(c) - 32*a^4*b^2*exp(d*x)*exp(c)))/d - (2*b^2*exp
(c + d*x))/(d + d*exp(2*c + 2*d*x)) + (16*b^2*exp(c + d*x))/(3*(d + 2*d*exp(2*c + 2*d*x) + d*exp(4*c + 4*d*x))
) + (64*b^2*exp(c + d*x))/(5*(d + 4*d*exp(2*c + 2*d*x) + 6*d*exp(4*c + 4*d*x) + 4*d*exp(6*c + 6*d*x) + d*exp(8
*c + 8*d*x))) - (2*a*b*exp(c + d*x))/(d + d*exp(2*c + 2*d*x)) + (4*a*b*exp(c + d*x))/(d + 2*d*exp(2*c + 2*d*x)
+ d*exp(4*c + 4*d*x)) - (a*b*(log(32*a^3*b^3*exp(d*x)*exp(c) - a^3*b^3*32i - a^5*b*32i + 32*a^5*b*exp(d*x)*ex
p(c))*1i - log(a^5*b*32i + a^3*b^3*32i + 32*a^3*b^3*exp(d*x)*exp(c) + 32*a^5*b*exp(d*x)*exp(c))*1i))/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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