### 3.64 $$\int \text{csch}^4(c+d x) (a+b \tanh ^3(c+d x))^2 \, dx$$

Optimal. Leaf size=97 $-\frac{a^2 \coth ^3(c+d x)}{3 d}+\frac{a^2 \coth (c+d x)}{d}-\frac{a b \tanh ^2(c+d x)}{d}+\frac{2 a b \log (\tanh (c+d x))}{d}-\frac{b^2 \tanh ^5(c+d x)}{5 d}+\frac{b^2 \tanh ^3(c+d x)}{3 d}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0980367, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.087, Rules used = {3663, 1802} $-\frac{a^2 \coth ^3(c+d x)}{3 d}+\frac{a^2 \coth (c+d x)}{d}-\frac{a b \tanh ^2(c+d x)}{d}+\frac{2 a b \log (\tanh (c+d x))}{d}-\frac{b^2 \tanh ^5(c+d x)}{5 d}+\frac{b^2 \tanh ^3(c+d x)}{3 d}$

Antiderivative was successfully veriﬁed.

[In]

Int[Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^3)^2,x]

[Out]

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

Rule 3663

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

Rule 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

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

Mathematica [A]  time = 0.185879, size = 147, normalized size = 1.52 $\frac{2 a^2 \coth (c+d x)}{3 d}-\frac{a^2 \coth (c+d x) \text{csch}^2(c+d x)}{3 d}+\frac{a b \text{sech}^2(c+d x)}{d}+\frac{2 a b \log (\sinh (c+d x))}{d}-\frac{2 a b \log (\cosh (c+d x))}{d}+\frac{2 b^2 \tanh (c+d x)}{15 d}-\frac{b^2 \tanh (c+d x) \text{sech}^4(c+d x)}{5 d}+\frac{b^2 \tanh (c+d x) \text{sech}^2(c+d x)}{15 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*a^2*Coth[c + d*x])/(3*d) - (a^2*Coth[c + d*x]*Csch[c + d*x]^2)/(3*d) - (2*a*b*Log[Cosh[c + d*x]])/d + (2*a*
b*Log[Sinh[c + d*x]])/d + (a*b*Sech[c + d*x]^2)/d + (2*b^2*Tanh[c + d*x])/(15*d) + (b^2*Sech[c + d*x]^2*Tanh[c
+ d*x])/(15*d) - (b^2*Sech[c + d*x]^4*Tanh[c + d*x])/(5*d)

________________________________________________________________________________________

Maple [A]  time = 0.083, size = 146, normalized size = 1.5 \begin{align*}{\frac{2\,{a}^{2}{\rm coth} \left (dx+c\right )}{3\,d}}-{\frac{{a}^{2}{\rm coth} \left (dx+c\right ) \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{3\,d}}+{\frac{ab}{d \left ( \cosh \left ( dx+c \right ) \right ) ^{2}}}+2\,{\frac{ab\ln \left ( \tanh \left ( dx+c \right ) \right ) }{d}}-{\frac{{b}^{2}\sinh \left ( dx+c \right ) }{4\,d \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}+{\frac{2\,{b}^{2}\tanh \left ( dx+c \right ) }{15\,d}}+{\frac{{b}^{2}\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{4}}{20\,d}}+{\frac{{b}^{2}\tanh \left ( dx+c \right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{15\,d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^4*(a+b*tanh(d*x+c)^3)^2,x)

[Out]

2/3*a^2*coth(d*x+c)/d-1/3/d*a^2*coth(d*x+c)*csch(d*x+c)^2+1/d*a*b/cosh(d*x+c)^2+2*a*b*ln(tanh(d*x+c))/d-1/4/d*
b^2*sinh(d*x+c)/cosh(d*x+c)^5+2/15*b^2*tanh(d*x+c)/d+1/20/d*b^2*tanh(d*x+c)*sech(d*x+c)^4+1/15/d*b^2*tanh(d*x+
c)*sech(d*x+c)^2

________________________________________________________________________________________

Maxima [B]  time = 1.58208, size = 632, normalized size = 6.52 \begin{align*} 2 \, a b{\left (\frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac{\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} + \frac{2 \, e^{\left (-2 \, d x - 2 \, 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{4}{15} \, b^{2}{\left (\frac{5 \, e^{\left (-2 \, d x - 2 \, 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{5 \, e^{\left (-4 \, d x - 4 \, 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 (-6 \, d x - 6 \, 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{1}{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{4}{3} \, a^{2}{\left (\frac{3 \, e^{\left (-2 \, d x - 2 \, 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{1}{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 )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^4*(a+b*tanh(d*x+c)^3)^2,x, algorithm="maxima")

[Out]

2*a*b*(log(e^(-d*x - c) + 1)/d + log(e^(-d*x - c) - 1)/d - log(e^(-2*d*x - 2*c) + 1)/d + 2*e^(-2*d*x - 2*c)/(d
*(2*e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c) + 1))) + 4/15*b^2*(5*e^(-2*d*x - 2*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)) - 5*e^(-4*d*x - 4*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^(-6*d*x - 6*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)) + 1/(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))) + 4/3*a^2*(3*e^(-2*d*x - 2*c)/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c)
+ e^(-6*d*x - 6*c) - 1)) - 1/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1)))

________________________________________________________________________________________

Fricas [B]  time = 2.81461, size = 11420, normalized size = 117.73 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^4*(a+b*tanh(d*x+c)^3)^2,x, algorithm="fricas")

[Out]

2/15*(30*a*b*cosh(d*x + c)^14 + 420*a*b*cosh(d*x + c)*sinh(d*x + c)^13 + 30*a*b*sinh(d*x + c)^14 - 30*(a^2 + b
^2)*cosh(d*x + c)^12 + 30*(91*a*b*cosh(d*x + c)^2 - a^2 - b^2)*sinh(d*x + c)^12 + 120*(91*a*b*cosh(d*x + c)^3
- 3*(a^2 + b^2)*cosh(d*x + c))*sinh(d*x + c)^11 - 10*(14*a^2 + 9*a*b - 10*b^2)*cosh(d*x + c)^10 + 10*(3003*a*b
*cosh(d*x + c)^4 - 198*(a^2 + b^2)*cosh(d*x + c)^2 - 14*a^2 - 9*a*b + 10*b^2)*sinh(d*x + c)^10 + 20*(3003*a*b*
cosh(d*x + c)^5 - 330*(a^2 + b^2)*cosh(d*x + c)^3 - 5*(14*a^2 + 9*a*b - 10*b^2)*cosh(d*x + c))*sinh(d*x + c)^9
- 10*(25*a^2 + 13*b^2)*cosh(d*x + c)^8 + 10*(9009*a*b*cosh(d*x + c)^6 - 1485*(a^2 + b^2)*cosh(d*x + c)^4 - 45
*(14*a^2 + 9*a*b - 10*b^2)*cosh(d*x + c)^2 - 25*a^2 - 13*b^2)*sinh(d*x + c)^8 + 80*(1287*a*b*cosh(d*x + c)^7 -
297*(a^2 + b^2)*cosh(d*x + c)^5 - 15*(14*a^2 + 9*a*b - 10*b^2)*cosh(d*x + c)^3 - (25*a^2 + 13*b^2)*cosh(d*x +
c))*sinh(d*x + c)^7 - 2*(100*a^2 - 45*a*b - 44*b^2)*cosh(d*x + c)^6 + 2*(45045*a*b*cosh(d*x + c)^8 - 13860*(a
^2 + b^2)*cosh(d*x + c)^6 - 1050*(14*a^2 + 9*a*b - 10*b^2)*cosh(d*x + c)^4 - 140*(25*a^2 + 13*b^2)*cosh(d*x +
c)^2 - 100*a^2 + 45*a*b + 44*b^2)*sinh(d*x + c)^6 + 4*(15015*a*b*cosh(d*x + c)^9 - 5940*(a^2 + b^2)*cosh(d*x +
c)^7 - 630*(14*a^2 + 9*a*b - 10*b^2)*cosh(d*x + c)^5 - 140*(25*a^2 + 13*b^2)*cosh(d*x + c)^3 - 3*(100*a^2 - 4
5*a*b - 44*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(25*a^2 + 17*b^2)*cosh(d*x + c)^4 + 2*(15015*a*b*cosh(d*x +
c)^10 - 7425*(a^2 + b^2)*cosh(d*x + c)^8 - 1050*(14*a^2 + 9*a*b - 10*b^2)*cosh(d*x + c)^6 - 350*(25*a^2 + 13*
b^2)*cosh(d*x + c)^4 - 15*(100*a^2 - 45*a*b - 44*b^2)*cosh(d*x + c)^2 - 25*a^2 - 17*b^2)*sinh(d*x + c)^4 + 8*(
1365*a*b*cosh(d*x + c)^11 - 825*(a^2 + b^2)*cosh(d*x + c)^9 - 150*(14*a^2 + 9*a*b - 10*b^2)*cosh(d*x + c)^7 -
70*(25*a^2 + 13*b^2)*cosh(d*x + c)^5 - 5*(100*a^2 - 45*a*b - 44*b^2)*cosh(d*x + c)^3 - (25*a^2 + 17*b^2)*cosh(
d*x + c))*sinh(d*x + c)^3 + 2*(10*a^2 - 15*a*b + 2*b^2)*cosh(d*x + c)^2 + 2*(1365*a*b*cosh(d*x + c)^12 - 990*(
a^2 + b^2)*cosh(d*x + c)^10 - 225*(14*a^2 + 9*a*b - 10*b^2)*cosh(d*x + c)^8 - 140*(25*a^2 + 13*b^2)*cosh(d*x +
c)^6 - 15*(100*a^2 - 45*a*b - 44*b^2)*cosh(d*x + c)^4 - 6*(25*a^2 + 17*b^2)*cosh(d*x + c)^2 + 10*a^2 - 15*a*b
+ 2*b^2)*sinh(d*x + c)^2 + 10*a^2 + 2*b^2 - 15*(a*b*cosh(d*x + c)^16 + 16*a*b*cosh(d*x + c)*sinh(d*x + c)^15
+ a*b*sinh(d*x + c)^16 + 2*a*b*cosh(d*x + c)^14 + 2*(60*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c)^14 - 2*a*b*co
sh(d*x + c)^12 + 28*(20*a*b*cosh(d*x + c)^3 + a*b*cosh(d*x + c))*sinh(d*x + c)^13 + 2*(910*a*b*cosh(d*x + c)^4
+ 91*a*b*cosh(d*x + c)^2 - a*b)*sinh(d*x + c)^12 - 6*a*b*cosh(d*x + c)^10 + 8*(546*a*b*cosh(d*x + c)^5 + 91*a
*b*cosh(d*x + c)^3 - 3*a*b*cosh(d*x + c))*sinh(d*x + c)^11 + 2*(4004*a*b*cosh(d*x + c)^6 + 1001*a*b*cosh(d*x +
c)^4 - 66*a*b*cosh(d*x + c)^2 - 3*a*b)*sinh(d*x + c)^10 + 4*(2860*a*b*cosh(d*x + c)^7 + 1001*a*b*cosh(d*x + c
)^5 - 110*a*b*cosh(d*x + c)^3 - 15*a*b*cosh(d*x + c))*sinh(d*x + c)^9 + 6*(2145*a*b*cosh(d*x + c)^8 + 1001*a*b
*cosh(d*x + c)^6 - 165*a*b*cosh(d*x + c)^4 - 45*a*b*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 6*a*b*cosh(d*x + c)^6 +
16*(715*a*b*cosh(d*x + c)^9 + 429*a*b*cosh(d*x + c)^7 - 99*a*b*cosh(d*x + c)^5 - 45*a*b*cosh(d*x + c)^3)*sinh
(d*x + c)^7 + 2*(4004*a*b*cosh(d*x + c)^10 + 3003*a*b*cosh(d*x + c)^8 - 924*a*b*cosh(d*x + c)^6 - 630*a*b*cosh
(d*x + c)^4 + 3*a*b)*sinh(d*x + c)^6 + 2*a*b*cosh(d*x + c)^4 + 4*(1092*a*b*cosh(d*x + c)^11 + 1001*a*b*cosh(d*
x + c)^9 - 396*a*b*cosh(d*x + c)^7 - 378*a*b*cosh(d*x + c)^5 + 9*a*b*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(910*a
*b*cosh(d*x + c)^12 + 1001*a*b*cosh(d*x + c)^10 - 495*a*b*cosh(d*x + c)^8 - 630*a*b*cosh(d*x + c)^6 + 45*a*b*c
osh(d*x + c)^2 + a*b)*sinh(d*x + c)^4 - 2*a*b*cosh(d*x + c)^2 + 8*(70*a*b*cosh(d*x + c)^13 + 91*a*b*cosh(d*x +
c)^11 - 55*a*b*cosh(d*x + c)^9 - 90*a*b*cosh(d*x + c)^7 + 15*a*b*cosh(d*x + c)^3 + a*b*cosh(d*x + c))*sinh(d*
x + c)^3 + 2*(60*a*b*cosh(d*x + c)^14 + 91*a*b*cosh(d*x + c)^12 - 66*a*b*cosh(d*x + c)^10 - 135*a*b*cosh(d*x +
c)^8 + 45*a*b*cosh(d*x + c)^4 + 6*a*b*cosh(d*x + c)^2 - a*b)*sinh(d*x + c)^2 - a*b + 4*(4*a*b*cosh(d*x + c)^1
5 + 7*a*b*cosh(d*x + c)^13 - 6*a*b*cosh(d*x + c)^11 - 15*a*b*cosh(d*x + c)^9 + 9*a*b*cosh(d*x + c)^5 + 2*a*b*c
osh(d*x + c)^3 - a*b*cosh(d*x + c))*sinh(d*x + c))*log(2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 15*(
a*b*cosh(d*x + c)^16 + 16*a*b*cosh(d*x + c)*sinh(d*x + c)^15 + a*b*sinh(d*x + c)^16 + 2*a*b*cosh(d*x + c)^14 +
2*(60*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c)^14 - 2*a*b*cosh(d*x + c)^12 + 28*(20*a*b*cosh(d*x + c)^3 + a*b
*cosh(d*x + c))*sinh(d*x + c)^13 + 2*(910*a*b*cosh(d*x + c)^4 + 91*a*b*cosh(d*x + c)^2 - a*b)*sinh(d*x + c)^12
- 6*a*b*cosh(d*x + c)^10 + 8*(546*a*b*cosh(d*x + c)^5 + 91*a*b*cosh(d*x + c)^3 - 3*a*b*cosh(d*x + c))*sinh(d*
x + c)^11 + 2*(4004*a*b*cosh(d*x + c)^6 + 1001*a*b*cosh(d*x + c)^4 - 66*a*b*cosh(d*x + c)^2 - 3*a*b)*sinh(d*x
+ c)^10 + 4*(2860*a*b*cosh(d*x + c)^7 + 1001*a*b*cosh(d*x + c)^5 - 110*a*b*cosh(d*x + c)^3 - 15*a*b*cosh(d*x +
c))*sinh(d*x + c)^9 + 6*(2145*a*b*cosh(d*x + c)^8 + 1001*a*b*cosh(d*x + c)^6 - 165*a*b*cosh(d*x + c)^4 - 45*a
*b*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 6*a*b*cosh(d*x + c)^6 + 16*(715*a*b*cosh(d*x + c)^9 + 429*a*b*cosh(d*x +
c)^7 - 99*a*b*cosh(d*x + c)^5 - 45*a*b*cosh(d*x + c)^3)*sinh(d*x + c)^7 + 2*(4004*a*b*cosh(d*x + c)^10 + 3003
*a*b*cosh(d*x + c)^8 - 924*a*b*cosh(d*x + c)^6 - 630*a*b*cosh(d*x + c)^4 + 3*a*b)*sinh(d*x + c)^6 + 2*a*b*cosh
(d*x + c)^4 + 4*(1092*a*b*cosh(d*x + c)^11 + 1001*a*b*cosh(d*x + c)^9 - 396*a*b*cosh(d*x + c)^7 - 378*a*b*cosh
(d*x + c)^5 + 9*a*b*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(910*a*b*cosh(d*x + c)^12 + 1001*a*b*cosh(d*x + c)^10 -
495*a*b*cosh(d*x + c)^8 - 630*a*b*cosh(d*x + c)^6 + 45*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c)^4 - 2*a*b*cos
h(d*x + c)^2 + 8*(70*a*b*cosh(d*x + c)^13 + 91*a*b*cosh(d*x + c)^11 - 55*a*b*cosh(d*x + c)^9 - 90*a*b*cosh(d*x
+ c)^7 + 15*a*b*cosh(d*x + c)^3 + a*b*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(60*a*b*cosh(d*x + c)^14 + 91*a*b*co
sh(d*x + c)^12 - 66*a*b*cosh(d*x + c)^10 - 135*a*b*cosh(d*x + c)^8 + 45*a*b*cosh(d*x + c)^4 + 6*a*b*cosh(d*x +
c)^2 - a*b)*sinh(d*x + c)^2 - a*b + 4*(4*a*b*cosh(d*x + c)^15 + 7*a*b*cosh(d*x + c)^13 - 6*a*b*cosh(d*x + c)^
11 - 15*a*b*cosh(d*x + c)^9 + 9*a*b*cosh(d*x + c)^5 + 2*a*b*cosh(d*x + c)^3 - a*b*cosh(d*x + c))*sinh(d*x + c)
)*log(2*sinh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 4*(105*a*b*cosh(d*x + c)^13 - 90*(a^2 + b^2)*cosh(d*x
+ c)^11 - 25*(14*a^2 + 9*a*b - 10*b^2)*cosh(d*x + c)^9 - 20*(25*a^2 + 13*b^2)*cosh(d*x + c)^7 - 3*(100*a^2 -
45*a*b - 44*b^2)*cosh(d*x + c)^5 - 2*(25*a^2 + 17*b^2)*cosh(d*x + c)^3 + (10*a^2 - 15*a*b + 2*b^2)*cosh(d*x +
c))*sinh(d*x + c))/(d*cosh(d*x + c)^16 + 16*d*cosh(d*x + c)*sinh(d*x + c)^15 + d*sinh(d*x + c)^16 + 2*d*cosh(d
*x + c)^14 + 2*(60*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^14 + 28*(20*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(
d*x + c)^13 - 2*d*cosh(d*x + c)^12 + 2*(910*d*cosh(d*x + c)^4 + 91*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^12 + 8
*(546*d*cosh(d*x + c)^5 + 91*d*cosh(d*x + c)^3 - 3*d*cosh(d*x + c))*sinh(d*x + c)^11 - 6*d*cosh(d*x + c)^10 +
2*(4004*d*cosh(d*x + c)^6 + 1001*d*cosh(d*x + c)^4 - 66*d*cosh(d*x + c)^2 - 3*d)*sinh(d*x + c)^10 + 4*(2860*d*
cosh(d*x + c)^7 + 1001*d*cosh(d*x + c)^5 - 110*d*cosh(d*x + c)^3 - 15*d*cosh(d*x + c))*sinh(d*x + c)^9 + 6*(21
45*d*cosh(d*x + c)^8 + 1001*d*cosh(d*x + c)^6 - 165*d*cosh(d*x + c)^4 - 45*d*cosh(d*x + c)^2)*sinh(d*x + c)^8
+ 16*(715*d*cosh(d*x + c)^9 + 429*d*cosh(d*x + c)^7 - 99*d*cosh(d*x + c)^5 - 45*d*cosh(d*x + c)^3)*sinh(d*x +
c)^7 + 6*d*cosh(d*x + c)^6 + 2*(4004*d*cosh(d*x + c)^10 + 3003*d*cosh(d*x + c)^8 - 924*d*cosh(d*x + c)^6 - 630
*d*cosh(d*x + c)^4 + 3*d)*sinh(d*x + c)^6 + 4*(1092*d*cosh(d*x + c)^11 + 1001*d*cosh(d*x + c)^9 - 396*d*cosh(d
*x + c)^7 - 378*d*cosh(d*x + c)^5 + 9*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*d*cosh(d*x + c)^4 + 2*(910*d*cosh(d
*x + c)^12 + 1001*d*cosh(d*x + c)^10 - 495*d*cosh(d*x + c)^8 - 630*d*cosh(d*x + c)^6 + 45*d*cosh(d*x + c)^2 +
d)*sinh(d*x + c)^4 + 8*(70*d*cosh(d*x + c)^13 + 91*d*cosh(d*x + c)^11 - 55*d*cosh(d*x + c)^9 - 90*d*cosh(d*x +
c)^7 + 15*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c)^3 - 2*d*cosh(d*x + c)^2 + 2*(60*d*cosh(d*x + c)^
14 + 91*d*cosh(d*x + c)^12 - 66*d*cosh(d*x + c)^10 - 135*d*cosh(d*x + c)^8 + 45*d*cosh(d*x + c)^4 + 6*d*cosh(d
*x + c)^2 - d)*sinh(d*x + c)^2 + 4*(4*d*cosh(d*x + c)^15 + 7*d*cosh(d*x + c)^13 - 6*d*cosh(d*x + c)^11 - 15*d*
cosh(d*x + c)^9 + 9*d*cosh(d*x + c)^5 + 2*d*cosh(d*x + c)^3 - d*cosh(d*x + c))*sinh(d*x + c) - d)

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)**4*(a+b*tanh(d*x+c)**3)**2,x)

[Out]

Integral((a + b*tanh(c + d*x)**3)**2*csch(c + d*x)**4, x)

________________________________________________________________________________________

Giac [B]  time = 1.69259, size = 336, normalized size = 3.46 \begin{align*} -\frac{60 \, a b \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) - 60 \, a b \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac{10 \,{\left (11 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 33 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 12 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 33 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 4 \, a^{2} - 11 \, a b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}} - \frac{137 \, a b e^{\left (10 \, d x + 10 \, c\right )} + 805 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 1730 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 120 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 1730 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 40 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 805 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 40 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 137 \, a b - 8 \, b^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}}{30 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^4*(a+b*tanh(d*x+c)^3)^2,x, algorithm="giac")

[Out]

-1/30*(60*a*b*log(e^(2*d*x + 2*c) + 1) - 60*a*b*log(abs(e^(2*d*x + 2*c) - 1)) + 10*(11*a*b*e^(6*d*x + 6*c) - 3
3*a*b*e^(4*d*x + 4*c) + 12*a^2*e^(2*d*x + 2*c) + 33*a*b*e^(2*d*x + 2*c) - 4*a^2 - 11*a*b)/(e^(2*d*x + 2*c) - 1
)^3 - (137*a*b*e^(10*d*x + 10*c) + 805*a*b*e^(8*d*x + 8*c) + 1730*a*b*e^(6*d*x + 6*c) - 120*b^2*e^(6*d*x + 6*c
) + 1730*a*b*e^(4*d*x + 4*c) + 40*b^2*e^(4*d*x + 4*c) + 805*a*b*e^(2*d*x + 2*c) - 40*b^2*e^(2*d*x + 2*c) + 137
*a*b - 8*b^2)/(e^(2*d*x + 2*c) + 1)^5)/d