### 3.55 $$\int \text{csch}^3(c+d x) (a+b \tanh ^3(c+d x)) \, dx$$

Optimal. Leaf size=71 $\frac{a \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{b \tanh (c+d x) \text{sech}(c+d x)}{2 d}$

[Out]

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

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Rubi [A]  time = 0.0904974, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.143, Rules used = {3666, 3768, 3770} $\frac{a \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a \coth (c+d x) \text{csch}(c+d x)}{2 d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{b \tanh (c+d x) \text{sech}(c+d x)}{2 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

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

Mathematica [A]  time = 0.0259812, size = 95, normalized size = 1.34 $-\frac{a \text{csch}^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{a \text{sech}^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{a \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}+\frac{b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac{b \tanh (c+d x) \text{sech}(c+d x)}{2 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(b*ArcTan[Sinh[c + d*x]])/(2*d) - (a*Csch[(c + d*x)/2]^2)/(8*d) - (a*Log[Tanh[(c + d*x)/2]])/(2*d) - (a*Sech[(
c + d*x)/2]^2)/(8*d) + (b*Sech[c + d*x]*Tanh[c + d*x])/(2*d)

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Maple [A]  time = 0.059, size = 62, normalized size = 0.9 \begin{align*} -{\frac{{\rm coth} \left (dx+c\right )a{\rm csch} \left (dx+c\right )}{2\,d}}+{\frac{a{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) }{d}}+{\frac{b{\rm sech} \left (dx+c\right )\tanh \left ( dx+c \right ) }{2\,d}}+{\frac{b\arctan \left ({{\rm e}^{dx+c}} \right ) }{d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*coth(d*x+c)*csch(d*x+c)/d+1/d*a*arctanh(exp(d*x+c))+1/2*b*sech(d*x+c)*tanh(d*x+c)/d+1/d*b*arctan(exp(d*
x+c))

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Maxima [B]  time = 1.53103, size = 211, normalized size = 2.97 \begin{align*} -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{1}{2} \, a{\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{2 \,{\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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))
) + 1/2*a*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d + 2*(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)))

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Fricas [B]  time = 2.79467, size = 3216, normalized size = 45.3 \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)^3*(a+b*tanh(d*x+c)^3),x, algorithm="fricas")

[Out]

-1/2*(2*(a - b)*cosh(d*x + c)^7 + 14*(a - b)*cosh(d*x + c)*sinh(d*x + c)^6 + 2*(a - b)*sinh(d*x + c)^7 + 6*(a
+ b)*cosh(d*x + c)^5 + 6*(7*(a - b)*cosh(d*x + c)^2 + a + b)*sinh(d*x + c)^5 + 10*(7*(a - b)*cosh(d*x + c)^3 +
3*(a + b)*cosh(d*x + c))*sinh(d*x + c)^4 + 6*(a - b)*cosh(d*x + c)^3 + 2*(35*(a - b)*cosh(d*x + c)^4 + 30*(a
+ b)*cosh(d*x + c)^2 + 3*a - 3*b)*sinh(d*x + c)^3 + 6*(7*(a - b)*cosh(d*x + c)^5 + 10*(a + b)*cosh(d*x + c)^3
+ 3*(a - b)*cosh(d*x + c))*sinh(d*x + c)^2 - 2*(b*cosh(d*x + c)^8 + 56*b*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*
b*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8*b*cosh(d*x + c)*sinh(d*x + c)^7 + b*sinh(d*x + c)^8 - 2*b*cosh(d*x + c)^
4 + 2*(35*b*cosh(d*x + c)^4 - b)*sinh(d*x + c)^4 + 8*(7*b*cosh(d*x + c)^5 - b*cosh(d*x + c))*sinh(d*x + c)^3 +
4*(7*b*cosh(d*x + c)^6 - 3*b*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*(b*cosh(d*x + c)^7 - b*cosh(d*x + c)^3)*sin
h(d*x + c) + b)*arctan(cosh(d*x + c) + sinh(d*x + c)) + 2*(a + b)*cosh(d*x + c) - (a*cosh(d*x + c)^8 + 56*a*co
sh(d*x + c)^3*sinh(d*x + c)^5 + 28*a*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8*a*cosh(d*x + c)*sinh(d*x + c)^7 + a*s
inh(d*x + c)^8 - 2*a*cosh(d*x + c)^4 + 2*(35*a*cosh(d*x + c)^4 - a)*sinh(d*x + c)^4 + 8*(7*a*cosh(d*x + c)^5 -
a*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*a*cosh(d*x + c)^6 - 3*a*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*(a*cosh(
d*x + c)^7 - a*cosh(d*x + c)^3)*sinh(d*x + c) + a)*log(cosh(d*x + c) + sinh(d*x + c) + 1) + (a*cosh(d*x + c)^8
+ 56*a*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*a*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8*a*cosh(d*x + c)*sinh(d*x +
c)^7 + a*sinh(d*x + c)^8 - 2*a*cosh(d*x + c)^4 + 2*(35*a*cosh(d*x + c)^4 - a)*sinh(d*x + c)^4 + 8*(7*a*cosh(d*
x + c)^5 - a*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*a*cosh(d*x + c)^6 - 3*a*cosh(d*x + c)^2)*sinh(d*x + c)^2 +
8*(a*cosh(d*x + c)^7 - a*cosh(d*x + c)^3)*sinh(d*x + c) + a)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*(7*(a
- b)*cosh(d*x + c)^6 + 15*(a + b)*cosh(d*x + c)^4 + 9*(a - b)*cosh(d*x + c)^2 + a + b)*sinh(d*x + c))/(d*cosh(
d*x + c)^8 + 56*d*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*d*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8*d*cosh(d*x + c)*s
inh(d*x + c)^7 + d*sinh(d*x + c)^8 - 2*d*cosh(d*x + c)^4 + 2*(35*d*cosh(d*x + c)^4 - d)*sinh(d*x + c)^4 + 8*(7
*d*cosh(d*x + c)^5 - d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*d*cosh(d*x + c)^6 - 3*d*cosh(d*x + c)^2)*sinh(d*x
+ c)^2 + 8*(d*cosh(d*x + c)^7 - d*cosh(d*x + c)^3)*sinh(d*x + c) + d)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.22179, size = 193, normalized size = 2.72 \begin{align*} \frac{2 \, b \arctan \left (e^{\left (d x + c\right )}\right ) + a \log \left (e^{\left (d x + c\right )} + 1\right ) - a \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac{2 \,{\left (a e^{\left (7 \, d x + 7 \, c\right )} - b e^{\left (7 \, d x + 7 \, c\right )} + 3 \, a e^{\left (5 \, d x + 5 \, c\right )} + 3 \, b e^{\left (5 \, d x + 5 \, c\right )} + 3 \, a e^{\left (3 \, d x + 3 \, c\right )} - 3 \, b e^{\left (3 \, d x + 3 \, c\right )} + a e^{\left (d x + c\right )} + b e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}^{2}}}{2 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*b*arctan(e^(d*x + c)) + a*log(e^(d*x + c) + 1) - a*log(abs(e^(d*x + c) - 1)) - 2*(a*e^(7*d*x + 7*c) - b
*e^(7*d*x + 7*c) + 3*a*e^(5*d*x + 5*c) + 3*b*e^(5*d*x + 5*c) + 3*a*e^(3*d*x + 3*c) - 3*b*e^(3*d*x + 3*c) + a*e
^(d*x + c) + b*e^(d*x + c))/(e^(4*d*x + 4*c) - 1)^2)/d