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

Optimal. Leaf size=77 \[ \frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a^2 \coth (c+d x) \text{csch}(c+d x)}{2 d}+2 a b x+\frac{b^2 \cosh ^3(c+d x)}{3 d}-\frac{b^2 \cosh (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.105334, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3220, 3768, 3770, 2633} \[ \frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a^2 \coth (c+d x) \text{csch}(c+d x)}{2 d}+2 a b x+\frac{b^2 \cosh ^3(c+d x)}{3 d}-\frac{b^2 \cosh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3220

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> Int[ExpandTr
ig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4]
|| GtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

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]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

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

Mathematica [A]  time = 0.0305111, size = 105, normalized size = 1.36 \[ -\frac{a^2 \text{csch}^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{a^2 \text{sech}^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{a^2 \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}+2 a b x-\frac{3 b^2 \cosh (c+d x)}{4 d}+\frac{b^2 \cosh (3 (c+d x))}{12 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*a*b*x - (3*b^2*Cosh[c + d*x])/(4*d) + (b^2*Cosh[3*(c + d*x)])/(12*d) - (a^2*Csch[(c + d*x)/2]^2)/(8*d) - (a^
2*Log[Tanh[(c + d*x)/2]])/(2*d) - (a^2*Sech[(c + d*x)/2]^2)/(8*d)

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Maple [A]  time = 0.059, size = 63, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( -{\frac{{\rm csch} \left (dx+c\right ){\rm coth} \left (dx+c\right )}{2}}+{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \right ) +2\,ab \left ( dx+c \right ) +{b}^{2} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \cosh \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a^2*(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp(d*x+c)))+2*a*b*(d*x+c)+b^2*(-2/3+1/3*sinh(d*x+c)^2)*cosh(d*
x+c))

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Maxima [B]  time = 1.22024, size = 205, normalized size = 2.66 \begin{align*} 2 \, a b x + \frac{1}{24} \, b^{2}{\left (\frac{e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac{9 \, e^{\left (d x + c\right )}}{d} - \frac{9 \, e^{\left (-d x - c\right )}}{d} + \frac{e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac{1}{2} \, a^{2}{\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*sinh(d*x+c)^3)^2,x, algorithm="maxima")

[Out]

2*a*b*x + 1/24*b^2*(e^(3*d*x + 3*c)/d - 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d*x - 3*c)/d) + 1/2*a^2*(lo
g(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 = 2.20708, size = 4246, normalized size = 55.14 \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*sinh(d*x+c)^3)^2,x, algorithm="fricas")

[Out]

1/24*(b^2*cosh(d*x + c)^10 + 10*b^2*cosh(d*x + c)*sinh(d*x + c)^9 + b^2*sinh(d*x + c)^10 + 48*a*b*d*x*cosh(d*x
+ c)^7 - 11*b^2*cosh(d*x + c)^8 - 96*a*b*d*x*cosh(d*x + c)^5 + (45*b^2*cosh(d*x + c)^2 - 11*b^2)*sinh(d*x + c
)^8 + 8*(15*b^2*cosh(d*x + c)^3 + 6*a*b*d*x - 11*b^2*cosh(d*x + c))*sinh(d*x + c)^7 + 48*a*b*d*x*cosh(d*x + c)
^3 - 2*(12*a^2 - 5*b^2)*cosh(d*x + c)^6 + 2*(105*b^2*cosh(d*x + c)^4 + 168*a*b*d*x*cosh(d*x + c) - 154*b^2*cos
h(d*x + c)^2 - 12*a^2 + 5*b^2)*sinh(d*x + c)^6 + 4*(63*b^2*cosh(d*x + c)^5 + 252*a*b*d*x*cosh(d*x + c)^2 - 154
*b^2*cosh(d*x + c)^3 - 24*a*b*d*x - 3*(12*a^2 - 5*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(12*a^2 - 5*b^2)*cos
h(d*x + c)^4 + 2*(105*b^2*cosh(d*x + c)^6 + 840*a*b*d*x*cosh(d*x + c)^3 - 385*b^2*cosh(d*x + c)^4 - 240*a*b*d*
x*cosh(d*x + c) - 15*(12*a^2 - 5*b^2)*cosh(d*x + c)^2 - 12*a^2 + 5*b^2)*sinh(d*x + c)^4 - 11*b^2*cosh(d*x + c)
^2 + 8*(15*b^2*cosh(d*x + c)^7 + 210*a*b*d*x*cosh(d*x + c)^4 - 77*b^2*cosh(d*x + c)^5 - 120*a*b*d*x*cosh(d*x +
c)^2 + 6*a*b*d*x - 5*(12*a^2 - 5*b^2)*cosh(d*x + c)^3 - (12*a^2 - 5*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + (45
*b^2*cosh(d*x + c)^8 + 1008*a*b*d*x*cosh(d*x + c)^5 - 308*b^2*cosh(d*x + c)^6 - 960*a*b*d*x*cosh(d*x + c)^3 +
144*a*b*d*x*cosh(d*x + c) - 30*(12*a^2 - 5*b^2)*cosh(d*x + c)^4 - 12*(12*a^2 - 5*b^2)*cosh(d*x + c)^2 - 11*b^2
)*sinh(d*x + c)^2 + b^2 + 12*(a^2*cosh(d*x + c)^7 + 7*a^2*cosh(d*x + c)*sinh(d*x + c)^6 + a^2*sinh(d*x + c)^7
- 2*a^2*cosh(d*x + c)^5 + (21*a^2*cosh(d*x + c)^2 - 2*a^2)*sinh(d*x + c)^5 + a^2*cosh(d*x + c)^3 + 5*(7*a^2*co
sh(d*x + c)^3 - 2*a^2*cosh(d*x + c))*sinh(d*x + c)^4 + (35*a^2*cosh(d*x + c)^4 - 20*a^2*cosh(d*x + c)^2 + a^2)
*sinh(d*x + c)^3 + (21*a^2*cosh(d*x + c)^5 - 20*a^2*cosh(d*x + c)^3 + 3*a^2*cosh(d*x + c))*sinh(d*x + c)^2 + (
7*a^2*cosh(d*x + c)^6 - 10*a^2*cosh(d*x + c)^4 + 3*a^2*cosh(d*x + c)^2)*sinh(d*x + c))*log(cosh(d*x + c) + sin
h(d*x + c) + 1) - 12*(a^2*cosh(d*x + c)^7 + 7*a^2*cosh(d*x + c)*sinh(d*x + c)^6 + a^2*sinh(d*x + c)^7 - 2*a^2*
cosh(d*x + c)^5 + (21*a^2*cosh(d*x + c)^2 - 2*a^2)*sinh(d*x + c)^5 + a^2*cosh(d*x + c)^3 + 5*(7*a^2*cosh(d*x +
c)^3 - 2*a^2*cosh(d*x + c))*sinh(d*x + c)^4 + (35*a^2*cosh(d*x + c)^4 - 20*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*
x + c)^3 + (21*a^2*cosh(d*x + c)^5 - 20*a^2*cosh(d*x + c)^3 + 3*a^2*cosh(d*x + c))*sinh(d*x + c)^2 + (7*a^2*co
sh(d*x + c)^6 - 10*a^2*cosh(d*x + c)^4 + 3*a^2*cosh(d*x + c)^2)*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x +
c) - 1) + 2*(5*b^2*cosh(d*x + c)^9 + 168*a*b*d*x*cosh(d*x + c)^6 - 44*b^2*cosh(d*x + c)^7 - 240*a*b*d*x*cosh(d
*x + c)^4 + 72*a*b*d*x*cosh(d*x + c)^2 - 6*(12*a^2 - 5*b^2)*cosh(d*x + c)^5 - 4*(12*a^2 - 5*b^2)*cosh(d*x + c)
^3 - 11*b^2*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^7 + 7*d*cosh(d*x + c)*sinh(d*x + c)^6 + d*sinh(d*x
+ c)^7 - 2*d*cosh(d*x + c)^5 + (21*d*cosh(d*x + c)^2 - 2*d)*sinh(d*x + c)^5 + 5*(7*d*cosh(d*x + c)^3 - 2*d*cos
h(d*x + c))*sinh(d*x + c)^4 + d*cosh(d*x + c)^3 + (35*d*cosh(d*x + c)^4 - 20*d*cosh(d*x + c)^2 + d)*sinh(d*x +
c)^3 + (21*d*cosh(d*x + c)^5 - 20*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^2 + (7*d*cosh(d*x + c)
^6 - 10*d*cosh(d*x + c)^4 + 3*d*cosh(d*x + c)^2)*sinh(d*x + c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.2288, size = 244, normalized size = 3.17 \begin{align*} \frac{2 \,{\left (d x + c\right )} a b}{d} + \frac{a^{2} \log \left (e^{\left (d x + c\right )} + 1\right )}{2 \, d} - \frac{a^{2} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{2 \, d} + \frac{b^{2} d^{2} e^{\left (3 \, d x + 3 \, c\right )} - 9 \, b^{2} d^{2} e^{\left (d x + c\right )}}{24 \, d^{3}} - \frac{{\left (11 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - b^{2} + 3 \,{\left (8 \, a^{2} + 3 \, b^{2}\right )} e^{\left (6 \, d x + 6 \, c\right )} +{\left (24 \, a^{2} - 19 \, b^{2}\right )} e^{\left (4 \, d x + 4 \, c\right )}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, d{\left (e^{\left (d x + c\right )} + 1\right )}^{2}{\left (e^{\left (d x + c\right )} - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(d*x + c)*a*b/d + 1/2*a^2*log(e^(d*x + c) + 1)/d - 1/2*a^2*log(abs(e^(d*x + c) - 1))/d + 1/24*(b^2*d^2*e^(3*
d*x + 3*c) - 9*b^2*d^2*e^(d*x + c))/d^3 - 1/24*(11*b^2*e^(2*d*x + 2*c) - b^2 + 3*(8*a^2 + 3*b^2)*e^(6*d*x + 6*
c) + (24*a^2 - 19*b^2)*e^(4*d*x + 4*c))*e^(-3*d*x - 3*c)/(d*(e^(d*x + c) + 1)^2*(e^(d*x + c) - 1)^2)