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

Optimal. Leaf size=39 \[ \frac{a \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a \coth (c+d x) \text{csch}(c+d x)}{2 d}+b x \]

[Out]

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

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Rubi [A]  time = 0.0626266, antiderivative size = 39, normalized size of antiderivative = 1., 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 = {3220, 3768, 3770} \[ \frac{a \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a \coth (c+d x) \text{csch}(c+d x)}{2 d}+b x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.0129128, size = 63, normalized size = 1.62 \[ -\frac{a \text{csch}^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{a \text{sech}^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{a \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}+b x \]

Antiderivative was successfully verified.

[In]

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

[Out]

b*x - (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)

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Maple [A]  time = 0.04, size = 37, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{{\rm csch} \left (dx+c\right ){\rm coth} \left (dx+c\right )}{2}}+{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \right ) + \left ( dx+c \right ) b \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),x)

[Out]

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

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Maxima [B]  time = 1.09883, size = 123, normalized size = 3.15 \begin{align*} b x + \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*sinh(d*x+c)^3),x, algorithm="maxima")

[Out]

b*x + 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 = 2.12639, size = 1413, normalized size = 36.23 \begin{align*} \frac{2 \, b d x \cosh \left (d x + c\right )^{4} + 2 \, b d x \sinh \left (d x + c\right )^{4} - 4 \, b d x \cosh \left (d x + c\right )^{2} - 2 \, a \cosh \left (d x + c\right )^{3} + 2 \,{\left (4 \, b d x \cosh \left (d x + c\right ) - a\right )} \sinh \left (d x + c\right )^{3} + 2 \, b d x + 2 \,{\left (6 \, b d x \cosh \left (d x + c\right )^{2} - 2 \, b d x - 3 \, a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 2 \, a \cosh \left (d x + c\right ) +{\left (a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} - 2 \, a \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, a \cosh \left (d x + c\right )^{2} - a\right )} \sinh \left (d x + c\right )^{2} + 4 \,{\left (a \cosh \left (d x + c\right )^{3} - a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) -{\left (a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} - 2 \, a \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, a \cosh \left (d x + c\right )^{2} - a\right )} \sinh \left (d x + c\right )^{2} + 4 \,{\left (a \cosh \left (d x + c\right )^{3} - a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 2 \,{\left (4 \, b d x \cosh \left (d x + c\right )^{3} - 4 \, b d x \cosh \left (d x + c\right ) - 3 \, a \cosh \left (d x + c\right )^{2} - a\right )} \sinh \left (d x + c\right )}{2 \,{\left (d \cosh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{4} - 2 \, d \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{2} + 4 \,{\left (d \cosh \left (d x + c\right )^{3} - d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d\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),x, algorithm="fricas")

[Out]

1/2*(2*b*d*x*cosh(d*x + c)^4 + 2*b*d*x*sinh(d*x + c)^4 - 4*b*d*x*cosh(d*x + c)^2 - 2*a*cosh(d*x + c)^3 + 2*(4*
b*d*x*cosh(d*x + c) - a)*sinh(d*x + c)^3 + 2*b*d*x + 2*(6*b*d*x*cosh(d*x + c)^2 - 2*b*d*x - 3*a*cosh(d*x + c))
*sinh(d*x + c)^2 - 2*a*cosh(d*x + c) + (a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c
)^4 - 2*a*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 - a)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - a*cosh(d*x +
c))*sinh(d*x + c) + a)*log(cosh(d*x + c) + sinh(d*x + c) + 1) - (a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*
x + c)^3 + a*sinh(d*x + c)^4 - 2*a*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 - a)*sinh(d*x + c)^2 + 4*(a*cosh(d
*x + c)^3 - a*cosh(d*x + c))*sinh(d*x + c) + a)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*(4*b*d*x*cosh(d*x +
c)^3 - 4*b*d*x*cosh(d*x + c) - 3*a*cosh(d*x + c)^2 - a)*sinh(d*x + c))/(d*cosh(d*x + c)^4 + 4*d*cosh(d*x + c)
*sinh(d*x + c)^3 + d*sinh(d*x + c)^4 - 2*d*cosh(d*x + c)^2 + 2*(3*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^2 + 4*(
d*cosh(d*x + c)^3 - d*cosh(d*x + c))*sinh(d*x + c) + d)

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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),x)

[Out]

Timed out

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Giac [B]  time = 1.16116, size = 108, normalized size = 2.77 \begin{align*} \frac{{\left (d x + c\right )} b}{d} + \frac{a \log \left (e^{\left (d x + c\right )} + 1\right )}{2 \, d} - \frac{a \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{2 \, d} - \frac{a e^{\left (3 \, d x + 3 \, c\right )} + a e^{\left (d x + c\right )}}{d{\left (e^{\left (2 \, d x + 2 \, 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),x, algorithm="giac")

[Out]

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