3.149 \(\int \text{csch}^4(c+d x) (a+b \sinh ^3(c+d x)) \, dx\)

Optimal. Leaf size=41 \[ -\frac{a \coth ^3(c+d x)}{3 d}+\frac{a \coth (c+d x)}{d}-\frac{b \tanh ^{-1}(\cosh (c+d x))}{d} \]

[Out]

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

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Rubi [A]  time = 0.0571616, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3220, 3770, 3767} \[ -\frac{a \coth ^3(c+d x)}{3 d}+\frac{a \coth (c+d x)}{d}-\frac{b \tanh ^{-1}(\cosh (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*ArcTanh[Cosh[c + d*x]])/d) + (a*Coth[c + d*x])/d - (a*Coth[c + d*x]^3)/(3*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 3770

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

Rule 3767

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

Rubi steps

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

Mathematica [A]  time = 0.0302817, size = 76, normalized size = 1.85 \[ \frac{2 a \coth (c+d x)}{3 d}-\frac{a \coth (c+d x) \text{csch}^2(c+d x)}{3 d}+\frac{b \log \left (\sinh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{b \log \left (\cosh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 36, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( a \left ({\frac{2}{3}}-{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{3}} \right ){\rm coth} \left (dx+c\right )-2\,b{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.16159, size = 177, normalized size = 4.32 \begin{align*} -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}\right )} + \frac{4}{3} \, a{\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*sinh(d*x+c)^3),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.02867, size = 1773, normalized size = 43.24 \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*sinh(d*x+c)^3),x, algorithm="fricas")

[Out]

-1/3*(12*a*cosh(d*x + c)^2 + 24*a*cosh(d*x + c)*sinh(d*x + c) + 12*a*sinh(d*x + c)^2 + 3*(b*cosh(d*x + c)^6 +
6*b*cosh(d*x + c)*sinh(d*x + c)^5 + b*sinh(d*x + c)^6 - 3*b*cosh(d*x + c)^4 + 3*(5*b*cosh(d*x + c)^2 - b)*sinh
(d*x + c)^4 + 4*(5*b*cosh(d*x + c)^3 - 3*b*cosh(d*x + c))*sinh(d*x + c)^3 + 3*b*cosh(d*x + c)^2 + 3*(5*b*cosh(
d*x + c)^4 - 6*b*cosh(d*x + c)^2 + b)*sinh(d*x + c)^2 + 6*(b*cosh(d*x + c)^5 - 2*b*cosh(d*x + c)^3 + b*cosh(d*
x + c))*sinh(d*x + c) - b)*log(cosh(d*x + c) + sinh(d*x + c) + 1) - 3*(b*cosh(d*x + c)^6 + 6*b*cosh(d*x + c)*s
inh(d*x + c)^5 + b*sinh(d*x + c)^6 - 3*b*cosh(d*x + c)^4 + 3*(5*b*cosh(d*x + c)^2 - b)*sinh(d*x + c)^4 + 4*(5*
b*cosh(d*x + c)^3 - 3*b*cosh(d*x + c))*sinh(d*x + c)^3 + 3*b*cosh(d*x + c)^2 + 3*(5*b*cosh(d*x + c)^4 - 6*b*co
sh(d*x + c)^2 + b)*sinh(d*x + c)^2 + 6*(b*cosh(d*x + c)^5 - 2*b*cosh(d*x + c)^3 + b*cosh(d*x + c))*sinh(d*x +
c) - b)*log(cosh(d*x + c) + sinh(d*x + c) - 1) - 4*a)/(d*cosh(d*x + c)^6 + 6*d*cosh(d*x + c)*sinh(d*x + c)^5 +
d*sinh(d*x + c)^6 - 3*d*cosh(d*x + c)^4 + 3*(5*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^4 + 4*(5*d*cosh(d*x + c)^
3 - 3*d*cosh(d*x + c))*sinh(d*x + c)^3 + 3*d*cosh(d*x + c)^2 + 3*(5*d*cosh(d*x + c)^4 - 6*d*cosh(d*x + c)^2 +
d)*sinh(d*x + c)^2 + 6*(d*cosh(d*x + c)^5 - 2*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)**4*(a+b*sinh(d*x+c)**3),x)

[Out]

Timed out

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Giac [A]  time = 1.1937, size = 88, normalized size = 2.15 \begin{align*} -\frac{b \log \left (e^{\left (d x + c\right )} + 1\right )}{d} + \frac{b \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{d} - \frac{4 \,{\left (3 \, a e^{\left (2 \, d x + 2 \, c\right )} - a\right )}}{3 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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