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

Optimal. Leaf size=42 \[ -\frac{a \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b \cosh ^3(c+d x)}{3 d}-\frac{b \cosh (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.0433106, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {3215, 1153, 206} \[ -\frac{a \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b \cosh ^3(c+d x)}{3 d}-\frac{b \cosh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3215

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0240189, size = 70, normalized size = 1.67 \[ \frac{a \log \left (\sinh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{a \log \left (\cosh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{3 b \cosh (c+d x)}{4 d}+\frac{b \cosh (3 (c+d x))}{12 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*Cosh[c + d*x])/(4*d) + (b*Cosh[3*(c + d*x)])/(12*d) - (a*Log[Cosh[c/2 + (d*x)/2]])/d + (a*Log[Sinh[c/2 +
(d*x)/2]])/d

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Maple [A]  time = 0.029, size = 36, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( -2\,a{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) +b \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)*(a+b*sinh(d*x+c)^4),x)

[Out]

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

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Maxima [A]  time = 1.16071, size = 96, normalized size = 2.29 \begin{align*} \frac{1}{24} \, b{\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{a \log \left (\tanh \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*b*(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) + a*log(tanh(1/2*d*x + 1/
2*c))/d

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Fricas [B]  time = 1.68997, size = 1100, normalized size = 26.19 \begin{align*} \frac{b \cosh \left (d x + c\right )^{6} + 6 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + b \sinh \left (d x + c\right )^{6} - 9 \, b \cosh \left (d x + c\right )^{4} + 3 \,{\left (5 \, b \cosh \left (d x + c\right )^{2} - 3 \, b\right )} \sinh \left (d x + c\right )^{4} + 4 \,{\left (5 \, b \cosh \left (d x + c\right )^{3} - 9 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 9 \, b \cosh \left (d x + c\right )^{2} + 3 \,{\left (5 \, b \cosh \left (d x + c\right )^{4} - 18 \, b \cosh \left (d x + c\right )^{2} - 3 \, b\right )} \sinh \left (d x + c\right )^{2} - 24 \,{\left (a \cosh \left (d x + c\right )^{3} + 3 \, a \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{3}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + 24 \,{\left (a \cosh \left (d x + c\right )^{3} + 3 \, a \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{3}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 6 \,{\left (b \cosh \left (d x + c\right )^{5} - 6 \, b \cosh \left (d x + c\right )^{3} - 3 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + b}{24 \,{\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + d \sinh \left (d x + c\right )^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(b*cosh(d*x + c)^6 + 6*b*cosh(d*x + c)*sinh(d*x + c)^5 + b*sinh(d*x + c)^6 - 9*b*cosh(d*x + c)^4 + 3*(5*b
*cosh(d*x + c)^2 - 3*b)*sinh(d*x + c)^4 + 4*(5*b*cosh(d*x + c)^3 - 9*b*cosh(d*x + c))*sinh(d*x + c)^3 - 9*b*co
sh(d*x + c)^2 + 3*(5*b*cosh(d*x + c)^4 - 18*b*cosh(d*x + c)^2 - 3*b)*sinh(d*x + c)^2 - 24*(a*cosh(d*x + c)^3 +
3*a*cosh(d*x + c)^2*sinh(d*x + c) + 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3)*log(cosh(d*x + c)
+ sinh(d*x + c) + 1) + 24*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)^2*sinh(d*x + c) + 3*a*cosh(d*x + c)*sinh(d*x
+ c)^2 + a*sinh(d*x + c)^3)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 6*(b*cosh(d*x + c)^5 - 6*b*cosh(d*x + c)^
3 - 3*b*cosh(d*x + c))*sinh(d*x + c) + b)/(d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c)^2*sinh(d*x + c) + 3*d*cosh(d*
x + c)*sinh(d*x + c)^2 + d*sinh(d*x + c)^3)

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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)*(a+b*sinh(d*x+c)**4),x)

[Out]

Timed out

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Giac [B]  time = 1.18077, size = 126, normalized size = 3. \begin{align*} -\frac{{\left (9 \, b e^{\left (2 \, d x + 2 \, c\right )} - b\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, d} - \frac{a \log \left (e^{\left (d x + c\right )} + 1\right )}{d} + \frac{a \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{d} + \frac{b d^{2} e^{\left (3 \, d x + 3 \, c\right )} - 9 \, b d^{2} e^{\left (d x + c\right )}}{24 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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