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

Optimal. Leaf size=31 \[ -\frac{a \coth ^3(c+d x)}{3 d}+\frac{a \coth (c+d x)}{d}+b x \]

[Out]

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

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Rubi [A]  time = 0.0498847, antiderivative size = 31, 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 = {3217, 1261, 207} \[ -\frac{a \coth ^3(c+d x)}{3 d}+\frac{a \coth (c+d x)}{d}+b x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3217

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

Rule 1261

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

Rule 207

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

Rubi steps

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

Mathematica [A]  time = 0.0154378, size = 40, normalized size = 1.29 \[ \frac{2 a \coth (c+d x)}{3 d}-\frac{a \coth (c+d x) \text{csch}^2(c+d x)}{3 d}+b x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.039, size = 33, normalized size = 1.1 \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 )+ \left ( dx+c \right ) b \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)^4),x)

[Out]

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

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Maxima [B]  time = 1.16455, size = 131, normalized size = 4.23 \begin{align*} b x + \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)^4),x, algorithm="maxima")

[Out]

b*x + 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 = 1.67282, size = 333, normalized size = 10.74 \begin{align*} \frac{2 \, a \cosh \left (d x + c\right )^{3} + 6 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} +{\left (3 \, b d x - 2 \, a\right )} \sinh \left (d x + c\right )^{3} - 6 \, a \cosh \left (d x + c\right ) - 3 \,{\left (3 \, b d x -{\left (3 \, b d x - 2 \, a\right )} \cosh \left (d x + c\right )^{2} - 2 \, a\right )} \sinh \left (d x + c\right )}{3 \,{\left (d \sinh \left (d x + c\right )^{3} + 3 \,{\left (d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\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)^4),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [A]  time = 1.16184, size = 61, normalized size = 1.97 \begin{align*} \frac{{\left (d x + c\right )} b}{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)^4),x, algorithm="giac")

[Out]

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