3.2.92 \(\int \text {csch}^4(c+d x) (a+b \sinh ^4(c+d x)) \, dx\) [192]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 31, normalized size of antiderivative = 1.00, 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 = {3296, 1275, 213} \begin {gather*} -\frac {a \coth ^3(c+d x)}{3 d}+\frac {a \coth (c+d x)}{d}+b x \end {gather*}

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 213

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

Rule 1275

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 3296

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]

Rubi steps

\begin {align*} \int \text {csch}^4(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx &=\frac {\text {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 {\text {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 \text {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*}

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Mathematica [A]
time = 0.01, size = 40, normalized size = 1.29 \begin {gather*} b x+\frac {2 a \coth (c+d x)}{3 d}-\frac {a \coth (c+d x) \text {csch}^2(c+d x)}{3 d} \end {gather*}

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 = 1.39, size = 37, normalized size = 1.19

method	result	size
risch	\(b x -\frac {4 a \left (3 \,{\mathrm e}^{2 d x +2 c}-1\right )}{3 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\)	\(37\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (29) = 58\).
time = 0.26, size = 97, normalized size = 3.13 \begin {gather*} 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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (29) = 58\).
time = 0.42, size = 129, normalized size = 4.16 \begin {gather*} \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 {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 4369 deep

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Giac [A]
time = 0.45, size = 45, normalized size = 1.45 \begin {gather*} \frac {3 \, {\left (d x + c\right )} b - \frac {4 \, {\left (3 \, a e^{\left (2 \, d x + 2 \, c\right )} - a\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{3 \, d} \end {gather*}

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]

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

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Mupad [B]
time = 0.70, size = 81, normalized size = 2.61 \begin {gather*} \frac {4\,a-12\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,b\,d\,x+9\,b\,d\,x\,{\mathrm {e}}^{2\,c+2\,d\,x}-9\,b\,d\,x\,{\mathrm {e}}^{4\,c+4\,d\,x}+3\,b\,d\,x\,{\mathrm {e}}^{6\,c+6\,d\,x}}{3\,d\,{\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*a - 12*a*exp(2*c + 2*d*x) - 3*b*d*x + 9*b*d*x*exp(2*c + 2*d*x) - 9*b*d*x*exp(4*c + 4*d*x) + 3*b*d*x*exp(6*c
+ 6*d*x))/(3*d*(exp(2*c + 2*d*x) - 1)^3)

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