3.1.9 \(\int \text {csch}^4(c+d x) (a+b \sinh ^2(c+d x)) \, dx\) [9]

Optimal. Leaf size=43 \[ \frac {(2 a-3 b) \coth (c+d x)}{3 d}-\frac {a \coth (c+d x) \text {csch}^2(c+d x)}{3 d} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3091, 3852, 8} \begin {gather*} \frac {(2 a-3 b) \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]

Int[Csch[c + d*x]^4*(a + b*Sinh[c + d*x]^2),x]

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3091

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A*Cos[e +
f*x]*((b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Dist[(A*(m + 2) + C*(m + 1))/(b^2*(m + 1)), Int[(b*Sin[e
+ f*x])^(m + 2), x], x] /; FreeQ[{b, e, f, A, C}, x] && LtQ[m, -1]

Rule 3852

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

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Mathematica [A]
time = 0.03, size = 49, normalized size = 1.14 \begin {gather*} \frac {2 a \coth (c+d x)}{3 d}-\frac {b \coth (c+d x)}{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]^2),x]

[Out]

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

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Maple [A]
time = 0.99, size = 62, normalized size = 1.44

method	result	size
risch	\(-\frac {2 \left (3 b \,{\mathrm e}^{4 d x +4 c}+6 a \,{\mathrm e}^{2 d x +2 c}-6 b \,{\mathrm e}^{2 d x +2 c}-2 a +3 b \right )}{3 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^4*(a+b*sinh(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

-2/3*(3*b*exp(4*d*x+4*c)+6*a*exp(2*d*x+2*c)-6*b*exp(2*d*x+2*c)-2*a+3*b)/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. 113 vs. \(2 (39) = 78\).
time = 0.27, size = 113, normalized size = 2.63 \begin {gather*} \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 )} + \frac {2 \, b}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\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)^2),x, algorithm="maxima")

[Out]

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))) + 2*b/(d*(e^(-2*d*x - 2*c) - 1))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (39) = 78\).
time = 0.42, size = 159, normalized size = 3.70 \begin {gather*} \frac {4 \, {\left ({\left (a - 3 \, b\right )} \cosh \left (d x + c\right )^{2} - 2 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a - 3 \, b\right )} \sinh \left (d x + c\right )^{2} - 3 \, a + 3 \, b\right )}}{3 \, {\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} - 4 \, d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, d \cosh \left (d x + c\right )^{2} - 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 ) + 3 \, d\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)^2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**2),x)

[Out]

Timed out

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Giac [A]
time = 0.42, size = 61, normalized size = 1.42 \begin {gather*} -\frac {2 \, {\left (3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a e^{\left (2 \, d x + 2 \, c\right )} - 6 \, b e^{\left (2 \, d x + 2 \, c\right )} - 2 \, a + 3 \, b\right )}}{3 \, d {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}} \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)^2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.60, size = 61, normalized size = 1.42 \begin {gather*} -\frac {2\,\left (3\,b-2\,a+6\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}-6\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}\right )}{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)^2)/sinh(c + d*x)^4,x)

[Out]

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

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