3.2.48 \(\int \text {csch}^3(c+d x) (a+b \sinh ^3(c+d x)) \, dx\) [148]

Optimal. Leaf size=39 \[ b x+\frac {a \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {a \coth (c+d x) \text {csch}(c+d x)}{2 d} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 39, 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 = {3299, 3853, 3855} \begin {gather*} \frac {a \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {a \coth (c+d x) \text {csch}(c+d x)}{2 d}+b x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3299

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 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

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

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 63, normalized size = 1.62 \begin {gather*} b x-\frac {a \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {a \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

b*x - (a*Csch[(c + d*x)/2]^2)/(8*d) - (a*Log[Tanh[(c + d*x)/2]])/(2*d) - (a*Sech[(c + d*x)/2]^2)/(8*d)

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Maple [A]
time = 2.06, size = 71, normalized size = 1.82

method	result	size
risch	\(b x -\frac {a \,{\mathrm e}^{d x +c} \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {a \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d}-\frac {a \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^3*(a+b*sinh(d*x+c)^3),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (35) = 70\).
time = 0.28, size = 91, normalized size = 2.33 \begin {gather*} b x + \frac {1}{2} \, a {\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} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 521 vs. \(2 (35) = 70\).
time = 0.45, size = 521, normalized size = 13.36 \begin {gather*} \frac {2 \, b d x \cosh \left (d x + c\right )^{4} + 2 \, b d x \sinh \left (d x + c\right )^{4} - 4 \, b d x \cosh \left (d x + c\right )^{2} - 2 \, a \cosh \left (d x + c\right )^{3} + 2 \, {\left (4 \, b d x \cosh \left (d x + c\right ) - a\right )} \sinh \left (d x + c\right )^{3} + 2 \, b d x + 2 \, {\left (6 \, b d x \cosh \left (d x + c\right )^{2} - 2 \, b d x - 3 \, a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 2 \, a \cosh \left (d x + c\right ) + {\left (a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} - 2 \, a \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} - a\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a \cosh \left (d x + c\right )^{3} - a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) - {\left (a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} - 2 \, a \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} - a\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a \cosh \left (d x + c\right )^{3} - a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 2 \, {\left (4 \, b d x \cosh \left (d x + c\right )^{3} - 4 \, b d x \cosh \left (d x + c\right ) - 3 \, a \cosh \left (d x + c\right )^{2} - a\right )} \sinh \left (d x + c\right )}{2 \, {\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} - 2 \, d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, d \cosh \left (d x + c\right )^{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 ) + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (35) = 70\).
time = 0.42, size = 73, normalized size = 1.87 \begin {gather*} \frac {2 \, {\left (d x + c\right )} b + a \log \left (e^{\left (d x + c\right )} + 1\right ) - a \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac {2 \, {\left (a e^{\left (3 \, d x + 3 \, c\right )} + a e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.11, size = 102, normalized size = 2.62 \begin {gather*} b\,x+\frac {\mathrm {atan}\left (\frac {a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^2}}\right )\,\sqrt {a^2}}{\sqrt {-d^2}}-\frac {a\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,a\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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