3.2.58 \(\int \text {csch}^5(c+d x) (a+b \sinh ^3(c+d x))^2 \, dx\) [158]

Optimal. Leaf size=90 \[ -\frac {3 a^2 \tanh ^{-1}(\cosh (c+d x))}{8 d}+\frac {b^2 \cosh (c+d x)}{d}-\frac {2 a b \coth (c+d x)}{d}+\frac {3 a^2 \coth (c+d x) \text {csch}(c+d x)}{8 d}-\frac {a^2 \coth (c+d x) \text {csch}^3(c+d x)}{4 d} \]

[Out]

-3/8*a^2*arctanh(cosh(d*x+c))/d+b^2*cosh(d*x+c)/d-2*a*b*coth(d*x+c)/d+3/8*a^2*coth(d*x+c)*csch(d*x+c)/d-1/4*a^
2*coth(d*x+c)*csch(d*x+c)^3/d

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Rubi [A]
time = 0.10, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3299, 3852, 8, 3853, 3855, 2718} \begin {gather*} -\frac {3 a^2 \tanh ^{-1}(\cosh (c+d x))}{8 d}-\frac {a^2 \coth (c+d x) \text {csch}^3(c+d x)}{4 d}+\frac {3 a^2 \coth (c+d x) \text {csch}(c+d x)}{8 d}-\frac {2 a b \coth (c+d x)}{d}+\frac {b^2 \cosh (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^5*(a + b*Sinh[c + d*x]^3)^2,x]

[Out]

(-3*a^2*ArcTanh[Cosh[c + d*x]])/(8*d) + (b^2*Cosh[c + d*x])/d - (2*a*b*Coth[c + d*x])/d + (3*a^2*Coth[c + d*x]
*Csch[c + d*x])/(8*d) - (a^2*Coth[c + d*x]*Csch[c + d*x]^3)/(4*d)

Rule 8

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

Rule 2718

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

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 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]

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}^5(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx &=i \int \left (-2 i a b \text {csch}^2(c+d x)-i a^2 \text {csch}^5(c+d x)-i b^2 \sinh (c+d x)\right ) \, dx\\ &=a^2 \int \text {csch}^5(c+d x) \, dx+(2 a b) \int \text {csch}^2(c+d x) \, dx+b^2 \int \sinh (c+d x) \, dx\\ &=\frac {b^2 \cosh (c+d x)}{d}-\frac {a^2 \coth (c+d x) \text {csch}^3(c+d x)}{4 d}-\frac {1}{4} \left (3 a^2\right ) \int \text {csch}^3(c+d x) \, dx-\frac {(2 i a b) \text {Subst}(\int 1 \, dx,x,-i \coth (c+d x))}{d}\\ &=\frac {b^2 \cosh (c+d x)}{d}-\frac {2 a b \coth (c+d x)}{d}+\frac {3 a^2 \coth (c+d x) \text {csch}(c+d x)}{8 d}-\frac {a^2 \coth (c+d x) \text {csch}^3(c+d x)}{4 d}+\frac {1}{8} \left (3 a^2\right ) \int \text {csch}(c+d x) \, dx\\ &=-\frac {3 a^2 \tanh ^{-1}(\cosh (c+d x))}{8 d}+\frac {b^2 \cosh (c+d x)}{d}-\frac {2 a b \coth (c+d x)}{d}+\frac {3 a^2 \coth (c+d x) \text {csch}(c+d x)}{8 d}-\frac {a^2 \coth (c+d x) \text {csch}^3(c+d x)}{4 d}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 149, normalized size = 1.66 \begin {gather*} \frac {b^2 \cosh (c) \cosh (d x)}{d}-\frac {2 a b \coth (c+d x)}{d}+\frac {3 a^2 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {a^2 \text {csch}^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {3 a^2 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {3 a^2 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {a^2 \text {sech}^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {b^2 \sinh (c) \sinh (d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^5*(a + b*Sinh[c + d*x]^3)^2,x]

[Out]

(b^2*Cosh[c]*Cosh[d*x])/d - (2*a*b*Coth[c + d*x])/d + (3*a^2*Csch[(c + d*x)/2]^2)/(32*d) - (a^2*Csch[(c + d*x)
/2]^4)/(64*d) + (3*a^2*Log[Tanh[(c + d*x)/2]])/(8*d) + (3*a^2*Sech[(c + d*x)/2]^2)/(32*d) + (a^2*Sech[(c + d*x
)/2]^4)/(64*d) + (b^2*Sinh[c]*Sinh[d*x])/d

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(170\) vs. \(2(84)=168\).
time = 2.23, size = 171, normalized size = 1.90

method	result	size
risch	\(\frac {{\mathrm e}^{d x +c} b^{2}}{2 d}+\frac {{\mathrm e}^{-d x -c} b^{2}}{2 d}+\frac {a \left (3 a \,{\mathrm e}^{7 d x +7 c}-16 b \,{\mathrm e}^{6 d x +6 c}-11 a \,{\mathrm e}^{5 d x +5 c}+48 b \,{\mathrm e}^{4 d x +4 c}-11 a \,{\mathrm e}^{3 d x +3 c}-48 b \,{\mathrm e}^{2 d x +2 c}+3 a \,{\mathrm e}^{d x +c}+16 b \right )}{4 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{4}}-\frac {3 a^{2} \ln \left ({\mathrm e}^{d x +c}+1\right )}{8 d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{8 d}\)	\(171\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^5*(a+b*sinh(d*x+c)^3)^2,x,method=_RETURNVERBOSE)

[Out]

1/2/d*exp(d*x+c)*b^2+1/2/d*exp(-d*x-c)*b^2+1/4*a*(3*a*exp(7*d*x+7*c)-16*b*exp(6*d*x+6*c)-11*a*exp(5*d*x+5*c)+4
8*b*exp(4*d*x+4*c)-11*a*exp(3*d*x+3*c)-48*b*exp(2*d*x+2*c)+3*a*exp(d*x+c)+16*b)/d/(exp(2*d*x+2*c)-1)^4-3/8*a^2
/d*ln(exp(d*x+c)+1)+3/8*a^2/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. 188 vs. \(2 (84) = 168\).
time = 0.27, size = 188, normalized size = 2.09 \begin {gather*} \frac {1}{2} \, b^{2} {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} - \frac {1}{8} \, a^{2} {\left (\frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (3 \, e^{\left (-d x - c\right )} - 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} + 3 \, e^{\left (-7 \, d x - 7 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} + \frac {4 \, a 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)^5*(a+b*sinh(d*x+c)^3)^2,x, algorithm="maxima")

[Out]

1/2*b^2*(e^(d*x + c)/d + e^(-d*x - c)/d) - 1/8*a^2*(3*log(e^(-d*x - c) + 1)/d - 3*log(e^(-d*x - c) - 1)/d + 2*
(3*e^(-d*x - c) - 11*e^(-3*d*x - 3*c) - 11*e^(-5*d*x - 5*c) + 3*e^(-7*d*x - 7*c))/(d*(4*e^(-2*d*x - 2*c) - 6*e
^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) - e^(-8*d*x - 8*c) - 1))) + 4*a*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. 2119 vs. \(2 (84) = 168\).
time = 0.59, size = 2119, normalized size = 23.54 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^5*(a+b*sinh(d*x+c)^3)^2,x, algorithm="fricas")

[Out]

1/8*(4*b^2*cosh(d*x + c)^10 + 40*b^2*cosh(d*x + c)*sinh(d*x + c)^9 + 4*b^2*sinh(d*x + c)^10 - 32*a*b*cosh(d*x
+ c)^7 + 6*(a^2 - 2*b^2)*cosh(d*x + c)^8 + 6*(30*b^2*cosh(d*x + c)^2 + a^2 - 2*b^2)*sinh(d*x + c)^8 + 16*(30*b
^2*cosh(d*x + c)^3 - 2*a*b + 3*(a^2 - 2*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 + 96*a*b*cosh(d*x + c)^5 - 2*(11*a
^2 - 4*b^2)*cosh(d*x + c)^6 + 2*(420*b^2*cosh(d*x + c)^4 - 112*a*b*cosh(d*x + c) + 84*(a^2 - 2*b^2)*cosh(d*x +
c)^2 - 11*a^2 + 4*b^2)*sinh(d*x + c)^6 + 12*(84*b^2*cosh(d*x + c)^5 - 56*a*b*cosh(d*x + c)^2 + 28*(a^2 - 2*b^
2)*cosh(d*x + c)^3 + 8*a*b - (11*a^2 - 4*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 96*a*b*cosh(d*x + c)^3 - 2*(11*
a^2 - 4*b^2)*cosh(d*x + c)^4 + 2*(420*b^2*cosh(d*x + c)^6 - 560*a*b*cosh(d*x + c)^3 + 210*(a^2 - 2*b^2)*cosh(d
*x + c)^4 + 240*a*b*cosh(d*x + c) - 15*(11*a^2 - 4*b^2)*cosh(d*x + c)^2 - 11*a^2 + 4*b^2)*sinh(d*x + c)^4 + 8*
(60*b^2*cosh(d*x + c)^7 - 140*a*b*cosh(d*x + c)^4 + 42*(a^2 - 2*b^2)*cosh(d*x + c)^5 + 120*a*b*cosh(d*x + c)^2
- 5*(11*a^2 - 4*b^2)*cosh(d*x + c)^3 - 12*a*b - (11*a^2 - 4*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 32*a*b*cosh
(d*x + c) + 6*(a^2 - 2*b^2)*cosh(d*x + c)^2 + 6*(30*b^2*cosh(d*x + c)^8 - 112*a*b*cosh(d*x + c)^5 + 28*(a^2 -
2*b^2)*cosh(d*x + c)^6 + 160*a*b*cosh(d*x + c)^3 - 5*(11*a^2 - 4*b^2)*cosh(d*x + c)^4 - 48*a*b*cosh(d*x + c) -
2*(11*a^2 - 4*b^2)*cosh(d*x + c)^2 + a^2 - 2*b^2)*sinh(d*x + c)^2 + 4*b^2 - 3*(a^2*cosh(d*x + c)^9 + 9*a^2*co
sh(d*x + c)*sinh(d*x + c)^8 + a^2*sinh(d*x + c)^9 - 4*a^2*cosh(d*x + c)^7 + 4*(9*a^2*cosh(d*x + c)^2 - a^2)*si
nh(d*x + c)^7 + 6*a^2*cosh(d*x + c)^5 + 28*(3*a^2*cosh(d*x + c)^3 - a^2*cosh(d*x + c))*sinh(d*x + c)^6 + 6*(21
*a^2*cosh(d*x + c)^4 - 14*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*x + c)^5 - 4*a^2*cosh(d*x + c)^3 + 2*(63*a^2*cosh(
d*x + c)^5 - 70*a^2*cosh(d*x + c)^3 + 15*a^2*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(21*a^2*cosh(d*x + c)^6 - 35*a
^2*cosh(d*x + c)^4 + 15*a^2*cosh(d*x + c)^2 - a^2)*sinh(d*x + c)^3 + a^2*cosh(d*x + c) + 12*(3*a^2*cosh(d*x +
c)^7 - 7*a^2*cosh(d*x + c)^5 + 5*a^2*cosh(d*x + c)^3 - a^2*cosh(d*x + c))*sinh(d*x + c)^2 + (9*a^2*cosh(d*x +
c)^8 - 28*a^2*cosh(d*x + c)^6 + 30*a^2*cosh(d*x + c)^4 - 12*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*x + c))*log(cosh
(d*x + c) + sinh(d*x + c) + 1) + 3*(a^2*cosh(d*x + c)^9 + 9*a^2*cosh(d*x + c)*sinh(d*x + c)^8 + a^2*sinh(d*x +
c)^9 - 4*a^2*cosh(d*x + c)^7 + 4*(9*a^2*cosh(d*x + c)^2 - a^2)*sinh(d*x + c)^7 + 6*a^2*cosh(d*x + c)^5 + 28*(
3*a^2*cosh(d*x + c)^3 - a^2*cosh(d*x + c))*sinh(d*x + c)^6 + 6*(21*a^2*cosh(d*x + c)^4 - 14*a^2*cosh(d*x + c)^
2 + a^2)*sinh(d*x + c)^5 - 4*a^2*cosh(d*x + c)^3 + 2*(63*a^2*cosh(d*x + c)^5 - 70*a^2*cosh(d*x + c)^3 + 15*a^2
*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(21*a^2*cosh(d*x + c)^6 - 35*a^2*cosh(d*x + c)^4 + 15*a^2*cosh(d*x + c)^2
- a^2)*sinh(d*x + c)^3 + a^2*cosh(d*x + c) + 12*(3*a^2*cosh(d*x + c)^7 - 7*a^2*cosh(d*x + c)^5 + 5*a^2*cosh(d*
x + c)^3 - a^2*cosh(d*x + c))*sinh(d*x + c)^2 + (9*a^2*cosh(d*x + c)^8 - 28*a^2*cosh(d*x + c)^6 + 30*a^2*cosh(
d*x + c)^4 - 12*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 4*(10*b^2*c
osh(d*x + c)^9 - 56*a*b*cosh(d*x + c)^6 + 12*(a^2 - 2*b^2)*cosh(d*x + c)^7 + 120*a*b*cosh(d*x + c)^4 - 3*(11*a
^2 - 4*b^2)*cosh(d*x + c)^5 - 72*a*b*cosh(d*x + c)^2 - 2*(11*a^2 - 4*b^2)*cosh(d*x + c)^3 + 8*a*b + 3*(a^2 - 2
*b^2)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^9 + 9*d*cosh(d*x + c)*sinh(d*x + c)^8 + d*sinh(d*x + c)^9
- 4*d*cosh(d*x + c)^7 + 4*(9*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^7 + 28*(3*d*cosh(d*x + c)^3 - d*cosh(d*x +
c))*sinh(d*x + c)^6 + 6*d*cosh(d*x + c)^5 + 6*(21*d*cosh(d*x + c)^4 - 14*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^
5 + 2*(63*d*cosh(d*x + c)^5 - 70*d*cosh(d*x + c)^3 + 15*d*cosh(d*x + c))*sinh(d*x + c)^4 - 4*d*cosh(d*x + c)^3
+ 4*(21*d*cosh(d*x + c)^6 - 35*d*cosh(d*x + c)^4 + 15*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^3 + 12*(3*d*cosh(d
*x + c)^7 - 7*d*cosh(d*x + c)^5 + 5*d*cosh(d*x + c)^3 - d*cosh(d*x + c))*sinh(d*x + c)^2 + d*cosh(d*x + c) + (
9*d*cosh(d*x + c)^8 - 28*d*cosh(d*x + c)^6 + 30*d*cosh(d*x + c)^4 - 12*d*cosh(d*x + c)^2 + d)*sinh(d*x + c))

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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)**5*(a+b*sinh(d*x+c)**3)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (84) = 168\).
time = 0.46, size = 172, normalized size = 1.91 \begin {gather*} \frac {4 \, b^{2} e^{\left (d x + c\right )} + 4 \, b^{2} e^{\left (-d x - c\right )} - 3 \, a^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) + 3 \, a^{2} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) + \frac {2 \, {\left (3 \, a^{2} e^{\left (7 \, d x + 7 \, c\right )} - 16 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 11 \, a^{2} e^{\left (5 \, d x + 5 \, c\right )} + 48 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 11 \, a^{2} e^{\left (3 \, d x + 3 \, c\right )} - 48 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a^{2} e^{\left (d x + c\right )} + 16 \, a b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{4}}}{8 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^5*(a+b*sinh(d*x+c)^3)^2,x, algorithm="giac")

[Out]

1/8*(4*b^2*e^(d*x + c) + 4*b^2*e^(-d*x - c) - 3*a^2*log(e^(d*x + c) + 1) + 3*a^2*log(abs(e^(d*x + c) - 1)) + 2
*(3*a^2*e^(7*d*x + 7*c) - 16*a*b*e^(6*d*x + 6*c) - 11*a^2*e^(5*d*x + 5*c) + 48*a*b*e^(4*d*x + 4*c) - 11*a^2*e^
(3*d*x + 3*c) - 48*a*b*e^(2*d*x + 2*c) + 3*a^2*e^(d*x + c) + 16*a*b)/(e^(2*d*x + 2*c) - 1)^4)/d

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Mupad [B]
time = 0.14, size = 355, normalized size = 3.94 \begin {gather*} \frac {\frac {3\,a^2\,{\mathrm {e}}^{c+d\,x}}{4\,d}-\frac {2\,a\,b}{d}}{{\mathrm {e}}^{2\,c+2\,d\,x}-1}-\frac {\frac {4\,a^2\,{\mathrm {e}}^{3\,c+3\,d\,x}}{d}-\frac {a\,b}{d}+\frac {3\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{d}-\frac {3\,a\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{d}+\frac {a\,b\,{\mathrm {e}}^{6\,c+6\,d\,x}}{d}}{6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}-\frac {\frac {2\,a^2\,{\mathrm {e}}^{c+d\,x}}{d}+\frac {a\,b}{d}-\frac {2\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{d}+\frac {a\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1}+\frac {b^2\,{\mathrm {e}}^{c+d\,x}}{2\,d}-\frac {3\,\mathrm {atan}\left (\frac {a^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^4}}\right )\,\sqrt {a^4}}{4\,\sqrt {-d^2}}+\frac {b^2\,{\mathrm {e}}^{-c-d\,x}}{2\,d}-\frac {a^2\,{\mathrm {e}}^{c+d\,x}}{2\,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)^2/sinh(c + d*x)^5,x)

[Out]

((3*a^2*exp(c + d*x))/(4*d) - (2*a*b)/d)/(exp(2*c + 2*d*x) - 1) - ((4*a^2*exp(3*c + 3*d*x))/d - (a*b)/d + (3*a
*b*exp(2*c + 2*d*x))/d - (3*a*b*exp(4*c + 4*d*x))/d + (a*b*exp(6*c + 6*d*x))/d)/(6*exp(4*c + 4*d*x) - 4*exp(2*
c + 2*d*x) - 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1) - ((2*a^2*exp(c + d*x))/d + (a*b)/d - (2*a*b*exp(2*c +
2*d*x))/d + (a*b*exp(4*c + 4*d*x))/d)/(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1) + (b^2
*exp(c + d*x))/(2*d) - (3*atan((a^2*exp(d*x)*exp(c)*(-d^2)^(1/2))/(d*(a^4)^(1/2)))*(a^4)^(1/2))/(4*(-d^2)^(1/2
)) + (b^2*exp(- c - d*x))/(2*d) - (a^2*exp(c + d*x))/(2*d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1))

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