3.1.18 \(\int (c+d x)^2 \sinh ^3(a+b x) \, dx\) [18]

Optimal. Leaf size=123 \[ -\frac {14 d^2 \cosh (a+b x)}{9 b^3}-\frac {2 (c+d x)^2 \cosh (a+b x)}{3 b}+\frac {2 d^2 \cosh ^3(a+b x)}{27 b^3}+\frac {4 d (c+d x) \sinh (a+b x)}{3 b^2}+\frac {(c+d x)^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {2 d (c+d x) \sinh ^3(a+b x)}{9 b^2} \]

[Out]

-14/9*d^2*cosh(b*x+a)/b^3-2/3*(d*x+c)^2*cosh(b*x+a)/b+2/27*d^2*cosh(b*x+a)^3/b^3+4/3*d*(d*x+c)*sinh(b*x+a)/b^2
+1/3*(d*x+c)^2*cosh(b*x+a)*sinh(b*x+a)^2/b-2/9*d*(d*x+c)*sinh(b*x+a)^3/b^2

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Rubi [A]
time = 0.09, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3392, 3377, 2718, 2713} \begin {gather*} \frac {2 d^2 \cosh ^3(a+b x)}{27 b^3}-\frac {14 d^2 \cosh (a+b x)}{9 b^3}-\frac {2 d (c+d x) \sinh ^3(a+b x)}{9 b^2}+\frac {4 d (c+d x) \sinh (a+b x)}{3 b^2}-\frac {2 (c+d x)^2 \cosh (a+b x)}{3 b}+\frac {(c+d x)^2 \sinh ^2(a+b x) \cosh (a+b x)}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2*Sinh[a + b*x]^3,x]

[Out]

(-14*d^2*Cosh[a + b*x])/(9*b^3) - (2*(c + d*x)^2*Cosh[a + b*x])/(3*b) + (2*d^2*Cosh[a + b*x]^3)/(27*b^3) + (4*
d*(c + d*x)*Sinh[a + b*x])/(3*b^2) + ((c + d*x)^2*Cosh[a + b*x]*Sinh[a + b*x]^2)/(3*b) - (2*d*(c + d*x)*Sinh[a
+ b*x]^3)/(9*b^2)

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2718

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

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3392

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*m*(c + d*x)^(m - 1)*((
b*Sin[e + f*x])^n/(f^2*n^2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[d^2*m*((m - 1)/(f^2*n^2)), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[b*(c + d*x)^m*Cos[e + f
*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rubi steps

\begin {align*} \int (c+d x)^2 \sinh ^3(a+b x) \, dx &=\frac {(c+d x)^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {2 d (c+d x) \sinh ^3(a+b x)}{9 b^2}-\frac {2}{3} \int (c+d x)^2 \sinh (a+b x) \, dx+\frac {\left (2 d^2\right ) \int \sinh ^3(a+b x) \, dx}{9 b^2}\\ &=-\frac {2 (c+d x)^2 \cosh (a+b x)}{3 b}+\frac {(c+d x)^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {2 d (c+d x) \sinh ^3(a+b x)}{9 b^2}+\frac {(4 d) \int (c+d x) \cosh (a+b x) \, dx}{3 b}-\frac {\left (2 d^2\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (a+b x)\right )}{9 b^3}\\ &=-\frac {2 d^2 \cosh (a+b x)}{9 b^3}-\frac {2 (c+d x)^2 \cosh (a+b x)}{3 b}+\frac {2 d^2 \cosh ^3(a+b x)}{27 b^3}+\frac {4 d (c+d x) \sinh (a+b x)}{3 b^2}+\frac {(c+d x)^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {2 d (c+d x) \sinh ^3(a+b x)}{9 b^2}-\frac {\left (4 d^2\right ) \int \sinh (a+b x) \, dx}{3 b^2}\\ &=-\frac {14 d^2 \cosh (a+b x)}{9 b^3}-\frac {2 (c+d x)^2 \cosh (a+b x)}{3 b}+\frac {2 d^2 \cosh ^3(a+b x)}{27 b^3}+\frac {4 d (c+d x) \sinh (a+b x)}{3 b^2}+\frac {(c+d x)^2 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {2 d (c+d x) \sinh ^3(a+b x)}{9 b^2}\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 86, normalized size = 0.70 \begin {gather*} \frac {-81 \left (2 d^2+b^2 (c+d x)^2\right ) \cosh (a+b x)+\left (2 d^2+9 b^2 (c+d x)^2\right ) \cosh (3 (a+b x))-6 b d (c+d x) (-27 \sinh (a+b x)+\sinh (3 (a+b x)))}{108 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-81*(2*d^2 + b^2*(c + d*x)^2)*Cosh[a + b*x] + (2*d^2 + 9*b^2*(c + d*x)^2)*Cosh[3*(a + b*x)] - 6*b*d*(c + d*x)
*(-27*Sinh[a + b*x] + Sinh[3*(a + b*x)]))/(108*b^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(342\) vs. \(2(111)=222\).
time = 0.42, size = 343, normalized size = 2.79

method	result	size
risch	\(\frac {\left (9 b^{2} d^{2} x^{2}+18 b^{2} c d x +9 b^{2} c^{2}-6 b \,d^{2} x -6 b c d +2 d^{2}\right ) {\mathrm e}^{3 b x +3 a}}{216 b^{3}}-\frac {3 \left (b^{2} d^{2} x^{2}+2 b^{2} c d x +b^{2} c^{2}-2 b \,d^{2} x -2 b c d +2 d^{2}\right ) {\mathrm e}^{b x +a}}{8 b^{3}}-\frac {3 \left (b^{2} d^{2} x^{2}+2 b^{2} c d x +b^{2} c^{2}+2 b \,d^{2} x +2 b c d +2 d^{2}\right ) {\mathrm e}^{-b x -a}}{8 b^{3}}+\frac {\left (9 b^{2} d^{2} x^{2}+18 b^{2} c d x +9 b^{2} c^{2}+6 b \,d^{2} x +6 b c d +2 d^{2}\right ) {\mathrm e}^{-3 b x -3 a}}{216 b^{3}}\)	\(231\)
default	\(\frac {-\frac {3 d^{2} \left (\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )}{4 b^{2}}+\frac {3 d^{2} a \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{2 b^{2}}-\frac {3 d^{2} a^{2} \cosh \left (b x +a \right )}{4 b^{2}}-\frac {3 c d \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )}{2 b}+\frac {3 c d a \cosh \left (b x +a \right )}{2 b}-\frac {3 c^{2} \cosh \left (b x +a \right )}{4}}{b}+\frac {\frac {d^{2} \left (\left (3 b x +3 a \right )^{2} \cosh \left (3 b x +3 a \right )-2 \left (3 b x +3 a \right ) \sinh \left (3 b x +3 a \right )+2 \cosh \left (3 b x +3 a \right )\right )}{b^{2}}-\frac {6 d^{2} a \left (\left (3 b x +3 a \right ) \cosh \left (3 b x +3 a \right )-\sinh \left (3 b x +3 a \right )\right )}{b^{2}}+\frac {9 d^{2} a^{2} \cosh \left (3 b x +3 a \right )}{b^{2}}+\frac {6 c d \left (\left (3 b x +3 a \right ) \cosh \left (3 b x +3 a \right )-\sinh \left (3 b x +3 a \right )\right )}{b}-\frac {18 c d a \cosh \left (3 b x +3 a \right )}{b}+9 c^{2} \cosh \left (3 b x +3 a \right )}{108 b}\)	\(343\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4/b*(-d^2/b^2*((b*x+a)^2*cosh(b*x+a)-2*(b*x+a)*sinh(b*x+a)+2*cosh(b*x+a))+2/b^2*d^2*a*((b*x+a)*cosh(b*x+a)-s
inh(b*x+a))-1/b^2*d^2*a^2*cosh(b*x+a)-2/b*c*d*((b*x+a)*cosh(b*x+a)-sinh(b*x+a))+2/b*c*d*a*cosh(b*x+a)-c^2*cosh
(b*x+a))+1/108/b*(1/b^2*d^2*((3*b*x+3*a)^2*cosh(3*b*x+3*a)-2*(3*b*x+3*a)*sinh(3*b*x+3*a)+2*cosh(3*b*x+3*a))-6/
b^2*d^2*a*((3*b*x+3*a)*cosh(3*b*x+3*a)-sinh(3*b*x+3*a))+9/b^2*d^2*a^2*cosh(3*b*x+3*a)+6/b*c*d*((3*b*x+3*a)*cos
h(3*b*x+3*a)-sinh(3*b*x+3*a))-18/b*c*d*a*cosh(3*b*x+3*a)+9*c^2*cosh(3*b*x+3*a))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (111) = 222\).
time = 0.28, size = 269, normalized size = 2.19 \begin {gather*} \frac {1}{36} \, c d {\left (\frac {{\left (3 \, b x e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{2}} - \frac {27 \, {\left (b x e^{a} - e^{a}\right )} e^{\left (b x\right )}}{b^{2}} - \frac {27 \, {\left (b x + 1\right )} e^{\left (-b x - a\right )}}{b^{2}} + \frac {{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{2}}\right )} + \frac {1}{24} \, c^{2} {\left (\frac {e^{\left (3 \, b x + 3 \, a\right )}}{b} - \frac {9 \, e^{\left (b x + a\right )}}{b} - \frac {9 \, e^{\left (-b x - a\right )}}{b} + \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{b}\right )} + \frac {1}{216} \, d^{2} {\left (\frac {{\left (9 \, b^{2} x^{2} e^{\left (3 \, a\right )} - 6 \, b x e^{\left (3 \, a\right )} + 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{3}} - \frac {81 \, {\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} e^{\left (b x\right )}}{b^{3}} - \frac {81 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x - a\right )}}{b^{3}} + \frac {{\left (9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*c*d*((3*b*x*e^(3*a) - e^(3*a))*e^(3*b*x)/b^2 - 27*(b*x*e^a - e^a)*e^(b*x)/b^2 - 27*(b*x + 1)*e^(-b*x - a)
/b^2 + (3*b*x + 1)*e^(-3*b*x - 3*a)/b^2) + 1/24*c^2*(e^(3*b*x + 3*a)/b - 9*e^(b*x + a)/b - 9*e^(-b*x - a)/b +
e^(-3*b*x - 3*a)/b) + 1/216*d^2*((9*b^2*x^2*e^(3*a) - 6*b*x*e^(3*a) + 2*e^(3*a))*e^(3*b*x)/b^3 - 81*(b^2*x^2*e
^a - 2*b*x*e^a + 2*e^a)*e^(b*x)/b^3 - 81*(b^2*x^2 + 2*b*x + 2)*e^(-b*x - a)/b^3 + (9*b^2*x^2 + 6*b*x + 2)*e^(-
3*b*x - 3*a)/b^3)

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Fricas [A]
time = 0.35, size = 199, normalized size = 1.62 \begin {gather*} \frac {{\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} + 2 \, d^{2}\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} + 2 \, d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 6 \, {\left (b d^{2} x + b c d\right )} \sinh \left (b x + a\right )^{3} - 81 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + 2 \, d^{2}\right )} \cosh \left (b x + a\right ) + 18 \, {\left (9 \, b d^{2} x + 9 \, b c d - {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )}{108 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/108*((9*b^2*d^2*x^2 + 18*b^2*c*d*x + 9*b^2*c^2 + 2*d^2)*cosh(b*x + a)^3 + 3*(9*b^2*d^2*x^2 + 18*b^2*c*d*x +
9*b^2*c^2 + 2*d^2)*cosh(b*x + a)*sinh(b*x + a)^2 - 6*(b*d^2*x + b*c*d)*sinh(b*x + a)^3 - 81*(b^2*d^2*x^2 + 2*b
^2*c*d*x + b^2*c^2 + 2*d^2)*cosh(b*x + a) + 18*(9*b*d^2*x + 9*b*c*d - (b*d^2*x + b*c*d)*cosh(b*x + a)^2)*sinh(
b*x + a))/b^3

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (121) = 242\).
time = 0.38, size = 284, normalized size = 2.31 \begin {gather*} \begin {cases} \frac {c^{2} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {2 c^{2} \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac {2 c d x \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {4 c d x \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac {d^{2} x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {2 d^{2} x^{2} \cosh ^{3}{\left (a + b x \right )}}{3 b} - \frac {14 c d \sinh ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {4 c d \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{2}} - \frac {14 d^{2} x \sinh ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {4 d^{2} x \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{2}} + \frac {14 d^{2} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{9 b^{3}} - \frac {40 d^{2} \cosh ^{3}{\left (a + b x \right )}}{27 b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \sinh ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((c**2*sinh(a + b*x)**2*cosh(a + b*x)/b - 2*c**2*cosh(a + b*x)**3/(3*b) + 2*c*d*x*sinh(a + b*x)**2*co
sh(a + b*x)/b - 4*c*d*x*cosh(a + b*x)**3/(3*b) + d**2*x**2*sinh(a + b*x)**2*cosh(a + b*x)/b - 2*d**2*x**2*cosh
(a + b*x)**3/(3*b) - 14*c*d*sinh(a + b*x)**3/(9*b**2) + 4*c*d*sinh(a + b*x)*cosh(a + b*x)**2/(3*b**2) - 14*d**
2*x*sinh(a + b*x)**3/(9*b**2) + 4*d**2*x*sinh(a + b*x)*cosh(a + b*x)**2/(3*b**2) + 14*d**2*sinh(a + b*x)**2*co
sh(a + b*x)/(9*b**3) - 40*d**2*cosh(a + b*x)**3/(27*b**3), Ne(b, 0)), ((c**2*x + c*d*x**2 + d**2*x**3/3)*sinh(
a)**3, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (111) = 222\).
time = 0.41, size = 230, normalized size = 1.87 \begin {gather*} \frac {{\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} - 6 \, b d^{2} x - 6 \, b c d + 2 \, d^{2}\right )} e^{\left (3 \, b x + 3 \, a\right )}}{216 \, b^{3}} - \frac {3 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, b d^{2} x - 2 \, b c d + 2 \, d^{2}\right )} e^{\left (b x + a\right )}}{8 \, b^{3}} - \frac {3 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + 2 \, b d^{2} x + 2 \, b c d + 2 \, d^{2}\right )} e^{\left (-b x - a\right )}}{8 \, b^{3}} + \frac {{\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} + 6 \, b d^{2} x + 6 \, b c d + 2 \, d^{2}\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{216 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/216*(9*b^2*d^2*x^2 + 18*b^2*c*d*x + 9*b^2*c^2 - 6*b*d^2*x - 6*b*c*d + 2*d^2)*e^(3*b*x + 3*a)/b^3 - 3/8*(b^2*
d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*b*d^2*x - 2*b*c*d + 2*d^2)*e^(b*x + a)/b^3 - 3/8*(b^2*d^2*x^2 + 2*b^2*c*d*
x + b^2*c^2 + 2*b*d^2*x + 2*b*c*d + 2*d^2)*e^(-b*x - a)/b^3 + 1/216*(9*b^2*d^2*x^2 + 18*b^2*c*d*x + 9*b^2*c^2
+ 6*b*d^2*x + 6*b*c*d + 2*d^2)*e^(-3*b*x - 3*a)/b^3

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Mupad [B]
time = 0.41, size = 184, normalized size = 1.50 \begin {gather*} -\frac {\frac {3\,d^2\,\mathrm {cosh}\left (a+b\,x\right )}{2}-\frac {d^2\,\mathrm {cosh}\left (3\,a+3\,b\,x\right )}{54}+\frac {3\,b^2\,c^2\,\mathrm {cosh}\left (a+b\,x\right )}{4}-\frac {b^2\,c^2\,\mathrm {cosh}\left (3\,a+3\,b\,x\right )}{12}+\frac {3\,b^2\,d^2\,x^2\,\mathrm {cosh}\left (a+b\,x\right )}{4}+\frac {b\,c\,d\,\mathrm {sinh}\left (3\,a+3\,b\,x\right )}{18}-\frac {3\,b\,d^2\,x\,\mathrm {sinh}\left (a+b\,x\right )}{2}-\frac {b^2\,d^2\,x^2\,\mathrm {cosh}\left (3\,a+3\,b\,x\right )}{12}+\frac {b\,d^2\,x\,\mathrm {sinh}\left (3\,a+3\,b\,x\right )}{18}-\frac {3\,b\,c\,d\,\mathrm {sinh}\left (a+b\,x\right )}{2}-\frac {b^2\,c\,d\,x\,\mathrm {cosh}\left (3\,a+3\,b\,x\right )}{6}+\frac {3\,b^2\,c\,d\,x\,\mathrm {cosh}\left (a+b\,x\right )}{2}}{b^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b*x)^3*(c + d*x)^2,x)

[Out]

-((3*d^2*cosh(a + b*x))/2 - (d^2*cosh(3*a + 3*b*x))/54 + (3*b^2*c^2*cosh(a + b*x))/4 - (b^2*c^2*cosh(3*a + 3*b
*x))/12 + (3*b^2*d^2*x^2*cosh(a + b*x))/4 + (b*c*d*sinh(3*a + 3*b*x))/18 - (3*b*d^2*x*sinh(a + b*x))/2 - (b^2*
d^2*x^2*cosh(3*a + 3*b*x))/12 + (b*d^2*x*sinh(3*a + 3*b*x))/18 - (3*b*c*d*sinh(a + b*x))/2 - (b^2*c*d*x*cosh(3
*a + 3*b*x))/6 + (3*b^2*c*d*x*cosh(a + b*x))/2)/b^3

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