3.2.89 \(\int \coth (c+d x) \sinh (a+b x) \, dx\) [189]

Optimal. Leaf size=117 \[ \frac {e^{-a-b x}}{2 b}+\frac {e^{a+b x}}{2 b}-\frac {e^{-a-b x} \, _2F_1\left (1,-\frac {b}{2 d};1-\frac {b}{2 d};e^{2 (c+d x)}\right )}{b}-\frac {e^{a+b x} \, _2F_1\left (1,\frac {b}{2 d};1+\frac {b}{2 d};e^{2 (c+d x)}\right )}{b} \]

[Out]

1/2*exp(-b*x-a)/b+1/2*exp(b*x+a)/b-exp(-b*x-a)*hypergeom([1, -1/2*b/d],[1-1/2*b/d],exp(2*d*x+2*c))/b-exp(b*x+a
)*hypergeom([1, 1/2*b/d],[1+1/2*b/d],exp(2*d*x+2*c))/b

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Rubi [A]
time = 0.08, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5722, 2225, 2283} \begin {gather*} -\frac {e^{-a-b x} \, _2F_1\left (1,-\frac {b}{2 d};1-\frac {b}{2 d};e^{2 (c+d x)}\right )}{b}-\frac {e^{a+b x} \, _2F_1\left (1,\frac {b}{2 d};\frac {b}{2 d}+1;e^{2 (c+d x)}\right )}{b}+\frac {e^{-a-b x}}{2 b}+\frac {e^{a+b x}}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[c + d*x]*Sinh[a + b*x],x]

[Out]

E^(-a - b*x)/(2*b) + E^(a + b*x)/(2*b) - (E^(-a - b*x)*Hypergeometric2F1[1, -1/2*b/d, 1 - b/(2*d), E^(2*(c + d
*x))])/b - (E^(a + b*x)*Hypergeometric2F1[1, b/(2*d), 1 + b/(2*d), E^(2*(c + d*x))])/b

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2283

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Simp
[a^p*(G^(h*(f + g*x))/(g*h*Log[G]))*Hypergeometric2F1[-p, g*h*(Log[G]/(d*e*Log[F])), g*h*(Log[G]/(d*e*Log[F]))
+ 1, Simplify[(-b/a)*F^(e*(c + d*x))]], x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && (ILtQ[p, 0] || G
tQ[a, 0])

Rule 5722

Int[Coth[(c_.) + (d_.)*(x_)]*Sinh[(a_.) + (b_.)*(x_)], x_Symbol] :> Int[-E^(-(a + b*x))/2 + E^(a + b*x)/2 + 1/
(E^(a + b*x)*(1 - E^(2*(c + d*x)))) - E^(a + b*x)/(1 - E^(2*(c + d*x))), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
^2 - d^2, 0]

Rubi steps

\begin {align*} \int \coth (c+d x) \sinh (a+b x) \, dx &=\int \left (-\frac {1}{2} e^{-a-b x}+\frac {1}{2} e^{a+b x}+\frac {e^{-a-b x}}{1-e^{2 (c+d x)}}-\frac {e^{a+b x}}{1-e^{2 (c+d x)}}\right ) \, dx\\ &=-\left (\frac {1}{2} \int e^{-a-b x} \, dx\right )+\frac {1}{2} \int e^{a+b x} \, dx+\int \frac {e^{-a-b x}}{1-e^{2 (c+d x)}} \, dx-\int \frac {e^{a+b x}}{1-e^{2 (c+d x)}} \, dx\\ &=\frac {e^{-a-b x}}{2 b}+\frac {e^{a+b x}}{2 b}-\frac {e^{-a-b x} \, _2F_1\left (1,-\frac {b}{2 d};1-\frac {b}{2 d};e^{2 (c+d x)}\right )}{b}-\frac {e^{a+b x} \, _2F_1\left (1,\frac {b}{2 d};1+\frac {b}{2 d};e^{2 (c+d x)}\right )}{b}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(240\) vs. \(2(117)=234\).
time = 4.51, size = 240, normalized size = 2.05 \begin {gather*} \frac {\cosh (a) \cosh (b x) \coth (c)}{b}+\frac {e^{-a+2 c-b x} \left (b e^{2 d x} \, _2F_1\left (1,1-\frac {b}{2 d};2-\frac {b}{2 d};e^{2 (c+d x)}\right )-(b-2 d) \, _2F_1\left (1,-\frac {b}{2 d};1-\frac {b}{2 d};e^{2 (c+d x)}\right )\right )}{b (b-2 d) \left (-1+e^{2 c}\right )}-\frac {e^{a+2 c} \left (-\frac {e^{(b+2 d) x} \, _2F_1\left (1,1+\frac {b}{2 d};2+\frac {b}{2 d};e^{2 (c+d x)}\right )}{b+2 d}+\frac {e^{b x} \, _2F_1\left (1,\frac {b}{2 d};1+\frac {b}{2 d};e^{2 (c+d x)}\right )}{b}\right )}{-1+e^{2 c}}+\frac {\coth (c) \sinh (a) \sinh (b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[c + d*x]*Sinh[a + b*x],x]

[Out]

(Cosh[a]*Cosh[b*x]*Coth[c])/b + (E^(-a + 2*c - b*x)*(b*E^(2*d*x)*Hypergeometric2F1[1, 1 - b/(2*d), 2 - b/(2*d)
, E^(2*(c + d*x))] - (b - 2*d)*Hypergeometric2F1[1, -1/2*b/d, 1 - b/(2*d), E^(2*(c + d*x))]))/(b*(b - 2*d)*(-1
+ E^(2*c))) - (E^(a + 2*c)*(-((E^((b + 2*d)*x)*Hypergeometric2F1[1, 1 + b/(2*d), 2 + b/(2*d), E^(2*(c + d*x))
])/(b + 2*d)) + (E^(b*x)*Hypergeometric2F1[1, b/(2*d), 1 + b/(2*d), E^(2*(c + d*x))])/b))/(-1 + E^(2*c)) + (Co
th[c]*Sinh[a]*Sinh[b*x])/b

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Maple [F]
time = 0.48, size = 0, normalized size = 0.00 \[\int \coth \left (d x +c \right ) \sinh \left (b x +a \right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*x+c)*sinh(b*x+a),x)

[Out]

int(coth(d*x+c)*sinh(b*x+a),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(e^(2*b*x + 2*a) + 1)*e^(-b*x - a)/b - 1/2*integrate((e^(2*b*x + 2*a) - 1)/(e^(b*x + d*x + a + c) + e^(b*x
+ a)), x) + 1/2*integrate((e^(2*b*x + 2*a) - 1)/(e^(b*x + d*x + a + c) - e^(b*x + a)), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(coth(d*x + c)*sinh(b*x + a), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sinh {\left (a + b x \right )} \coth {\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sinh(a + b*x)*coth(c + d*x), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \mathrm {coth}\left (c+d\,x\right )\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(c + d*x)*sinh(a + b*x),x)

[Out]

int(coth(c + d*x)*sinh(a + b*x), x)

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