3.289 \(\int \sinh (\frac{a}{c+d x}) \, dx\)

Optimal. Leaf size=36 \[ \frac{(c+d x) \sinh \left (\frac{a}{c+d x}\right )}{d}-\frac{a \text{Chi}\left (\frac{a}{c+d x}\right )}{d} \]

[Out]

-((a*CoshIntegral[a/(c + d*x)])/d) + ((c + d*x)*Sinh[a/(c + d*x)])/d

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Rubi [A]  time = 0.0467654, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5310, 5302, 3297, 3301} \[ \frac{(c+d x) \sinh \left (\frac{a}{c+d x}\right )}{d}-\frac{a \text{Chi}\left (\frac{a}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a/(c + d*x)],x]

[Out]

-((a*CoshIntegral[a/(c + d*x)])/d) + ((c + d*x)*Sinh[a/(c + d*x)])/d

Rule 5310

Int[((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(u_)^(n_)])^(p_.), x_Symbol] :> Dist[1/Coefficient[u, x, 1], Subst[Int[(
a + b*Sinh[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d, n}, x] && IntegerQ[p] && LinearQ[u, x] && NeQ[u,
x]

Rule 5302

Int[((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> -Subst[Int[(a + b*Sinh[c + d/x^n])^p/x^2
, x], x, 1/x] /; FreeQ[{a, b, c, d}, x] && ILtQ[n, 0] && IntegerQ[p]

Rule 3297

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

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rubi steps

\begin{align*} \int \sinh \left (\frac{a}{c+d x}\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \sinh \left (\frac{a}{x}\right ) \, dx,x,c+d x\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sinh (a x)}{x^2} \, dx,x,\frac{1}{c+d x}\right )}{d}\\ &=\frac{(c+d x) \sinh \left (\frac{a}{c+d x}\right )}{d}-\frac{a \operatorname{Subst}\left (\int \frac{\cosh (a x)}{x} \, dx,x,\frac{1}{c+d x}\right )}{d}\\ &=-\frac{a \text{Chi}\left (\frac{a}{c+d x}\right )}{d}+\frac{(c+d x) \sinh \left (\frac{a}{c+d x}\right )}{d}\\ \end{align*}

Mathematica [A]  time = 0.0195202, size = 36, normalized size = 1. \[ \frac{(c+d x) \sinh \left (\frac{a}{c+d x}\right )}{d}-\frac{a \text{Chi}\left (\frac{a}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a/(c + d*x)],x]

[Out]

-((a*CoshIntegral[a/(c + d*x)])/d) + ((c + d*x)*Sinh[a/(c + d*x)])/d

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Maple [A]  time = 0.009, size = 38, normalized size = 1.1 \begin{align*} -{\frac{a}{d} \left ( -{\frac{dx+c}{a}\sinh \left ({\frac{a}{dx+c}} \right ) }+{\it Chi} \left ({\frac{a}{dx+c}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d*a*(-1/a*(d*x+c)*sinh(a/(d*x+c))+Chi(a/(d*x+c)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a d \int \frac{x e^{\left (\frac{a}{d x + c}\right )}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\,{d x} + \frac{1}{2} \, a d \int \frac{x e^{\left (-\frac{a}{d x + c}\right )}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\,{d x} + \frac{1}{2} \, x e^{\left (\frac{a}{d x + c}\right )} - \frac{1}{2} \, x e^{\left (-\frac{a}{d x + c}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.18674, size = 109, normalized size = 3.03 \begin{align*} -\frac{a{\rm Ei}\left (\frac{a}{d x + c}\right ) + a{\rm Ei}\left (-\frac{a}{d x + c}\right ) - 2 \,{\left (d x + c\right )} \sinh \left (\frac{a}{d x + c}\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a*Ei(a/(d*x + c)) + a*Ei(-a/(d*x + c)) - 2*(d*x + c)*sinh(a/(d*x + c)))/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a/(d*x+c)),x)

[Out]

Integral(sinh(a/(c + d*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh \left (\frac{a}{d x + c}\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sinh(a/(d*x + c)), x)