3.3.89 \(\int \sinh (\frac {a}{c+d x}) \, dx\) [289]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5418, 5410, 3378, 3382} \begin {gather*} \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} \end {gather*}

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 3378

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 3382

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]

Rule 5410

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 5418

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]

Rubi steps

\begin {align*} \int \sinh \left (\frac {a}{c+d x}\right ) \, dx &=\frac {\text {Subst}\left (\int \sinh \left (\frac {a}{x}\right ) \, dx,x,c+d x\right )}{d}\\ &=-\frac {\text {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 \text {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*}

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Mathematica [A]
time = 0.02, size = 36, normalized size = 1.00 \begin {gather*} -\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 {gather*}

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 = 0.89, size = 38, normalized size = 1.06

method	result	size
derivativedivides	\(-\frac {a \left (-\frac {\left (d x +c \right ) \sinh \left (\frac {a}{d x +c}\right )}{a}+\hyperbolicCosineIntegral \left (\frac {a}{d x +c}\right )\right )}{d}\)	\(38\)
default	\(-\frac {a \left (-\frac {\left (d x +c \right ) \sinh \left (\frac {a}{d x +c}\right )}{a}+\hyperbolicCosineIntegral \left (\frac {a}{d x +c}\right )\right )}{d}\)	\(38\)
risch	\(-\frac {{\mathrm e}^{-\frac {a}{d x +c}} x}{2}-\frac {{\mathrm e}^{-\frac {a}{d x +c}} c}{2 d}+\frac {a \expIntegral \left (1, \frac {a}{d x +c}\right )}{2 d}+\frac {{\mathrm e}^{\frac {a}{d x +c}} x}{2}+\frac {{\mathrm e}^{\frac {a}{d x +c}} c}{2 d}+\frac {a \expIntegral \left (1, -\frac {a}{d x +c}\right )}{2 d}\)	\(97\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(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 = 0.44, size = 48, normalized size = 1.33 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sinh {\left (\frac {a}{c + d x} \right )}\, dx \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (36) = 72\).
time = 0.42, size = 102, normalized size = 2.83 \begin {gather*} -\frac {{\left (\frac {a^{3} {\rm Ei}\left (\frac {a}{d x + c}\right )}{d x + c} - a^{2} e^{\left (\frac {a}{d x + c}\right )}\right )} {\left (d x + c\right )}}{2 \, a^{2} d} - \frac {{\left (\frac {a^{3} {\rm Ei}\left (-\frac {a}{d x + c}\right )}{d x + c} + a^{2} e^{\left (-\frac {a}{d x + c}\right )}\right )} {\left (d x + c\right )}}{2 \, a^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a^3*Ei(a/(d*x + c))/(d*x + c) - a^2*e^(a/(d*x + c)))*(d*x + c)/(a^2*d) - 1/2*(a^3*Ei(-a/(d*x + c))/(d*x
+ c) + a^2*e^(-a/(d*x + c)))*(d*x + c)/(a^2*d)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \mathrm {sinh}\left (\frac {a}{c+d\,x}\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a/(c + d*x)),x)

[Out]

int(sinh(a/(c + d*x)), x)

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