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

Optimal result

Integrand size = 10, antiderivative size = 36 \[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=-\frac {a \text {Chi}\left (\frac {a}{c+d x}\right )}{d}+\frac {(c+d x) \sinh \left (\frac {a}{c+d x}\right )}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5418, 5410, 3378, 3382} \[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=\frac {(c+d x) \sinh \left (\frac {a}{c+d x}\right )}{d}-\frac {a \text {Chi}\left (\frac {a}{c+d x}\right )}{d} \]

[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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=-\frac {a \text {Chi}\left (\frac {a}{c+d x}\right )}{d}+\frac {(c+d x) \sinh \left (\frac {a}{c+d x}\right )}{d} \]

[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

Maple [A] (verified)

Time = 1.64 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06

method	result	size
derivativedivides	\(-\frac {a \left (-\frac {\left (d x +c \right ) \sinh \left (\frac {a}{d x +c}\right )}{a}+\operatorname {Chi}\left (\frac {a}{d x +c}\right )\right )}{d}\)	\(38\)
default	\(-\frac {a \left (-\frac {\left (d x +c \right ) \sinh \left (\frac {a}{d x +c}\right )}{a}+\operatorname {Chi}\left (\frac {a}{d x +c}\right )\right )}{d}\)	\(38\)
risch	\(-\frac {{\mathrm e}^{-\frac {a}{d x +c}} x}{2}-\frac {{\mathrm e}^{-\frac {a}{d x +c}} c}{2 d}+\frac {a \,\operatorname {Ei}_{1}\left (\frac {a}{d x +c}\right )}{2 d}+\frac {{\mathrm e}^{\frac {a}{d x +c}} x}{2}+\frac {{\mathrm e}^{\frac {a}{d x +c}} c}{2 d}+\frac {a \,\operatorname {Ei}_{1}\left (-\frac {a}{d x +c}\right )}{2 d}\)	\(99\)

[In]

int(sinh(1/(d*x+c)*a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.33 \[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=-\frac {a {\rm Ei}\left (\frac {a}{d x + c}\right ) + a {\rm Ei}\left (-\frac {a}{d x + c}\right ) - 2 \, {\left (d x + c\right )} \sinh \left (\frac {a}{d x + c}\right )}{2 \, d} \]

[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

Sympy [F]

\[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=\int \sinh {\left (\frac {a}{c + d x} \right )}\, dx \]

[In]

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

[Out]

Integral(sinh(a/(c + d*x)), x)

Maxima [F]

\[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=\int { \sinh \left (\frac {a}{d x + c}\right ) \,d x } \]

[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))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (36) = 72\).

Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.83 \[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=-\frac {{\left (\frac {a^{3} {\rm Ei}\left (\frac {a}{d x + c}\right )}{d x + c} - a^{2} e^{\left (\frac {a}{d x + c}\right )}\right )} {\left (d x + c\right )}}{2 \, a^{2} d} - \frac {{\left (\frac {a^{3} {\rm Ei}\left (-\frac {a}{d x + c}\right )}{d x + c} + a^{2} e^{\left (-\frac {a}{d x + c}\right )}\right )} {\left (d x + c\right )}}{2 \, a^{2} d} \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \sinh \left (\frac {a}{c+d x}\right ) \, dx=\int \mathrm {sinh}\left (\frac {a}{c+d\,x}\right ) \,d x \]

[In]

int(sinh(a/(c + d*x)),x)

[Out]

int(sinh(a/(c + d*x)), x)