\(\int \text {Chi}(b x) \, dx\) [73]

Optimal result

Integrand size = 4, antiderivative size = 16 \[ \int \text {Chi}(b x) \, dx=x \text {Chi}(b x)-\frac {\sinh (b x)}{b} \]

[Out]

x*Chi(b*x)-sinh(b*x)/b

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6664} \[ \int \text {Chi}(b x) \, dx=x \text {Chi}(b x)-\frac {\sinh (b x)}{b} \]

[In]

Int[CoshIntegral[b*x],x]

[Out]

x*CoshIntegral[b*x] - Sinh[b*x]/b

Rule 6664

Int[CoshIntegral[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(CoshIntegral[a + b*x]/b), x] - Simp[Sinh[a
+ b*x]/b, x] /; FreeQ[{a, b}, x]

Rubi steps \begin{align*} \text {integral}& = x \text {Chi}(b x)-\frac {\sinh (b x)}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \text {Chi}(b x) \, dx=x \text {Chi}(b x)-\frac {\sinh (b x)}{b} \]

[In]

Integrate[CoshIntegral[b*x],x]

[Out]

x*CoshIntegral[b*x] - Sinh[b*x]/b

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06

method	result	size
parts	\(x \,\operatorname {Chi}\left (b x \right )-\frac {\sinh \left (b x \right )}{b}\)	\(17\)
derivativedivides	\(\frac {\operatorname {Chi}\left (b x \right ) b x -\sinh \left (b x \right )}{b}\)	\(19\)
default	\(\frac {\operatorname {Chi}\left (b x \right ) b x -\sinh \left (b x \right )}{b}\)	\(19\)

[In]

int(Chi(b*x),x,method=_RETURNVERBOSE)

[Out]

x*Chi(b*x)-sinh(b*x)/b

Fricas [F]

\[ \int \text {Chi}(b x) \, dx=\int { {\rm Chi}\left (b x\right ) \,d x } \]

[In]

integrate(Chi(b*x),x, algorithm="fricas")

[Out]

integral(cosh_integral(b*x), x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (12) = 24\).

Time = 0.99 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.94 \[ \int \text {Chi}(b x) \, dx=- x \log {\left (b x \right )} + \frac {x \log {\left (b^{2} x^{2} \right )}}{2} + x \operatorname {Chi}\left (b x\right ) - \frac {\sinh {\left (b x \right )}}{b} \]

[In]

integrate(Chi(b*x),x)

[Out]

-x*log(b*x) + x*log(b**2*x**2)/2 + x*Chi(b*x) - sinh(b*x)/b

Maxima [F]

\[ \int \text {Chi}(b x) \, dx=\int { {\rm Chi}\left (b x\right ) \,d x } \]

[In]

integrate(Chi(b*x),x, algorithm="maxima")

[Out]

integrate(Chi(b*x), x)

Giac [F]

\[ \int \text {Chi}(b x) \, dx=\int { {\rm Chi}\left (b x\right ) \,d x } \]

[In]

integrate(Chi(b*x),x, algorithm="giac")

[Out]

integrate(Chi(b*x), x)

Mupad [F(-1)]

Timed out. \[ \int \text {Chi}(b x) \, dx=x\,\mathrm {coshint}\left (b\,x\right )-\frac {\mathrm {sinh}\left (b\,x\right )}{b} \]

[In]

int(coshint(b*x),x)

[Out]

x*coshint(b*x) - sinh(b*x)/b