\(\int \text {Chi}(a+b x) \, dx\) [89]

Optimal result

Integrand size = 6, antiderivative size = 27 \[ \int \text {Chi}(a+b x) \, dx=\frac {(a+b x) \text {Chi}(a+b x)}{b}-\frac {\sinh (a+b x)}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6664} \[ \int \text {Chi}(a+b x) \, dx=\frac {(a+b x) \text {Chi}(a+b x)}{b}-\frac {\sinh (a+b x)}{b} \]

[In]

Int[CoshIntegral[a + b*x],x]

[Out]

((a + b*x)*CoshIntegral[a + b*x])/b - Sinh[a + 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}& = \frac {(a+b x) \text {Chi}(a+b x)}{b}-\frac {\sinh (a+b x)}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \text {Chi}(a+b x) \, dx=\frac {a \text {Chi}(a+b x)}{b}+x \text {Chi}(a+b x)-\frac {\cosh (b x) \sinh (a)}{b}-\frac {\cosh (a) \sinh (b x)}{b} \]

[In]

Integrate[CoshIntegral[a + b*x],x]

[Out]

(a*CoshIntegral[a + b*x])/b + x*CoshIntegral[a + b*x] - (Cosh[b*x]*Sinh[a])/b - (Cosh[a]*Sinh[b*x])/b

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(\frac {\operatorname {Chi}\left (b x +a \right ) \left (b x +a \right )-\sinh \left (b x +a \right )}{b}\)	\(26\)
default	\(\frac {\operatorname {Chi}\left (b x +a \right ) \left (b x +a \right )-\sinh \left (b x +a \right )}{b}\)	\(26\)
parts	\(x \,\operatorname {Chi}\left (b x +a \right )-\frac {-a \,\operatorname {Chi}\left (b x +a \right )+\sinh \left (b x +a \right )}{b}\)	\(31\)

[In]

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

[Out]

1/b*(Chi(b*x+a)*(b*x+a)-sinh(b*x+a))

Fricas [F]

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

[In]

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

[Out]

integral(cosh_integral(b*x + a), x)

Sympy [F]

\[ \int \text {Chi}(a+b x) \, dx=\int \operatorname {Chi}\left (a + b x\right )\, dx \]

[In]

integrate(Chi(b*x+a),x)

[Out]

Integral(Chi(a + b*x), x)

Maxima [F]

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

[In]

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

[Out]

integrate(Chi(b*x + a), x)

Giac [F]

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

[In]

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

[Out]

integrate(Chi(b*x + a), x)

Mupad [F(-1)]

Timed out. \[ \int \text {Chi}(a+b x) \, dx=x\,\mathrm {coshint}\left (a+b\,x\right )-\frac {{\mathrm {e}}^{a+b\,x}}{2\,b}+\frac {{\mathrm {e}}^{-a-b\,x}}{2\,b}+\frac {a\,\mathrm {coshint}\left (a+b\,x\right )}{b} \]

[In]

int(coshint(a + b*x),x)

[Out]

x*coshint(a + b*x) - exp(a + b*x)/(2*b) + exp(- a - b*x)/(2*b) + (a*coshint(a + b*x))/b