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

Optimal. Leaf size=27 \[ \frac {(a+b x) \text {Chi}(a+b x)}{b}-\frac {\sinh (a+b x)}{b} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6664} \begin {gather*} \frac {(a+b x) \text {Chi}(a+b x)}{b}-\frac {\sinh (a+b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \text {Chi}(a+b x) \, dx &=\frac {(a+b x) \text {Chi}(a+b x)}{b}-\frac {\sinh (a+b x)}{b}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 42, normalized size = 1.56 \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[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

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Maple [A]
time = 0.12, size = 26, normalized size = 0.96

method	result	size
derivativedivides	\(\frac {\hyperbolicCosineIntegral \left (b x +a \right ) \left (b x +a \right )-\sinh \left (b x +a \right )}{b}\)	\(26\)
default	\(\frac {\hyperbolicCosineIntegral \left (b x +a \right ) \left (b x +a \right )-\sinh \left (b x +a \right )}{b}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(Chi(b*x+a),x, algorithm="maxima")

[Out]

integrate(Chi(b*x + a), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cosh_integral(b*x + a), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {Chi}\left (a + b x\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(Chi(a + b*x), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Chi(b*x + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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