3.2.25 \(\int \text {Chi}(a+b x) \sinh (a+b x) \, dx\) [125]

Optimal. Leaf size=46 \[ \frac {\cosh (a+b x) \text {Chi}(a+b x)}{b}-\frac {\text {Chi}(2 a+2 b x)}{2 b}-\frac {\log (a+b x)}{2 b} \]

[Out]

-1/2*Chi(2*b*x+2*a)/b+Chi(b*x+a)*cosh(b*x+a)/b-1/2*ln(b*x+a)/b

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Rubi [A]
time = 0.05, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6682, 3393, 3382} \begin {gather*} -\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\text {Chi}(a+b x) \cosh (a+b x)}{b}-\frac {\log (a+b x)}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[CoshIntegral[a + b*x]*Sinh[a + b*x],x]

[Out]

(Cosh[a + b*x]*CoshIntegral[a + b*x])/b - CoshIntegral[2*a + 2*b*x]/(2*b) - Log[a + b*x]/(2*b)

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 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 6682

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

Rubi steps

\begin {align*} \int \text {Chi}(a+b x) \sinh (a+b x) \, dx &=\frac {\cosh (a+b x) \text {Chi}(a+b x)}{b}-\int \frac {\cosh ^2(a+b x)}{a+b x} \, dx\\ &=\frac {\cosh (a+b x) \text {Chi}(a+b x)}{b}-\int \left (\frac {1}{2 (a+b x)}+\frac {\cosh (2 a+2 b x)}{2 (a+b x)}\right ) \, dx\\ &=\frac {\cosh (a+b x) \text {Chi}(a+b x)}{b}-\frac {\log (a+b x)}{2 b}-\frac {1}{2} \int \frac {\cosh (2 a+2 b x)}{a+b x} \, dx\\ &=\frac {\cosh (a+b x) \text {Chi}(a+b x)}{b}-\frac {\text {Chi}(2 a+2 b x)}{2 b}-\frac {\log (a+b x)}{2 b}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 45, normalized size = 0.98 \begin {gather*} \frac {\cosh (a+b x) \text {Chi}(a+b x)}{b}-\frac {\text {Chi}(2 (a+b x))}{2 b}-\frac {\log (a+b x)}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[CoshIntegral[a + b*x]*Sinh[a + b*x],x]

[Out]

(Cosh[a + b*x]*CoshIntegral[a + b*x])/b - CoshIntegral[2*(a + b*x)]/(2*b) - Log[a + b*x]/(2*b)

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Maple [A]
time = 0.29, size = 38, normalized size = 0.83

method	result	size
derivativedivides	\(\frac {\hyperbolicCosineIntegral \left (b x +a \right ) \cosh \left (b x +a \right )-\frac {\ln \left (b x +a \right )}{2}-\frac {\hyperbolicCosineIntegral \left (2 b x +2 a \right )}{2}}{b}\)	\(38\)
default	\(\frac {\hyperbolicCosineIntegral \left (b x +a \right ) \cosh \left (b x +a \right )-\frac {\ln \left (b x +a \right )}{2}-\frac {\hyperbolicCosineIntegral \left (2 b x +2 a \right )}{2}}{b}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(Chi(b*x+a)*cosh(b*x+a)-1/2*ln(b*x+a)-1/2*Chi(2*b*x+2*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)*sinh(b*x+a),x, algorithm="maxima")

[Out]

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

[Out]

integral(cosh_integral(b*x + a)*sinh(b*x + a), x)

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

[Out]

Integral(sinh(a + b*x)*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)*sinh(b*x+a),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \mathrm {coshint}\left (a+b\,x\right )\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coshint(a + b*x)*sinh(a + b*x),x)

[Out]

int(coshint(a + b*x)*sinh(a + b*x), x)

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