\(\int \frac {\cosh (b x) \text {Chi}(b x)}{x^2} \, dx\) [108]

Optimal result

Integrand size = 12, antiderivative size = 12 \[ \int \frac {\cosh (b x) \text {Chi}(b x)}{x^2} \, dx=-\frac {\cosh ^2(b x)}{x}-\frac {\cosh (b x) \text {Chi}(b x)}{x}+b \text {Shi}(2 b x)+b \text {Int}\left (\frac {\text {Chi}(b x) \sinh (b x)}{x},x\right ) \]

[Out]

b*CannotIntegrate(Chi(b*x)*sinh(b*x)/x,x)-Chi(b*x)*cosh(b*x)/x-cosh(b*x)^2/x+b*Shi(2*b*x)

Rubi [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cosh (b x) \text {Chi}(b x)}{x^2} \, dx=\int \frac {\cosh (b x) \text {Chi}(b x)}{x^2} \, dx \]

[In]

Int[(Cosh[b*x]*CoshIntegral[b*x])/x^2,x]

[Out]

-(Cosh[b*x]^2/x) - (Cosh[b*x]*CoshIntegral[b*x])/x + b*SinhIntegral[2*b*x] + b*Defer[Int][(CoshIntegral[b*x]*S
inh[b*x])/x, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cosh (b x) \text {Chi}(b x)}{x}+b \int \frac {\cosh ^2(b x)}{b x^2} \, dx+b \int \frac {\text {Chi}(b x) \sinh (b x)}{x} \, dx \\ & = -\frac {\cosh (b x) \text {Chi}(b x)}{x}+b \int \frac {\text {Chi}(b x) \sinh (b x)}{x} \, dx+\int \frac {\cosh ^2(b x)}{x^2} \, dx \\ & = -\frac {\cosh ^2(b x)}{x}-\frac {\cosh (b x) \text {Chi}(b x)}{x}+(2 i b) \int -\frac {i \sinh (2 b x)}{2 x} \, dx+b \int \frac {\text {Chi}(b x) \sinh (b x)}{x} \, dx \\ & = -\frac {\cosh ^2(b x)}{x}-\frac {\cosh (b x) \text {Chi}(b x)}{x}+b \int \frac {\text {Chi}(b x) \sinh (b x)}{x} \, dx+b \int \frac {\sinh (2 b x)}{x} \, dx \\ & = -\frac {\cosh ^2(b x)}{x}-\frac {\cosh (b x) \text {Chi}(b x)}{x}+b \text {Shi}(2 b x)+b \int \frac {\text {Chi}(b x) \sinh (b x)}{x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Chi}(b x)}{x^2} \, dx=\int \frac {\cosh (b x) \text {Chi}(b x)}{x^2} \, dx \]

[In]

Integrate[(Cosh[b*x]*CoshIntegral[b*x])/x^2,x]

[Out]

Integrate[(Cosh[b*x]*CoshIntegral[b*x])/x^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00

\[\int \frac {\operatorname {Chi}\left (b x \right ) \cosh \left (b x \right )}{x^{2}}d x\]

[In]

int(Chi(b*x)*cosh(b*x)/x^2,x)

[Out]

int(Chi(b*x)*cosh(b*x)/x^2,x)

Fricas [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Chi}(b x)}{x^2} \, dx=\int { \frac {{\rm Chi}\left (b x\right ) \cosh \left (b x\right )}{x^{2}} \,d x } \]

[In]

integrate(Chi(b*x)*cosh(b*x)/x^2,x, algorithm="fricas")

[Out]

integral(cosh(b*x)*cosh_integral(b*x)/x^2, x)

Sympy [N/A]

Not integrable

Time = 3.11 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Chi}(b x)}{x^2} \, dx=\int \frac {\cosh {\left (b x \right )} \operatorname {Chi}\left (b x\right )}{x^{2}}\, dx \]

[In]

integrate(Chi(b*x)*cosh(b*x)/x**2,x)

[Out]

Integral(cosh(b*x)*Chi(b*x)/x**2, x)

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Chi}(b x)}{x^2} \, dx=\int { \frac {{\rm Chi}\left (b x\right ) \cosh \left (b x\right )}{x^{2}} \,d x } \]

[In]

integrate(Chi(b*x)*cosh(b*x)/x^2,x, algorithm="maxima")

[Out]

integrate(Chi(b*x)*cosh(b*x)/x^2, x)

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Chi}(b x)}{x^2} \, dx=\int { \frac {{\rm Chi}\left (b x\right ) \cosh \left (b x\right )}{x^{2}} \,d x } \]

[In]

integrate(Chi(b*x)*cosh(b*x)/x^2,x, algorithm="giac")

[Out]

integrate(Chi(b*x)*cosh(b*x)/x^2, x)

Mupad [N/A]

Not integrable

Time = 4.64 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\cosh (b x) \text {Chi}(b x)}{x^2} \, dx=\int \frac {\mathrm {coshint}\left (b\,x\right )\,\mathrm {cosh}\left (b\,x\right )}{x^2} \,d x \]

[In]

int((coshint(b*x)*cosh(b*x))/x^2,x)

[Out]

int((coshint(b*x)*cosh(b*x))/x^2, x)