\(\int \frac {\text {Chi}(b x) \sinh (b x)}{x^3} \, dx\) [114]

Optimal result

Integrand size = 12, antiderivative size = 12 \[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^3} \, dx=-\frac {b \cosh ^2(b x)}{2 x}-\frac {b \cosh (2 b x)}{4 x}-\frac {b \cosh (b x) \text {Chi}(b x)}{2 x}-\frac {\text {Chi}(b x) \sinh (b x)}{2 x^2}-\frac {\sinh (2 b x)}{8 x^2}+b^2 \text {Shi}(2 b x)+\frac {1}{2} b^2 \text {Int}\left (\frac {\text {Chi}(b x) \sinh (b x)}{x},x\right ) \]

[Out]

1/2*b^2*CannotIntegrate(Chi(b*x)*sinh(b*x)/x,x)-1/2*b*Chi(b*x)*cosh(b*x)/x-1/2*b*cosh(b*x)^2/x-1/4*b*cosh(2*b*
x)/x+b^2*Shi(2*b*x)-1/2*Chi(b*x)*sinh(b*x)/x^2-1/8*sinh(2*b*x)/x^2

Rubi [N/A]

Not integrable

Time = 0.15 (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 = {} \[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^3} \, dx=\int \frac {\text {Chi}(b x) \sinh (b x)}{x^3} \, dx \]

[In]

Int[(CoshIntegral[b*x]*Sinh[b*x])/x^3,x]

[Out]

-1/2*(b*Cosh[b*x]^2)/x - (b*Cosh[2*b*x])/(4*x) - (b*Cosh[b*x]*CoshIntegral[b*x])/(2*x) - (CoshIntegral[b*x]*Si
nh[b*x])/(2*x^2) - Sinh[2*b*x]/(8*x^2) + b^2*SinhIntegral[2*b*x] + (b^2*Defer[Int][(CoshIntegral[b*x]*Sinh[b*x
])/x, x])/2

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Chi}(b x) \sinh (b x)}{2 x^2}+\frac {1}{2} b \int \frac {\cosh (b x) \text {Chi}(b x)}{x^2} \, dx+\frac {1}{2} b \int \frac {\cosh (b x) \sinh (b x)}{b x^3} \, dx \\ & = -\frac {b \cosh (b x) \text {Chi}(b x)}{2 x}-\frac {\text {Chi}(b x) \sinh (b x)}{2 x^2}+\frac {1}{2} \int \frac {\cosh (b x) \sinh (b x)}{x^3} \, dx+\frac {1}{2} b^2 \int \frac {\cosh ^2(b x)}{b x^2} \, dx+\frac {1}{2} b^2 \int \frac {\text {Chi}(b x) \sinh (b x)}{x} \, dx \\ & = -\frac {b \cosh (b x) \text {Chi}(b x)}{2 x}-\frac {\text {Chi}(b x) \sinh (b x)}{2 x^2}+\frac {1}{2} \int \frac {\sinh (2 b x)}{2 x^3} \, dx+\frac {1}{2} b \int \frac {\cosh ^2(b x)}{x^2} \, dx+\frac {1}{2} b^2 \int \frac {\text {Chi}(b x) \sinh (b x)}{x} \, dx \\ & = -\frac {b \cosh ^2(b x)}{2 x}-\frac {b \cosh (b x) \text {Chi}(b x)}{2 x}-\frac {\text {Chi}(b x) \sinh (b x)}{2 x^2}+\frac {1}{4} \int \frac {\sinh (2 b x)}{x^3} \, dx+\left (i b^2\right ) \int -\frac {i \sinh (2 b x)}{2 x} \, dx+\frac {1}{2} b^2 \int \frac {\text {Chi}(b x) \sinh (b x)}{x} \, dx \\ & = -\frac {b \cosh ^2(b x)}{2 x}-\frac {b \cosh (b x) \text {Chi}(b x)}{2 x}-\frac {\text {Chi}(b x) \sinh (b x)}{2 x^2}-\frac {\sinh (2 b x)}{8 x^2}+\frac {1}{4} b \int \frac {\cosh (2 b x)}{x^2} \, dx+\frac {1}{2} b^2 \int \frac {\text {Chi}(b x) \sinh (b x)}{x} \, dx+\frac {1}{2} b^2 \int \frac {\sinh (2 b x)}{x} \, dx \\ & = -\frac {b \cosh ^2(b x)}{2 x}-\frac {b \cosh (2 b x)}{4 x}-\frac {b \cosh (b x) \text {Chi}(b x)}{2 x}-\frac {\text {Chi}(b x) \sinh (b x)}{2 x^2}-\frac {\sinh (2 b x)}{8 x^2}+\frac {1}{2} b^2 \text {Shi}(2 b x)+\frac {1}{2} b^2 \int \frac {\text {Chi}(b x) \sinh (b x)}{x} \, dx+\frac {1}{2} b^2 \int \frac {\sinh (2 b x)}{x} \, dx \\ & = -\frac {b \cosh ^2(b x)}{2 x}-\frac {b \cosh (2 b x)}{4 x}-\frac {b \cosh (b x) \text {Chi}(b x)}{2 x}-\frac {\text {Chi}(b x) \sinh (b x)}{2 x^2}-\frac {\sinh (2 b x)}{8 x^2}+b^2 \text {Shi}(2 b x)+\frac {1}{2} b^2 \int \frac {\text {Chi}(b x) \sinh (b x)}{x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^3} \, dx=\int \frac {\text {Chi}(b x) \sinh (b x)}{x^3} \, dx \]

[In]

Integrate[(CoshIntegral[b*x]*Sinh[b*x])/x^3,x]

[Out]

Integrate[(CoshIntegral[b*x]*Sinh[b*x])/x^3, x]

Maple [N/A] (verified)

Not integrable

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

\[\int \frac {\operatorname {Chi}\left (b x \right ) \sinh \left (b x \right )}{x^{3}}d x\]

[In]

int(Chi(b*x)*sinh(b*x)/x^3,x)

[Out]

int(Chi(b*x)*sinh(b*x)/x^3,x)

Fricas [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^3} \, dx=\int { \frac {{\rm Chi}\left (b x\right ) \sinh \left (b x\right )}{x^{3}} \,d x } \]

[In]

integrate(Chi(b*x)*sinh(b*x)/x^3,x, algorithm="fricas")

[Out]

integral(cosh_integral(b*x)*sinh(b*x)/x^3, x)

Sympy [N/A]

Not integrable

Time = 3.64 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^3} \, dx=\int \frac {\sinh {\left (b x \right )} \operatorname {Chi}\left (b x\right )}{x^{3}}\, dx \]

[In]

integrate(Chi(b*x)*sinh(b*x)/x**3,x)

[Out]

Integral(sinh(b*x)*Chi(b*x)/x**3, x)

Maxima [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^3} \, dx=\int { \frac {{\rm Chi}\left (b x\right ) \sinh \left (b x\right )}{x^{3}} \,d x } \]

[In]

integrate(Chi(b*x)*sinh(b*x)/x^3,x, algorithm="maxima")

[Out]

integrate(Chi(b*x)*sinh(b*x)/x^3, x)

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^3} \, dx=\int { \frac {{\rm Chi}\left (b x\right ) \sinh \left (b x\right )}{x^{3}} \,d x } \]

[In]

integrate(Chi(b*x)*sinh(b*x)/x^3,x, algorithm="giac")

[Out]

integrate(Chi(b*x)*sinh(b*x)/x^3, x)

Mupad [N/A]

Not integrable

Time = 4.87 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\text {Chi}(b x) \sinh (b x)}{x^3} \, dx=\int \frac {\mathrm {coshint}\left (b\,x\right )\,\mathrm {sinh}\left (b\,x\right )}{x^3} \,d x \]

[In]

int((coshint(b*x)*sinh(b*x))/x^3,x)

[Out]

int((coshint(b*x)*sinh(b*x))/x^3, x)