Integrand size = 12, antiderivative size = 66 \[ \int \frac {\sinh \left (a+b x^2\right )}{x^2} \, dx=\frac {1}{2} \sqrt {b} e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} x\right )+\frac {1}{2} \sqrt {b} e^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x\right )-\frac {\sinh \left (a+b x^2\right )}{x} \] Output:
1/2*b^(1/2)*Pi^(1/2)*erf(b^(1/2)*x)/exp(a)+1/2*b^(1/2)*exp(a)*Pi^(1/2)*erf i(b^(1/2)*x)-sinh(b*x^2+a)/x
Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.06 \[ \int \frac {\sinh \left (a+b x^2\right )}{x^2} \, dx=\frac {\sqrt {b} \sqrt {\pi } x \text {erf}\left (\sqrt {b} x\right ) (\cosh (a)-\sinh (a))+\sqrt {b} \sqrt {\pi } x \text {erfi}\left (\sqrt {b} x\right ) (\cosh (a)+\sinh (a))-2 \sinh \left (a+b x^2\right )}{2 x} \] Input:
Integrate[Sinh[a + b*x^2]/x^2,x]
Output:
(Sqrt[b]*Sqrt[Pi]*x*Erf[Sqrt[b]*x]*(Cosh[a] - Sinh[a]) + Sqrt[b]*Sqrt[Pi]* x*Erfi[Sqrt[b]*x]*(Cosh[a] + Sinh[a]) - 2*Sinh[a + b*x^2])/(2*x)
Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5849, 5822, 2633, 2634}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sinh \left (a+b x^2\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 5849 |
\(\displaystyle 2 b \int \cosh \left (b x^2+a\right )dx-\frac {\sinh \left (a+b x^2\right )}{x}\) |
\(\Big \downarrow \) 5822 |
\(\displaystyle 2 b \left (\frac {1}{2} \int e^{-b x^2-a}dx+\frac {1}{2} \int e^{b x^2+a}dx\right )-\frac {\sinh \left (a+b x^2\right )}{x}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle 2 b \left (\frac {1}{2} \int e^{-b x^2-a}dx+\frac {\sqrt {\pi } e^a \text {erfi}\left (\sqrt {b} x\right )}{4 \sqrt {b}}\right )-\frac {\sinh \left (a+b x^2\right )}{x}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle 2 b \left (\frac {\sqrt {\pi } e^{-a} \text {erf}\left (\sqrt {b} x\right )}{4 \sqrt {b}}+\frac {\sqrt {\pi } e^a \text {erfi}\left (\sqrt {b} x\right )}{4 \sqrt {b}}\right )-\frac {\sinh \left (a+b x^2\right )}{x}\) |
Input:
Int[Sinh[a + b*x^2]/x^2,x]
Output:
2*b*((Sqrt[Pi]*Erf[Sqrt[b]*x])/(4*Sqrt[b]*E^a) + (E^a*Sqrt[Pi]*Erfi[Sqrt[b ]*x])/(4*Sqrt[b])) - Sinh[a + b*x^2]/x
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[Cosh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Simp[1/2 Int[E^(c + d*x^n ), x], x] + Simp[1/2 Int[E^(-c - d*x^n), x], x] /; FreeQ[{c, d}, x] && IG tQ[n, 1]
Int[((e_.)*(x_))^(m_)*Sinh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Simp[(e*x )^(m + 1)*(Sinh[c + d*x^n]/(e*(m + 1))), x] - Simp[d*(n/(e^n*(m + 1))) In t[(e*x)^(m + n)*Cosh[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0 ] && LtQ[m, -1]
Time = 0.12 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.06
method | result | size |
risch | \(\frac {{\mathrm e}^{-a} {\mathrm e}^{-b \,x^{2}}}{2 x}+\frac {\sqrt {b}\, \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, x \right ) {\mathrm e}^{-a}}{2}-\frac {{\mathrm e}^{a} {\mathrm e}^{b \,x^{2}}}{2 x}+\frac {{\mathrm e}^{a} b \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b}\, x \right )}{2 \sqrt {-b}}\) | \(70\) |
meijerg | \(\frac {i \sinh \left (a \right ) \sqrt {\pi }\, b \sqrt {2}\, \left (-\frac {2 \sqrt {2}\, {\mathrm e}^{b \,x^{2}}}{\sqrt {\pi }\, x \sqrt {i b}}-\frac {2 \sqrt {2}\, {\mathrm e}^{-b \,x^{2}}}{\sqrt {\pi }\, x \sqrt {i b}}-\frac {2 \sqrt {2}\, \sqrt {b}\, \operatorname {erf}\left (\sqrt {b}\, x \right )}{\sqrt {i b}}+\frac {2 \sqrt {2}\, \sqrt {b}\, \operatorname {erfi}\left (\sqrt {b}\, x \right )}{\sqrt {i b}}\right )}{8 \sqrt {i b}}+\frac {\cosh \left (a \right ) \sqrt {\pi }\, b \sqrt {2}\, \left (\frac {2 \sqrt {2}\, \sqrt {i b}\, {\mathrm e}^{-b \,x^{2}}}{\sqrt {\pi }\, x b}-\frac {2 \sqrt {2}\, \sqrt {i b}\, {\mathrm e}^{b \,x^{2}}}{\sqrt {\pi }\, x b}+\frac {2 \sqrt {i b}\, \sqrt {2}\, \operatorname {erf}\left (\sqrt {b}\, x \right )}{\sqrt {b}}+\frac {2 \sqrt {i b}\, \sqrt {2}\, \operatorname {erfi}\left (\sqrt {b}\, x \right )}{\sqrt {b}}\right )}{8 \sqrt {i b}}\) | \(219\) |
Input:
int(sinh(b*x^2+a)/x^2,x,method=_RETURNVERBOSE)
Output:
1/2/exp(a)/x*exp(-b*x^2)+1/2*b^(1/2)*Pi^(1/2)*erf(b^(1/2)*x)/exp(a)-1/2*ex p(a)*exp(b*x^2)/x+1/2*exp(a)*b*Pi^(1/2)/(-b)^(1/2)*erf((-b)^(1/2)*x)
Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (48) = 96\).
Time = 0.09 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.79 \[ \int \frac {\sinh \left (a+b x^2\right )}{x^2} \, dx=-\frac {\sqrt {\pi } {\left (x \cosh \left (b x^{2} + a\right ) \cosh \left (a\right ) + x \cosh \left (b x^{2} + a\right ) \sinh \left (a\right ) + {\left (x \cosh \left (a\right ) + x \sinh \left (a\right )\right )} \sinh \left (b x^{2} + a\right )\right )} \sqrt {-b} \operatorname {erf}\left (\sqrt {-b} x\right ) - \sqrt {\pi } {\left (x \cosh \left (b x^{2} + a\right ) \cosh \left (a\right ) - x \cosh \left (b x^{2} + a\right ) \sinh \left (a\right ) + {\left (x \cosh \left (a\right ) - x \sinh \left (a\right )\right )} \sinh \left (b x^{2} + a\right )\right )} \sqrt {b} \operatorname {erf}\left (\sqrt {b} x\right ) + \cosh \left (b x^{2} + a\right )^{2} + 2 \, \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right ) + \sinh \left (b x^{2} + a\right )^{2} - 1}{2 \, {\left (x \cosh \left (b x^{2} + a\right ) + x \sinh \left (b x^{2} + a\right )\right )}} \] Input:
integrate(sinh(b*x^2+a)/x^2,x, algorithm="fricas")
Output:
-1/2*(sqrt(pi)*(x*cosh(b*x^2 + a)*cosh(a) + x*cosh(b*x^2 + a)*sinh(a) + (x *cosh(a) + x*sinh(a))*sinh(b*x^2 + a))*sqrt(-b)*erf(sqrt(-b)*x) - sqrt(pi) *(x*cosh(b*x^2 + a)*cosh(a) - x*cosh(b*x^2 + a)*sinh(a) + (x*cosh(a) - x*s inh(a))*sinh(b*x^2 + a))*sqrt(b)*erf(sqrt(b)*x) + cosh(b*x^2 + a)^2 + 2*co sh(b*x^2 + a)*sinh(b*x^2 + a) + sinh(b*x^2 + a)^2 - 1)/(x*cosh(b*x^2 + a) + x*sinh(b*x^2 + a))
\[ \int \frac {\sinh \left (a+b x^2\right )}{x^2} \, dx=\int \frac {\sinh {\left (a + b x^{2} \right )}}{x^{2}}\, dx \] Input:
integrate(sinh(b*x**2+a)/x**2,x)
Output:
Integral(sinh(a + b*x**2)/x**2, x)
Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.82 \[ \int \frac {\sinh \left (a+b x^2\right )}{x^2} \, dx=\frac {1}{2} \, {\left (\frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {b} x\right ) e^{\left (-a\right )}}{\sqrt {b}} + \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {-b} x\right ) e^{a}}{\sqrt {-b}}\right )} b - \frac {\sinh \left (b x^{2} + a\right )}{x} \] Input:
integrate(sinh(b*x^2+a)/x^2,x, algorithm="maxima")
Output:
1/2*(sqrt(pi)*erf(sqrt(b)*x)*e^(-a)/sqrt(b) + sqrt(pi)*erf(sqrt(-b)*x)*e^a /sqrt(-b))*b - sinh(b*x^2 + a)/x
\[ \int \frac {\sinh \left (a+b x^2\right )}{x^2} \, dx=\int { \frac {\sinh \left (b x^{2} + a\right )}{x^{2}} \,d x } \] Input:
integrate(sinh(b*x^2+a)/x^2,x, algorithm="giac")
Output:
integrate(sinh(b*x^2 + a)/x^2, x)
Timed out. \[ \int \frac {\sinh \left (a+b x^2\right )}{x^2} \, dx=\int \frac {\mathrm {sinh}\left (b\,x^2+a\right )}{x^2} \,d x \] Input:
int(sinh(a + b*x^2)/x^2,x)
Output:
int(sinh(a + b*x^2)/x^2, x)
\[ \int \frac {\sinh \left (a+b x^2\right )}{x^2} \, dx=\int \frac {\sinh \left (b \,x^{2}+a \right )}{x^{2}}d x \] Input:
int(sinh(b*x^2+a)/x^2,x)
Output:
int(sinh(a + b*x**2)/x**2,x)