Integrand size = 10, antiderivative size = 125 \[ \int \cosh ^3\left (a+b x^2\right ) \, dx=\frac {3 e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} x\right )}{16 \sqrt {b}}+\frac {e^{-3 a} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {b} x\right )}{16 \sqrt {b}}+\frac {3 e^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x\right )}{16 \sqrt {b}}+\frac {e^{3 a} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )}{16 \sqrt {b}} \]
1/48*erf(x*3^(1/2)*b^(1/2))*3^(1/2)*Pi^(1/2)/exp(3*a)/b^(1/2)+1/48*exp(3*a )*erfi(x*3^(1/2)*b^(1/2))*3^(1/2)*Pi^(1/2)/b^(1/2)+3/16*erf(x*b^(1/2))*Pi^ (1/2)/exp(a)/b^(1/2)+3/16*exp(a)*erfi(x*b^(1/2))*Pi^(1/2)/b^(1/2)
Time = 0.10 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.09 \[ \int \cosh ^3\left (a+b x^2\right ) \, dx=\frac {\sqrt {\frac {\pi }{3}} \left (3 \sqrt {3} \cosh (a) \text {erfi}\left (\sqrt {b} x\right )+\cosh (3 a) \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )+3 \sqrt {3} \text {erf}\left (\sqrt {b} x\right ) (\cosh (a)-\sinh (a))+3 \sqrt {3} \text {erfi}\left (\sqrt {b} x\right ) \sinh (a)+\text {erf}\left (\sqrt {3} \sqrt {b} x\right ) (\cosh (3 a)-\sinh (3 a))+\text {erfi}\left (\sqrt {3} \sqrt {b} x\right ) \sinh (3 a)\right )}{16 \sqrt {b}} \]
(Sqrt[Pi/3]*(3*Sqrt[3]*Cosh[a]*Erfi[Sqrt[b]*x] + Cosh[3*a]*Erfi[Sqrt[3]*Sq rt[b]*x] + 3*Sqrt[3]*Erf[Sqrt[b]*x]*(Cosh[a] - Sinh[a]) + 3*Sqrt[3]*Erfi[S qrt[b]*x]*Sinh[a] + Erf[Sqrt[3]*Sqrt[b]*x]*(Cosh[3*a] - Sinh[3*a]) + Erfi[ Sqrt[3]*Sqrt[b]*x]*Sinh[3*a]))/(16*Sqrt[b])
Time = 0.28 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5824, 2009}
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.
\(\displaystyle \int \cosh ^3\left (a+b x^2\right ) \, dx\) |
\(\Big \downarrow \) 5824 |
\(\displaystyle \int \left (\frac {3}{4} \cosh \left (a+b x^2\right )+\frac {1}{4} \cosh \left (3 a+3 b x^2\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 \sqrt {\pi } e^{-a} \text {erf}\left (\sqrt {b} x\right )}{16 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{3}} e^{-3 a} \text {erf}\left (\sqrt {3} \sqrt {b} x\right )}{16 \sqrt {b}}+\frac {3 \sqrt {\pi } e^a \text {erfi}\left (\sqrt {b} x\right )}{16 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{3}} e^{3 a} \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )}{16 \sqrt {b}}\) |
(3*Sqrt[Pi]*Erf[Sqrt[b]*x])/(16*Sqrt[b]*E^a) + (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqr t[b]*x])/(16*Sqrt[b]*E^(3*a)) + (3*E^a*Sqrt[Pi]*Erfi[Sqrt[b]*x])/(16*Sqrt[ b]) + (E^(3*a)*Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[b]*x])/(16*Sqrt[b])
3.1.18.3.1 Defintions of rubi rules used
Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_), x_Symbol] :> Int[Ex pandTrigReduce[(a + b*Cosh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 1] && IGtQ[p, 1]
Time = 0.10 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.69
method | result | size |
risch | \(\frac {{\mathrm e}^{-3 a} \sqrt {\pi }\, \sqrt {3}\, \operatorname {erf}\left (x \sqrt {3}\, \sqrt {b}\right )}{48 \sqrt {b}}+\frac {3 \,\operatorname {erf}\left (x \sqrt {b}\right ) \sqrt {\pi }\, {\mathrm e}^{-a}}{16 \sqrt {b}}+\frac {{\mathrm e}^{3 a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-3 b}\, x \right )}{16 \sqrt {-3 b}}+\frac {3 \,{\mathrm e}^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b}\, x \right )}{16 \sqrt {-b}}\) | \(86\) |
1/48/exp(a)^3*Pi^(1/2)*3^(1/2)/b^(1/2)*erf(x*3^(1/2)*b^(1/2))+3/16*erf(x*b ^(1/2))*Pi^(1/2)/exp(a)/b^(1/2)+1/16*exp(a)^3*Pi^(1/2)/(-3*b)^(1/2)*erf((- 3*b)^(1/2)*x)+3/16*exp(a)*Pi^(1/2)/(-b)^(1/2)*erf((-b)^(1/2)*x)
Time = 0.26 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.90 \[ \int \cosh ^3\left (a+b x^2\right ) \, dx=-\frac {\sqrt {3} \sqrt {\pi } \sqrt {-b} {\left (\cosh \left (3 \, a\right ) + \sinh \left (3 \, a\right )\right )} \operatorname {erf}\left (\sqrt {3} \sqrt {-b} x\right ) - \sqrt {3} \sqrt {\pi } \sqrt {b} {\left (\cosh \left (3 \, a\right ) - \sinh \left (3 \, a\right )\right )} \operatorname {erf}\left (\sqrt {3} \sqrt {b} x\right ) + 9 \, \sqrt {\pi } \sqrt {-b} {\left (\cosh \left (a\right ) + \sinh \left (a\right )\right )} \operatorname {erf}\left (\sqrt {-b} x\right ) - 9 \, \sqrt {\pi } \sqrt {b} {\left (\cosh \left (a\right ) - \sinh \left (a\right )\right )} \operatorname {erf}\left (\sqrt {b} x\right )}{48 \, b} \]
-1/48*(sqrt(3)*sqrt(pi)*sqrt(-b)*(cosh(3*a) + sinh(3*a))*erf(sqrt(3)*sqrt( -b)*x) - sqrt(3)*sqrt(pi)*sqrt(b)*(cosh(3*a) - sinh(3*a))*erf(sqrt(3)*sqrt (b)*x) + 9*sqrt(pi)*sqrt(-b)*(cosh(a) + sinh(a))*erf(sqrt(-b)*x) - 9*sqrt( pi)*sqrt(b)*(cosh(a) - sinh(a))*erf(sqrt(b)*x))/b
\[ \int \cosh ^3\left (a+b x^2\right ) \, dx=\int \cosh ^{3}{\left (a + b x^{2} \right )}\, dx \]
Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.73 \[ \int \cosh ^3\left (a+b x^2\right ) \, dx=\frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\sqrt {3} \sqrt {-b} x\right ) e^{\left (3 \, a\right )}}{48 \, \sqrt {-b}} + \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\sqrt {3} \sqrt {b} x\right ) e^{\left (-3 \, a\right )}}{48 \, \sqrt {b}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {b} x\right ) e^{\left (-a\right )}}{16 \, \sqrt {b}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {-b} x\right ) e^{a}}{16 \, \sqrt {-b}} \]
1/48*sqrt(3)*sqrt(pi)*erf(sqrt(3)*sqrt(-b)*x)*e^(3*a)/sqrt(-b) + 1/48*sqrt (3)*sqrt(pi)*erf(sqrt(3)*sqrt(b)*x)*e^(-3*a)/sqrt(b) + 3/16*sqrt(pi)*erf(s qrt(b)*x)*e^(-a)/sqrt(b) + 3/16*sqrt(pi)*erf(sqrt(-b)*x)*e^a/sqrt(-b)
Time = 0.25 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.76 \[ \int \cosh ^3\left (a+b x^2\right ) \, dx=-\frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {3} \sqrt {-b} x\right ) e^{\left (3 \, a\right )}}{48 \, \sqrt {-b}} - \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {3} \sqrt {b} x\right ) e^{\left (-3 \, a\right )}}{48 \, \sqrt {b}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} x\right ) e^{\left (-a\right )}}{16 \, \sqrt {b}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {-b} x\right ) e^{a}}{16 \, \sqrt {-b}} \]
-1/48*sqrt(3)*sqrt(pi)*erf(-sqrt(3)*sqrt(-b)*x)*e^(3*a)/sqrt(-b) - 1/48*sq rt(3)*sqrt(pi)*erf(-sqrt(3)*sqrt(b)*x)*e^(-3*a)/sqrt(b) - 3/16*sqrt(pi)*er f(-sqrt(b)*x)*e^(-a)/sqrt(b) - 3/16*sqrt(pi)*erf(-sqrt(-b)*x)*e^a/sqrt(-b)
Timed out. \[ \int \cosh ^3\left (a+b x^2\right ) \, dx=\int {\mathrm {cosh}\left (b\,x^2+a\right )}^3 \,d x \]