Integrand size = 14, antiderivative size = 160 \[ \int x^2 \sinh ^3\left (a+b x^2\right ) \, dx=-\frac {3 x \cosh \left (a+b x^2\right )}{8 b}+\frac {x \cosh \left (3 a+3 b x^2\right )}{24 b}+\frac {3 e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} x\right )}{32 b^{3/2}}-\frac {e^{-3 a} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {b} x\right )}{96 b^{3/2}}+\frac {3 e^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x\right )}{32 b^{3/2}}-\frac {e^{3 a} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )}{96 b^{3/2}} \]
-3/8*x*cosh(b*x^2+a)/b+1/24*x*cosh(3*b*x^2+3*a)/b-1/288*erf(x*3^(1/2)*b^(1 /2))*3^(1/2)*Pi^(1/2)/b^(3/2)/exp(3*a)-1/288*exp(3*a)*erfi(x*3^(1/2)*b^(1/ 2))*3^(1/2)*Pi^(1/2)/b^(3/2)+3/32*erf(x*b^(1/2))*Pi^(1/2)/b^(3/2)/exp(a)+3 /32*exp(a)*erfi(x*b^(1/2))*Pi^(1/2)/b^(3/2)
Time = 0.23 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.15 \[ \int x^2 \sinh ^3\left (a+b x^2\right ) \, dx=\frac {-108 \sqrt {b} x \cosh \left (a+b x^2\right )+12 \sqrt {b} x \cosh \left (3 \left (a+b x^2\right )\right )+27 \sqrt {\pi } \cosh (a) \text {erfi}\left (\sqrt {b} x\right )-\sqrt {3 \pi } \cosh (3 a) \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )+27 \sqrt {\pi } \text {erf}\left (\sqrt {b} x\right ) (\cosh (a)-\sinh (a))+27 \sqrt {\pi } \text {erfi}\left (\sqrt {b} x\right ) \sinh (a)-\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {b} x\right ) \sinh (3 a)+\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {b} x\right ) (-\cosh (3 a)+\sinh (3 a))}{288 b^{3/2}} \]
(-108*Sqrt[b]*x*Cosh[a + b*x^2] + 12*Sqrt[b]*x*Cosh[3*(a + b*x^2)] + 27*Sq rt[Pi]*Cosh[a]*Erfi[Sqrt[b]*x] - Sqrt[3*Pi]*Cosh[3*a]*Erfi[Sqrt[3]*Sqrt[b] *x] + 27*Sqrt[Pi]*Erf[Sqrt[b]*x]*(Cosh[a] - Sinh[a]) + 27*Sqrt[Pi]*Erfi[Sq rt[b]*x]*Sinh[a] - Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[b]*x]*Sinh[3*a] + Sqrt[3*P i]*Erf[Sqrt[3]*Sqrt[b]*x]*(-Cosh[3*a] + Sinh[3*a]))/(288*b^(3/2))
Time = 0.33 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5863, 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.
\(\displaystyle \int x^2 \sinh ^3\left (a+b x^2\right ) \, dx\) |
\(\Big \downarrow \) 5863 |
\(\displaystyle \int \left (\frac {1}{4} x^2 \sinh \left (3 a+3 b x^2\right )-\frac {3}{4} x^2 \sinh \left (a+b x^2\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 \sqrt {\pi } e^{-a} \text {erf}\left (\sqrt {b} x\right )}{32 b^{3/2}}-\frac {\sqrt {\frac {\pi }{3}} e^{-3 a} \text {erf}\left (\sqrt {3} \sqrt {b} x\right )}{96 b^{3/2}}+\frac {3 \sqrt {\pi } e^a \text {erfi}\left (\sqrt {b} x\right )}{32 b^{3/2}}-\frac {\sqrt {\frac {\pi }{3}} e^{3 a} \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )}{96 b^{3/2}}-\frac {3 x \cosh \left (a+b x^2\right )}{8 b}+\frac {x \cosh \left (3 a+3 b x^2\right )}{24 b}\) |
(-3*x*Cosh[a + b*x^2])/(8*b) + (x*Cosh[3*a + 3*b*x^2])/(24*b) + (3*Sqrt[Pi ]*Erf[Sqrt[b]*x])/(32*b^(3/2)*E^a) - (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[b]*x])/( 96*b^(3/2)*E^(3*a)) + (3*E^a*Sqrt[Pi]*Erfi[Sqrt[b]*x])/(32*b^(3/2)) - (E^( 3*a)*Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[b]*x])/(96*b^(3/2))
3.1.16.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_), x_Symbol] :> Int[ExpandTrigReduce[(e*x)^m, (a + b*Sinh[c + d*x^n])^p, x], x ] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 1] && IGtQ[n, 0]
Time = 0.67 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.98
method | result | size |
risch | \(\frac {{\mathrm e}^{-3 a} x \,{\mathrm e}^{-3 x^{2} b}}{48 b}-\frac {{\mathrm e}^{-3 a} \sqrt {\pi }\, \sqrt {3}\, \operatorname {erf}\left (x \sqrt {3}\, \sqrt {b}\right )}{288 b^{\frac {3}{2}}}-\frac {3 \,{\mathrm e}^{-a} x \,{\mathrm e}^{-x^{2} b}}{16 b}+\frac {3 \,\operatorname {erf}\left (x \sqrt {b}\right ) \sqrt {\pi }\, {\mathrm e}^{-a}}{32 b^{\frac {3}{2}}}-\frac {3 \,{\mathrm e}^{a} {\mathrm e}^{x^{2} b} x}{16 b}+\frac {3 \,{\mathrm e}^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b}\, x \right )}{32 b \sqrt {-b}}+\frac {{\mathrm e}^{3 a} x \,{\mathrm e}^{3 x^{2} b}}{48 b}-\frac {{\mathrm e}^{3 a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-3 b}\, x \right )}{96 b \sqrt {-3 b}}\) | \(157\) |
1/48/exp(a)^3/b*x*exp(-3*x^2*b)-1/288/exp(a)^3/b^(3/2)*Pi^(1/2)*3^(1/2)*er f(x*3^(1/2)*b^(1/2))-3/16/exp(a)/b*x*exp(-x^2*b)+3/32*erf(x*b^(1/2))*Pi^(1 /2)/b^(3/2)/exp(a)-3/16*exp(a)*exp(x^2*b)*x/b+3/32*exp(a)/b*Pi^(1/2)/(-b)^ (1/2)*erf((-b)^(1/2)*x)+1/48*exp(a)^3/b*x*exp(3*x^2*b)-1/96*exp(a)^3/b*Pi^ (1/2)/(-3*b)^(1/2)*erf((-3*b)^(1/2)*x)
Leaf count of result is larger than twice the leaf count of optimal. 904 vs. \(2 (114) = 228\).
Time = 0.26 (sec) , antiderivative size = 904, normalized size of antiderivative = 5.65 \[ \int x^2 \sinh ^3\left (a+b x^2\right ) \, dx=\text {Too large to display} \]
1/288*(6*b*x*cosh(b*x^2 + a)^6 + 36*b*x*cosh(b*x^2 + a)*sinh(b*x^2 + a)^5 + 6*b*x*sinh(b*x^2 + a)^6 - 54*b*x*cosh(b*x^2 + a)^4 + 18*(5*b*x*cosh(b*x^ 2 + a)^2 - 3*b*x)*sinh(b*x^2 + a)^4 - 54*b*x*cosh(b*x^2 + a)^2 + 24*(5*b*x *cosh(b*x^2 + a)^3 - 9*b*x*cosh(b*x^2 + a))*sinh(b*x^2 + a)^3 + sqrt(3)*sq rt(pi)*(cosh(b*x^2 + a)^3*cosh(3*a) + (cosh(3*a) + sinh(3*a))*sinh(b*x^2 + a)^3 + cosh(b*x^2 + a)^3*sinh(3*a) + 3*(cosh(b*x^2 + a)*cosh(3*a) + cosh( b*x^2 + a)*sinh(3*a))*sinh(b*x^2 + a)^2 + 3*(cosh(b*x^2 + a)^2*cosh(3*a) + cosh(b*x^2 + a)^2*sinh(3*a))*sinh(b*x^2 + a))*sqrt(-b)*erf(sqrt(3)*sqrt(- b)*x) - sqrt(3)*sqrt(pi)*(cosh(b*x^2 + a)^3*cosh(3*a) + (cosh(3*a) - sinh( 3*a))*sinh(b*x^2 + a)^3 - cosh(b*x^2 + a)^3*sinh(3*a) + 3*(cosh(b*x^2 + a) *cosh(3*a) - cosh(b*x^2 + a)*sinh(3*a))*sinh(b*x^2 + a)^2 + 3*(cosh(b*x^2 + a)^2*cosh(3*a) - cosh(b*x^2 + a)^2*sinh(3*a))*sinh(b*x^2 + a))*sqrt(b)*e rf(sqrt(3)*sqrt(b)*x) - 27*sqrt(pi)*(cosh(b*x^2 + a)^3*cosh(a) + (cosh(a) + sinh(a))*sinh(b*x^2 + a)^3 + cosh(b*x^2 + a)^3*sinh(a) + 3*(cosh(b*x^2 + a)*cosh(a) + cosh(b*x^2 + a)*sinh(a))*sinh(b*x^2 + a)^2 + 3*(cosh(b*x^2 + a)^2*cosh(a) + cosh(b*x^2 + a)^2*sinh(a))*sinh(b*x^2 + a))*sqrt(-b)*erf(s qrt(-b)*x) + 27*sqrt(pi)*(cosh(b*x^2 + a)^3*cosh(a) + (cosh(a) - sinh(a))* sinh(b*x^2 + a)^3 - cosh(b*x^2 + a)^3*sinh(a) + 3*(cosh(b*x^2 + a)*cosh(a) - cosh(b*x^2 + a)*sinh(a))*sinh(b*x^2 + a)^2 + 3*(cosh(b*x^2 + a)^2*cosh( a) - cosh(b*x^2 + a)^2*sinh(a))*sinh(b*x^2 + a))*sqrt(b)*erf(sqrt(b)*x)...
\[ \int x^2 \sinh ^3\left (a+b x^2\right ) \, dx=\int x^{2} \sinh ^{3}{\left (a + b x^{2} \right )}\, dx \]
Time = 0.30 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.01 \[ \int x^2 \sinh ^3\left (a+b x^2\right ) \, dx=-\frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\sqrt {3} \sqrt {-b} x\right ) e^{\left (3 \, a\right )}}{288 \, \sqrt {-b} b} - \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\sqrt {3} \sqrt {b} x\right ) e^{\left (-3 \, a\right )}}{288 \, b^{\frac {3}{2}}} + \frac {x e^{\left (3 \, b x^{2} + 3 \, a\right )}}{48 \, b} - \frac {3 \, x e^{\left (b x^{2} + a\right )}}{16 \, b} - \frac {3 \, x e^{\left (-b x^{2} - a\right )}}{16 \, b} + \frac {x e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{48 \, b} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {b} x\right ) e^{\left (-a\right )}}{32 \, b^{\frac {3}{2}}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {-b} x\right ) e^{a}}{32 \, \sqrt {-b} b} \]
-1/288*sqrt(3)*sqrt(pi)*erf(sqrt(3)*sqrt(-b)*x)*e^(3*a)/(sqrt(-b)*b) - 1/2 88*sqrt(3)*sqrt(pi)*erf(sqrt(3)*sqrt(b)*x)*e^(-3*a)/b^(3/2) + 1/48*x*e^(3* b*x^2 + 3*a)/b - 3/16*x*e^(b*x^2 + a)/b - 3/16*x*e^(-b*x^2 - a)/b + 1/48*x *e^(-3*b*x^2 - 3*a)/b + 3/32*sqrt(pi)*erf(sqrt(b)*x)*e^(-a)/b^(3/2) + 3/32 *sqrt(pi)*erf(sqrt(-b)*x)*e^a/(sqrt(-b)*b)
Time = 0.29 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.04 \[ \int x^2 \sinh ^3\left (a+b x^2\right ) \, dx=\frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {3} \sqrt {-b} x\right ) e^{\left (3 \, a\right )}}{288 \, \sqrt {-b} b} + \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {3} \sqrt {b} x\right ) e^{\left (-3 \, a\right )}}{288 \, b^{\frac {3}{2}}} + \frac {x e^{\left (3 \, b x^{2} + 3 \, a\right )}}{48 \, b} - \frac {3 \, x e^{\left (b x^{2} + a\right )}}{16 \, b} - \frac {3 \, x e^{\left (-b x^{2} - a\right )}}{16 \, b} + \frac {x e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{48 \, b} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} x\right ) e^{\left (-a\right )}}{32 \, b^{\frac {3}{2}}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {-b} x\right ) e^{a}}{32 \, \sqrt {-b} b} \]
1/288*sqrt(3)*sqrt(pi)*erf(-sqrt(3)*sqrt(-b)*x)*e^(3*a)/(sqrt(-b)*b) + 1/2 88*sqrt(3)*sqrt(pi)*erf(-sqrt(3)*sqrt(b)*x)*e^(-3*a)/b^(3/2) + 1/48*x*e^(3 *b*x^2 + 3*a)/b - 3/16*x*e^(b*x^2 + a)/b - 3/16*x*e^(-b*x^2 - a)/b + 1/48* x*e^(-3*b*x^2 - 3*a)/b - 3/32*sqrt(pi)*erf(-sqrt(b)*x)*e^(-a)/b^(3/2) - 3/ 32*sqrt(pi)*erf(-sqrt(-b)*x)*e^a/(sqrt(-b)*b)
Timed out. \[ \int x^2 \sinh ^3\left (a+b x^2\right ) \, dx=\int x^2\,{\mathrm {sinh}\left (b\,x^2+a\right )}^3 \,d x \]