Integrand size = 20, antiderivative size = 128 \[ \int f^{a+c x^2} \sinh ^2\left (d+f x^2\right ) \, dx=-\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{-2 d} f^a \sqrt {\pi } \text {erf}\left (x \sqrt {2 f-c \log (f)}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {e^{2 d} f^a \sqrt {\pi } \text {erfi}\left (x \sqrt {2 f+c \log (f)}\right )}{8 \sqrt {2 f+c \log (f)}} \]
-1/4*f^a*erfi(x*c^(1/2)*ln(f)^(1/2))*Pi^(1/2)/c^(1/2)/ln(f)^(1/2)+1/8*f^a* erf(x*(2*f-c*ln(f))^(1/2))*Pi^(1/2)/exp(2*d)/(2*f-c*ln(f))^(1/2)+1/8*exp(2 *d)*f^a*erfi(x*(2*f+c*ln(f))^(1/2))*Pi^(1/2)/(2*f+c*ln(f))^(1/2)
Time = 0.38 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.40 \[ \int f^{a+c x^2} \sinh ^2\left (d+f x^2\right ) \, dx=\frac {f^a \sqrt {\pi } \left (\text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right ) \left (8 f^2-2 c^2 \log ^2(f)\right )+\sqrt {c} \sqrt {\log (f)} \left (\text {erf}\left (x \sqrt {2 f-c \log (f)}\right ) \sqrt {2 f-c \log (f)} (2 f+c \log (f)) (-\cosh (2 d)+\sinh (2 d))-\text {erfi}\left (x \sqrt {2 f+c \log (f)}\right ) (2 f-c \log (f)) \sqrt {2 f+c \log (f)} (\cosh (2 d)+\sinh (2 d))\right )\right )}{8 \sqrt {c} \sqrt {\log (f)} \left (-4 f^2+c^2 \log ^2(f)\right )} \]
(f^a*Sqrt[Pi]*(Erfi[Sqrt[c]*x*Sqrt[Log[f]]]*(8*f^2 - 2*c^2*Log[f]^2) + Sqr t[c]*Sqrt[Log[f]]*(Erf[x*Sqrt[2*f - c*Log[f]]]*Sqrt[2*f - c*Log[f]]*(2*f + c*Log[f])*(-Cosh[2*d] + Sinh[2*d]) - Erfi[x*Sqrt[2*f + c*Log[f]]]*(2*f - c*Log[f])*Sqrt[2*f + c*Log[f]]*(Cosh[2*d] + Sinh[2*d]))))/(8*Sqrt[c]*Sqrt[ Log[f]]*(-4*f^2 + c^2*Log[f]^2))
Time = 0.42 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6038, 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.
\(\displaystyle \int f^{a+c x^2} \sinh ^2\left (d+f x^2\right ) \, dx\) |
\(\Big \downarrow \) 6038 |
\(\displaystyle \int \left (\frac {1}{4} e^{-2 d-2 f x^2} f^{a+c x^2}+\frac {1}{4} e^{2 d+2 f x^2} f^{a+c x^2}-\frac {1}{2} f^{a+c x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {\pi } e^{-2 d} f^a \text {erf}\left (x \sqrt {2 f-c \log (f)}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {\sqrt {\pi } e^{2 d} f^a \text {erfi}\left (x \sqrt {c \log (f)+2 f}\right )}{8 \sqrt {c \log (f)+2 f}}-\frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}\) |
-1/4*(f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(Sqrt[c]*Sqrt[Log[f]]) + (f^a*Sqrt[Pi]*Erf[x*Sqrt[2*f - c*Log[f]]])/(8*E^(2*d)*Sqrt[2*f - c*Log[f]] ) + (E^(2*d)*f^a*Sqrt[Pi]*Erfi[x*Sqrt[2*f + c*Log[f]]])/(8*Sqrt[2*f + c*Lo g[f]])
3.4.52.3.1 Defintions of rubi rules used
Int[(F_)^(u_)*Sinh[v_]^(n_.), x_Symbol] :> Int[ExpandTrigToExp[F^u, Sinh[v] ^n, x], x] /; FreeQ[F, x] && (LinearQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[ v, x] || PolyQ[v, x, 2]) && IGtQ[n, 0]
Time = 0.31 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.79
method | result | size |
risch | \(\frac {f^{a} {\mathrm e}^{-2 d} \sqrt {\pi }\, \operatorname {erf}\left (x \sqrt {2 f -c \ln \left (f \right )}\right )}{8 \sqrt {2 f -c \ln \left (f \right )}}+\frac {f^{a} {\mathrm e}^{2 d} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-c \ln \left (f \right )-2 f}\, x \right )}{8 \sqrt {-c \ln \left (f \right )-2 f}}-\frac {f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-c \ln \left (f \right )}\, x \right )}{4 \sqrt {-c \ln \left (f \right )}}\) | \(101\) |
1/8*f^a*exp(-2*d)*Pi^(1/2)/(2*f-c*ln(f))^(1/2)*erf(x*(2*f-c*ln(f))^(1/2))+ 1/8*f^a*exp(2*d)*Pi^(1/2)/(-c*ln(f)-2*f)^(1/2)*erf((-c*ln(f)-2*f)^(1/2)*x) -1/4*f^a*Pi^(1/2)/(-c*ln(f))^(1/2)*erf((-c*ln(f))^(1/2)*x)
Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (98) = 196\).
Time = 0.29 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.98 \[ \int f^{a+c x^2} \sinh ^2\left (d+f x^2\right ) \, dx=-\frac {{\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 2 \, c f \log \left (f\right )\right )} \cosh \left (a \log \left (f\right ) - 2 \, d\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 2 \, c f \log \left (f\right )\right )} \sinh \left (a \log \left (f\right ) - 2 \, d\right )\right )} \sqrt {-c \log \left (f\right ) + 2 \, f} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + 2 \, f} x\right ) + {\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 2 \, c f \log \left (f\right )\right )} \cosh \left (a \log \left (f\right ) + 2 \, d\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 2 \, c f \log \left (f\right )\right )} \sinh \left (a \log \left (f\right ) + 2 \, d\right )\right )} \sqrt {-c \log \left (f\right ) - 2 \, f} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - 2 \, f} x\right ) - 2 \, {\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 4 \, f^{2}\right )} \cosh \left (a \log \left (f\right )\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 4 \, f^{2}\right )} \sinh \left (a \log \left (f\right )\right )\right )} \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x\right )}{8 \, {\left (c^{3} \log \left (f\right )^{3} - 4 \, c f^{2} \log \left (f\right )\right )}} \]
-1/8*((sqrt(pi)*(c^2*log(f)^2 + 2*c*f*log(f))*cosh(a*log(f) - 2*d) + sqrt( pi)*(c^2*log(f)^2 + 2*c*f*log(f))*sinh(a*log(f) - 2*d))*sqrt(-c*log(f) + 2 *f)*erf(sqrt(-c*log(f) + 2*f)*x) + (sqrt(pi)*(c^2*log(f)^2 - 2*c*f*log(f)) *cosh(a*log(f) + 2*d) + sqrt(pi)*(c^2*log(f)^2 - 2*c*f*log(f))*sinh(a*log( f) + 2*d))*sqrt(-c*log(f) - 2*f)*erf(sqrt(-c*log(f) - 2*f)*x) - 2*(sqrt(pi )*(c^2*log(f)^2 - 4*f^2)*cosh(a*log(f)) + sqrt(pi)*(c^2*log(f)^2 - 4*f^2)* sinh(a*log(f)))*sqrt(-c*log(f))*erf(sqrt(-c*log(f))*x))/(c^3*log(f)^3 - 4* c*f^2*log(f))
\[ \int f^{a+c x^2} \sinh ^2\left (d+f x^2\right ) \, dx=\int f^{a + c x^{2}} \sinh ^{2}{\left (d + f x^{2} \right )}\, dx \]
Time = 0.18 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.78 \[ \int f^{a+c x^2} \sinh ^2\left (d+f x^2\right ) \, dx=\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - 2 \, f} x\right ) e^{\left (2 \, d\right )}}{8 \, \sqrt {-c \log \left (f\right ) - 2 \, f}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + 2 \, f} x\right ) e^{\left (-2 \, d\right )}}{8 \, \sqrt {-c \log \left (f\right ) + 2 \, f}} - \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x\right )}{4 \, \sqrt {-c \log \left (f\right )}} \]
1/8*sqrt(pi)*f^a*erf(sqrt(-c*log(f) - 2*f)*x)*e^(2*d)/sqrt(-c*log(f) - 2*f ) + 1/8*sqrt(pi)*f^a*erf(sqrt(-c*log(f) + 2*f)*x)*e^(-2*d)/sqrt(-c*log(f) + 2*f) - 1/4*sqrt(pi)*f^a*erf(sqrt(-c*log(f))*x)/sqrt(-c*log(f))
Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.84 \[ \int f^{a+c x^2} \sinh ^2\left (d+f x^2\right ) \, dx=\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-c \log \left (f\right )} x\right )}{4 \, \sqrt {-c \log \left (f\right )}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right ) - 2 \, f} x\right ) e^{\left (a \log \left (f\right ) + 2 \, d\right )}}{8 \, \sqrt {-c \log \left (f\right ) - 2 \, f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right ) + 2 \, f} x\right ) e^{\left (a \log \left (f\right ) - 2 \, d\right )}}{8 \, \sqrt {-c \log \left (f\right ) + 2 \, f}} \]
1/4*sqrt(pi)*f^a*erf(-sqrt(-c*log(f))*x)/sqrt(-c*log(f)) - 1/8*sqrt(pi)*er f(-sqrt(-c*log(f) - 2*f)*x)*e^(a*log(f) + 2*d)/sqrt(-c*log(f) - 2*f) - 1/8 *sqrt(pi)*erf(-sqrt(-c*log(f) + 2*f)*x)*e^(a*log(f) - 2*d)/sqrt(-c*log(f) + 2*f)
Timed out. \[ \int f^{a+c x^2} \sinh ^2\left (d+f x^2\right ) \, dx=\int f^{c\,x^2+a}\,{\mathrm {sinh}\left (f\,x^2+d\right )}^2 \,d x \]