Integrand size = 23, antiderivative size = 183 \[ \int f^{a+c x^2} \sinh ^2\left (d+e x+f x^2\right ) \, dx=-\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{-2 d+\frac {e^2}{2 f-c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {e+x (2 f-c \log (f))}{\sqrt {2 f-c \log (f)}}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {e^{2 d-\frac {e^2}{2 f+c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e+x (2 f+c \log (f))}{\sqrt {2 f+c \log (f)}}\right )}{8 \sqrt {2 f+c \log (f)}} \] Output:
-1/4*f^a*Pi^(1/2)*erfi(c^(1/2)*x*ln(f)^(1/2))/c^(1/2)/ln(f)^(1/2)+1/8*exp( -2*d+e^2/(2*f-c*ln(f)))*f^a*Pi^(1/2)*erf((e+x*(2*f-c*ln(f)))/(2*f-c*ln(f)) ^(1/2))/(2*f-c*ln(f))^(1/2)+1/8*exp(2*d-e^2/(2*f+c*ln(f)))*f^a*Pi^(1/2)*er fi((e+x*(2*f+c*ln(f)))/(2*f+c*ln(f))^(1/2))/(2*f+c*ln(f))^(1/2)
Time = 0.98 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.41 \[ \int f^{a+c x^2} \sinh ^2\left (d+e x+f x^2\right ) \, dx=\frac {e^{\frac {e^2}{2 f-c \log (f)}} f^a \sqrt {\pi } \left (2 e^{\frac {e^2}{-2 f+c \log (f)}} \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right ) \left (4 f^2-c^2 \log ^2(f)\right )-\sqrt {c} \sqrt {\log (f)} \left (\text {erf}\left (\frac {e+2 f x-c x \log (f)}{\sqrt {2 f-c \log (f)}}\right ) \sqrt {2 f-c \log (f)} (2 f+c \log (f)) (\cosh (2 d)-\sinh (2 d))+e^{\frac {4 e^2 f}{-4 f^2+c^2 \log ^2(f)}} \text {erfi}\left (\frac {e+2 f x+c x \log (f)}{\sqrt {2 f+c \log (f)}}\right ) (2 f-c \log (f)) \sqrt {2 f+c \log (f)} (\cosh (2 d)+\sinh (2 d))\right )\right )}{8 \sqrt {c} \sqrt {\log (f)} \left (-4 f^2+c^2 \log ^2(f)\right )} \] Input:
Integrate[f^(a + c*x^2)*Sinh[d + e*x + f*x^2]^2,x]
Output:
(E^(e^2/(2*f - c*Log[f]))*f^a*Sqrt[Pi]*(2*E^(e^2/(-2*f + c*Log[f]))*Erfi[S qrt[c]*x*Sqrt[Log[f]]]*(4*f^2 - c^2*Log[f]^2) - Sqrt[c]*Sqrt[Log[f]]*(Erf[ (e + 2*f*x - c*x*Log[f])/Sqrt[2*f - c*Log[f]]]*Sqrt[2*f - c*Log[f]]*(2*f + c*Log[f])*(Cosh[2*d] - Sinh[2*d]) + E^((4*e^2*f)/(-4*f^2 + c^2*Log[f]^2)) *Erfi[(e + 2*f*x + c*x*Log[f])/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 = 0.61 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.
\(\displaystyle \int f^{a+c x^2} \sinh ^2\left (d+e x+f x^2\right ) \, dx\) |
\(\Big \downarrow \) 6038 |
\(\displaystyle \int \left (\frac {1}{4} f^{a+c x^2} e^{-2 d-2 e x-2 f x^2}+\frac {1}{4} f^{a+c x^2} e^{2 d+2 e x+2 f x^2}-\frac {1}{2} f^{a+c x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {\pi } f^a e^{\frac {e^2}{2 f-c \log (f)}-2 d} \text {erf}\left (\frac {x (2 f-c \log (f))+e}{\sqrt {2 f-c \log (f)}}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {\sqrt {\pi } f^a e^{2 d-\frac {e^2}{c \log (f)+2 f}} \text {erfi}\left (\frac {x (c \log (f)+2 f)+e}{\sqrt {c \log (f)+2 f}}\right )}{8 \sqrt {c \log (f)+2 f}}-\frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}\) |
Input:
Int[f^(a + c*x^2)*Sinh[d + e*x + f*x^2]^2,x]
Output:
-1/4*(f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(Sqrt[c]*Sqrt[Log[f]]) + (E^(-2*d + e^2/(2*f - c*Log[f]))*f^a*Sqrt[Pi]*Erf[(e + x*(2*f - c*Log[f])) /Sqrt[2*f - c*Log[f]]])/(8*Sqrt[2*f - c*Log[f]]) + (E^(2*d - e^2/(2*f + c* Log[f]))*f^a*Sqrt[Pi]*Erfi[(e + x*(2*f + c*Log[f]))/Sqrt[2*f + c*Log[f]]]) /(8*Sqrt[2*f + c*Log[f]])
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 = 0.53 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.97
method | result | size |
risch | \(\frac {\operatorname {erf}\left (x \sqrt {2 f -c \ln \left (f \right )}+\frac {e}{\sqrt {2 f -c \ln \left (f \right )}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {2 d \ln \left (f \right ) c -4 d f +e^{2}}{c \ln \left (f \right )-2 f}}}{8 \sqrt {2 f -c \ln \left (f \right )}}-\frac {\operatorname {erf}\left (-\sqrt {-c \ln \left (f \right )-2 f}\, x +\frac {e}{\sqrt {-c \ln \left (f \right )-2 f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {2 d \ln \left (f \right ) c +4 d f -e^{2}}{2 f +c \ln \left (f \right )}}}{8 \sqrt {-c \ln \left (f \right )-2 f}}-\frac {f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-c \ln \left (f \right )}\, x \right )}{4 \sqrt {-c \ln \left (f \right )}}\) | \(177\) |
Input:
int(f^(c*x^2+a)*sinh(f*x^2+e*x+d)^2,x,method=_RETURNVERBOSE)
Output:
1/8*erf(x*(2*f-c*ln(f))^(1/2)+e/(2*f-c*ln(f))^(1/2))/(2*f-c*ln(f))^(1/2)*P i^(1/2)*f^a*exp(-(2*d*ln(f)*c-4*d*f+e^2)/(c*ln(f)-2*f))-1/8*erf(-(-c*ln(f) -2*f)^(1/2)*x+e/(-c*ln(f)-2*f)^(1/2))/(-c*ln(f)-2*f)^(1/2)*Pi^(1/2)*f^a*ex p((2*d*ln(f)*c+4*d*f-e^2)/(2*f+c*ln(f)))-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. 422 vs. \(2 (155) = 310\).
Time = 0.11 (sec) , antiderivative size = 422, normalized size of antiderivative = 2.31 \[ \int f^{a+c x^2} \sinh ^2\left (d+e x+f x^2\right ) \, dx=\frac {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 ) - {\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 2 \, c f \log \left (f\right )\right )} \cosh \left (\frac {a c \log \left (f\right )^{2} - e^{2} + 4 \, d f - 2 \, {\left (c d + a f\right )} \log \left (f\right )}{c \log \left (f\right ) - 2 \, f}\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 2 \, c f \log \left (f\right )\right )} \sinh \left (\frac {a c \log \left (f\right )^{2} - e^{2} + 4 \, d f - 2 \, {\left (c d + a f\right )} \log \left (f\right )}{c \log \left (f\right ) - 2 \, f}\right )\right )} \sqrt {-c \log \left (f\right ) + 2 \, f} \operatorname {erf}\left (\frac {{\left (c x \log \left (f\right ) - 2 \, f x - e\right )} \sqrt {-c \log \left (f\right ) + 2 \, f}}{c \log \left (f\right ) - 2 \, f}\right ) - {\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 2 \, c f \log \left (f\right )\right )} \cosh \left (\frac {a c \log \left (f\right )^{2} - e^{2} + 4 \, d f + 2 \, {\left (c d + a f\right )} \log \left (f\right )}{c \log \left (f\right ) + 2 \, f}\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 2 \, c f \log \left (f\right )\right )} \sinh \left (\frac {a c \log \left (f\right )^{2} - e^{2} + 4 \, d f + 2 \, {\left (c d + a f\right )} \log \left (f\right )}{c \log \left (f\right ) + 2 \, f}\right )\right )} \sqrt {-c \log \left (f\right ) - 2 \, f} \operatorname {erf}\left (\frac {{\left (c x \log \left (f\right ) + 2 \, f x + e\right )} \sqrt {-c \log \left (f\right ) - 2 \, f}}{c \log \left (f\right ) + 2 \, f}\right )}{8 \, {\left (c^{3} \log \left (f\right )^{3} - 4 \, c f^{2} \log \left (f\right )\right )}} \] Input:
integrate(f^(c*x^2+a)*sinh(f*x^2+e*x+d)^2,x, algorithm="fricas")
Output:
1/8*(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) - (s qrt(pi)*(c^2*log(f)^2 + 2*c*f*log(f))*cosh((a*c*log(f)^2 - e^2 + 4*d*f - 2 *(c*d + a*f)*log(f))/(c*log(f) - 2*f)) + sqrt(pi)*(c^2*log(f)^2 + 2*c*f*lo g(f))*sinh((a*c*log(f)^2 - e^2 + 4*d*f - 2*(c*d + a*f)*log(f))/(c*log(f) - 2*f)))*sqrt(-c*log(f) + 2*f)*erf((c*x*log(f) - 2*f*x - e)*sqrt(-c*log(f) + 2*f)/(c*log(f) - 2*f)) - (sqrt(pi)*(c^2*log(f)^2 - 2*c*f*log(f))*cosh((a *c*log(f)^2 - e^2 + 4*d*f + 2*(c*d + a*f)*log(f))/(c*log(f) + 2*f)) + sqrt (pi)*(c^2*log(f)^2 - 2*c*f*log(f))*sinh((a*c*log(f)^2 - e^2 + 4*d*f + 2*(c *d + a*f)*log(f))/(c*log(f) + 2*f)))*sqrt(-c*log(f) - 2*f)*erf((c*x*log(f) + 2*f*x + e)*sqrt(-c*log(f) - 2*f)/(c*log(f) + 2*f)))/(c^3*log(f)^3 - 4*c *f^2*log(f))
\[ \int f^{a+c x^2} \sinh ^2\left (d+e x+f x^2\right ) \, dx=\int f^{a + c x^{2}} \sinh ^{2}{\left (d + e x + f x^{2} \right )}\, dx \] Input:
integrate(f**(c*x**2+a)*sinh(f*x**2+e*x+d)**2,x)
Output:
Integral(f**(a + c*x**2)*sinh(d + e*x + f*x**2)**2, x)
Time = 0.04 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.88 \[ \int f^{a+c x^2} \sinh ^2\left (d+e x+f x^2\right ) \, dx=\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - 2 \, f} x - \frac {e}{\sqrt {-c \log \left (f\right ) - 2 \, f}}\right ) e^{\left (2 \, d - \frac {e^{2}}{c \log \left (f\right ) + 2 \, f}\right )}}{8 \, \sqrt {-c \log \left (f\right ) - 2 \, f}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + 2 \, f} x + \frac {e}{\sqrt {-c \log \left (f\right ) + 2 \, f}}\right ) e^{\left (-2 \, d - \frac {e^{2}}{c \log \left (f\right ) - 2 \, f}\right )}}{8 \, \sqrt {-c \log \left (f\right ) + 2 \, f}} - \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x\right )}{4 \, \sqrt {-c \log \left (f\right )}} \] Input:
integrate(f^(c*x^2+a)*sinh(f*x^2+e*x+d)^2,x, algorithm="maxima")
Output:
1/8*sqrt(pi)*f^a*erf(sqrt(-c*log(f) - 2*f)*x - e/sqrt(-c*log(f) - 2*f))*e^ (2*d - e^2/(c*log(f) + 2*f))/sqrt(-c*log(f) - 2*f) + 1/8*sqrt(pi)*f^a*erf( sqrt(-c*log(f) + 2*f)*x + e/sqrt(-c*log(f) + 2*f))*e^(-2*d - e^2/(c*log(f) - 2*f))/sqrt(-c*log(f) + 2*f) - 1/4*sqrt(pi)*f^a*erf(sqrt(-c*log(f))*x)/s qrt(-c*log(f))
Time = 0.14 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.08 \[ \int f^{a+c x^2} \sinh ^2\left (d+e x+f x^2\right ) \, dx=\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-c \log \left (f\right )} x\right )}{4 \, \sqrt {-c \log \left (f\right )}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right ) - 2 \, f} {\left (x + \frac {e}{c \log \left (f\right ) + 2 \, f}\right )}\right ) e^{\left (\frac {a c \log \left (f\right )^{2} + 2 \, c d \log \left (f\right ) + 2 \, a f \log \left (f\right ) - e^{2} + 4 \, d f}{c \log \left (f\right ) + 2 \, f}\right )}}{8 \, \sqrt {-c \log \left (f\right ) - 2 \, f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right ) + 2 \, f} {\left (x - \frac {e}{c \log \left (f\right ) - 2 \, f}\right )}\right ) e^{\left (\frac {a c \log \left (f\right )^{2} - 2 \, c d \log \left (f\right ) - 2 \, a f \log \left (f\right ) - e^{2} + 4 \, d f}{c \log \left (f\right ) - 2 \, f}\right )}}{8 \, \sqrt {-c \log \left (f\right ) + 2 \, f}} \] Input:
integrate(f^(c*x^2+a)*sinh(f*x^2+e*x+d)^2,x, algorithm="giac")
Output:
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/(c*log(f) + 2*f)))*e^((a*c*log(f)^2 + 2*c* d*log(f) + 2*a*f*log(f) - e^2 + 4*d*f)/(c*log(f) + 2*f))/sqrt(-c*log(f) - 2*f) - 1/8*sqrt(pi)*erf(-sqrt(-c*log(f) + 2*f)*(x - e/(c*log(f) - 2*f)))*e ^((a*c*log(f)^2 - 2*c*d*log(f) - 2*a*f*log(f) - e^2 + 4*d*f)/(c*log(f) - 2 *f))/sqrt(-c*log(f) + 2*f)
Timed out. \[ \int f^{a+c x^2} \sinh ^2\left (d+e x+f x^2\right ) \, dx=\int f^{c\,x^2+a}\,{\mathrm {sinh}\left (f\,x^2+e\,x+d\right )}^2 \,d x \] Input:
int(f^(a + c*x^2)*sinh(d + e*x + f*x^2)^2,x)
Output:
int(f^(a + c*x^2)*sinh(d + e*x + f*x^2)^2, x)
\[ \int f^{a+c x^2} \sinh ^2\left (d+e x+f x^2\right ) \, dx=f^{a} \left (\int f^{c \,x^{2}} \sinh \left (f \,x^{2}+e x +d \right )^{2}d x \right ) \] Input:
int(f^(c*x^2+a)*sinh(f*x^2+e*x+d)^2,x)
Output:
f**a*int(f**(c*x**2)*sinh(d + e*x + f*x**2)**2,x)