Integrand size = 16, antiderivative size = 110 \[ \int f^{a+b x} \sinh \left (d+f x^2\right ) \, dx=-\frac {1}{4} e^{-d+\frac {b^2 \log ^2(f)}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erf}\left (\frac {2 f x-b \log (f)}{2 \sqrt {f}}\right )+\frac {1}{4} e^{d-\frac {b^2 \log ^2(f)}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \text {erfi}\left (\frac {2 f x+b \log (f)}{2 \sqrt {f}}\right ) \] Output:
-1/4*exp(-d+1/4*b^2*ln(f)^2/f)*f^(-1/2+a)*Pi^(1/2)*erf(1/2*(2*f*x-b*ln(f)) /f^(1/2))+1/4*exp(d-1/4*b^2*ln(f)^2/f)*f^(-1/2+a)*Pi^(1/2)*erfi(1/2*(2*f*x +b*ln(f))/f^(1/2))
Time = 0.10 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.94 \[ \int f^{a+b x} \sinh \left (d+f x^2\right ) \, dx=\frac {1}{4} e^{-\frac {b^2 \log ^2(f)}{4 f}} f^{-\frac {1}{2}+a} \sqrt {\pi } \left (-e^{\frac {b^2 \log ^2(f)}{2 f}} \text {erf}\left (\frac {2 f x-b \log (f)}{2 \sqrt {f}}\right ) (\cosh (d)-\sinh (d))+\text {erfi}\left (\frac {2 f x+b \log (f)}{2 \sqrt {f}}\right ) (\cosh (d)+\sinh (d))\right ) \] Input:
Integrate[f^(a + b*x)*Sinh[d + f*x^2],x]
Output:
(f^(-1/2 + a)*Sqrt[Pi]*(-(E^((b^2*Log[f]^2)/(2*f))*Erf[(2*f*x - b*Log[f])/ (2*Sqrt[f])]*(Cosh[d] - Sinh[d])) + Erfi[(2*f*x + b*Log[f])/(2*Sqrt[f])]*( Cosh[d] + Sinh[d])))/(4*E^((b^2*Log[f]^2)/(4*f)))
Time = 0.37 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.
\(\displaystyle \int f^{a+b x} \sinh \left (d+f x^2\right ) \, dx\) |
\(\Big \downarrow \) 6038 |
\(\displaystyle \int \left (\frac {1}{2} e^{d+f x^2} f^{a+b x}-\frac {1}{2} e^{-d-f x^2} f^{a+b x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \sqrt {\pi } f^{a-\frac {1}{2}} e^{d-\frac {b^2 \log ^2(f)}{4 f}} \text {erfi}\left (\frac {b \log (f)+2 f x}{2 \sqrt {f}}\right )-\frac {1}{4} \sqrt {\pi } f^{a-\frac {1}{2}} e^{\frac {b^2 \log ^2(f)}{4 f}-d} \text {erf}\left (\frac {2 f x-b \log (f)}{2 \sqrt {f}}\right )\) |
Input:
Int[f^(a + b*x)*Sinh[d + f*x^2],x]
Output:
-1/4*(E^(-d + (b^2*Log[f]^2)/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[(2*f*x - b*L og[f])/(2*Sqrt[f])]) + (E^(d - (b^2*Log[f]^2)/(4*f))*f^(-1/2 + a)*Sqrt[Pi] *Erfi[(2*f*x + b*Log[f])/(2*Sqrt[f])])/4
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 = 0.14 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.91
method | result | size |
risch | \(-\frac {\operatorname {erf}\left (-\sqrt {-f}\, x +\frac {\ln \left (f \right ) b}{2 \sqrt {-f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}-4 d f}{4 f}}}{4 \sqrt {-f}}+\frac {\operatorname {erf}\left (-\sqrt {f}\, x +\frac {\ln \left (f \right ) b}{2 \sqrt {f}}\right ) \sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {b^{2} \ln \left (f \right )^{2}-4 d f}{4 f}}}{4 \sqrt {f}}\) | \(100\) |
Input:
int(f^(b*x+a)*sinh(f*x^2+d),x,method=_RETURNVERBOSE)
Output:
-1/4*erf(-(-f)^(1/2)*x+1/2*ln(f)*b/(-f)^(1/2))/(-f)^(1/2)*Pi^(1/2)*f^a*exp (-1/4*(b^2*ln(f)^2-4*d*f)/f)+1/4*erf(-f^(1/2)*x+1/2*ln(f)*b/f^(1/2))/f^(1/ 2)*Pi^(1/2)*f^a*exp(1/4*(b^2*ln(f)^2-4*d*f)/f)
Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (84) = 168\).
Time = 0.09 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.94 \[ \int f^{a+b x} \sinh \left (d+f x^2\right ) \, dx=-\frac {\sqrt {\pi } \sqrt {-f} \cosh \left (\frac {b^{2} \log \left (f\right )^{2} - 4 \, a f \log \left (f\right ) - 4 \, d f}{4 \, f}\right ) \operatorname {erf}\left (\frac {{\left (2 \, f x + b \log \left (f\right )\right )} \sqrt {-f}}{2 \, f}\right ) - \sqrt {\pi } \sqrt {f} \cosh \left (\frac {b^{2} \log \left (f\right )^{2} + 4 \, a f \log \left (f\right ) - 4 \, d f}{4 \, f}\right ) \operatorname {erf}\left (-\frac {2 \, f x - b \log \left (f\right )}{2 \, \sqrt {f}}\right ) - \sqrt {\pi } \sqrt {f} \operatorname {erf}\left (-\frac {2 \, f x - b \log \left (f\right )}{2 \, \sqrt {f}}\right ) \sinh \left (\frac {b^{2} \log \left (f\right )^{2} + 4 \, a f \log \left (f\right ) - 4 \, d f}{4 \, f}\right ) - \sqrt {\pi } \sqrt {-f} \operatorname {erf}\left (\frac {{\left (2 \, f x + b \log \left (f\right )\right )} \sqrt {-f}}{2 \, f}\right ) \sinh \left (\frac {b^{2} \log \left (f\right )^{2} - 4 \, a f \log \left (f\right ) - 4 \, d f}{4 \, f}\right )}{4 \, f} \] Input:
integrate(f^(b*x+a)*sinh(f*x^2+d),x, algorithm="fricas")
Output:
-1/4*(sqrt(pi)*sqrt(-f)*cosh(1/4*(b^2*log(f)^2 - 4*a*f*log(f) - 4*d*f)/f)* erf(1/2*(2*f*x + b*log(f))*sqrt(-f)/f) - sqrt(pi)*sqrt(f)*cosh(1/4*(b^2*lo g(f)^2 + 4*a*f*log(f) - 4*d*f)/f)*erf(-1/2*(2*f*x - b*log(f))/sqrt(f)) - s qrt(pi)*sqrt(f)*erf(-1/2*(2*f*x - b*log(f))/sqrt(f))*sinh(1/4*(b^2*log(f)^ 2 + 4*a*f*log(f) - 4*d*f)/f) - sqrt(pi)*sqrt(-f)*erf(1/2*(2*f*x + b*log(f) )*sqrt(-f)/f)*sinh(1/4*(b^2*log(f)^2 - 4*a*f*log(f) - 4*d*f)/f))/f
\[ \int f^{a+b x} \sinh \left (d+f x^2\right ) \, dx=\int f^{a + b x} \sinh {\left (d + f x^{2} \right )}\, dx \] Input:
integrate(f**(b*x+a)*sinh(f*x**2+d),x)
Output:
Integral(f**(a + b*x)*sinh(d + f*x**2), x)
Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.82 \[ \int f^{a+b x} \sinh \left (d+f x^2\right ) \, dx=-\frac {1}{4} \, \sqrt {\pi } f^{a - \frac {1}{2}} \operatorname {erf}\left (\sqrt {f} x - \frac {b \log \left (f\right )}{2 \, \sqrt {f}}\right ) e^{\left (\frac {b^{2} \log \left (f\right )^{2}}{4 \, f} - d\right )} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-f} x - \frac {b \log \left (f\right )}{2 \, \sqrt {-f}}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2}}{4 \, f} + d\right )}}{4 \, \sqrt {-f}} \] Input:
integrate(f^(b*x+a)*sinh(f*x^2+d),x, algorithm="maxima")
Output:
-1/4*sqrt(pi)*f^(a - 1/2)*erf(sqrt(f)*x - 1/2*b*log(f)/sqrt(f))*e^(1/4*b^2 *log(f)^2/f - d) + 1/4*sqrt(pi)*f^a*erf(sqrt(-f)*x - 1/2*b*log(f)/sqrt(-f) )*e^(-1/4*b^2*log(f)^2/f + d)/sqrt(-f)
Time = 0.13 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.96 \[ \int f^{a+b x} \sinh \left (d+f x^2\right ) \, dx=\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {f} {\left (2 \, x - \frac {b \log \left (f\right )}{f}\right )}\right ) e^{\left (\frac {b^{2} \log \left (f\right )^{2} + 4 \, a f \log \left (f\right ) - 4 \, d f}{4 \, f}\right )}}{4 \, \sqrt {f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-f} {\left (2 \, x + \frac {b \log \left (f\right )}{f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a f \log \left (f\right ) - 4 \, d f}{4 \, f}\right )}}{4 \, \sqrt {-f}} \] Input:
integrate(f^(b*x+a)*sinh(f*x^2+d),x, algorithm="giac")
Output:
1/4*sqrt(pi)*erf(-1/2*sqrt(f)*(2*x - b*log(f)/f))*e^(1/4*(b^2*log(f)^2 + 4 *a*f*log(f) - 4*d*f)/f)/sqrt(f) - 1/4*sqrt(pi)*erf(-1/2*sqrt(-f)*(2*x + b* log(f)/f))*e^(-1/4*(b^2*log(f)^2 - 4*a*f*log(f) - 4*d*f)/f)/sqrt(-f)
Timed out. \[ \int f^{a+b x} \sinh \left (d+f x^2\right ) \, dx=\int f^{a+b\,x}\,\mathrm {sinh}\left (f\,x^2+d\right ) \,d x \] Input:
int(f^(a + b*x)*sinh(d + f*x^2),x)
Output:
int(f^(a + b*x)*sinh(d + f*x^2), x)
\[ \int f^{a+b x} \sinh \left (d+f x^2\right ) \, dx=f^{a} \left (\int f^{b x} \sinh \left (f \,x^{2}+d \right )d x \right ) \] Input:
int(f^(b*x+a)*sinh(f*x^2+d),x)
Output:
f**a*int(f**(b*x)*sinh(d + f*x**2),x)