3.4.63 \(\int f^{a+b x+c x^2} \sinh (d+e x+f x^2) \, dx\) [363]

Optimal. Leaf size=161 \[ -\frac {e^{-d+\frac {(e-b \log (f))^2}{4 (f-c \log (f))}} f^a \sqrt {\pi } \text {Erf}\left (\frac {e-b \log (f)+2 x (f-c \log (f))}{2 \sqrt {f-c \log (f)}}\right )}{4 \sqrt {f-c \log (f)}}+\frac {e^{d-\frac {(e+b \log (f))^2}{4 (f+c \log (f))}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {e+b \log (f)+2 x (f+c \log (f))}{2 \sqrt {f+c \log (f)}}\right )}{4 \sqrt {f+c \log (f)}} \]

[Out]

-1/4*exp(-d+1/4*(e-b*ln(f))^2/(f-c*ln(f)))*f^a*erf(1/2*(e-b*ln(f)+2*x*(f-c*ln(f)))/(f-c*ln(f))^(1/2))*Pi^(1/2)
/(f-c*ln(f))^(1/2)+1/4*exp(d-1/4*(e+b*ln(f))^2/(f+c*ln(f)))*f^a*erfi(1/2*(e+b*ln(f)+2*x*(f+c*ln(f)))/(f+c*ln(f
))^(1/2))*Pi^(1/2)/(f+c*ln(f))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.37, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5623, 2325, 2266, 2236, 2235} \begin {gather*} \frac {\sqrt {\pi } f^a e^{d-\frac {(b \log (f)+e)^2}{4 (c \log (f)+f)}} \text {Erfi}\left (\frac {b \log (f)+2 x (c \log (f)+f)+e}{2 \sqrt {c \log (f)+f}}\right )}{4 \sqrt {c \log (f)+f}}-\frac {\sqrt {\pi } f^a e^{\frac {(e-b \log (f))^2}{4 (f-c \log (f))}-d} \text {Erf}\left (\frac {-b \log (f)+2 x (f-c \log (f))+e}{2 \sqrt {f-c \log (f)}}\right )}{4 \sqrt {f-c \log (f)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x + c*x^2)*Sinh[d + e*x + f*x^2],x]

[Out]

-1/4*(E^(-d + (e - b*Log[f])^2/(4*(f - c*Log[f])))*f^a*Sqrt[Pi]*Erf[(e - b*Log[f] + 2*x*(f - c*Log[f]))/(2*Sqr
t[f - c*Log[f]])])/Sqrt[f - c*Log[f]] + (E^(d - (e + b*Log[f])^2/(4*(f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(e + b*
Log[f] + 2*x*(f + c*Log[f]))/(2*Sqrt[f + c*Log[f]])])/(4*Sqrt[f + c*Log[f]])

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 2266

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2325

Int[(u_.)*(F_)^(v_)*(G_)^(w_), x_Symbol] :> With[{z = v*Log[F] + w*Log[G]}, Int[u*NormalizeIntegrand[E^z, x],
x] /; BinomialQ[z, x] || (PolynomialQ[z, x] && LeQ[Exponent[z, x], 2])] /; FreeQ[{F, G}, x]

Rule 5623

Int[(F_)^(u_)*Sinh[v_]^(n_.), x_Symbol] :> Int[ExpandTrigToExp[F^u, Sinh[v]^n, x], x] /; FreeQ[F, x] && (Linea
rQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[n, 0]

Rubi steps

\begin {align*} \int f^{a+b x+c x^2} \sinh \left (d+e x+f x^2\right ) \, dx &=\int \left (-\frac {1}{2} e^{-d-e x-f x^2} f^{a+b x+c x^2}+\frac {1}{2} e^{d+e x+f x^2} f^{a+b x+c x^2}\right ) \, dx\\ &=-\left (\frac {1}{2} \int e^{-d-e x-f x^2} f^{a+b x+c x^2} \, dx\right )+\frac {1}{2} \int e^{d+e x+f x^2} f^{a+b x+c x^2} \, dx\\ &=-\left (\frac {1}{2} \int \exp \left (-d+a \log (f)-x (e-b \log (f))-x^2 (f-c \log (f))\right ) \, dx\right )+\frac {1}{2} \int \exp \left (d+a \log (f)+x (e+b \log (f))+x^2 (f+c \log (f))\right ) \, dx\\ &=-\left (\frac {1}{2} \left (e^{-d+\frac {(e-b \log (f))^2}{4 (f-c \log (f))}} f^a\right ) \int \exp \left (\frac {(-e+b \log (f)+2 x (-f+c \log (f)))^2}{4 (-f+c \log (f))}\right ) \, dx\right )+\frac {1}{2} \left (e^{d-\frac {(e+b \log (f))^2}{4 (f+c \log (f))}} f^a\right ) \int \exp \left (\frac {(e+b \log (f)+2 x (f+c \log (f)))^2}{4 (f+c \log (f))}\right ) \, dx\\ &=-\frac {e^{-d+\frac {(e-b \log (f))^2}{4 (f-c \log (f))}} f^a \sqrt {\pi } \text {erf}\left (\frac {e-b \log (f)+2 x (f-c \log (f))}{2 \sqrt {f-c \log (f)}}\right )}{4 \sqrt {f-c \log (f)}}+\frac {e^{d-\frac {(e+b \log (f))^2}{4 (f+c \log (f))}} f^a \sqrt {\pi } \text {erfi}\left (\frac {e+b \log (f)+2 x (f+c \log (f))}{2 \sqrt {f+c \log (f)}}\right )}{4 \sqrt {f+c \log (f)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.06, size = 252, normalized size = 1.57 \begin {gather*} \frac {e^{-\frac {e^2+b^2 \log ^2(f)}{4 (f+c \log (f))}} f^{a+\frac {b e f}{-f^2+c^2 \log ^2(f)}} \sqrt {\pi } \left (-e^{\frac {f \left (e^2+b^2 \log ^2(f)\right )}{2 \left (f^2-c^2 \log ^2(f)\right )}} f^{\frac {b e}{2 (f+c \log (f))}} \text {Erf}\left (\frac {e+2 f x-(b+2 c x) \log (f)}{2 \sqrt {f-c \log (f)}}\right ) \sqrt {f-c \log (f)} (f+c \log (f)) (\cosh (d)-\sinh (d))+f^{\frac {b e}{2 f-2 c \log (f)}} \text {Erfi}\left (\frac {e+2 f x+(b+2 c x) \log (f)}{2 \sqrt {f+c \log (f)}}\right ) (f-c \log (f)) \sqrt {f+c \log (f)} (\cosh (d)+\sinh (d))\right )}{4 \left (f^2-c^2 \log ^2(f)\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[f^(a + b*x + c*x^2)*Sinh[d + e*x + f*x^2],x]

[Out]

(f^(a + (b*e*f)/(-f^2 + c^2*Log[f]^2))*Sqrt[Pi]*(-(E^((f*(e^2 + b^2*Log[f]^2))/(2*(f^2 - c^2*Log[f]^2)))*f^((b
*e)/(2*(f + c*Log[f])))*Erf[(e + 2*f*x - (b + 2*c*x)*Log[f])/(2*Sqrt[f - c*Log[f]])]*Sqrt[f - c*Log[f]]*(f + c
*Log[f])*(Cosh[d] - Sinh[d])) + f^((b*e)/(2*f - 2*c*Log[f]))*Erfi[(e + 2*f*x + (b + 2*c*x)*Log[f])/(2*Sqrt[f +
c*Log[f]])]*(f - c*Log[f])*Sqrt[f + c*Log[f]]*(Cosh[d] + Sinh[d])))/(4*E^((e^2 + b^2*Log[f]^2)/(4*(f + c*Log[
f])))*(f^2 - c^2*Log[f]^2))

________________________________________________________________________________________

Maple [A]
time = 0.92, size = 186, normalized size = 1.16

method	result	size
risch	\(-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}+2 \ln \left (f \right ) b e -4 d \ln \left (f \right ) c -4 d f +e^{2}}{4 \left (f +c \ln \left (f \right )\right )}} \erf \left (-\sqrt {-c \ln \left (f \right )-f}\, x +\frac {e +b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )-f}}\right )}{4 \sqrt {-c \ln \left (f \right )-f}}+\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}-2 \ln \left (f \right ) b e +4 d \ln \left (f \right ) c -4 d f +e^{2}}{4 \left (-f +c \ln \left (f \right )\right )}} \erf \left (-x \sqrt {f -c \ln \left (f \right )}+\frac {b \ln \left (f \right )-e}{2 \sqrt {f -c \ln \left (f \right )}}\right )}{4 \sqrt {f -c \ln \left (f \right )}}\)	\(186\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c*x^2+b*x+a)*sinh(f*x^2+e*x+d),x,method=_RETURNVERBOSE)

[Out]

-1/4*Pi^(1/2)*f^a*exp(-1/4*(b^2*ln(f)^2+2*ln(f)*b*e-4*d*ln(f)*c-4*d*f+e^2)/(f+c*ln(f)))/(-c*ln(f)-f)^(1/2)*erf
(-(-c*ln(f)-f)^(1/2)*x+1/2*(e+b*ln(f))/(-c*ln(f)-f)^(1/2))+1/4*Pi^(1/2)*f^a*exp(-1/4*(b^2*ln(f)^2-2*ln(f)*b*e+
4*d*ln(f)*c-4*d*f+e^2)/(-f+c*ln(f)))/(f-c*ln(f))^(1/2)*erf(-x*(f-c*ln(f))^(1/2)+1/2*(b*ln(f)-e)/(f-c*ln(f))^(1
/2))

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 155, normalized size = 0.96 \begin {gather*} \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - f} x - \frac {b \log \left (f\right ) + e}{2 \, \sqrt {-c \log \left (f\right ) - f}}\right ) e^{\left (-\frac {{\left (b \log \left (f\right ) + e\right )}^{2}}{4 \, {\left (c \log \left (f\right ) + f\right )}} + d\right )}}{4 \, \sqrt {-c \log \left (f\right ) - f}} - \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + f} x - \frac {b \log \left (f\right ) - e}{2 \, \sqrt {-c \log \left (f\right ) + f}}\right ) e^{\left (-\frac {{\left (b \log \left (f\right ) - e\right )}^{2}}{4 \, {\left (c \log \left (f\right ) - f\right )}} - d\right )}}{4 \, \sqrt {-c \log \left (f\right ) + f}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*x^2+b*x+a)*sinh(f*x^2+e*x+d),x, algorithm="maxima")

[Out]

1/4*sqrt(pi)*f^a*erf(sqrt(-c*log(f) - f)*x - 1/2*(b*log(f) + e)/sqrt(-c*log(f) - f))*e^(-1/4*(b*log(f) + e)^2/
(c*log(f) + f) + d)/sqrt(-c*log(f) - f) - 1/4*sqrt(pi)*f^a*erf(sqrt(-c*log(f) + f)*x - 1/2*(b*log(f) - e)/sqrt
(-c*log(f) + f))*e^(-1/4*(b*log(f) - e)^2/(c*log(f) - f) - d)/sqrt(-c*log(f) + f)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 437 vs. \(2 (143) = 286\).
time = 0.40, size = 437, normalized size = 2.71 \begin {gather*} \frac {{\left (\sqrt {\pi } {\left (c \log \left (f\right ) + f\right )} \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 4 \, d f + \cosh \left (1\right )^{2} + 2 \, {\left (2 \, c d + 2 \, a f - b \cosh \left (1\right ) - b \sinh \left (1\right )\right )} \log \left (f\right ) + 2 \, \cosh \left (1\right ) \sinh \left (1\right ) + \sinh \left (1\right )^{2}}{4 \, {\left (c \log \left (f\right ) - f\right )}}\right ) + \sqrt {\pi } {\left (c \log \left (f\right ) + f\right )} \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 4 \, d f + \cosh \left (1\right )^{2} + 2 \, {\left (2 \, c d + 2 \, a f - b \cosh \left (1\right ) - b \sinh \left (1\right )\right )} \log \left (f\right ) + 2 \, \cosh \left (1\right ) \sinh \left (1\right ) + \sinh \left (1\right )^{2}}{4 \, {\left (c \log \left (f\right ) - f\right )}}\right )\right )} \sqrt {-c \log \left (f\right ) + f} \operatorname {erf}\left (-\frac {{\left (2 \, f x - {\left (2 \, c x + b\right )} \log \left (f\right ) + \cosh \left (1\right ) + \sinh \left (1\right )\right )} \sqrt {-c \log \left (f\right ) + f}}{2 \, {\left (c \log \left (f\right ) - f\right )}}\right ) - {\left (\sqrt {\pi } {\left (c \log \left (f\right ) - f\right )} \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 4 \, d f + \cosh \left (1\right )^{2} - 2 \, {\left (2 \, c d + 2 \, a f - b \cosh \left (1\right ) - b \sinh \left (1\right )\right )} \log \left (f\right ) + 2 \, \cosh \left (1\right ) \sinh \left (1\right ) + \sinh \left (1\right )^{2}}{4 \, {\left (c \log \left (f\right ) + f\right )}}\right ) + \sqrt {\pi } {\left (c \log \left (f\right ) - f\right )} \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 4 \, d f + \cosh \left (1\right )^{2} - 2 \, {\left (2 \, c d + 2 \, a f - b \cosh \left (1\right ) - b \sinh \left (1\right )\right )} \log \left (f\right ) + 2 \, \cosh \left (1\right ) \sinh \left (1\right ) + \sinh \left (1\right )^{2}}{4 \, {\left (c \log \left (f\right ) + f\right )}}\right )\right )} \sqrt {-c \log \left (f\right ) - f} \operatorname {erf}\left (\frac {{\left (2 \, f x + {\left (2 \, c x + b\right )} \log \left (f\right ) + \cosh \left (1\right ) + \sinh \left (1\right )\right )} \sqrt {-c \log \left (f\right ) - f}}{2 \, {\left (c \log \left (f\right ) + f\right )}}\right )}{4 \, {\left (c^{2} \log \left (f\right )^{2} - f^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*x^2+b*x+a)*sinh(f*x^2+e*x+d),x, algorithm="fricas")

[Out]

1/4*((sqrt(pi)*(c*log(f) + f)*cosh(-1/4*((b^2 - 4*a*c)*log(f)^2 - 4*d*f + cosh(1)^2 + 2*(2*c*d + 2*a*f - b*cos
h(1) - b*sinh(1))*log(f) + 2*cosh(1)*sinh(1) + sinh(1)^2)/(c*log(f) - f)) + sqrt(pi)*(c*log(f) + f)*sinh(-1/4*
((b^2 - 4*a*c)*log(f)^2 - 4*d*f + cosh(1)^2 + 2*(2*c*d + 2*a*f - b*cosh(1) - b*sinh(1))*log(f) + 2*cosh(1)*sin
h(1) + sinh(1)^2)/(c*log(f) - f)))*sqrt(-c*log(f) + f)*erf(-1/2*(2*f*x - (2*c*x + b)*log(f) + cosh(1) + sinh(1
))*sqrt(-c*log(f) + f)/(c*log(f) - f)) - (sqrt(pi)*(c*log(f) - f)*cosh(-1/4*((b^2 - 4*a*c)*log(f)^2 - 4*d*f +
cosh(1)^2 - 2*(2*c*d + 2*a*f - b*cosh(1) - b*sinh(1))*log(f) + 2*cosh(1)*sinh(1) + sinh(1)^2)/(c*log(f) + f))
+ sqrt(pi)*(c*log(f) - f)*sinh(-1/4*((b^2 - 4*a*c)*log(f)^2 - 4*d*f + cosh(1)^2 - 2*(2*c*d + 2*a*f - b*cosh(1)
- b*sinh(1))*log(f) + 2*cosh(1)*sinh(1) + sinh(1)^2)/(c*log(f) + f)))*sqrt(-c*log(f) - f)*erf(1/2*(2*f*x + (2
*c*x + b)*log(f) + cosh(1) + sinh(1))*sqrt(-c*log(f) - f)/(c*log(f) + f)))/(c^2*log(f)^2 - f^2)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int f^{a + b x + c x^{2}} \sinh {\left (d + e x + f x^{2} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(c*x**2+b*x+a)*sinh(f*x**2+e*x+d),x)

[Out]

Integral(f**(a + b*x + c*x**2)*sinh(d + e*x + f*x**2), x)

________________________________________________________________________________________

Giac [A]
time = 0.41, size = 207, normalized size = 1.29 \begin {gather*} -\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) - f} {\left (2 \, x + \frac {b \log \left (f\right ) + e}{c \log \left (f\right ) + f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) + 2 \, b e \log \left (f\right ) - 4 \, a f \log \left (f\right ) + e^{2} - 4 \, d f}{4 \, {\left (c \log \left (f\right ) + f\right )}}\right )}}{4 \, \sqrt {-c \log \left (f\right ) - f}} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) + f} {\left (2 \, x + \frac {b \log \left (f\right ) - e}{c \log \left (f\right ) - f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) - 2 \, b e \log \left (f\right ) + 4 \, a f \log \left (f\right ) + e^{2} - 4 \, d f}{4 \, {\left (c \log \left (f\right ) - f\right )}}\right )}}{4 \, \sqrt {-c \log \left (f\right ) + f}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*x^2+b*x+a)*sinh(f*x^2+e*x+d),x, algorithm="giac")

[Out]

-1/4*sqrt(pi)*erf(-1/2*sqrt(-c*log(f) - f)*(2*x + (b*log(f) + e)/(c*log(f) + f)))*e^(-1/4*(b^2*log(f)^2 - 4*a*
c*log(f)^2 - 4*c*d*log(f) + 2*b*e*log(f) - 4*a*f*log(f) + e^2 - 4*d*f)/(c*log(f) + f))/sqrt(-c*log(f) - f) + 1
/4*sqrt(pi)*erf(-1/2*sqrt(-c*log(f) + f)*(2*x + (b*log(f) - e)/(c*log(f) - f)))*e^(-1/4*(b^2*log(f)^2 - 4*a*c*
log(f)^2 + 4*c*d*log(f) - 2*b*e*log(f) + 4*a*f*log(f) + e^2 - 4*d*f)/(c*log(f) - f))/sqrt(-c*log(f) + f)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int f^{c\,x^2+b\,x+a}\,\mathrm {sinh}\left (f\,x^2+e\,x+d\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b*x + c*x^2)*sinh(d + e*x + f*x^2),x)

[Out]

int(f^(a + b*x + c*x^2)*sinh(d + e*x + f*x^2), x)

________________________________________________________________________________________