3.4.61 \(\int f^{a+b x+c x^2} \sinh ^2(d+f x^2) \, dx\) [361]

Optimal. Leaf size=225 \[ -\frac {f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {Erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {e^{-2 d+\frac {b^2 \log ^2(f)}{8 f-4 c \log (f)}} f^a \sqrt {\pi } \text {Erf}\left (\frac {b \log (f)-2 x (2 f-c \log (f))}{2 \sqrt {2 f-c \log (f)}}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {e^{2 d-\frac {b^2 \log ^2(f)}{8 f+4 c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {b \log (f)+2 x (2 f+c \log (f))}{2 \sqrt {2 f+c \log (f)}}\right )}{8 \sqrt {2 f+c \log (f)}} \]

[Out]

-1/4*f^(a-1/4*b^2/c)*erfi(1/2*(2*c*x+b)*ln(f)^(1/2)/c^(1/2))*Pi^(1/2)/c^(1/2)/ln(f)^(1/2)-1/8*exp(-2*d+b^2*ln(
f)^2/(8*f-4*c*ln(f)))*f^a*erf(1/2*(b*ln(f)-2*x*(2*f-c*ln(f)))/(2*f-c*ln(f))^(1/2))*Pi^(1/2)/(2*f-c*ln(f))^(1/2
)+1/8*exp(2*d-b^2*ln(f)^2/(8*f+4*c*ln(f)))*f^a*erfi(1/2*(b*ln(f)+2*x*(2*f+c*ln(f)))/(2*f+c*ln(f))^(1/2))*Pi^(1
/2)/(2*f+c*ln(f))^(1/2)

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Rubi [A]
time = 0.29, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5623, 2266, 2235, 2325, 2236} \begin {gather*} -\frac {\sqrt {\pi } f^a e^{\frac {b^2 \log ^2(f)}{8 f-4 c \log (f)}-2 d} \text {Erf}\left (\frac {b \log (f)-2 x (2 f-c \log (f))}{2 \sqrt {2 f-c \log (f)}}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {\sqrt {\pi } f^a e^{2 d-\frac {b^2 \log ^2(f)}{4 c \log (f)+8 f}} \text {Erfi}\left (\frac {b \log (f)+2 x (c \log (f)+2 f)}{2 \sqrt {c \log (f)+2 f}}\right )}{8 \sqrt {c \log (f)+2 f}}-\frac {\sqrt {\pi } f^{a-\frac {b^2}{4 c}} \text {Erfi}\left (\frac {\sqrt {\log (f)} (b+2 c x)}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(Sqrt[c]*Sqrt[Log[f]]) - (E^(-2
*d + (b^2*Log[f]^2)/(8*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(b*Log[f] - 2*x*(2*f - c*Log[f]))/(2*Sqrt[2*f - c*Log
[f]])])/(8*Sqrt[2*f - c*Log[f]]) + (E^(2*d - (b^2*Log[f]^2)/(8*f + 4*c*Log[f]))*f^a*Sqrt[Pi]*Erfi[(b*Log[f] +
2*x*(2*f + c*Log[f]))/(2*Sqrt[2*f + c*Log[f]])])/(8*Sqrt[2*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 ^2\left (d+f x^2\right ) \, dx &=\int \left (-\frac {1}{2} f^{a+b x+c x^2}+\frac {1}{4} e^{-2 d-2 f x^2} f^{a+b x+c x^2}+\frac {1}{4} e^{2 d+2 f x^2} f^{a+b x+c x^2}\right ) \, dx\\ &=\frac {1}{4} \int e^{-2 d-2 f x^2} f^{a+b x+c x^2} \, dx+\frac {1}{4} \int e^{2 d+2 f x^2} f^{a+b x+c x^2} \, dx-\frac {1}{2} \int f^{a+b x+c x^2} \, dx\\ &=\frac {1}{4} \int \exp \left (-2 d+a \log (f)+b x \log (f)-x^2 (2 f-c \log (f))\right ) \, dx+\frac {1}{4} \int \exp \left (2 d+a \log (f)+b x \log (f)+x^2 (2 f+c \log (f))\right ) \, dx-\frac {1}{2} f^{a-\frac {b^2}{4 c}} \int f^{\frac {(b+2 c x)^2}{4 c}} \, dx\\ &=-\frac {f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {1}{4} \left (e^{-2 d+\frac {b^2 \log ^2(f)}{8 f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(b \log (f)+2 x (-2 f+c \log (f)))^2}{4 (-2 f+c \log (f))}\right ) \, dx+\frac {1}{4} \left (e^{2 d-\frac {b^2 \log ^2(f)}{8 f+4 c \log (f)}} f^a\right ) \int \exp \left (\frac {(b \log (f)+2 x (2 f+c \log (f)))^2}{4 (2 f+c \log (f))}\right ) \, dx\\ &=-\frac {f^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{4 \sqrt {c} \sqrt {\log (f)}}-\frac {e^{-2 d+\frac {b^2 \log ^2(f)}{8 f-4 c \log (f)}} f^a \sqrt {\pi } \text {erf}\left (\frac {b \log (f)-2 x (2 f-c \log (f))}{2 \sqrt {2 f-c \log (f)}}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {e^{2 d-\frac {b^2 \log ^2(f)}{8 f+4 c \log (f)}} f^a \sqrt {\pi } \text {erfi}\left (\frac {b \log (f)+2 x (2 f+c \log (f))}{2 \sqrt {2 f+c \log (f)}}\right )}{8 \sqrt {2 f+c \log (f)}}\\ \end {align*}

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Mathematica [A]
time = 1.59, size = 257, normalized size = 1.14 \begin {gather*} \frac {1}{8} f^a \sqrt {\pi } \left (-\frac {2 f^{-\frac {b^2}{4 c}} \text {Erfi}\left (\frac {(b+2 c x) \sqrt {\log (f)}}{2 \sqrt {c}}\right )}{\sqrt {c} \sqrt {\log (f)}}-\frac {e^{-\frac {b^2 \log ^2(f)}{8 f+4 c \log (f)}} \left (e^{\frac {b^2 f \log ^2(f)}{4 f^2-c^2 \log ^2(f)}} \text {Erf}\left (\frac {4 f x-(b+2 c x) \log (f)}{2 \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 (\frac {4 f x+(b+2 c x) \log (f)}{2 \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 )}{-4 f^2+c^2 \log ^2(f)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(f^a*Sqrt[Pi]*((-2*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])])/(Sqrt[c]*f^(b^2/(4*c))*Sqrt[Log[f]]) - (E^((b
^2*f*Log[f]^2)/(4*f^2 - c^2*Log[f]^2))*Erf[(4*f*x - (b + 2*c*x)*Log[f])/(2*Sqrt[2*f - c*Log[f]])]*Sqrt[2*f - c
*Log[f]]*(2*f + c*Log[f])*(Cosh[2*d] - Sinh[2*d]) + Erfi[(4*f*x + (b + 2*c*x)*Log[f])/(2*Sqrt[2*f + c*Log[f]])
]*(2*f - c*Log[f])*Sqrt[2*f + c*Log[f]]*(Cosh[2*d] + Sinh[2*d]))/(E^((b^2*Log[f]^2)/(8*f + 4*c*Log[f]))*(-4*f^
2 + c^2*Log[f]^2))))/8

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Maple [A]
time = 1.49, size = 217, normalized size = 0.96

method	result	size
risch	\(-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}+8 d \ln \left (f \right ) c -16 d f}{4 \left (-2 f +c \ln \left (f \right )\right )}} \erf \left (-x \sqrt {2 f -c \ln \left (f \right )}+\frac {\ln \left (f \right ) b}{2 \sqrt {2 f -c \ln \left (f \right )}}\right )}{8 \sqrt {2 f -c \ln \left (f \right )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}-8 d \ln \left (f \right ) c -16 d f}{4 \left (2 f +c \ln \left (f \right )\right )}} \erf \left (-\sqrt {-c \ln \left (f \right )-2 f}\, x +\frac {\ln \left (f \right ) b}{2 \sqrt {-c \ln \left (f \right )-2 f}}\right )}{8 \sqrt {-c \ln \left (f \right )-2 f}}+\frac {\sqrt {\pi }\, f^{a} f^{-\frac {b^{2}}{4 c}} \erf \left (-\sqrt {-c \ln \left (f \right )}\, x +\frac {b \ln \left (f \right )}{2 \sqrt {-c \ln \left (f \right )}}\right )}{4 \sqrt {-c \ln \left (f \right )}}\)	\(217\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*Pi^(1/2)*f^a*exp(-1/4*(b^2*ln(f)^2+8*d*ln(f)*c-16*d*f)/(-2*f+c*ln(f)))/(2*f-c*ln(f))^(1/2)*erf(-x*(2*f-c*
ln(f))^(1/2)+1/2*ln(f)*b/(2*f-c*ln(f))^(1/2))-1/8*Pi^(1/2)*f^a*exp(-1/4*(b^2*ln(f)^2-8*d*ln(f)*c-16*d*f)/(2*f+
c*ln(f)))/(-c*ln(f)-2*f)^(1/2)*erf(-(-c*ln(f)-2*f)^(1/2)*x+1/2*ln(f)*b/(-c*ln(f)-2*f)^(1/2))+1/4*Pi^(1/2)*f^a*
f^(-1/4*b^2/c)/(-c*ln(f))^(1/2)*erf(-(-c*ln(f))^(1/2)*x+1/2*b*ln(f)/(-c*ln(f))^(1/2))

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Maxima [A]
time = 0.28, size = 199, normalized size = 0.88 \begin {gather*} \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - 2 \, f} x - \frac {b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right ) - 2 \, f}}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2}}{4 \, {\left (c \log \left (f\right ) + 2 \, f\right )}} + 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 - \frac {b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right ) + 2 \, f}}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2}}{4 \, {\left (c \log \left (f\right ) - 2 \, f\right )}} - 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 - \frac {b \log \left (f\right )}{2 \, \sqrt {-c \log \left (f\right )}}\right )}{4 \, \sqrt {-c \log \left (f\right )} f^{\frac {b^{2}}{4 \, c}}} \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+d)^2,x, algorithm="maxima")

[Out]

1/8*sqrt(pi)*f^a*erf(sqrt(-c*log(f) - 2*f)*x - 1/2*b*log(f)/sqrt(-c*log(f) - 2*f))*e^(-1/4*b^2*log(f)^2/(c*log
(f) + 2*f) + 2*d)/sqrt(-c*log(f) - 2*f) + 1/8*sqrt(pi)*f^a*erf(sqrt(-c*log(f) + 2*f)*x - 1/2*b*log(f)/sqrt(-c*
log(f) + 2*f))*e^(-1/4*b^2*log(f)^2/(c*log(f) - 2*f) - 2*d)/sqrt(-c*log(f) + 2*f) - 1/4*sqrt(pi)*f^a*erf(sqrt(
-c*log(f))*x - 1/2*b*log(f)/sqrt(-c*log(f)))/(sqrt(-c*log(f))*f^(1/4*b^2/c))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (185) = 370\).
time = 0.37, size = 466, normalized size = 2.07 \begin {gather*} -\frac {{\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 2 \, c f \log \left (f\right )\right )} \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 16 \, d f + 8 \, {\left (c d + a f\right )} \log \left (f\right )}{4 \, {\left (c \log \left (f\right ) - 2 \, f\right )}}\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 2 \, c f \log \left (f\right )\right )} \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 16 \, d f + 8 \, {\left (c d + a f\right )} \log \left (f\right )}{4 \, {\left (c \log \left (f\right ) - 2 \, f\right )}}\right )\right )} \sqrt {-c \log \left (f\right ) + 2 \, f} \operatorname {erf}\left (-\frac {{\left (4 \, f x - {\left (2 \, c x + b\right )} \log \left (f\right )\right )} \sqrt {-c \log \left (f\right ) + 2 \, f}}{2 \, {\left (c \log \left (f\right ) - 2 \, f\right )}}\right ) + {\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 2 \, c f \log \left (f\right )\right )} \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 16 \, d f - 8 \, {\left (c d + a f\right )} \log \left (f\right )}{4 \, {\left (c \log \left (f\right ) + 2 \, f\right )}}\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 2 \, c f \log \left (f\right )\right )} \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )^{2} - 16 \, d f - 8 \, {\left (c d + a f\right )} \log \left (f\right )}{4 \, {\left (c \log \left (f\right ) + 2 \, f\right )}}\right )\right )} \sqrt {-c \log \left (f\right ) - 2 \, f} \operatorname {erf}\left (\frac {{\left (4 \, f x + {\left (2 \, c x + b\right )} \log \left (f\right )\right )} \sqrt {-c \log \left (f\right ) - 2 \, f}}{2 \, {\left (c \log \left (f\right ) + 2 \, f\right )}}\right ) - 2 \, {\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 4 \, f^{2}\right )} \cosh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )}{4 \, c}\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 4 \, f^{2}\right )} \sinh \left (-\frac {{\left (b^{2} - 4 \, a c\right )} \log \left (f\right )}{4 \, c}\right )\right )} \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c \log \left (f\right )}}{2 \, c}\right )}{8 \, {\left (c^{3} \log \left (f\right )^{3} - 4 \, c f^{2} \log \left (f\right )\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+d)^2,x, algorithm="fricas")

[Out]

-1/8*((sqrt(pi)*(c^2*log(f)^2 + 2*c*f*log(f))*cosh(-1/4*((b^2 - 4*a*c)*log(f)^2 - 16*d*f + 8*(c*d + a*f)*log(f
))/(c*log(f) - 2*f)) + sqrt(pi)*(c^2*log(f)^2 + 2*c*f*log(f))*sinh(-1/4*((b^2 - 4*a*c)*log(f)^2 - 16*d*f + 8*(
c*d + a*f)*log(f))/(c*log(f) - 2*f)))*sqrt(-c*log(f) + 2*f)*erf(-1/2*(4*f*x - (2*c*x + b)*log(f))*sqrt(-c*log(
f) + 2*f)/(c*log(f) - 2*f)) + (sqrt(pi)*(c^2*log(f)^2 - 2*c*f*log(f))*cosh(-1/4*((b^2 - 4*a*c)*log(f)^2 - 16*d
*f - 8*(c*d + a*f)*log(f))/(c*log(f) + 2*f)) + sqrt(pi)*(c^2*log(f)^2 - 2*c*f*log(f))*sinh(-1/4*((b^2 - 4*a*c)
*log(f)^2 - 16*d*f - 8*(c*d + a*f)*log(f))/(c*log(f) + 2*f)))*sqrt(-c*log(f) - 2*f)*erf(1/2*(4*f*x + (2*c*x +
b)*log(f))*sqrt(-c*log(f) - 2*f)/(c*log(f) + 2*f)) - 2*(sqrt(pi)*(c^2*log(f)^2 - 4*f^2)*cosh(-1/4*(b^2 - 4*a*c
)*log(f)/c) + sqrt(pi)*(c^2*log(f)^2 - 4*f^2)*sinh(-1/4*(b^2 - 4*a*c)*log(f)/c))*sqrt(-c*log(f))*erf(1/2*(2*c*
x + b)*sqrt(-c*log(f))/c))/(c^3*log(f)^3 - 4*c*f^2*log(f))

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

[Out]

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

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Giac [A]
time = 0.42, size = 239, normalized size = 1.06 \begin {gather*} -\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) - 2 \, f} {\left (2 \, x + \frac {b \log \left (f\right )}{c \log \left (f\right ) + 2 \, f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} - 8 \, c d \log \left (f\right ) - 8 \, a f \log \left (f\right ) - 16 \, d f}{4 \, {\left (c \log \left (f\right ) + 2 \, f\right )}}\right )}}{8 \, \sqrt {-c \log \left (f\right ) - 2 \, f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right ) + 2 \, f} {\left (2 \, x + \frac {b \log \left (f\right )}{c \log \left (f\right ) - 2 \, f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} + 8 \, c d \log \left (f\right ) + 8 \, a f \log \left (f\right ) - 16 \, d f}{4 \, {\left (c \log \left (f\right ) - 2 \, f\right )}}\right )}}{8 \, \sqrt {-c \log \left (f\right ) + 2 \, f}} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c \log \left (f\right )} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right ) - 4 \, a c \log \left (f\right )}{4 \, c}\right )}}{4 \, \sqrt {-c \log \left (f\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+d)^2,x, algorithm="giac")

[Out]

-1/8*sqrt(pi)*erf(-1/2*sqrt(-c*log(f) - 2*f)*(2*x + b*log(f)/(c*log(f) + 2*f)))*e^(-1/4*(b^2*log(f)^2 - 4*a*c*
log(f)^2 - 8*c*d*log(f) - 8*a*f*log(f) - 16*d*f)/(c*log(f) + 2*f))/sqrt(-c*log(f) - 2*f) - 1/8*sqrt(pi)*erf(-1
/2*sqrt(-c*log(f) + 2*f)*(2*x + b*log(f)/(c*log(f) - 2*f)))*e^(-1/4*(b^2*log(f)^2 - 4*a*c*log(f)^2 + 8*c*d*log
(f) + 8*a*f*log(f) - 16*d*f)/(c*log(f) - 2*f))/sqrt(-c*log(f) + 2*f) + 1/4*sqrt(pi)*erf(-1/2*sqrt(-c*log(f))*(
2*x + b/c))*e^(-1/4*(b^2*log(f) - 4*a*c*log(f))/c)/sqrt(-c*log(f))

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

[Out]

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

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