3.4.55 \(\int f^{a+c x^2} \sinh ^2(d+e x+f x^2) \, dx\) [355]

Optimal. Leaf size=183 \[ -\frac {f^a \sqrt {\pi } \text {Erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{-2 d+\frac {e^2}{2 f-c \log (f)}} f^a \sqrt {\pi } \text {Erf}\left (\frac {e+x (2 f-c \log (f))}{\sqrt {2 f-c \log (f)}}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {e^{2 d-\frac {e^2}{2 f+c \log (f)}} f^a \sqrt {\pi } \text {Erfi}\left (\frac {e+x (2 f+c \log (f))}{\sqrt {2 f+c \log (f)}}\right )}{8 \sqrt {2 f+c \log (f)}} \]

[Out]

-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*exp(-2*d+e^2/(2*f-c*ln(f)))*f^a*erf((e+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-e^2/(2*f+c*ln(f)))*f^a*erfi((e+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 = 0.25, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5623, 2235, 2325, 2266, 2236} \begin {gather*} \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)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-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]])

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

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Mathematica [A]
time = 1.02, size = 258, normalized size = 1.41 \begin {gather*} \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 )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(E^(e^2/(2*f - c*Log[f]))*f^a*Sqrt[Pi]*(2*E^(e^2/(-2*f + c*Log[f]))*Erfi[Sqrt[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])/Sqr
t[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))

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Maple [A]
time = 1.56, size = 177, normalized size = 0.97

method	result	size
risch	\(\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {2 d \ln \left (f \right ) c -4 d f +e^{2}}{-2 f +c \ln \left (f \right )}} \erf \left (x \sqrt {2 f -c \ln \left (f \right )}+\frac {e}{\sqrt {2 f -c \ln \left (f \right )}}\right )}{8 \sqrt {2 f -c \ln \left (f \right )}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {2 d \ln \left (f \right ) c +4 d f -e^{2}}{2 f +c \ln \left (f \right )}} \erf \left (-\sqrt {-c \ln \left (f \right )-2 f}\, x +\frac {e}{\sqrt {-c \ln \left (f \right )-2 f}}\right )}{8 \sqrt {-c \ln \left (f \right )-2 f}}-\frac {f^{a} \sqrt {\pi }\, \erf \left (\sqrt {-c \ln \left (f \right )}\, x \right )}{4 \sqrt {-c \ln \left (f \right )}}\)	\(177\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 161, normalized size = 0.88 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)/sqrt(-c*log(f))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 482 vs. \(2 (155) = 310\).
time = 0.41, size = 482, normalized size = 2.63 \begin {gather*} \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} + 4 \, d f - \cosh \left (1\right )^{2} - 2 \, {\left (c d + a f\right )} \log \left (f\right ) - 2 \, \cosh \left (1\right ) \sinh \left (1\right ) - \sinh \left (1\right )^{2}}{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} + 4 \, d f - \cosh \left (1\right )^{2} - 2 \, {\left (c d + a f\right )} \log \left (f\right ) - 2 \, \cosh \left (1\right ) \sinh \left (1\right ) - \sinh \left (1\right )^{2}}{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 - \cosh \left (1\right ) - \sinh \left (1\right )\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} + 4 \, d f - \cosh \left (1\right )^{2} + 2 \, {\left (c d + a f\right )} \log \left (f\right ) - 2 \, \cosh \left (1\right ) \sinh \left (1\right ) - \sinh \left (1\right )^{2}}{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} + 4 \, d f - \cosh \left (1\right )^{2} + 2 \, {\left (c d + a f\right )} \log \left (f\right ) - 2 \, \cosh \left (1\right ) \sinh \left (1\right ) - \sinh \left (1\right )^{2}}{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 + \cosh \left (1\right ) + \sinh \left (1\right )\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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) - (sqrt(pi)*(c^2*log(f)^2 + 2*c*f*log(f))*cosh((a*c*log(f)^2 + 4*d*f - cosh(
1)^2 - 2*(c*d + a*f)*log(f) - 2*cosh(1)*sinh(1) - sinh(1)^2)/(c*log(f) - 2*f)) + sqrt(pi)*(c^2*log(f)^2 + 2*c*
f*log(f))*sinh((a*c*log(f)^2 + 4*d*f - cosh(1)^2 - 2*(c*d + a*f)*log(f) - 2*cosh(1)*sinh(1) - sinh(1)^2)/(c*lo
g(f) - 2*f)))*sqrt(-c*log(f) + 2*f)*erf((c*x*log(f) - 2*f*x - cosh(1) - sinh(1))*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 + 4*d*f - cosh(1)^2 + 2*(c*d + a*f)*lo
g(f) - 2*cosh(1)*sinh(1) - sinh(1)^2)/(c*log(f) + 2*f)) + sqrt(pi)*(c^2*log(f)^2 - 2*c*f*log(f))*sinh((a*c*log
(f)^2 + 4*d*f - cosh(1)^2 + 2*(c*d + a*f)*log(f) - 2*cosh(1)*sinh(1) - sinh(1)^2)/(c*log(f) + 2*f)))*sqrt(-c*l
og(f) - 2*f)*erf((c*x*log(f) + 2*f*x + cosh(1) + sinh(1))*sqrt(-c*log(f) - 2*f)/(c*log(f) + 2*f)))/(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 + c x^{2}} \sinh ^{2}{\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+a)*sinh(f*x**2+e*x+d)**2,x)

[Out]

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

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Giac [A]
time = 0.44, size = 198, normalized size = 1.08 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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