3.352 \(\int f^{a+c x^2} \sinh ^2(d+f x^2) \, dx\)

Optimal. Leaf size=128 \[ \frac{\sqrt{\pi } e^{-2 d} f^a \text{Erf}\left (x \sqrt{2 f-c \log (f)}\right )}{8 \sqrt{2 f-c \log (f)}}+\frac{\sqrt{\pi } e^{2 d} f^a \text{Erfi}\left (x \sqrt{c \log (f)+2 f}\right )}{8 \sqrt{c \log (f)+2 f}}-\frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )}{4 \sqrt{c} \sqrt{\log (f)}} \]

[Out]

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

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Rubi [A]  time = 0.203509, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5512, 2204, 2287, 2205} \[ \frac{\sqrt{\pi } e^{-2 d} f^a \text{Erf}\left (x \sqrt{2 f-c \log (f)}\right )}{8 \sqrt{2 f-c \log (f)}}+\frac{\sqrt{\pi } e^{2 d} f^a \text{Erfi}\left (x \sqrt{c \log (f)+2 f}\right )}{8 \sqrt{c \log (f)+2 f}}-\frac{\sqrt{\pi } f^a \text{Erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )}{4 \sqrt{c} \sqrt{\log (f)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5512

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]

Rule 2204

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 2287

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 2205

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]

Rubi steps

\begin{align*} \int f^{a+c x^2} \sinh ^2\left (d+f x^2\right ) \, dx &=\int \left (-\frac{1}{2} f^{a+c x^2}+\frac{1}{4} e^{-2 d-2 f x^2} f^{a+c x^2}+\frac{1}{4} e^{2 d+2 f x^2} f^{a+c x^2}\right ) \, dx\\ &=\frac{1}{4} \int e^{-2 d-2 f x^2} f^{a+c x^2} \, dx+\frac{1}{4} \int e^{2 d+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 e^{-2 d+a \log (f)-x^2 (2 f-c \log (f))} \, dx+\frac{1}{4} \int e^{2 d+a \log (f)+x^2 (2 f+c \log (f))} \, 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} f^a \sqrt{\pi } \text{erf}\left (x \sqrt{2 f-c \log (f)}\right )}{8 \sqrt{2 f-c \log (f)}}+\frac{e^{2 d} f^a \sqrt{\pi } \text{erfi}\left (x \sqrt{2 f+c \log (f)}\right )}{8 \sqrt{2 f+c \log (f)}}\\ \end{align*}

Mathematica [A]  time = 0.53445, size = 179, normalized size = 1.4 \[ \frac{\sqrt{\pi } f^a \left (\left (8 f^2-2 c^2 \log ^2(f)\right ) \text{Erfi}\left (\sqrt{c} x \sqrt{\log (f)}\right )+\sqrt{c} \sqrt{\log (f)} \left (\sqrt{2 f-c \log (f)} (c \log (f)+2 f) (\sinh (2 d)-\cosh (2 d)) \text{Erf}\left (x \sqrt{2 f-c \log (f)}\right )-(2 f-c \log (f)) \sqrt{c \log (f)+2 f} (\sinh (2 d)+\cosh (2 d)) \text{Erfi}\left (x \sqrt{c \log (f)+2 f}\right )\right )\right )}{8 \sqrt{c} \sqrt{\log (f)} \left (c^2 \log ^2(f)-4 f^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.098, size = 101, normalized size = 0.8 \begin{align*}{\frac{\sqrt{\pi }{f}^{a}{{\rm e}^{-2\,d}}}{8}{\it Erf} \left ( x\sqrt{2\,f-c\ln \left ( f \right ) } \right ){\frac{1}{\sqrt{2\,f-c\ln \left ( f \right ) }}}}+{\frac{\sqrt{\pi }{f}^{a}{{\rm e}^{2\,d}}}{8}{\it Erf} \left ( \sqrt{-c\ln \left ( f \right ) -2\,f}x \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -2\,f}}}}-{\frac{\sqrt{\pi }{f}^{a}}{4}{\it Erf} \left ( \sqrt{-c\ln \left ( f \right ) }x \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*Pi^(1/2)*f^a*exp(-2*d)/(2*f-c*ln(f))^(1/2)*erf(x*(2*f-c*ln(f))^(1/2))+1/8*Pi^(1/2)*f^a*exp(2*d)/(-c*ln(f)-
2*f)^(1/2)*erf((-c*ln(f)-2*f)^(1/2)*x)-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 = 1.06907, size = 135, normalized size = 1.05 \begin{align*} \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) - 2 \, f} x\right ) e^{\left (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\right ) e^{\left (-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\right )}{4 \, \sqrt{-c \log \left (f\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*x^2+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)*e^(2*d)/sqrt(-c*log(f) - 2*f) + 1/8*sqrt(pi)*f^a*erf(sqrt(-c*log
(f) + 2*f)*x)*e^(-2*d)/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]  time = 1.91296, size = 720, normalized size = 5.62 \begin{align*} -\frac{{\left (\sqrt{\pi }{\left (c^{2} \log \left (f\right )^{2} + 2 \, c f \log \left (f\right )\right )} \cosh \left (a \log \left (f\right ) - 2 \, d\right ) + \sqrt{\pi }{\left (c^{2} \log \left (f\right )^{2} + 2 \, c f \log \left (f\right )\right )} \sinh \left (a \log \left (f\right ) - 2 \, d\right )\right )} \sqrt{-c \log \left (f\right ) + 2 \, f} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) + 2 \, f} x\right ) +{\left (\sqrt{\pi }{\left (c^{2} \log \left (f\right )^{2} - 2 \, c f \log \left (f\right )\right )} \cosh \left (a \log \left (f\right ) + 2 \, d\right ) + \sqrt{\pi }{\left (c^{2} \log \left (f\right )^{2} - 2 \, c f \log \left (f\right )\right )} \sinh \left (a \log \left (f\right ) + 2 \, d\right )\right )} \sqrt{-c \log \left (f\right ) - 2 \, f} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) - 2 \, f} x\right ) - 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 )}{8 \,{\left (c^{3} \log \left (f\right )^{3} - 4 \, c f^{2} \log \left (f\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*x^2+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(a*log(f) - 2*d) + sqrt(pi)*(c^2*log(f)^2 + 2*c*f*log(f))*si
nh(a*log(f) - 2*d))*sqrt(-c*log(f) + 2*f)*erf(sqrt(-c*log(f) + 2*f)*x) + (sqrt(pi)*(c^2*log(f)^2 - 2*c*f*log(f
))*cosh(a*log(f) + 2*d) + sqrt(pi)*(c^2*log(f)^2 - 2*c*f*log(f))*sinh(a*log(f) + 2*d))*sqrt(-c*log(f) - 2*f)*e
rf(sqrt(-c*log(f) - 2*f)*x) - 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))/(c^3*log(f)^3 - 4*c*f^2*log(f))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.27607, size = 144, normalized size = 1.12 \begin{align*} \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} x\right ) e^{\left (a \log \left (f\right ) + 2 \, d\right )}}{8 \, \sqrt{-c \log \left (f\right ) - 2 \, f}} - \frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-c \log \left (f\right ) + 2 \, f} x\right ) e^{\left (a \log \left (f\right ) - 2 \, d\right )}}{8 \, \sqrt{-c \log \left (f\right ) + 2 \, f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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