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

Optimal. Leaf size=81 \[ \frac{\sqrt{\pi } e^d f^a \text{Erfi}\left (x \sqrt{c \log (f)+f}\right )}{4 \sqrt{c \log (f)+f}}-\frac{\sqrt{\pi } e^{-d} f^a \text{Erf}\left (x \sqrt{f-c \log (f)}\right )}{4 \sqrt{f-c \log (f)}} \]

[Out]

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

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Rubi [A]  time = 0.170314, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5512, 2287, 2205, 2204} \[ \frac{\sqrt{\pi } e^d f^a \text{Erfi}\left (x \sqrt{c \log (f)+f}\right )}{4 \sqrt{c \log (f)+f}}-\frac{\sqrt{\pi } e^{-d} f^a \text{Erf}\left (x \sqrt{f-c \log (f)}\right )}{4 \sqrt{f-c \log (f)}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + c*x^2)*Sinh[d + f*x^2],x]

[Out]

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

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]

Rubi steps

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

Mathematica [A]  time = 0.341473, size = 76, normalized size = 0.94 \[ \frac{1}{4} \sqrt{\pi } f^a \left (\frac{(\sinh (d)+\cosh (d)) \text{Erfi}\left (x \sqrt{c \log (f)+f}\right )}{\sqrt{c \log (f)+f}}-\frac{(\cosh (d)-\sinh (d)) \text{Erf}\left (x \sqrt{f-c \log (f)}\right )}{\sqrt{f-c \log (f)}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.084, size = 70, normalized size = 0.9 \begin{align*}{\frac{\sqrt{\pi }{f}^{a}{{\rm e}^{d}}}{4}{\it Erf} \left ( \sqrt{-c\ln \left ( f \right ) -f}x \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -f}}}}-{\frac{\sqrt{\pi }{f}^{a}{{\rm e}^{-d}}}{4}{\it Erf} \left ( x\sqrt{f-c\ln \left ( f \right ) } \right ){\frac{1}{\sqrt{f-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),x)

[Out]

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

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Maxima [A]  time = 1.08368, size = 93, normalized size = 1.15 \begin{align*} -\frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) + f} x\right ) e^{\left (-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\right ) e^{d}}{4 \, \sqrt{-c \log \left (f\right ) - 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),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.90379, size = 410, normalized size = 5.06 \begin{align*} \frac{{\left (\sqrt{\pi }{\left (c \log \left (f\right ) + f\right )} \cosh \left (a \log \left (f\right ) - d\right ) + \sqrt{\pi }{\left (c \log \left (f\right ) + f\right )} \sinh \left (a \log \left (f\right ) - d\right )\right )} \sqrt{-c \log \left (f\right ) + f} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) + f} x\right ) -{\left (\sqrt{\pi }{\left (c \log \left (f\right ) - f\right )} \cosh \left (a \log \left (f\right ) + d\right ) + \sqrt{\pi }{\left (c \log \left (f\right ) - f\right )} \sinh \left (a \log \left (f\right ) + d\right )\right )} \sqrt{-c \log \left (f\right ) - f} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) - f} x\right )}{4 \,{\left (c^{2} \log \left (f\right )^{2} - f^{2}\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),x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [A]  time = 1.288, size = 101, normalized size = 1.25 \begin{align*} -\frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-c \log \left (f\right ) - f} x\right ) e^{\left (a \log \left (f\right ) + d\right )}}{4 \, \sqrt{-c \log \left (f\right ) - f}} + \frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-c \log \left (f\right ) + f} x\right ) e^{\left (a \log \left (f\right ) - d\right )}}{4 \, \sqrt{-c \log \left (f\right ) + 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),x, algorithm="giac")

[Out]

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