3.344 \(\int f^{a+b x} \sinh ^3(d+f x^2) \, dx\)

Optimal. Leaf size=239 \[ \frac{3}{16} \sqrt{\pi } f^{a-\frac{1}{2}} e^{\frac{b^2 \log ^2(f)}{4 f}-d} \text{Erf}\left (\frac{2 f x-b \log (f)}{2 \sqrt{f}}\right )-\frac{1}{16} \sqrt{\frac{\pi }{3}} f^{a-\frac{1}{2}} e^{\frac{b^2 \log ^2(f)}{12 f}-3 d} \text{Erf}\left (\frac{6 f x-b \log (f)}{2 \sqrt{3} \sqrt{f}}\right )-\frac{3}{16} \sqrt{\pi } f^{a-\frac{1}{2}} e^{d-\frac{b^2 \log ^2(f)}{4 f}} \text{Erfi}\left (\frac{b \log (f)+2 f x}{2 \sqrt{f}}\right )+\frac{1}{16} \sqrt{\frac{\pi }{3}} f^{a-\frac{1}{2}} e^{3 d-\frac{b^2 \log ^2(f)}{12 f}} \text{Erfi}\left (\frac{b \log (f)+6 f x}{2 \sqrt{3} \sqrt{f}}\right ) \]

[Out]

(3*E^(-d + (b^2*Log[f]^2)/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[(2*f*x - b*Log[f])/(2*Sqrt[f])])/16 - (E^(-3*d + (b
^2*Log[f]^2)/(12*f))*f^(-1/2 + a)*Sqrt[Pi/3]*Erf[(6*f*x - b*Log[f])/(2*Sqrt[3]*Sqrt[f])])/16 - (3*E^(d - (b^2*
Log[f]^2)/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erfi[(2*f*x + b*Log[f])/(2*Sqrt[f])])/16 + (E^(3*d - (b^2*Log[f]^2)/(12
*f))*f^(-1/2 + a)*Sqrt[Pi/3]*Erfi[(6*f*x + b*Log[f])/(2*Sqrt[3]*Sqrt[f])])/16

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Rubi [A]  time = 0.286135, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {5512, 2287, 2234, 2205, 2204} \[ \frac{3}{16} \sqrt{\pi } f^{a-\frac{1}{2}} e^{\frac{b^2 \log ^2(f)}{4 f}-d} \text{Erf}\left (\frac{2 f x-b \log (f)}{2 \sqrt{f}}\right )-\frac{1}{16} \sqrt{\frac{\pi }{3}} f^{a-\frac{1}{2}} e^{\frac{b^2 \log ^2(f)}{12 f}-3 d} \text{Erf}\left (\frac{6 f x-b \log (f)}{2 \sqrt{3} \sqrt{f}}\right )-\frac{3}{16} \sqrt{\pi } f^{a-\frac{1}{2}} e^{d-\frac{b^2 \log ^2(f)}{4 f}} \text{Erfi}\left (\frac{b \log (f)+2 f x}{2 \sqrt{f}}\right )+\frac{1}{16} \sqrt{\frac{\pi }{3}} f^{a-\frac{1}{2}} e^{3 d-\frac{b^2 \log ^2(f)}{12 f}} \text{Erfi}\left (\frac{b \log (f)+6 f x}{2 \sqrt{3} \sqrt{f}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x)*Sinh[d + f*x^2]^3,x]

[Out]

(3*E^(-d + (b^2*Log[f]^2)/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[(2*f*x - b*Log[f])/(2*Sqrt[f])])/16 - (E^(-3*d + (b
^2*Log[f]^2)/(12*f))*f^(-1/2 + a)*Sqrt[Pi/3]*Erf[(6*f*x - b*Log[f])/(2*Sqrt[3]*Sqrt[f])])/16 - (3*E^(d - (b^2*
Log[f]^2)/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erfi[(2*f*x + b*Log[f])/(2*Sqrt[f])])/16 + (E^(3*d - (b^2*Log[f]^2)/(12
*f))*f^(-1/2 + a)*Sqrt[Pi/3]*Erfi[(6*f*x + b*Log[f])/(2*Sqrt[3]*Sqrt[f])])/16

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 2234

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 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+b x} \sinh ^3\left (d+f x^2\right ) \, dx &=\int \left (-\frac{1}{8} e^{-3 d-3 f x^2} f^{a+b x}+\frac{3}{8} e^{-d-f x^2} f^{a+b x}-\frac{3}{8} e^{d+f x^2} f^{a+b x}+\frac{1}{8} e^{3 d+3 f x^2} f^{a+b x}\right ) \, dx\\ &=-\left (\frac{1}{8} \int e^{-3 d-3 f x^2} f^{a+b x} \, dx\right )+\frac{1}{8} \int e^{3 d+3 f x^2} f^{a+b x} \, dx+\frac{3}{8} \int e^{-d-f x^2} f^{a+b x} \, dx-\frac{3}{8} \int e^{d+f x^2} f^{a+b x} \, dx\\ &=-\left (\frac{1}{8} \int e^{-3 d-3 f x^2+a \log (f)+b x \log (f)} \, dx\right )+\frac{1}{8} \int e^{3 d+3 f x^2+a \log (f)+b x \log (f)} \, dx+\frac{3}{8} \int e^{-d-f x^2+a \log (f)+b x \log (f)} \, dx-\frac{3}{8} \int e^{d+f x^2+a \log (f)+b x \log (f)} \, dx\\ &=-\left (\frac{1}{8} \left (3 e^{d-\frac{b^2 \log ^2(f)}{4 f}} f^a\right ) \int e^{\frac{(2 f x+b \log (f))^2}{4 f}} \, dx\right )+\frac{1}{8} \left (e^{3 d-\frac{b^2 \log ^2(f)}{12 f}} f^a\right ) \int e^{\frac{(6 f x+b \log (f))^2}{12 f}} \, dx-\frac{1}{8} \left (e^{-3 d+\frac{b^2 \log ^2(f)}{12 f}} f^a\right ) \int e^{-\frac{(-6 f x+b \log (f))^2}{12 f}} \, dx+\frac{1}{8} \left (3 e^{-d+\frac{b^2 \log ^2(f)}{4 f}} f^a\right ) \int e^{-\frac{(-2 f x+b \log (f))^2}{4 f}} \, dx\\ &=\frac{3}{16} e^{-d+\frac{b^2 \log ^2(f)}{4 f}} f^{-\frac{1}{2}+a} \sqrt{\pi } \text{erf}\left (\frac{2 f x-b \log (f)}{2 \sqrt{f}}\right )-\frac{1}{16} e^{-3 d+\frac{b^2 \log ^2(f)}{12 f}} f^{-\frac{1}{2}+a} \sqrt{\frac{\pi }{3}} \text{erf}\left (\frac{6 f x-b \log (f)}{2 \sqrt{3} \sqrt{f}}\right )-\frac{3}{16} e^{d-\frac{b^2 \log ^2(f)}{4 f}} f^{-\frac{1}{2}+a} \sqrt{\pi } \text{erfi}\left (\frac{2 f x+b \log (f)}{2 \sqrt{f}}\right )+\frac{1}{16} e^{3 d-\frac{b^2 \log ^2(f)}{12 f}} f^{-\frac{1}{2}+a} \sqrt{\frac{\pi }{3}} \text{erfi}\left (\frac{6 f x+b \log (f)}{2 \sqrt{3} \sqrt{f}}\right )\\ \end{align*}

Mathematica [A]  time = 0.417246, size = 287, normalized size = 1.2 \[ \frac{1}{16} \sqrt{\frac{\pi }{3}} f^{a-\frac{1}{2}} e^{-\frac{b^2 \log ^2(f)}{4 f}} \left (3 \sqrt{3} e^{\frac{b^2 \log ^2(f)}{2 f}} (\cosh (d)-\sinh (d)) \text{Erf}\left (\frac{2 f x-b \log (f)}{2 \sqrt{f}}\right )-e^{\frac{b^2 \log ^2(f)}{3 f}} (\cosh (3 d)-\sinh (3 d)) \text{Erf}\left (\frac{6 f x-b \log (f)}{2 \sqrt{3} \sqrt{f}}\right )+\sinh (3 d) e^{\frac{b^2 \log ^2(f)}{6 f}} \text{Erfi}\left (\frac{b \log (f)+6 f x}{2 \sqrt{3} \sqrt{f}}\right )+\cosh (3 d) e^{\frac{b^2 \log ^2(f)}{6 f}} \text{Erfi}\left (\frac{b \log (f)+6 f x}{2 \sqrt{3} \sqrt{f}}\right )-3 \sqrt{3} \sinh (d) \text{Erfi}\left (\frac{b \log (f)+2 f x}{2 \sqrt{f}}\right )-3 \sqrt{3} \cosh (d) \text{Erfi}\left (\frac{b \log (f)+2 f x}{2 \sqrt{f}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x)*Sinh[d + f*x^2]^3,x]

[Out]

(f^(-1/2 + a)*Sqrt[Pi/3]*(-3*Sqrt[3]*Cosh[d]*Erfi[(2*f*x + b*Log[f])/(2*Sqrt[f])] + E^((b^2*Log[f]^2)/(6*f))*C
osh[3*d]*Erfi[(6*f*x + b*Log[f])/(2*Sqrt[3]*Sqrt[f])] + 3*Sqrt[3]*E^((b^2*Log[f]^2)/(2*f))*Erf[(2*f*x - b*Log[
f])/(2*Sqrt[f])]*(Cosh[d] - Sinh[d]) - 3*Sqrt[3]*Erfi[(2*f*x + b*Log[f])/(2*Sqrt[f])]*Sinh[d] - E^((b^2*Log[f]
^2)/(3*f))*Erf[(6*f*x - b*Log[f])/(2*Sqrt[3]*Sqrt[f])]*(Cosh[3*d] - Sinh[3*d]) + E^((b^2*Log[f]^2)/(6*f))*Erfi
[(6*f*x + b*Log[f])/(2*Sqrt[3]*Sqrt[f])]*Sinh[3*d]))/(16*E^((b^2*Log[f]^2)/(4*f)))

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Maple [A]  time = 0.207, size = 207, normalized size = 0.9 \begin{align*} -{\frac{\sqrt{\pi }{f}^{a}}{16}{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}-36\,df}{12\,f}}}}{\it Erf} \left ( -\sqrt{-3\,f}x+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-3\,f}}}} \right ){\frac{1}{\sqrt{-3\,f}}}}+{\frac{\sqrt{\pi }{f}^{a}\sqrt{3}}{48}{{\rm e}^{{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}-36\,df}{12\,f}}}}{\it Erf} \left ( -\sqrt{3}\sqrt{f}x+{\frac{b\ln \left ( f \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{f}}}} \right ){\frac{1}{\sqrt{f}}}}-{\frac{3\,\sqrt{\pi }{f}^{a}}{16}{{\rm e}^{{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}-4\,df}{4\,f}}}}{\it Erf} \left ( -\sqrt{f}x+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{f}}}} \right ){\frac{1}{\sqrt{f}}}}+{\frac{3\,\sqrt{\pi }{f}^{a}}{16}{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}-4\,df}{4\,f}}}}{\it Erf} \left ( -\sqrt{-f}x+{\frac{b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-f}}}} \right ){\frac{1}{\sqrt{-f}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(b*x+a)*sinh(f*x^2+d)^3,x)

[Out]

-1/16*Pi^(1/2)*f^a*exp(-1/12*(ln(f)^2*b^2-36*d*f)/f)/(-3*f)^(1/2)*erf(-(-3*f)^(1/2)*x+1/2*ln(f)*b/(-3*f)^(1/2)
)+1/48*Pi^(1/2)*f^a*exp(1/12*(ln(f)^2*b^2-36*d*f)/f)*3^(1/2)/f^(1/2)*erf(-3^(1/2)*f^(1/2)*x+1/6*ln(f)*b*3^(1/2
)/f^(1/2))-3/16*Pi^(1/2)*f^a*exp(1/4*(ln(f)^2*b^2-4*d*f)/f)/f^(1/2)*erf(-f^(1/2)*x+1/2*ln(f)*b/f^(1/2))+3/16*P
i^(1/2)*f^a*exp(-1/4*(ln(f)^2*b^2-4*d*f)/f)/(-f)^(1/2)*erf(-(-f)^(1/2)*x+1/2*ln(f)*b/(-f)^(1/2))

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Maxima [A]  time = 1.61419, size = 270, normalized size = 1.13 \begin{align*} \frac{3}{16} \, \sqrt{\pi } f^{a - \frac{1}{2}} \operatorname{erf}\left (\sqrt{f} x - \frac{b \log \left (f\right )}{2 \, \sqrt{f}}\right ) e^{\left (\frac{b^{2} \log \left (f\right )^{2}}{4 \, f} - d\right )} - \frac{\sqrt{3} \sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{3} \sqrt{f} x - \frac{\sqrt{3} b \log \left (f\right )}{6 \, \sqrt{f}}\right ) e^{\left (\frac{b^{2} \log \left (f\right )^{2}}{12 \, f} - 3 \, d\right )}}{48 \, \sqrt{f}} + \frac{\sqrt{3} \sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{3} \sqrt{-f} x - \frac{\sqrt{3} b \log \left (f\right )}{6 \, \sqrt{-f}}\right ) e^{\left (-\frac{b^{2} \log \left (f\right )^{2}}{12 \, f} + 3 \, d\right )}}{48 \, \sqrt{-f}} - \frac{3 \, \sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-f} x - \frac{b \log \left (f\right )}{2 \, \sqrt{-f}}\right ) e^{\left (-\frac{b^{2} \log \left (f\right )^{2}}{4 \, f} + d\right )}}{16 \, \sqrt{-f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x+a)*sinh(f*x^2+d)^3,x, algorithm="maxima")

[Out]

3/16*sqrt(pi)*f^(a - 1/2)*erf(sqrt(f)*x - 1/2*b*log(f)/sqrt(f))*e^(1/4*b^2*log(f)^2/f - d) - 1/48*sqrt(3)*sqrt
(pi)*f^a*erf(sqrt(3)*sqrt(f)*x - 1/6*sqrt(3)*b*log(f)/sqrt(f))*e^(1/12*b^2*log(f)^2/f - 3*d)/sqrt(f) + 1/48*sq
rt(3)*sqrt(pi)*f^a*erf(sqrt(3)*sqrt(-f)*x - 1/6*sqrt(3)*b*log(f)/sqrt(-f))*e^(-1/12*b^2*log(f)^2/f + 3*d)/sqrt
(-f) - 3/16*sqrt(pi)*f^a*erf(sqrt(-f)*x - 1/2*b*log(f)/sqrt(-f))*e^(-1/4*b^2*log(f)^2/f + d)/sqrt(-f)

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Fricas [B]  time = 1.91625, size = 1287, normalized size = 5.38 \begin{align*} -\frac{\sqrt{3} \sqrt{\pi } \sqrt{-f} \cosh \left (\frac{b^{2} \log \left (f\right )^{2} - 12 \, a f \log \left (f\right ) - 36 \, d f}{12 \, f}\right ) \operatorname{erf}\left (\frac{\sqrt{3}{\left (6 \, f x + b \log \left (f\right )\right )} \sqrt{-f}}{6 \, f}\right ) - \sqrt{3} \sqrt{\pi } \sqrt{f} \cosh \left (\frac{b^{2} \log \left (f\right )^{2} + 12 \, a f \log \left (f\right ) - 36 \, d f}{12 \, f}\right ) \operatorname{erf}\left (-\frac{\sqrt{3}{\left (6 \, f x - b \log \left (f\right )\right )}}{6 \, \sqrt{f}}\right ) - \sqrt{3} \sqrt{\pi } \sqrt{f} \operatorname{erf}\left (-\frac{\sqrt{3}{\left (6 \, f x - b \log \left (f\right )\right )}}{6 \, \sqrt{f}}\right ) \sinh \left (\frac{b^{2} \log \left (f\right )^{2} + 12 \, a f \log \left (f\right ) - 36 \, d f}{12 \, f}\right ) - \sqrt{3} \sqrt{\pi } \sqrt{-f} \operatorname{erf}\left (\frac{\sqrt{3}{\left (6 \, f x + b \log \left (f\right )\right )} \sqrt{-f}}{6 \, f}\right ) \sinh \left (\frac{b^{2} \log \left (f\right )^{2} - 12 \, a f \log \left (f\right ) - 36 \, d f}{12 \, f}\right ) - 9 \, \sqrt{\pi } \sqrt{-f} \cosh \left (\frac{b^{2} \log \left (f\right )^{2} - 4 \, a f \log \left (f\right ) - 4 \, d f}{4 \, f}\right ) \operatorname{erf}\left (\frac{{\left (2 \, f x + b \log \left (f\right )\right )} \sqrt{-f}}{2 \, f}\right ) + 9 \, \sqrt{\pi } \sqrt{f} \cosh \left (\frac{b^{2} \log \left (f\right )^{2} + 4 \, a f \log \left (f\right ) - 4 \, d f}{4 \, f}\right ) \operatorname{erf}\left (-\frac{2 \, f x - b \log \left (f\right )}{2 \, \sqrt{f}}\right ) + 9 \, \sqrt{\pi } \sqrt{f} \operatorname{erf}\left (-\frac{2 \, f x - b \log \left (f\right )}{2 \, \sqrt{f}}\right ) \sinh \left (\frac{b^{2} \log \left (f\right )^{2} + 4 \, a f \log \left (f\right ) - 4 \, d f}{4 \, f}\right ) + 9 \, \sqrt{\pi } \sqrt{-f} \operatorname{erf}\left (\frac{{\left (2 \, f x + b \log \left (f\right )\right )} \sqrt{-f}}{2 \, f}\right ) \sinh \left (\frac{b^{2} \log \left (f\right )^{2} - 4 \, a f \log \left (f\right ) - 4 \, d f}{4 \, f}\right )}{48 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x+a)*sinh(f*x^2+d)^3,x, algorithm="fricas")

[Out]

-1/48*(sqrt(3)*sqrt(pi)*sqrt(-f)*cosh(1/12*(b^2*log(f)^2 - 12*a*f*log(f) - 36*d*f)/f)*erf(1/6*sqrt(3)*(6*f*x +
b*log(f))*sqrt(-f)/f) - sqrt(3)*sqrt(pi)*sqrt(f)*cosh(1/12*(b^2*log(f)^2 + 12*a*f*log(f) - 36*d*f)/f)*erf(-1/
6*sqrt(3)*(6*f*x - b*log(f))/sqrt(f)) - sqrt(3)*sqrt(pi)*sqrt(f)*erf(-1/6*sqrt(3)*(6*f*x - b*log(f))/sqrt(f))*
sinh(1/12*(b^2*log(f)^2 + 12*a*f*log(f) - 36*d*f)/f) - sqrt(3)*sqrt(pi)*sqrt(-f)*erf(1/6*sqrt(3)*(6*f*x + b*lo
g(f))*sqrt(-f)/f)*sinh(1/12*(b^2*log(f)^2 - 12*a*f*log(f) - 36*d*f)/f) - 9*sqrt(pi)*sqrt(-f)*cosh(1/4*(b^2*log
(f)^2 - 4*a*f*log(f) - 4*d*f)/f)*erf(1/2*(2*f*x + b*log(f))*sqrt(-f)/f) + 9*sqrt(pi)*sqrt(f)*cosh(1/4*(b^2*log
(f)^2 + 4*a*f*log(f) - 4*d*f)/f)*erf(-1/2*(2*f*x - b*log(f))/sqrt(f)) + 9*sqrt(pi)*sqrt(f)*erf(-1/2*(2*f*x - b
*log(f))/sqrt(f))*sinh(1/4*(b^2*log(f)^2 + 4*a*f*log(f) - 4*d*f)/f) + 9*sqrt(pi)*sqrt(-f)*erf(1/2*(2*f*x + b*l
og(f))*sqrt(-f)/f)*sinh(1/4*(b^2*log(f)^2 - 4*a*f*log(f) - 4*d*f)/f))/f

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + b x} \sinh ^{3}{\left (d + f x^{2} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(b*x+a)*sinh(f*x**2+d)**3,x)

[Out]

Integral(f**(a + b*x)*sinh(d + f*x**2)**3, x)

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Giac [A]  time = 1.33959, size = 301, normalized size = 1.26 \begin{align*} \frac{\sqrt{3} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{6} \, \sqrt{3} \sqrt{f}{\left (6 \, x - \frac{b \log \left (f\right )}{f}\right )}\right ) e^{\left (\frac{b^{2} \log \left (f\right )^{2} + 12 \, a f \log \left (f\right ) - 36 \, d f}{12 \, f}\right )}}{48 \, \sqrt{f}} - \frac{\sqrt{3} \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{6} \, \sqrt{3} \sqrt{-f}{\left (6 \, x + \frac{b \log \left (f\right )}{f}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right )^{2} - 12 \, a f \log \left (f\right ) - 36 \, d f}{12 \, f}\right )}}{48 \, \sqrt{-f}} - \frac{3 \, \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{f}{\left (2 \, x - \frac{b \log \left (f\right )}{f}\right )}\right ) e^{\left (\frac{b^{2} \log \left (f\right )^{2} + 4 \, a f \log \left (f\right ) - 4 \, d f}{4 \, f}\right )}}{16 \, \sqrt{f}} + \frac{3 \, \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-f}{\left (2 \, x + \frac{b \log \left (f\right )}{f}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right )^{2} - 4 \, a f \log \left (f\right ) - 4 \, d f}{4 \, f}\right )}}{16 \, \sqrt{-f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x+a)*sinh(f*x^2+d)^3,x, algorithm="giac")

[Out]

1/48*sqrt(3)*sqrt(pi)*erf(-1/6*sqrt(3)*sqrt(f)*(6*x - b*log(f)/f))*e^(1/12*(b^2*log(f)^2 + 12*a*f*log(f) - 36*
d*f)/f)/sqrt(f) - 1/48*sqrt(3)*sqrt(pi)*erf(-1/6*sqrt(3)*sqrt(-f)*(6*x + b*log(f)/f))*e^(-1/12*(b^2*log(f)^2 -
12*a*f*log(f) - 36*d*f)/f)/sqrt(-f) - 3/16*sqrt(pi)*erf(-1/2*sqrt(f)*(2*x - b*log(f)/f))*e^(1/4*(b^2*log(f)^2
+ 4*a*f*log(f) - 4*d*f)/f)/sqrt(f) + 3/16*sqrt(pi)*erf(-1/2*sqrt(-f)*(2*x + b*log(f)/f))*e^(-1/4*(b^2*log(f)^
2 - 4*a*f*log(f) - 4*d*f)/f)/sqrt(-f)