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

Optimal. Leaf size=257 \[ \frac{3}{16} \sqrt{\pi } f^{a-\frac{1}{2}} e^{\frac{(e-b \log (f))^2}{4 f}-d} \text{Erf}\left (\frac{-b \log (f)+e+2 f x}{2 \sqrt{f}}\right )-\frac{1}{16} \sqrt{\frac{\pi }{3}} f^{a-\frac{1}{2}} e^{\frac{(3 e-b \log (f))^2}{12 f}-3 d} \text{Erf}\left (\frac{-b \log (f)+3 e+6 f x}{2 \sqrt{3} \sqrt{f}}\right )-\frac{3}{16} \sqrt{\pi } f^{a-\frac{1}{2}} e^{d-\frac{(b \log (f)+e)^2}{4 f}} \text{Erfi}\left (\frac{b \log (f)+e+2 f x}{2 \sqrt{f}}\right )+\frac{1}{16} \sqrt{\frac{\pi }{3}} f^{a-\frac{1}{2}} e^{3 d-\frac{(b \log (f)+3 e)^2}{12 f}} \text{Erfi}\left (\frac{b \log (f)+3 e+6 f x}{2 \sqrt{3} \sqrt{f}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.476691, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {5512, 2287, 2234, 2205, 2204} \[ \frac{3}{16} \sqrt{\pi } f^{a-\frac{1}{2}} e^{\frac{(e-b \log (f))^2}{4 f}-d} \text{Erf}\left (\frac{-b \log (f)+e+2 f x}{2 \sqrt{f}}\right )-\frac{1}{16} \sqrt{\frac{\pi }{3}} f^{a-\frac{1}{2}} e^{\frac{(3 e-b \log (f))^2}{12 f}-3 d} \text{Erf}\left (\frac{-b \log (f)+3 e+6 f x}{2 \sqrt{3} \sqrt{f}}\right )-\frac{3}{16} \sqrt{\pi } f^{a-\frac{1}{2}} e^{d-\frac{(b \log (f)+e)^2}{4 f}} \text{Erfi}\left (\frac{b \log (f)+e+2 f x}{2 \sqrt{f}}\right )+\frac{1}{16} \sqrt{\frac{\pi }{3}} f^{a-\frac{1}{2}} e^{3 d-\frac{(b \log (f)+3 e)^2}{12 f}} \text{Erfi}\left (\frac{b \log (f)+3 e+6 f x}{2 \sqrt{3} \sqrt{f}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*E^(-d + (e - b*Log[f])^2/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erf[(e + 2*f*x - b*Log[f])/(2*Sqrt[f])])/16 - (E^(-3*
d + (3*e - b*Log[f])^2/(12*f))*f^(-1/2 + a)*Sqrt[Pi/3]*Erf[(3*e + 6*f*x - b*Log[f])/(2*Sqrt[3]*Sqrt[f])])/16 -
(3*E^(d - (e + b*Log[f])^2/(4*f))*f^(-1/2 + a)*Sqrt[Pi]*Erfi[(e + 2*f*x + b*Log[f])/(2*Sqrt[f])])/16 + (E^(3*
d - (3*e + b*Log[f])^2/(12*f))*f^(-1/2 + a)*Sqrt[Pi/3]*Erfi[(3*e + 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+e x+f x^2\right ) \, dx &=\int \left (-\frac{1}{8} e^{-3 \left (d+e x+f x^2\right )} f^{a+b x}+\frac{3}{8} \exp \left (2 d+2 e x+2 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x}-\frac{3}{8} \exp \left (4 d+4 e x+4 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x}+\frac{1}{8} \exp \left (6 d+6 e x+6 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x}\right ) \, dx\\ &=-\left (\frac{1}{8} \int e^{-3 \left (d+e x+f x^2\right )} f^{a+b x} \, dx\right )+\frac{1}{8} \int \exp \left (6 d+6 e x+6 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x} \, dx+\frac{3}{8} \int \exp \left (2 d+2 e x+2 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x} \, dx-\frac{3}{8} \int \exp \left (4 d+4 e x+4 f x^2-3 \left (d+e x+f x^2\right )\right ) f^{a+b x} \, dx\\ &=-\left (\frac{1}{8} \int \exp \left (-3 d-3 f x^2+a \log (f)-x (3 e-b \log (f))\right ) \, dx\right )+\frac{1}{8} \int \exp \left (3 d+3 f x^2+a \log (f)+x (3 e+b \log (f))\right ) \, dx+\frac{3}{8} \int e^{-d-f x^2+a \log (f)-x (e-b \log (f))} \, dx-\frac{3}{8} \int e^{d+f x^2+a \log (f)+x (e+b \log (f))} \, dx\\ &=\frac{1}{8} \left (3 e^{-d+\frac{(e-b \log (f))^2}{4 f}} f^a\right ) \int e^{-\frac{(-e-2 f x+b \log (f))^2}{4 f}} \, dx-\frac{1}{8} \left (e^{-3 d+\frac{(3 e-b \log (f))^2}{12 f}} f^a\right ) \int e^{-\frac{(-3 e-6 f x+b \log (f))^2}{12 f}} \, dx-\frac{1}{8} \left (3 e^{d-\frac{(e+b \log (f))^2}{4 f}} f^a\right ) \int e^{\frac{(e+2 f x+b \log (f))^2}{4 f}} \, dx+\frac{1}{8} \left (e^{3 d-\frac{(3 e+b \log (f))^2}{12 f}} f^a\right ) \int e^{\frac{(3 e+6 f x+b \log (f))^2}{12 f}} \, dx\\ &=\frac{3}{16} e^{-d+\frac{(e-b \log (f))^2}{4 f}} f^{-\frac{1}{2}+a} \sqrt{\pi } \text{erf}\left (\frac{e+2 f x-b \log (f)}{2 \sqrt{f}}\right )-\frac{1}{16} e^{-3 d+\frac{(3 e-b \log (f))^2}{12 f}} f^{-\frac{1}{2}+a} \sqrt{\frac{\pi }{3}} \text{erf}\left (\frac{3 e+6 f x-b \log (f)}{2 \sqrt{3} \sqrt{f}}\right )-\frac{3}{16} e^{d-\frac{(e+b \log (f))^2}{4 f}} f^{-\frac{1}{2}+a} \sqrt{\pi } \text{erfi}\left (\frac{e+2 f x+b \log (f)}{2 \sqrt{f}}\right )+\frac{1}{16} e^{3 d-\frac{(3 e+b \log (f))^2}{12 f}} f^{-\frac{1}{2}+a} \sqrt{\frac{\pi }{3}} \text{erfi}\left (\frac{3 e+6 f x+b \log (f)}{2 \sqrt{3} \sqrt{f}}\right )\\ \end{align*}

Mathematica [A]  time = 0.767054, size = 354, normalized size = 1.38 \[ \frac{1}{16} \sqrt{\frac{\pi }{3}} f^{a-\frac{b e+f}{2 f}} e^{-\frac{b^2 \log ^2(f)+3 e^2}{4 f}} \left (3 \sqrt{3} (\cosh (d)-\sinh (d)) e^{\frac{b^2 \log ^2(f)+2 e^2}{2 f}} \text{Erf}\left (\frac{-b \log (f)+e+2 f x}{2 \sqrt{f}}\right )-(\cosh (3 d)-\sinh (3 d)) e^{\frac{2 b^2 \log ^2(f)+9 e^2}{6 f}} \text{Erf}\left (\frac{-b \log (f)+3 e+6 f x}{2 \sqrt{3} \sqrt{f}}\right )+\sinh (3 d) e^{\frac{b^2 \log ^2(f)}{6 f}} \text{Erfi}\left (\frac{b \log (f)+3 e+6 f x}{2 \sqrt{3} \sqrt{f}}\right )+\cosh (3 d) e^{\frac{b^2 \log ^2(f)}{6 f}} \text{Erfi}\left (\frac{b \log (f)+3 e+6 f x}{2 \sqrt{3} \sqrt{f}}\right )-3 \sqrt{3} \sinh (d) e^{\frac{e^2}{2 f}} \text{Erfi}\left (\frac{b \log (f)+e+2 f x}{2 \sqrt{f}}\right )-3 \sqrt{3} \cosh (d) e^{\frac{e^2}{2 f}} \text{Erfi}\left (\frac{b \log (f)+e+2 f x}{2 \sqrt{f}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(f^(a - (b*e + f)/(2*f))*Sqrt[Pi/3]*(-3*Sqrt[3]*E^(e^2/(2*f))*Cosh[d]*Erfi[(e + 2*f*x + b*Log[f])/(2*Sqrt[f])]
+ E^((b^2*Log[f]^2)/(6*f))*Cosh[3*d]*Erfi[(3*e + 6*f*x + b*Log[f])/(2*Sqrt[3]*Sqrt[f])] + 3*Sqrt[3]*E^((2*e^2
+ b^2*Log[f]^2)/(2*f))*Erf[(e + 2*f*x - b*Log[f])/(2*Sqrt[f])]*(Cosh[d] - Sinh[d]) - 3*Sqrt[3]*E^(e^2/(2*f))*
Erfi[(e + 2*f*x + b*Log[f])/(2*Sqrt[f])]*Sinh[d] - E^((9*e^2 + 2*b^2*Log[f]^2)/(6*f))*Erf[(3*e + 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[(3*e + 6*f*x + b*Log[f])/(2*
Sqrt[3]*Sqrt[f])]*Sinh[3*d]))/(16*E^((3*e^2 + b^2*Log[f]^2)/(4*f)))

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Maple [A]  time = 0.179, size = 265, normalized size = 1. \begin{align*} -{\frac{\sqrt{\pi }{f}^{a}}{16}{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}+6\,\ln \left ( f \right ) be-36\,df+9\,{e}^{2}}{12\,f}}}}{\it Erf} \left ( -\sqrt{-3\,f}x+{\frac{3\,e+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}-6\,\ln \left ( f \right ) be-36\,df+9\,{e}^{2}}{12\,f}}}}{\it Erf} \left ( -\sqrt{3}\sqrt{f}x+{\frac{ \left ( b\ln \left ( f \right ) -3\,e \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}-2\,\ln \left ( f \right ) be-4\,df+{e}^{2}}{4\,f}}}}{\it Erf} \left ( -\sqrt{f}x+{\frac{b\ln \left ( f \right ) -e}{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}+2\,\ln \left ( f \right ) be-4\,df+{e}^{2}}{4\,f}}}}{\it Erf} \left ( -\sqrt{-f}x+{\frac{e+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+e*x+d)^3,x)

[Out]

-1/16*Pi^(1/2)*f^a*exp(-1/12*(ln(f)^2*b^2+6*ln(f)*b*e-36*d*f+9*e^2)/f)/(-3*f)^(1/2)*erf(-(-3*f)^(1/2)*x+1/2*(3
*e+b*ln(f))/(-3*f)^(1/2))+1/48*Pi^(1/2)*f^a*exp(1/12*(ln(f)^2*b^2-6*ln(f)*b*e-36*d*f+9*e^2)/f)*3^(1/2)/f^(1/2)
*erf(-3^(1/2)*f^(1/2)*x+1/6*(b*ln(f)-3*e)*3^(1/2)/f^(1/2))-3/16*Pi^(1/2)*f^a*exp(1/4*(ln(f)^2*b^2-2*ln(f)*b*e-
4*d*f+e^2)/f)/f^(1/2)*erf(-f^(1/2)*x+1/2*(b*ln(f)-e)/f^(1/2))+3/16*Pi^(1/2)*f^a*exp(-1/4*(ln(f)^2*b^2+2*ln(f)*
b*e-4*d*f+e^2)/f)/(-f)^(1/2)*erf(-(-f)^(1/2)*x+1/2*(e+b*ln(f))/(-f)^(1/2))

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Maxima [A]  time = 1.58316, size = 308, normalized size = 1.2 \begin{align*} \frac{\sqrt{3} \sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{3} \sqrt{-f} x - \frac{\sqrt{3}{\left (b \log \left (f\right ) + 3 \, e\right )}}{6 \, \sqrt{-f}}\right ) e^{\left (3 \, d - \frac{{\left (b \log \left (f\right ) + 3 \, e\right )}^{2}}{12 \, f}\right )}}{48 \, \sqrt{-f}} + \frac{3}{16} \, \sqrt{\pi } f^{a - \frac{1}{2}} \operatorname{erf}\left (\sqrt{f} x - \frac{b \log \left (f\right ) - e}{2 \, \sqrt{f}}\right ) e^{\left (-d + \frac{{\left (b \log \left (f\right ) - e\right )}^{2}}{4 \, f}\right )} - \frac{\sqrt{3} \sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{3} \sqrt{f} x - \frac{\sqrt{3}{\left (b \log \left (f\right ) - 3 \, e\right )}}{6 \, \sqrt{f}}\right ) e^{\left (-3 \, d + \frac{{\left (b \log \left (f\right ) - 3 \, e\right )}^{2}}{12 \, f}\right )}}{48 \, \sqrt{f}} - \frac{3 \, \sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-f} x - \frac{b \log \left (f\right ) + e}{2 \, \sqrt{-f}}\right ) e^{\left (d - \frac{{\left (b \log \left (f\right ) + e\right )}^{2}}{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+e*x+d)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.94115, size = 1519, normalized size = 5.91 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.30275, size = 385, normalized size = 1.5 \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 ) - 3 \, e}{f}\right )}\right ) e^{\left (\frac{b^{2} \log \left (f\right )^{2} + 12 \, a f \log \left (f\right ) - 6 \, b e \log \left (f\right ) - 36 \, d f + 9 \, e^{2}}{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 ) + 3 \, e}{f}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right )^{2} - 12 \, a f \log \left (f\right ) + 6 \, b e \log \left (f\right ) - 36 \, d f + 9 \, e^{2}}{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 ) - e}{f}\right )}\right ) e^{\left (\frac{b^{2} \log \left (f\right )^{2} + 4 \, a f \log \left (f\right ) - 2 \, b e \log \left (f\right ) - 4 \, d f + e^{2}}{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 ) + e}{f}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right )^{2} - 4 \, a f \log \left (f\right ) + 2 \, b e \log \left (f\right ) - 4 \, d f + e^{2}}{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+e*x+d)^3,x, algorithm="giac")

[Out]

1/48*sqrt(3)*sqrt(pi)*erf(-1/6*sqrt(3)*sqrt(f)*(6*x - (b*log(f) - 3*e)/f))*e^(1/12*(b^2*log(f)^2 + 12*a*f*log(
f) - 6*b*e*log(f) - 36*d*f + 9*e^2)/f)/sqrt(f) - 1/48*sqrt(3)*sqrt(pi)*erf(-1/6*sqrt(3)*sqrt(-f)*(6*x + (b*log
(f) + 3*e)/f))*e^(-1/12*(b^2*log(f)^2 - 12*a*f*log(f) + 6*b*e*log(f) - 36*d*f + 9*e^2)/f)/sqrt(-f) - 3/16*sqrt
(pi)*erf(-1/2*sqrt(f)*(2*x - (b*log(f) - e)/f))*e^(1/4*(b^2*log(f)^2 + 4*a*f*log(f) - 2*b*e*log(f) - 4*d*f + e
^2)/f)/sqrt(f) + 3/16*sqrt(pi)*erf(-1/2*sqrt(-f)*(2*x + (b*log(f) + e)/f))*e^(-1/4*(b^2*log(f)^2 - 4*a*f*log(f
) + 2*b*e*log(f) - 4*d*f + e^2)/f)/sqrt(-f)