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

Optimal. Leaf size=344 \[ -\frac{\sqrt{\pi } f^a \exp \left (\frac{(3 e-b \log (f))^2}{12 f-4 c \log (f)}-3 d\right ) \text{Erf}\left (\frac{-b \log (f)+2 x (3 f-c \log (f))+3 e}{2 \sqrt{3 f-c \log (f)}}\right )}{16 \sqrt{3 f-c \log (f)}}+\frac{3 \sqrt{\pi } f^a e^{\frac{(e-b \log (f))^2}{4 (f-c \log (f))}-d} \text{Erf}\left (\frac{-b \log (f)+2 x (f-c \log (f))+e}{2 \sqrt{f-c \log (f)}}\right )}{16 \sqrt{f-c \log (f)}}+\frac{\sqrt{\pi } f^a \exp \left (3 d-\frac{(b \log (f)+3 e)^2}{4 (c \log (f)+3 f)}\right ) \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+3 f)+3 e}{2 \sqrt{c \log (f)+3 f}}\right )}{16 \sqrt{c \log (f)+3 f}}-\frac{3 \sqrt{\pi } f^a e^{d-\frac{(b \log (f)+e)^2}{4 (c \log (f)+f)}} \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+f)+e}{2 \sqrt{c \log (f)+f}}\right )}{16 \sqrt{c \log (f)+f}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.779412, antiderivative size = 344, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {5512, 2287, 2234, 2205, 2204} \[ -\frac{\sqrt{\pi } f^a \exp \left (\frac{(3 e-b \log (f))^2}{12 f-4 c \log (f)}-3 d\right ) \text{Erf}\left (\frac{-b \log (f)+2 x (3 f-c \log (f))+3 e}{2 \sqrt{3 f-c \log (f)}}\right )}{16 \sqrt{3 f-c \log (f)}}+\frac{3 \sqrt{\pi } f^a e^{\frac{(e-b \log (f))^2}{4 (f-c \log (f))}-d} \text{Erf}\left (\frac{-b \log (f)+2 x (f-c \log (f))+e}{2 \sqrt{f-c \log (f)}}\right )}{16 \sqrt{f-c \log (f)}}+\frac{\sqrt{\pi } f^a \exp \left (3 d-\frac{(b \log (f)+3 e)^2}{4 (c \log (f)+3 f)}\right ) \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+3 f)+3 e}{2 \sqrt{c \log (f)+3 f}}\right )}{16 \sqrt{c \log (f)+3 f}}-\frac{3 \sqrt{\pi } f^a e^{d-\frac{(b \log (f)+e)^2}{4 (c \log (f)+f)}} \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+f)+e}{2 \sqrt{c \log (f)+f}}\right )}{16 \sqrt{c \log (f)+f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [B]  time = 6.64703, size = 2991, normalized size = 8.69 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(f^a*Sqrt[Pi]*((27*f^3*Cosh[d]*Erf[(e + 2*f*x - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[f - c*Log[f]])]*Sqrt[f - c*Lo
g[f]])/E^((-e^2 + 2*b*e*Log[f] - b^2*Log[f]^2)/(4*(f - c*Log[f]))) + (27*c*f^2*Cosh[d]*Erf[(e + 2*f*x - b*Log[
f] - 2*c*x*Log[f])/(2*Sqrt[f - c*Log[f]])]*Log[f]*Sqrt[f - c*Log[f]])/E^((-e^2 + 2*b*e*Log[f] - b^2*Log[f]^2)/
(4*(f - c*Log[f]))) - (3*c^2*f*Cosh[d]*Erf[(e + 2*f*x - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[f - c*Log[f]])]*Log[f
]^2*Sqrt[f - c*Log[f]])/E^((-e^2 + 2*b*e*Log[f] - b^2*Log[f]^2)/(4*(f - c*Log[f]))) - (3*c^3*Cosh[d]*Erf[(e +
2*f*x - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[f - c*Log[f]])]*Log[f]^3*Sqrt[f - c*Log[f]])/E^((-e^2 + 2*b*e*Log[f]
- b^2*Log[f]^2)/(4*(f - c*Log[f]))) - (3*f^3*Cosh[3*d]*Erf[(3*e + 6*f*x - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[3*f
- c*Log[f]])]*Sqrt[3*f - c*Log[f]])/E^((-9*e^2 + 6*b*e*Log[f] - b^2*Log[f]^2)/(4*(3*f - c*Log[f]))) - (c*f^2*
Cosh[3*d]*Erf[(3*e + 6*f*x - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[3*f - c*Log[f]])]*Log[f]*Sqrt[3*f - c*Log[f]])/E
^((-9*e^2 + 6*b*e*Log[f] - b^2*Log[f]^2)/(4*(3*f - c*Log[f]))) + (3*c^2*f*Cosh[3*d]*Erf[(3*e + 6*f*x - b*Log[f
] - 2*c*x*Log[f])/(2*Sqrt[3*f - c*Log[f]])]*Log[f]^2*Sqrt[3*f - c*Log[f]])/E^((-9*e^2 + 6*b*e*Log[f] - b^2*Log
[f]^2)/(4*(3*f - c*Log[f]))) + (c^3*Cosh[3*d]*Erf[(3*e + 6*f*x - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[3*f - c*Log[
f]])]*Log[f]^3*Sqrt[3*f - c*Log[f]])/E^((-9*e^2 + 6*b*e*Log[f] - b^2*Log[f]^2)/(4*(3*f - c*Log[f]))) - (27*f^3
*Cosh[d]*Erfi[(e + 2*f*x + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[f + c*Log[f]])]*Sqrt[f + c*Log[f]])/E^((e^2 + 2*b*
e*Log[f] + b^2*Log[f]^2)/(4*(f + c*Log[f]))) + (27*c*f^2*Cosh[d]*Erfi[(e + 2*f*x + b*Log[f] + 2*c*x*Log[f])/(2
*Sqrt[f + c*Log[f]])]*Log[f]*Sqrt[f + c*Log[f]])/E^((e^2 + 2*b*e*Log[f] + b^2*Log[f]^2)/(4*(f + c*Log[f]))) +
(3*c^2*f*Cosh[d]*Erfi[(e + 2*f*x + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[f + c*Log[f]])]*Log[f]^2*Sqrt[f + c*Log[f]
])/E^((e^2 + 2*b*e*Log[f] + b^2*Log[f]^2)/(4*(f + c*Log[f]))) - (3*c^3*Cosh[d]*Erfi[(e + 2*f*x + b*Log[f] + 2*
c*x*Log[f])/(2*Sqrt[f + c*Log[f]])]*Log[f]^3*Sqrt[f + c*Log[f]])/E^((e^2 + 2*b*e*Log[f] + b^2*Log[f]^2)/(4*(f
+ c*Log[f]))) + (3*f^3*Cosh[3*d]*Erfi[(3*e + 6*f*x + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[3*f + c*Log[f]])]*Sqrt[3
*f + c*Log[f]])/E^((9*e^2 + 6*b*e*Log[f] + b^2*Log[f]^2)/(4*(3*f + c*Log[f]))) - (c*f^2*Cosh[3*d]*Erfi[(3*e +
6*f*x + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[3*f + c*Log[f]])]*Log[f]*Sqrt[3*f + c*Log[f]])/E^((9*e^2 + 6*b*e*Log[
f] + b^2*Log[f]^2)/(4*(3*f + c*Log[f]))) - (3*c^2*f*Cosh[3*d]*Erfi[(3*e + 6*f*x + b*Log[f] + 2*c*x*Log[f])/(2*
Sqrt[3*f + c*Log[f]])]*Log[f]^2*Sqrt[3*f + c*Log[f]])/E^((9*e^2 + 6*b*e*Log[f] + b^2*Log[f]^2)/(4*(3*f + c*Log
[f]))) + (c^3*Cosh[3*d]*Erfi[(3*e + 6*f*x + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[3*f + c*Log[f]])]*Log[f]^3*Sqrt[3
*f + c*Log[f]])/E^((9*e^2 + 6*b*e*Log[f] + b^2*Log[f]^2)/(4*(3*f + c*Log[f]))) - (27*f^3*Erf[(e + 2*f*x - b*Lo
g[f] - 2*c*x*Log[f])/(2*Sqrt[f - c*Log[f]])]*Sqrt[f - c*Log[f]]*Sinh[d])/E^((-e^2 + 2*b*e*Log[f] - b^2*Log[f]^
2)/(4*(f - c*Log[f]))) - (27*c*f^2*Erf[(e + 2*f*x - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[f - c*Log[f]])]*Log[f]*Sq
rt[f - c*Log[f]]*Sinh[d])/E^((-e^2 + 2*b*e*Log[f] - b^2*Log[f]^2)/(4*(f - c*Log[f]))) + (3*c^2*f*Erf[(e + 2*f*
x - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[f - c*Log[f]])]*Log[f]^2*Sqrt[f - c*Log[f]]*Sinh[d])/E^((-e^2 + 2*b*e*Log
[f] - b^2*Log[f]^2)/(4*(f - c*Log[f]))) + (3*c^3*Erf[(e + 2*f*x - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[f - c*Log[f
]])]*Log[f]^3*Sqrt[f - c*Log[f]]*Sinh[d])/E^((-e^2 + 2*b*e*Log[f] - b^2*Log[f]^2)/(4*(f - c*Log[f]))) - (27*f^
3*Erfi[(e + 2*f*x + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[f + c*Log[f]])]*Sqrt[f + c*Log[f]]*Sinh[d])/E^((e^2 + 2*b
*e*Log[f] + b^2*Log[f]^2)/(4*(f + c*Log[f]))) + (27*c*f^2*Erfi[(e + 2*f*x + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[f
+ c*Log[f]])]*Log[f]*Sqrt[f + c*Log[f]]*Sinh[d])/E^((e^2 + 2*b*e*Log[f] + b^2*Log[f]^2)/(4*(f + c*Log[f]))) +
(3*c^2*f*Erfi[(e + 2*f*x + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[f + c*Log[f]])]*Log[f]^2*Sqrt[f + c*Log[f]]*Sinh[
d])/E^((e^2 + 2*b*e*Log[f] + b^2*Log[f]^2)/(4*(f + c*Log[f]))) - (3*c^3*Erfi[(e + 2*f*x + b*Log[f] + 2*c*x*Log
[f])/(2*Sqrt[f + c*Log[f]])]*Log[f]^3*Sqrt[f + c*Log[f]]*Sinh[d])/E^((e^2 + 2*b*e*Log[f] + b^2*Log[f]^2)/(4*(f
+ c*Log[f]))) + (3*f^3*Erf[(3*e + 6*f*x - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[3*f - c*Log[f]])]*Sqrt[3*f - c*Log
[f]]*Sinh[3*d])/E^((-9*e^2 + 6*b*e*Log[f] - b^2*Log[f]^2)/(4*(3*f - c*Log[f]))) + (c*f^2*Erf[(3*e + 6*f*x - b*
Log[f] - 2*c*x*Log[f])/(2*Sqrt[3*f - c*Log[f]])]*Log[f]*Sqrt[3*f - c*Log[f]]*Sinh[3*d])/E^((-9*e^2 + 6*b*e*Log
[f] - b^2*Log[f]^2)/(4*(3*f - c*Log[f]))) - (3*c^2*f*Erf[(3*e + 6*f*x - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[3*f -
c*Log[f]])]*Log[f]^2*Sqrt[3*f - c*Log[f]]*Sinh[3*d])/E^((-9*e^2 + 6*b*e*Log[f] - b^2*Log[f]^2)/(4*(3*f - c*Lo
g[f]))) - (c^3*Erf[(3*e + 6*f*x - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[3*f - c*Log[f]])]*Log[f]^3*Sqrt[3*f - c*Log
[f]]*Sinh[3*d])/E^((-9*e^2 + 6*b*e*Log[f] - b^2*Log[f]^2)/(4*(3*f - c*Log[f]))) + (3*f^3*Erfi[(3*e + 6*f*x + b
*Log[f] + 2*c*x*Log[f])/(2*Sqrt[3*f + c*Log[f]])]*Sqrt[3*f + c*Log[f]]*Sinh[3*d])/E^((9*e^2 + 6*b*e*Log[f] + b
^2*Log[f]^2)/(4*(3*f + c*Log[f]))) - (c*f^2*Erfi[(3*e + 6*f*x + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[3*f + c*Log[f
]])]*Log[f]*Sqrt[3*f + c*Log[f]]*Sinh[3*d])/E^((9*e^2 + 6*b*e*Log[f] + b^2*Log[f]^2)/(4*(3*f + c*Log[f]))) - (
3*c^2*f*Erfi[(3*e + 6*f*x + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[3*f + c*Log[f]])]*Log[f]^2*Sqrt[3*f + c*Log[f]]*S
inh[3*d])/E^((9*e^2 + 6*b*e*Log[f] + b^2*Log[f]^2)/(4*(3*f + c*Log[f]))) + (c^3*Erfi[(3*e + 6*f*x + b*Log[f] +
2*c*x*Log[f])/(2*Sqrt[3*f + c*Log[f]])]*Log[f]^3*Sqrt[3*f + c*Log[f]]*Sinh[3*d])/E^((9*e^2 + 6*b*e*Log[f] + b
^2*Log[f]^2)/(4*(3*f + c*Log[f])))))/(16*(f - c*Log[f])*(3*f - c*Log[f])*(f + c*Log[f])*(3*f + c*Log[f]))

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Maple [A]  time = 0.209, size = 384, normalized size = 1.1 \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-12\,d\ln \left ( f \right ) c-36\,df+9\,{e}^{2}}{4\,c\ln \left ( f \right ) +12\,f}}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -3\,f}x+{\frac{3\,e+b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) -3\,f}}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -3\,f}}}}+{\frac{\sqrt{\pi }{f}^{a}}{16}{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}-6\,\ln \left ( f \right ) be+12\,d\ln \left ( f \right ) c-36\,df+9\,{e}^{2}}{4\,c\ln \left ( f \right ) -12\,f}}}}{\it Erf} \left ( -x\sqrt{3\,f-c\ln \left ( f \right ) }+{\frac{b\ln \left ( f \right ) -3\,e}{2}{\frac{1}{\sqrt{3\,f-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{3\,f-c\ln \left ( f \right ) }}}}-{\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\,d\ln \left ( f \right ) c-4\,df+{e}^{2}}{4\,c\ln \left ( f \right ) -4\,f}}}}{\it Erf} \left ( -x\sqrt{f-c\ln \left ( f \right ) }+{\frac{b\ln \left ( f \right ) -e}{2}{\frac{1}{\sqrt{f-c\ln \left ( f \right ) }}}} \right ){\frac{1}{\sqrt{f-c\ln \left ( f \right ) }}}}+{\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\,d\ln \left ( f \right ) c-4\,df+{e}^{2}}{4\,c\ln \left ( f \right ) +4\,f}}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -f}x+{\frac{e+b\ln \left ( f \right ) }{2}{\frac{1}{\sqrt{-c\ln \left ( f \right ) -f}}}} \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -f}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.12418, size = 425, normalized size = 1.24 \begin{align*} \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) - 3 \, f} x - \frac{b \log \left (f\right ) + 3 \, e}{2 \, \sqrt{-c \log \left (f\right ) - 3 \, f}}\right ) e^{\left (-\frac{{\left (b \log \left (f\right ) + 3 \, e\right )}^{2}}{4 \,{\left (c \log \left (f\right ) + 3 \, f\right )}} + 3 \, d\right )}}{16 \, \sqrt{-c \log \left (f\right ) - 3 \, f}} - \frac{3 \, \sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) - f} x - \frac{b \log \left (f\right ) + e}{2 \, \sqrt{-c \log \left (f\right ) - f}}\right ) e^{\left (-\frac{{\left (b \log \left (f\right ) + e\right )}^{2}}{4 \,{\left (c \log \left (f\right ) + f\right )}} + d\right )}}{16 \, \sqrt{-c \log \left (f\right ) - f}} + \frac{3 \, \sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) + f} x - \frac{b \log \left (f\right ) - e}{2 \, \sqrt{-c \log \left (f\right ) + f}}\right ) e^{\left (-\frac{{\left (b \log \left (f\right ) - e\right )}^{2}}{4 \,{\left (c \log \left (f\right ) - f\right )}} - d\right )}}{16 \, \sqrt{-c \log \left (f\right ) + f}} - \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) + 3 \, f} x - \frac{b \log \left (f\right ) - 3 \, e}{2 \, \sqrt{-c \log \left (f\right ) + 3 \, f}}\right ) e^{\left (-\frac{{\left (b \log \left (f\right ) - 3 \, e\right )}^{2}}{4 \,{\left (c \log \left (f\right ) - 3 \, f\right )}} - 3 \, d\right )}}{16 \, \sqrt{-c \log \left (f\right ) + 3 \, f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.14624, size = 2453, normalized size = 7.13 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*((sqrt(pi)*(c^3*log(f)^3 + 3*c^2*f*log(f)^2 - c*f^2*log(f) - 3*f^3)*cosh(-1/4*((b^2 - 4*a*c)*log(f)^2 + 9
*e^2 - 36*d*f + 6*(2*c*d - b*e + 2*a*f)*log(f))/(c*log(f) - 3*f)) + sqrt(pi)*(c^3*log(f)^3 + 3*c^2*f*log(f)^2
- c*f^2*log(f) - 3*f^3)*sinh(-1/4*((b^2 - 4*a*c)*log(f)^2 + 9*e^2 - 36*d*f + 6*(2*c*d - b*e + 2*a*f)*log(f))/(
c*log(f) - 3*f)))*sqrt(-c*log(f) + 3*f)*erf(-1/2*(6*f*x - (2*c*x + b)*log(f) + 3*e)*sqrt(-c*log(f) + 3*f)/(c*l
og(f) - 3*f)) - 3*(sqrt(pi)*(c^3*log(f)^3 + c^2*f*log(f)^2 - 9*c*f^2*log(f) - 9*f^3)*cosh(-1/4*((b^2 - 4*a*c)*
log(f)^2 + e^2 - 4*d*f + 2*(2*c*d - b*e + 2*a*f)*log(f))/(c*log(f) - f)) + sqrt(pi)*(c^3*log(f)^3 + c^2*f*log(
f)^2 - 9*c*f^2*log(f) - 9*f^3)*sinh(-1/4*((b^2 - 4*a*c)*log(f)^2 + e^2 - 4*d*f + 2*(2*c*d - b*e + 2*a*f)*log(f
))/(c*log(f) - f)))*sqrt(-c*log(f) + f)*erf(-1/2*(2*f*x - (2*c*x + b)*log(f) + e)*sqrt(-c*log(f) + f)/(c*log(f
) - f)) + 3*(sqrt(pi)*(c^3*log(f)^3 - c^2*f*log(f)^2 - 9*c*f^2*log(f) + 9*f^3)*cosh(-1/4*((b^2 - 4*a*c)*log(f)
^2 + e^2 - 4*d*f - 2*(2*c*d - b*e + 2*a*f)*log(f))/(c*log(f) + f)) + sqrt(pi)*(c^3*log(f)^3 - c^2*f*log(f)^2 -
9*c*f^2*log(f) + 9*f^3)*sinh(-1/4*((b^2 - 4*a*c)*log(f)^2 + e^2 - 4*d*f - 2*(2*c*d - b*e + 2*a*f)*log(f))/(c*
log(f) + f)))*sqrt(-c*log(f) - f)*erf(1/2*(2*f*x + (2*c*x + b)*log(f) + e)*sqrt(-c*log(f) - f)/(c*log(f) + f))
- (sqrt(pi)*(c^3*log(f)^3 - 3*c^2*f*log(f)^2 - c*f^2*log(f) + 3*f^3)*cosh(-1/4*((b^2 - 4*a*c)*log(f)^2 + 9*e^
2 - 36*d*f - 6*(2*c*d - b*e + 2*a*f)*log(f))/(c*log(f) + 3*f)) + sqrt(pi)*(c^3*log(f)^3 - 3*c^2*f*log(f)^2 - c
*f^2*log(f) + 3*f^3)*sinh(-1/4*((b^2 - 4*a*c)*log(f)^2 + 9*e^2 - 36*d*f - 6*(2*c*d - b*e + 2*a*f)*log(f))/(c*l
og(f) + 3*f)))*sqrt(-c*log(f) - 3*f)*erf(1/2*(6*f*x + (2*c*x + b)*log(f) + 3*e)*sqrt(-c*log(f) - 3*f)/(c*log(f
) + 3*f)))/(c^4*log(f)^4 - 10*c^2*f^2*log(f)^2 + 9*f^4)

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

[Out]

Timed out

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Giac [A]  time = 1.32295, size = 582, normalized size = 1.69 \begin{align*} -\frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right ) - 3 \, f}{\left (2 \, x + \frac{b \log \left (f\right ) + 3 \, e}{c \log \left (f\right ) + 3 \, f}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} - 12 \, c d \log \left (f\right ) - 12 \, a f \log \left (f\right ) + 6 \, b e \log \left (f\right ) - 36 \, d f + 9 \, e^{2}}{4 \,{\left (c \log \left (f\right ) + 3 \, f\right )}}\right )}}{16 \, \sqrt{-c \log \left (f\right ) - 3 \, f}} + \frac{3 \, \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right ) - f}{\left (2 \, x + \frac{b \log \left (f\right ) + e}{c \log \left (f\right ) + f}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} - 4 \, c d \log \left (f\right ) - 4 \, a f \log \left (f\right ) + 2 \, b e \log \left (f\right ) - 4 \, d f + e^{2}}{4 \,{\left (c \log \left (f\right ) + f\right )}}\right )}}{16 \, \sqrt{-c \log \left (f\right ) - f}} - \frac{3 \, \sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right ) + f}{\left (2 \, x + \frac{b \log \left (f\right ) - e}{c \log \left (f\right ) - f}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} + 4 \, c d \log \left (f\right ) + 4 \, a f \log \left (f\right ) - 2 \, b e \log \left (f\right ) - 4 \, d f + e^{2}}{4 \,{\left (c \log \left (f\right ) - f\right )}}\right )}}{16 \, \sqrt{-c \log \left (f\right ) + f}} + \frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c \log \left (f\right ) + 3 \, f}{\left (2 \, x + \frac{b \log \left (f\right ) - 3 \, e}{c \log \left (f\right ) - 3 \, f}\right )}\right ) e^{\left (-\frac{b^{2} \log \left (f\right )^{2} - 4 \, a c \log \left (f\right )^{2} + 12 \, c d \log \left (f\right ) + 12 \, a f \log \left (f\right ) - 6 \, b e \log \left (f\right ) - 36 \, d f + 9 \, e^{2}}{4 \,{\left (c \log \left (f\right ) - 3 \, f\right )}}\right )}}{16 \, \sqrt{-c \log \left (f\right ) + 3 \, f}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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