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

Optimal. Leaf size=323 \[ -\frac{3 \sqrt{\pi } f^a e^{\frac{b^2 \log ^2(f)}{4 f-4 c \log (f)}-d} \text{Erf}\left (\frac{b \log (f)-2 x (f-c \log (f))}{2 \sqrt{f-c \log (f)}}\right )}{16 \sqrt{f-c \log (f)}}+\frac{\sqrt{\pi } f^a e^{\frac{b^2 \log ^2(f)}{12 f-4 c \log (f)}-3 d} \text{Erf}\left (\frac{b \log (f)-2 x (3 f-c \log (f))}{2 \sqrt{3 f-c \log (f)}}\right )}{16 \sqrt{3 f-c \log (f)}}-\frac{3 \sqrt{\pi } f^a e^{d-\frac{b^2 \log ^2(f)}{4 (c \log (f)+f)}} \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+f)}{2 \sqrt{c \log (f)+f}}\right )}{16 \sqrt{c \log (f)+f}}+\frac{\sqrt{\pi } f^a e^{3 d-\frac{b^2 \log ^2(f)}{4 (c \log (f)+3 f)}} \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+3 f)}{2 \sqrt{c \log (f)+3 f}}\right )}{16 \sqrt{c \log (f)+3 f}} \]

[Out]

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

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Rubi [A]  time = 0.530021, antiderivative size = 323, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {5512, 2287, 2234, 2205, 2204} \[ -\frac{3 \sqrt{\pi } f^a e^{\frac{b^2 \log ^2(f)}{4 f-4 c \log (f)}-d} \text{Erf}\left (\frac{b \log (f)-2 x (f-c \log (f))}{2 \sqrt{f-c \log (f)}}\right )}{16 \sqrt{f-c \log (f)}}+\frac{\sqrt{\pi } f^a e^{\frac{b^2 \log ^2(f)}{12 f-4 c \log (f)}-3 d} \text{Erf}\left (\frac{b \log (f)-2 x (3 f-c \log (f))}{2 \sqrt{3 f-c \log (f)}}\right )}{16 \sqrt{3 f-c \log (f)}}-\frac{3 \sqrt{\pi } f^a e^{d-\frac{b^2 \log ^2(f)}{4 (c \log (f)+f)}} \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+f)}{2 \sqrt{c \log (f)+f}}\right )}{16 \sqrt{c \log (f)+f}}+\frac{\sqrt{\pi } f^a e^{3 d-\frac{b^2 \log ^2(f)}{4 (c \log (f)+3 f)}} \text{Erfi}\left (\frac{b \log (f)+2 x (c \log (f)+3 f)}{2 \sqrt{c \log (f)+3 f}}\right )}{16 \sqrt{c \log (f)+3 f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*E^(-d + (b^2*Log[f]^2)/(4*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(b*Log[f] - 2*x*(f - c*Log[f]))/(2*Sqrt[f - c*
Log[f]])])/(16*Sqrt[f - c*Log[f]]) + (E^(-3*d + (b^2*Log[f]^2)/(12*f - 4*c*Log[f]))*f^a*Sqrt[Pi]*Erf[(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 - (b^2*Log[f]^2)/(4*(f
+ c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(b*Log[f] + 2*x*(f + c*Log[f]))/(2*Sqrt[f + c*Log[f]])])/(16*Sqrt[f + c*Log[f
]]) + (E^(3*d - (b^2*Log[f]^2)/(4*(3*f + c*Log[f])))*f^a*Sqrt[Pi]*Erfi[(b*Log[f] + 2*x*(3*f + c*Log[f]))/(2*Sq
rt[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+f x^2\right ) \, dx &=\int \left (-\frac{1}{8} e^{-3 d-3 f x^2} f^{a+b x+c x^2}+\frac{3}{8} e^{-d-f x^2} f^{a+b x+c x^2}-\frac{3}{8} e^{d+f x^2} f^{a+b x+c x^2}+\frac{1}{8} e^{3 d+3 f x^2} f^{a+b x+c x^2}\right ) \, dx\\ &=-\left (\frac{1}{8} \int e^{-3 d-3 f x^2} f^{a+b x+c x^2} \, dx\right )+\frac{1}{8} \int e^{3 d+3 f x^2} f^{a+b x+c x^2} \, dx+\frac{3}{8} \int e^{-d-f x^2} f^{a+b x+c x^2} \, dx-\frac{3}{8} \int e^{d+f x^2} f^{a+b x+c x^2} \, dx\\ &=-\left (\frac{1}{8} \int \exp \left (-3 d+a \log (f)+b x \log (f)-x^2 (3 f-c \log (f))\right ) \, dx\right )+\frac{1}{8} \int \exp \left (3 d+a \log (f)+b x \log (f)+x^2 (3 f+c \log (f))\right ) \, dx+\frac{3}{8} \int \exp \left (-d+a \log (f)+b x \log (f)-x^2 (f-c \log (f))\right ) \, dx-\frac{3}{8} \int \exp \left (d+a \log (f)+b x \log (f)+x^2 (f+c \log (f))\right ) \, dx\\ &=\frac{1}{8} \left (3 e^{-d+\frac{b^2 \log ^2(f)}{4 f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(b \log (f)+2 x (-f+c \log (f)))^2}{4 (-f+c \log (f))}\right ) \, dx-\frac{1}{8} \left (e^{-3 d+\frac{b^2 \log ^2(f)}{12 f-4 c \log (f)}} f^a\right ) \int \exp \left (\frac{(b \log (f)+2 x (-3 f+c \log (f)))^2}{4 (-3 f+c \log (f))}\right ) \, dx-\frac{1}{8} \left (3 e^{d-\frac{b^2 \log ^2(f)}{4 (f+c \log (f))}} f^a\right ) \int \exp \left (\frac{(b \log (f)+2 x (f+c \log (f)))^2}{4 (f+c \log (f))}\right ) \, dx+\frac{1}{8} \left (e^{3 d-\frac{b^2 \log ^2(f)}{4 (3 f+c \log (f))}} f^a\right ) \int \exp \left (\frac{(b \log (f)+2 x (3 f+c \log (f)))^2}{4 (3 f+c \log (f))}\right ) \, dx\\ &=-\frac{3 e^{-d+\frac{b^2 \log ^2(f)}{4 f-4 c \log (f)}} f^a \sqrt{\pi } \text{erf}\left (\frac{b \log (f)-2 x (f-c \log (f))}{2 \sqrt{f-c \log (f)}}\right )}{16 \sqrt{f-c \log (f)}}+\frac{e^{-3 d+\frac{b^2 \log ^2(f)}{12 f-4 c \log (f)}} f^a \sqrt{\pi } \text{erf}\left (\frac{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{b^2 \log ^2(f)}{4 (f+c \log (f))}} f^a \sqrt{\pi } \text{erfi}\left (\frac{b \log (f)+2 x (f+c \log (f))}{2 \sqrt{f+c \log (f)}}\right )}{16 \sqrt{f+c \log (f)}}+\frac{e^{3 d-\frac{b^2 \log ^2(f)}{4 (3 f+c \log (f))}} f^a \sqrt{\pi } \text{erfi}\left (\frac{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 = 6.49887, size = 2511, normalized size = 7.77 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(f^a*Sqrt[Pi]*(27*E^((b^2*Log[f]^2)/(4*(f - c*Log[f])))*f^3*Cosh[d]*Erf[(2*f*x - b*Log[f] - 2*c*x*Log[f])/(2*S
qrt[f - c*Log[f]])]*Sqrt[f - c*Log[f]] + 27*c*E^((b^2*Log[f]^2)/(4*(f - c*Log[f])))*f^2*Cosh[d]*Erf[(2*f*x - b
*Log[f] - 2*c*x*Log[f])/(2*Sqrt[f - c*Log[f]])]*Log[f]*Sqrt[f - c*Log[f]] - 3*c^2*E^((b^2*Log[f]^2)/(4*(f - c*
Log[f])))*f*Cosh[d]*Erf[(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]]
- 3*c^3*E^((b^2*Log[f]^2)/(4*(f - c*Log[f])))*Cosh[d]*Erf[(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]] - 3*E^((b^2*Log[f]^2)/(4*(3*f - c*Log[f])))*f^3*Cosh[3*d]*Erf[(6*f*x - b*Log
[f] - 2*c*x*Log[f])/(2*Sqrt[3*f - c*Log[f]])]*Sqrt[3*f - c*Log[f]] - c*E^((b^2*Log[f]^2)/(4*(3*f - c*Log[f])))
*f^2*Cosh[3*d]*Erf[(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]] + 3
*c^2*E^((b^2*Log[f]^2)/(4*(3*f - c*Log[f])))*f*Cosh[3*d]*Erf[(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]] + c^3*E^((b^2*Log[f]^2)/(4*(3*f - c*Log[f])))*Cosh[3*d]*Erf[(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]] - (27*f^3*Cosh[d]*Erfi[(2*f*x
+ b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[f + c*Log[f]])]*Sqrt[f + c*Log[f]])/E^((b^2*Log[f]^2)/(4*(f + c*Log[f])))
+ (27*c*f^2*Cosh[d]*Erfi[(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^((b^2*Log[f]^2)/(4*(f + c*Log[f]))) + (3*c^2*f*Cosh[d]*Erfi[(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^((b^2*Log[f]^2)/(4*(f + c*Log[f]))) - (3*c^3*Cosh[d]*Erfi[(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^((b^2*Log[f]^2)/(4*(f + c*Log[f]
))) + (3*f^3*Cosh[3*d]*Erfi[(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^((b^2*Log[f]^2)/(4*(3*f + c*Log[f]))) - (c*f^2*Cosh[3*d]*Erfi[(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^((b^2*Log[f]^2)/(4*(3*f + c*Log[f]))) - (3*c^2*f*Cosh[3*d]*Erfi[(
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^((b^2*Log[f]^2)/(4
*(3*f + c*Log[f]))) + (c^3*Cosh[3*d]*Erfi[(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^((b^2*Log[f]^2)/(4*(3*f + c*Log[f]))) - 27*E^((b^2*Log[f]^2)/(4*(f - c*Log[f])))*f^3*
Erf[(2*f*x - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[f - c*Log[f]])]*Sqrt[f - c*Log[f]]*Sinh[d] - 27*c*E^((b^2*Log[f]
^2)/(4*(f - c*Log[f])))*f^2*Erf[(2*f*x - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[f - c*Log[f]])]*Log[f]*Sqrt[f - c*Lo
g[f]]*Sinh[d] + 3*c^2*E^((b^2*Log[f]^2)/(4*(f - c*Log[f])))*f*Erf[(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] + 3*c^3*E^((b^2*Log[f]^2)/(4*(f - c*Log[f])))*Erf[(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] - (27*f^3*Erfi[(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^((b^2*Log[f]^2)/(4*(f + c*Log[f])
)) + (27*c*f^2*Erfi[(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^((b^2*Log[f]^2)/(4*(f + c*Log[f]))) + (3*c^2*f*Erfi[(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^((b^2*Log[f]^2)/(4*(f + c*Log[f]))) - (3*c^3*Erfi[(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^((b^2*Log[f]^2)/(4*(f + c*Log
[f]))) + 3*E^((b^2*Log[f]^2)/(4*(3*f - c*Log[f])))*f^3*Erf[(6*f*x - b*Log[f] - 2*c*x*Log[f])/(2*Sqrt[3*f - c*L
og[f]])]*Sqrt[3*f - c*Log[f]]*Sinh[3*d] + c*E^((b^2*Log[f]^2)/(4*(3*f - c*Log[f])))*f^2*Erf[(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] - 3*c^2*E^((b^2*Log[f]^2)/(4*(
3*f - c*Log[f])))*f*Erf[(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] - c^3*E^((b^2*Log[f]^2)/(4*(3*f - c*Log[f])))*Erf[(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] + (3*f^3*Erfi[(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^((b^2*Log[f]^2)/(4*(3*f + c*Log[f]))) - (c*f^2*Erfi[(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^((b^2*Log[f]^2)
/(4*(3*f + c*Log[f]))) - (3*c^2*f*Erfi[(6*f*x + b*Log[f] + 2*c*x*Log[f])/(2*Sqrt[3*f + c*Log[f]])]*Log[f]^2*Sq
rt[3*f + c*Log[f]]*Sinh[3*d])/E^((b^2*Log[f]^2)/(4*(3*f + c*Log[f]))) + (c^3*Erfi[(6*f*x + b*Log[f] + 2*c*x*Lo
g[f])/(2*Sqrt[3*f + c*Log[f]])]*Log[f]^3*Sqrt[3*f + c*Log[f]]*Sinh[3*d])/E^((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 = 0.248, size = 326, normalized size = 1. \begin{align*} -{\frac{\sqrt{\pi }{f}^{a}}{16}{{\rm e}^{-{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}-12\,d\ln \left ( f \right ) c-36\,df}{4\,c\ln \left ( f \right ) +12\,f}}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -3\,f}x+{\frac{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}+12\,d\ln \left ( f \right ) c-36\,df}{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 ) }{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}+4\,d\ln \left ( f \right ) c-4\,df}{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 ) }{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}-4\,d\ln \left ( f \right ) c-4\,df}{4\,c\ln \left ( f \right ) +4\,f}}}}{\it Erf} \left ( -\sqrt{-c\ln \left ( f \right ) -f}x+{\frac{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+d)^3,x)

[Out]

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

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Maxima [A]  time = 1.12138, size = 387, normalized size = 1.2 \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 )}{2 \, \sqrt{-c \log \left (f\right ) - 3 \, f}}\right ) e^{\left (-\frac{b^{2} \log \left (f\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 )}{2 \, \sqrt{-c \log \left (f\right ) - f}}\right ) e^{\left (-\frac{b^{2} \log \left (f\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 )}{2 \, \sqrt{-c \log \left (f\right ) + f}}\right ) e^{\left (-\frac{b^{2} \log \left (f\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 )}{2 \, \sqrt{-c \log \left (f\right ) + 3 \, f}}\right ) e^{\left (-\frac{b^{2} \log \left (f\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+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)/sqrt(-c*log(f) - 3*f))*e^(-1/4*b^2*log(f)^2/(c*lo
g(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)/sqrt(-c*
log(f) - f))*e^(-1/4*b^2*log(f)^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)/sqrt(-c*log(f) + f))*e^(-1/4*b^2*log(f)^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)/sqrt(-c*log(f) + 3*f))*e^(-1/4*b^2*log(f)^2/(c*log
(f) - 3*f) - 3*d)/sqrt(-c*log(f) + 3*f)

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Fricas [B]  time = 2.01982, size = 2248, normalized size = 6.96 \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+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 - 3
6*d*f + 12*(c*d + 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 - 36*d*f + 12*(c*d + 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))*sqrt(-c*log(f) + 3*f)/(c*log(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 - 4*d*f + 4*(c*d + 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 - 4*d*f + 4*(c*d + a*f)*log(f))/(c*log(f) - f)))*sqrt(-c*log(f) + f)*erf(-1/2*(2*f*x - (2*c*x + b)*lo
g(f))*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 - 4*d*f - 4*(c*d + 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 - 4*d*f - 4*(c*d + 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))*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 - 36*d*f
- 12*(c*d + 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 - 36*d*f - 12*(c*d + 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))*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+d)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.24609, size = 498, normalized size = 1.54 \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 )}{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 ) - 36 \, d f}{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 )}{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 ) - 4 \, d f}{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 )}{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 ) - 4 \, d f}{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 )}{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 ) - 36 \, d f}{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+d)^3,x, algorithm="giac")

[Out]

-1/16*sqrt(pi)*erf(-1/2*sqrt(-c*log(f) - 3*f)*(2*x + b*log(f)/(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) - 36*d*f)/(c*log(f) + 3*f))/sqrt(-c*log(f) - 3*f) + 3/16*sqrt(pi)*er
f(-1/2*sqrt(-c*log(f) - f)*(2*x + b*log(f)/(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) - 4*d*f)/(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)/(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) - 4*d*f)/
(c*log(f) - f))/sqrt(-c*log(f) + f) + 1/16*sqrt(pi)*erf(-1/2*sqrt(-c*log(f) + 3*f)*(2*x + b*log(f)/(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) - 36*d*f)/(c*log(f) - 3*f))/sqr
t(-c*log(f) + 3*f)