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

Optimal. Leaf size=171 \[ \frac{3 \sqrt{\pi } e^{-d} f^a \text{Erf}\left (x \sqrt{f-c \log (f)}\right )}{16 \sqrt{f-c \log (f)}}-\frac{\sqrt{\pi } e^{-3 d} f^a \text{Erf}\left (x \sqrt{3 f-c \log (f)}\right )}{16 \sqrt{3 f-c \log (f)}}-\frac{3 \sqrt{\pi } e^d f^a \text{Erfi}\left (x \sqrt{c \log (f)+f}\right )}{16 \sqrt{c \log (f)+f}}+\frac{\sqrt{\pi } e^{3 d} f^a \text{Erfi}\left (x \sqrt{c \log (f)+3 f}\right )}{16 \sqrt{c \log (f)+3 f}} \]

[Out]

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

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Rubi [A]  time = 0.300349, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5512, 2287, 2205, 2204} \[ \frac{3 \sqrt{\pi } e^{-d} f^a \text{Erf}\left (x \sqrt{f-c \log (f)}\right )}{16 \sqrt{f-c \log (f)}}-\frac{\sqrt{\pi } e^{-3 d} f^a \text{Erf}\left (x \sqrt{3 f-c \log (f)}\right )}{16 \sqrt{3 f-c \log (f)}}-\frac{3 \sqrt{\pi } e^d f^a \text{Erfi}\left (x \sqrt{c \log (f)+f}\right )}{16 \sqrt{c \log (f)+f}}+\frac{\sqrt{\pi } e^{3 d} f^a \text{Erfi}\left (x \sqrt{c \log (f)+3 f}\right )}{16 \sqrt{c \log (f)+3 f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 1.20654, size = 272, normalized size = 1.59 \[ \frac{\sqrt{\pi } f^a \left (3 \sqrt{f-c \log (f)} \left (-c^2 f \log ^2(f)-c^3 \log ^3(f)+9 c f^2 \log (f)+9 f^3\right ) (\cosh (d)-\sinh (d)) \text{Erf}\left (x \sqrt{f-c \log (f)}\right )-(f-c \log (f)) \left (\sqrt{3 f-c \log (f)} \left (c^2 \log ^2(f)+4 c f \log (f)+3 f^2\right ) (\cosh (3 d)-\sinh (3 d)) \text{Erf}\left (x \sqrt{3 f-c \log (f)}\right )+(3 f-c \log (f)) \left (3 \sqrt{c \log (f)+f} (c \log (f)+3 f) (\sinh (d)+\cosh (d)) \text{Erfi}\left (x \sqrt{c \log (f)+f}\right )-(c \log (f)+f) \sqrt{c \log (f)+3 f} (\sinh (3 d)+\cosh (3 d)) \text{Erfi}\left (x \sqrt{c \log (f)+3 f}\right )\right )\right )\right )}{16 \left (-10 c^2 f^2 \log ^2(f)+c^4 \log ^4(f)+9 f^4\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(f^a*Sqrt[Pi]*(3*Erf[x*Sqrt[f - c*Log[f]]]*Sqrt[f - c*Log[f]]*(9*f^3 + 9*c*f^2*Log[f] - c^2*f*Log[f]^2 - c^3*L
og[f]^3)*(Cosh[d] - Sinh[d]) - (f - c*Log[f])*(Erf[x*Sqrt[3*f - c*Log[f]]]*Sqrt[3*f - c*Log[f]]*(3*f^2 + 4*c*f
*Log[f] + c^2*Log[f]^2)*(Cosh[3*d] - Sinh[3*d]) + (3*f - c*Log[f])*(3*Erfi[x*Sqrt[f + c*Log[f]]]*Sqrt[f + c*Lo
g[f]]*(3*f + c*Log[f])*(Cosh[d] + Sinh[d]) - Erfi[x*Sqrt[3*f + c*Log[f]]]*(f + c*Log[f])*Sqrt[3*f + c*Log[f]]*
(Cosh[3*d] + Sinh[3*d])))))/(16*(9*f^4 - 10*c^2*f^2*Log[f]^2 + c^4*Log[f]^4))

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Maple [A]  time = 0.125, size = 144, normalized size = 0.8 \begin{align*}{\frac{\sqrt{\pi }{f}^{a}{{\rm e}^{3\,d}}}{16}{\it Erf} \left ( \sqrt{-c\ln \left ( f \right ) -3\,f}x \right ){\frac{1}{\sqrt{-c\ln \left ( f \right ) -3\,f}}}}-{\frac{\sqrt{\pi }{f}^{a}{{\rm e}^{-3\,d}}}{16}{\it Erf} \left ( x\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}{{\rm e}^{-d}}}{16}{\it Erf} \left ( x\sqrt{f-c\ln \left ( f \right ) } \right ){\frac{1}{\sqrt{f-c\ln \left ( f \right ) }}}}-{\frac{3\,\sqrt{\pi }{f}^{a}{{\rm e}^{d}}}{16}{\it Erf} \left ( \sqrt{-c\ln \left ( f \right ) -f}x \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+a)*sinh(f*x^2+d)^3,x)

[Out]

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

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Maxima [A]  time = 1.07553, size = 193, normalized size = 1.13 \begin{align*} \frac{\sqrt{\pi } f^{a} \operatorname{erf}\left (\sqrt{-c \log \left (f\right ) - 3 \, f} x\right ) e^{\left (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\right ) e^{\left (-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\right ) e^{\left (-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\right ) e^{d}}{16 \, \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)^3,x, algorithm="maxima")

[Out]

1/16*sqrt(pi)*f^a*erf(sqrt(-c*log(f) - 3*f)*x)*e^(3*d)/sqrt(-c*log(f) - 3*f) + 3/16*sqrt(pi)*f^a*erf(sqrt(-c*l
og(f) + f)*x)*e^(-d)/sqrt(-c*log(f) + f) - 1/16*sqrt(pi)*f^a*erf(sqrt(-c*log(f) + 3*f)*x)*e^(-3*d)/sqrt(-c*log
(f) + 3*f) - 3/16*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 = 2.08304, size = 1300, normalized size = 7.6 \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+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(a*log(f) - 3*d) + sqrt(pi)*(c^3*
log(f)^3 + 3*c^2*f*log(f)^2 - c*f^2*log(f) - 3*f^3)*sinh(a*log(f) - 3*d))*sqrt(-c*log(f) + 3*f)*erf(sqrt(-c*lo
g(f) + 3*f)*x) - 3*(sqrt(pi)*(c^3*log(f)^3 + c^2*f*log(f)^2 - 9*c*f^2*log(f) - 9*f^3)*cosh(a*log(f) - d) + sqr
t(pi)*(c^3*log(f)^3 + c^2*f*log(f)^2 - 9*c*f^2*log(f) - 9*f^3)*sinh(a*log(f) - d))*sqrt(-c*log(f) + f)*erf(sqr
t(-c*log(f) + f)*x) + 3*(sqrt(pi)*(c^3*log(f)^3 - c^2*f*log(f)^2 - 9*c*f^2*log(f) + 9*f^3)*cosh(a*log(f) + d)
+ sqrt(pi)*(c^3*log(f)^3 - c^2*f*log(f)^2 - 9*c*f^2*log(f) + 9*f^3)*sinh(a*log(f) + d))*sqrt(-c*log(f) - f)*er
f(sqrt(-c*log(f) - f)*x) - (sqrt(pi)*(c^3*log(f)^3 - 3*c^2*f*log(f)^2 - c*f^2*log(f) + 3*f^3)*cosh(a*log(f) +
3*d) + sqrt(pi)*(c^3*log(f)^3 - 3*c^2*f*log(f)^2 - c*f^2*log(f) + 3*f^3)*sinh(a*log(f) + 3*d))*sqrt(-c*log(f)
- 3*f)*erf(sqrt(-c*log(f) - 3*f)*x))/(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+a)*sinh(f*x**2+d)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.31044, size = 209, normalized size = 1.22 \begin{align*} -\frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-c \log \left (f\right ) - 3 \, f} x\right ) e^{\left (a \log \left (f\right ) + 3 \, d\right )}}{16 \, \sqrt{-c \log \left (f\right ) - 3 \, f}} + \frac{3 \, \sqrt{\pi } \operatorname{erf}\left (-\sqrt{-c \log \left (f\right ) - f} x\right ) e^{\left (a \log \left (f\right ) + d\right )}}{16 \, \sqrt{-c \log \left (f\right ) - f}} - \frac{3 \, \sqrt{\pi } \operatorname{erf}\left (-\sqrt{-c \log \left (f\right ) + f} x\right ) e^{\left (a \log \left (f\right ) - d\right )}}{16 \, \sqrt{-c \log \left (f\right ) + f}} + \frac{\sqrt{\pi } \operatorname{erf}\left (-\sqrt{-c \log \left (f\right ) + 3 \, f} x\right ) e^{\left (a \log \left (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+a)*sinh(f*x^2+d)^3,x, algorithm="giac")

[Out]

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