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

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]

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

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Rubi [A]  time = 0.30, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5513, 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)*Cosh[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 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]

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 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 5513

Int[Cosh[v_]^(n_.)*(F_)^(u_), x_Symbol] :> Int[ExpandTrigToExp[F^u, Cosh[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]

Rubi steps

\begin {align*} \int f^{a+c x^2} \cosh ^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\\ &=\frac {1}{8} \int e^{-3 d-3 f x^2} f^{a+c x^2} \, dx+\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\\ &=\frac {1}{8} \int e^{-3 d+a \log (f)-x^2 (3 f-c \log (f))} \, dx+\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*}

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Mathematica [A]  time = 1.24, size = 270, normalized size = 1.58 \[ \frac {\sqrt {\pi } f^a \left ((f-c \log (f)) \left (\sqrt {3 f-c \log (f)} \left (c^2 \log ^2(f)+4 c f \log (f)+3 f^2\right ) (\cosh (3 d)-\sinh (3 d)) \text {erf}\left (x \sqrt {3 f-c \log (f)}\right )+(3 f-c \log (f)) \left (3 \sqrt {c \log (f)+f} (c \log (f)+3 f) (\sinh (d)+\cosh (d)) \text {erfi}\left (x \sqrt {c \log (f)+f}\right )+(c \log (f)+f) \sqrt {c \log (f)+3 f} (\sinh (3 d)+\cosh (3 d)) \text {erfi}\left (x \sqrt {c \log (f)+3 f}\right )\right )\right )+3 \sqrt {f-c \log (f)} \left (-c^3 \log ^3(f)-c^2 f \log ^2(f)+9 c f^2 \log (f)+9 f^3\right ) (\cosh (d)-\sinh (d)) \text {erf}\left (x \sqrt {f-c \log (f)}\right )\right )}{16 \left (c^4 \log ^4(f)-10 c^2 f^2 \log ^2(f)+9 f^4\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + c*x^2)*Cosh[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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fricas [B]  time = 0.69, size = 491, normalized size = 2.87 \[ -\frac {{\left (\sqrt {\pi } {\left (c^{3} \log \relax (f)^{3} + 3 \, c^{2} f \log \relax (f)^{2} - c f^{2} \log \relax (f) - 3 \, f^{3}\right )} \cosh \left (a \log \relax (f) - 3 \, d\right ) + \sqrt {\pi } {\left (c^{3} \log \relax (f)^{3} + 3 \, c^{2} f \log \relax (f)^{2} - c f^{2} \log \relax (f) - 3 \, f^{3}\right )} \sinh \left (a \log \relax (f) - 3 \, d\right )\right )} \sqrt {-c \log \relax (f) + 3 \, f} \operatorname {erf}\left (\sqrt {-c \log \relax (f) + 3 \, f} x\right ) + 3 \, {\left (\sqrt {\pi } {\left (c^{3} \log \relax (f)^{3} + c^{2} f \log \relax (f)^{2} - 9 \, c f^{2} \log \relax (f) - 9 \, f^{3}\right )} \cosh \left (a \log \relax (f) - d\right ) + \sqrt {\pi } {\left (c^{3} \log \relax (f)^{3} + c^{2} f \log \relax (f)^{2} - 9 \, c f^{2} \log \relax (f) - 9 \, f^{3}\right )} \sinh \left (a \log \relax (f) - d\right )\right )} \sqrt {-c \log \relax (f) + f} \operatorname {erf}\left (\sqrt {-c \log \relax (f) + f} x\right ) + 3 \, {\left (\sqrt {\pi } {\left (c^{3} \log \relax (f)^{3} - c^{2} f \log \relax (f)^{2} - 9 \, c f^{2} \log \relax (f) + 9 \, f^{3}\right )} \cosh \left (a \log \relax (f) + d\right ) + \sqrt {\pi } {\left (c^{3} \log \relax (f)^{3} - c^{2} f \log \relax (f)^{2} - 9 \, c f^{2} \log \relax (f) + 9 \, f^{3}\right )} \sinh \left (a \log \relax (f) + d\right )\right )} \sqrt {-c \log \relax (f) - f} \operatorname {erf}\left (\sqrt {-c \log \relax (f) - f} x\right ) + {\left (\sqrt {\pi } {\left (c^{3} \log \relax (f)^{3} - 3 \, c^{2} f \log \relax (f)^{2} - c f^{2} \log \relax (f) + 3 \, f^{3}\right )} \cosh \left (a \log \relax (f) + 3 \, d\right ) + \sqrt {\pi } {\left (c^{3} \log \relax (f)^{3} - 3 \, c^{2} f \log \relax (f)^{2} - c f^{2} \log \relax (f) + 3 \, f^{3}\right )} \sinh \left (a \log \relax (f) + 3 \, d\right )\right )} \sqrt {-c \log \relax (f) - 3 \, f} \operatorname {erf}\left (\sqrt {-c \log \relax (f) - 3 \, f} x\right )}{16 \, {\left (c^{4} \log \relax (f)^{4} - 10 \, c^{2} f^{2} \log \relax (f)^{2} + 9 \, f^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*x^2+a)*cosh(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*l
og(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) + sq
rt(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(sq
rt(-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)*e
rf(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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giac [A]  time = 0.16, size = 155, normalized size = 0.91 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \relax (f) - 3 \, f} x\right ) e^{\left (a \log \relax (f) + 3 \, d\right )}}{16 \, \sqrt {-c \log \relax (f) - 3 \, f}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \relax (f) - f} x\right ) e^{\left (a \log \relax (f) + d\right )}}{16 \, \sqrt {-c \log \relax (f) - f}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \relax (f) + f} x\right ) e^{\left (a \log \relax (f) - d\right )}}{16 \, \sqrt {-c \log \relax (f) + f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \relax (f) + 3 \, f} x\right ) e^{\left (a \log \relax (f) - 3 \, d\right )}}{16 \, \sqrt {-c \log \relax (f) + 3 \, f}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*x^2+a)*cosh(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)

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maple [A]  time = 0.37, size = 144, normalized size = 0.84 \[ \frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-3 d} \erf \left (x \sqrt {3 f -c \ln \relax (f )}\right )}{16 \sqrt {3 f -c \ln \relax (f )}}+\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{3 d} \erf \left (\sqrt {-c \ln \relax (f )-3 f}\, x \right )}{16 \sqrt {-c \ln \relax (f )-3 f}}+\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{-d} \erf \left (x \sqrt {f -c \ln \relax (f )}\right )}{16 \sqrt {f -c \ln \relax (f )}}+\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{d} \erf \left (\sqrt {-c \ln \relax (f )-f}\, x \right )}{16 \sqrt {-c \ln \relax (f )-f}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c*x^2+a)*cosh(f*x^2+d)^3,x)

[Out]

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))+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)+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 = 0.33, size = 143, normalized size = 0.84 \[ \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f) - 3 \, f} x\right ) e^{\left (3 \, d\right )}}{16 \, \sqrt {-c \log \relax (f) - 3 \, f}} + \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f) + f} x\right ) e^{\left (-d\right )}}{16 \, \sqrt {-c \log \relax (f) + f}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f) + 3 \, f} x\right ) e^{\left (-3 \, d\right )}}{16 \, \sqrt {-c \log \relax (f) + 3 \, f}} + \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \relax (f) - f} x\right ) e^{d}}{16 \, \sqrt {-c \log \relax (f) - f}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*x^2+a)*cosh(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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int f^{c\,x^2+a}\,{\mathrm {cosh}\left (f\,x^2+d\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + c*x^2)*cosh(d + f*x^2)^3,x)

[Out]

int(f^(a + c*x^2)*cosh(d + f*x^2)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + c x^{2}} \cosh ^{3}{\left (d + f x^{2} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(c*x**2+a)*cosh(f*x**2+d)**3,x)

[Out]

Integral(f**(a + c*x**2)*cosh(d + f*x**2)**3, x)

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