3.18 \(\int \cosh ^3(a+b x^2) \, dx\)

Optimal. Leaf size=125 \[ \frac {3 \sqrt {\pi } e^{-a} \text {erf}\left (\sqrt {b} x\right )}{16 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{3}} e^{-3 a} \text {erf}\left (\sqrt {3} \sqrt {b} x\right )}{16 \sqrt {b}}+\frac {3 \sqrt {\pi } e^a \text {erfi}\left (\sqrt {b} x\right )}{16 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{3}} e^{3 a} \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )}{16 \sqrt {b}} \]

[Out]

1/48*erf(x*3^(1/2)*b^(1/2))*3^(1/2)*Pi^(1/2)/exp(3*a)/b^(1/2)+1/48*exp(3*a)*erfi(x*3^(1/2)*b^(1/2))*3^(1/2)*Pi
^(1/2)/b^(1/2)+3/16*erf(x*b^(1/2))*Pi^(1/2)/exp(a)/b^(1/2)+3/16*exp(a)*erfi(x*b^(1/2))*Pi^(1/2)/b^(1/2)

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Rubi [A]  time = 0.07, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5301, 5299, 2204, 2205} \[ \frac {3 \sqrt {\pi } e^{-a} \text {Erf}\left (\sqrt {b} x\right )}{16 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{3}} e^{-3 a} \text {Erf}\left (\sqrt {3} \sqrt {b} x\right )}{16 \sqrt {b}}+\frac {3 \sqrt {\pi } e^a \text {Erfi}\left (\sqrt {b} x\right )}{16 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{3}} e^{3 a} \text {Erfi}\left (\sqrt {3} \sqrt {b} x\right )}{16 \sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x^2]^3,x]

[Out]

(3*Sqrt[Pi]*Erf[Sqrt[b]*x])/(16*Sqrt[b]*E^a) + (Sqrt[Pi/3]*Erf[Sqrt[3]*Sqrt[b]*x])/(16*Sqrt[b]*E^(3*a)) + (3*E
^a*Sqrt[Pi]*Erfi[Sqrt[b]*x])/(16*Sqrt[b]) + (E^(3*a)*Sqrt[Pi/3]*Erfi[Sqrt[3]*Sqrt[b]*x])/(16*Sqrt[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]

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 5299

Int[Cosh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[1/2, Int[E^(c + d*x^n), x], x] + Dist[1/2, Int[E^(-c - d*
x^n), x], x] /; FreeQ[{c, d}, x] && IGtQ[n, 1]

Rule 5301

Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_), x_Symbol] :> Int[ExpandTrigReduce[(a + b*Cosh[c + d*x^
n])^p, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 1] && IGtQ[p, 1]

Rubi steps

\begin {align*} \int \cosh ^3\left (a+b x^2\right ) \, dx &=\int \left (\frac {3}{4} \cosh \left (a+b x^2\right )+\frac {1}{4} \cosh \left (3 a+3 b x^2\right )\right ) \, dx\\ &=\frac {1}{4} \int \cosh \left (3 a+3 b x^2\right ) \, dx+\frac {3}{4} \int \cosh \left (a+b x^2\right ) \, dx\\ &=\frac {1}{8} \int e^{-3 a-3 b x^2} \, dx+\frac {1}{8} \int e^{3 a+3 b x^2} \, dx+\frac {3}{8} \int e^{-a-b x^2} \, dx+\frac {3}{8} \int e^{a+b x^2} \, dx\\ &=\frac {3 e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} x\right )}{16 \sqrt {b}}+\frac {e^{-3 a} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {b} x\right )}{16 \sqrt {b}}+\frac {3 e^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x\right )}{16 \sqrt {b}}+\frac {e^{3 a} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )}{16 \sqrt {b}}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 136, normalized size = 1.09 \[ \frac {\sqrt {\frac {\pi }{3}} \left (3 \sqrt {3} (\cosh (a)-\sinh (a)) \text {erf}\left (\sqrt {b} x\right )+(\cosh (3 a)-\sinh (3 a)) \text {erf}\left (\sqrt {3} \sqrt {b} x\right )+3 \sqrt {3} \sinh (a) \text {erfi}\left (\sqrt {b} x\right )+\sinh (3 a) \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )+3 \sqrt {3} \cosh (a) \text {erfi}\left (\sqrt {b} x\right )+\cosh (3 a) \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )\right )}{16 \sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x^2]^3,x]

[Out]

(Sqrt[Pi/3]*(3*Sqrt[3]*Cosh[a]*Erfi[Sqrt[b]*x] + Cosh[3*a]*Erfi[Sqrt[3]*Sqrt[b]*x] + 3*Sqrt[3]*Erf[Sqrt[b]*x]*
(Cosh[a] - Sinh[a]) + 3*Sqrt[3]*Erfi[Sqrt[b]*x]*Sinh[a] + Erf[Sqrt[3]*Sqrt[b]*x]*(Cosh[3*a] - Sinh[3*a]) + Erf
i[Sqrt[3]*Sqrt[b]*x]*Sinh[3*a]))/(16*Sqrt[b])

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fricas [A]  time = 0.55, size = 113, normalized size = 0.90 \[ -\frac {\sqrt {3} \sqrt {\pi } \sqrt {-b} {\left (\cosh \left (3 \, a\right ) + \sinh \left (3 \, a\right )\right )} \operatorname {erf}\left (\sqrt {3} \sqrt {-b} x\right ) - \sqrt {3} \sqrt {\pi } \sqrt {b} {\left (\cosh \left (3 \, a\right ) - \sinh \left (3 \, a\right )\right )} \operatorname {erf}\left (\sqrt {3} \sqrt {b} x\right ) + 9 \, \sqrt {\pi } \sqrt {-b} {\left (\cosh \relax (a) + \sinh \relax (a)\right )} \operatorname {erf}\left (\sqrt {-b} x\right ) - 9 \, \sqrt {\pi } \sqrt {b} {\left (\cosh \relax (a) - \sinh \relax (a)\right )} \operatorname {erf}\left (\sqrt {b} x\right )}{48 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x^2+a)^3,x, algorithm="fricas")

[Out]

-1/48*(sqrt(3)*sqrt(pi)*sqrt(-b)*(cosh(3*a) + sinh(3*a))*erf(sqrt(3)*sqrt(-b)*x) - sqrt(3)*sqrt(pi)*sqrt(b)*(c
osh(3*a) - sinh(3*a))*erf(sqrt(3)*sqrt(b)*x) + 9*sqrt(pi)*sqrt(-b)*(cosh(a) + sinh(a))*erf(sqrt(-b)*x) - 9*sqr
t(pi)*sqrt(b)*(cosh(a) - sinh(a))*erf(sqrt(b)*x))/b

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giac [A]  time = 0.12, size = 95, normalized size = 0.76 \[ -\frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {3} \sqrt {-b} x\right ) e^{\left (3 \, a\right )}}{48 \, \sqrt {-b}} - \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {3} \sqrt {b} x\right ) e^{\left (-3 \, a\right )}}{48 \, \sqrt {b}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} x\right ) e^{\left (-a\right )}}{16 \, \sqrt {b}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {-b} x\right ) e^{a}}{16 \, \sqrt {-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x^2+a)^3,x, algorithm="giac")

[Out]

-1/48*sqrt(3)*sqrt(pi)*erf(-sqrt(3)*sqrt(-b)*x)*e^(3*a)/sqrt(-b) - 1/48*sqrt(3)*sqrt(pi)*erf(-sqrt(3)*sqrt(b)*
x)*e^(-3*a)/sqrt(b) - 3/16*sqrt(pi)*erf(-sqrt(b)*x)*e^(-a)/sqrt(b) - 3/16*sqrt(pi)*erf(-sqrt(-b)*x)*e^a/sqrt(-
b)

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maple [A]  time = 0.18, size = 86, normalized size = 0.69 \[ \frac {{\mathrm e}^{-3 a} \sqrt {\pi }\, \sqrt {3}\, \erf \left (x \sqrt {3}\, \sqrt {b}\right )}{48 \sqrt {b}}+\frac {3 \erf \left (x \sqrt {b}\right ) \sqrt {\pi }\, {\mathrm e}^{-a}}{16 \sqrt {b}}+\frac {{\mathrm e}^{3 a} \sqrt {\pi }\, \erf \left (\sqrt {-3 b}\, x \right )}{16 \sqrt {-3 b}}+\frac {3 \,{\mathrm e}^{a} \sqrt {\pi }\, \erf \left (\sqrt {-b}\, x \right )}{16 \sqrt {-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(b*x^2+a)^3,x)

[Out]

1/48*exp(-3*a)*Pi^(1/2)*3^(1/2)/b^(1/2)*erf(x*3^(1/2)*b^(1/2))+3/16*erf(x*b^(1/2))*Pi^(1/2)*exp(-a)/b^(1/2)+1/
16*exp(3*a)*Pi^(1/2)/(-3*b)^(1/2)*erf((-3*b)^(1/2)*x)+3/16*exp(a)*Pi^(1/2)/(-b)^(1/2)*erf((-b)^(1/2)*x)

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maxima [A]  time = 0.40, size = 91, normalized size = 0.73 \[ \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\sqrt {3} \sqrt {-b} x\right ) e^{\left (3 \, a\right )}}{48 \, \sqrt {-b}} + \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\sqrt {3} \sqrt {b} x\right ) e^{\left (-3 \, a\right )}}{48 \, \sqrt {b}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {b} x\right ) e^{\left (-a\right )}}{16 \, \sqrt {b}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {-b} x\right ) e^{a}}{16 \, \sqrt {-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x^2+a)^3,x, algorithm="maxima")

[Out]

1/48*sqrt(3)*sqrt(pi)*erf(sqrt(3)*sqrt(-b)*x)*e^(3*a)/sqrt(-b) + 1/48*sqrt(3)*sqrt(pi)*erf(sqrt(3)*sqrt(b)*x)*
e^(-3*a)/sqrt(b) + 3/16*sqrt(pi)*erf(sqrt(b)*x)*e^(-a)/sqrt(b) + 3/16*sqrt(pi)*erf(sqrt(-b)*x)*e^a/sqrt(-b)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x^2)^3,x)

[Out]

int(cosh(a + b*x^2)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x**2+a)**3,x)

[Out]

Integral(cosh(a + b*x**2)**3, x)

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