3.1.18 \(\int \sinh ^3(a+b x^2) \, dx\) [18]

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

[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)*P
i^(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 = 0.05, 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 = {5408, 5406, 2235, 2236} \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[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 2235

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 2236

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 5406

Int[Sinh[(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 5408

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

Rubi steps

\begin {align*} \int \sinh ^3\left (a+b x^2\right ) \, dx &=\int \left (-\frac {3}{4} \sinh \left (a+b x^2\right )+\frac {1}{4} \sinh \left (3 a+3 b x^2\right )\right ) \, dx\\ &=\frac {1}{4} \int \sinh \left (3 a+3 b x^2\right ) \, dx-\frac {3}{4} \int \sinh \left (a+b x^2\right ) \, dx\\ &=-\left (\frac {1}{8} \int e^{-3 a-3 b x^2} \, dx\right )+\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 = 0.09, size = 136, normalized size = 1.09 \begin {gather*} \frac {\sqrt {\frac {\pi }{3}} \left (-3 \sqrt {3} \cosh (a) \text {Erfi}\left (\sqrt {b} x\right )+\cosh (3 a) \text {Erfi}\left (\sqrt {3} \sqrt {b} x\right )+3 \sqrt {3} \text {Erf}\left (\sqrt {b} x\right ) (\cosh (a)-\sinh (a))-3 \sqrt {3} \text {Erfi}\left (\sqrt {b} x\right ) \sinh (a)+\text {Erfi}\left (\sqrt {3} \sqrt {b} x\right ) \sinh (3 a)+\text {Erf}\left (\sqrt {3} \sqrt {b} x\right ) (-\cosh (3 a)+\sinh (3 a))\right )}{16 \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[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] + Erfi[Sqrt[3]*Sqrt[b]*x]*Sinh[3*a] + Erf[Sqrt[3]*Sqr
t[b]*x]*(-Cosh[3*a] + Sinh[3*a])))/(16*Sqrt[b])

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Maple [A]
time = 0.88, size = 86, normalized size = 0.69

method	result	size
risch	\(-\frac {{\mathrm e}^{-3 a} \sqrt {\pi }\, \sqrt {3}\, \erf \left (x \sqrt {3}\, \sqrt {b}\right )}{48 \sqrt {b}}+\frac {3 \erf \left (x \sqrt {b}\right ) \sqrt {\pi }\, {\mathrm e}^{-a}}{16 \sqrt {b}}+\frac {{\mathrm e}^{3 a} \sqrt {\pi }\, \erf \left (\sqrt {-3 b}\, x \right )}{16 \sqrt {-3 b}}-\frac {3 \,{\mathrm e}^{a} \sqrt {\pi }\, \erf \left (\sqrt {-b}\, x \right )}{16 \sqrt {-b}}\)	\(86\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(b*x^2+a)^3,x,method=_RETURNVERBOSE)

[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 = 0.47, size = 91, normalized size = 0.73 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(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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Fricas [A]
time = 0.42, size = 112, normalized size = 0.90 \begin {gather*} -\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 \left (a\right ) + \sinh \left (a\right )\right )} \operatorname {erf}\left (\sqrt {-b} x\right ) - 9 \, \sqrt {\pi } \sqrt {b} {\left (\cosh \left (a\right ) - \sinh \left (a\right )\right )} \operatorname {erf}\left (\sqrt {b} x\right )}{48 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sinh ^{3}{\left (a + b x^{2} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sinh(a + b*x**2)**3, x)

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Giac [A]
time = 0.46, size = 95, normalized size = 0.76 \begin {gather*} -\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {sinh}\left (b\,x^2+a\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b*x^2)^3,x)

[Out]

int(sinh(a + b*x^2)^3, x)

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