3.1.20 \(\int \frac {\sinh ^3(a+b x^2)}{x^2} \, dx\) [20]

Optimal. Leaf size=136 \[ -\frac {3}{8} \sqrt {b} e^{-a} \sqrt {\pi } \text {Erf}\left (\sqrt {b} x\right )+\frac {1}{8} \sqrt {b} e^{-3 a} \sqrt {3 \pi } \text {Erf}\left (\sqrt {3} \sqrt {b} x\right )-\frac {3}{8} \sqrt {b} e^a \sqrt {\pi } \text {Erfi}\left (\sqrt {b} x\right )+\frac {1}{8} \sqrt {b} e^{3 a} \sqrt {3 \pi } \text {Erfi}\left (\sqrt {3} \sqrt {b} x\right )-\frac {\sinh ^3\left (a+b x^2\right )}{x} \]

[Out]

-sinh(b*x^2+a)^3/x-3/8*erf(x*b^(1/2))*b^(1/2)*Pi^(1/2)/exp(a)-3/8*exp(a)*erfi(x*b^(1/2))*b^(1/2)*Pi^(1/2)+1/8*
erf(x*3^(1/2)*b^(1/2))*b^(1/2)*3^(1/2)*Pi^(1/2)/exp(3*a)+1/8*exp(3*a)*erfi(x*3^(1/2)*b^(1/2))*b^(1/2)*3^(1/2)*
Pi^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5438, 5737, 5407, 2235, 2236} \begin {gather*} -\frac {3}{8} \sqrt {\pi } e^{-a} \sqrt {b} \text {Erf}\left (\sqrt {b} x\right )+\frac {1}{8} \sqrt {3 \pi } e^{-3 a} \sqrt {b} \text {Erf}\left (\sqrt {3} \sqrt {b} x\right )-\frac {3}{8} \sqrt {\pi } e^a \sqrt {b} \text {Erfi}\left (\sqrt {b} x\right )+\frac {1}{8} \sqrt {3 \pi } e^{3 a} \sqrt {b} \text {Erfi}\left (\sqrt {3} \sqrt {b} x\right )-\frac {\sinh ^3\left (a+b x^2\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x^2]^3/x^2,x]

[Out]

(-3*Sqrt[b]*Sqrt[Pi]*Erf[Sqrt[b]*x])/(8*E^a) + (Sqrt[b]*Sqrt[3*Pi]*Erf[Sqrt[3]*Sqrt[b]*x])/(8*E^(3*a)) - (3*Sq
rt[b]*E^a*Sqrt[Pi]*Erfi[Sqrt[b]*x])/8 + (Sqrt[b]*E^(3*a)*Sqrt[3*Pi]*Erfi[Sqrt[3]*Sqrt[b]*x])/8 - Sinh[a + b*x^
2]^3/x

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 5407

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 5438

Int[(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_)]^(p_), x_Symbol] :> Simp[-Sinh[a + b*x^n]^p/((n - 1)*x^(n - 1)), x
] + Dist[b*n*(p/(n - 1)), Int[Sinh[a + b*x^n]^(p - 1)*Cosh[a + b*x^n], x], x] /; FreeQ[{a, b}, x] && IntegersQ
[n, p] && EqQ[m + n, 0] && GtQ[p, 1] && NeQ[n, 1]

Rule 5737

Int[Cosh[w_]^(q_.)*Sinh[v_]^(p_.), x_Symbol] :> Int[ExpandTrigReduce[Sinh[v]^p*Cosh[w]^q, x], x] /; IGtQ[p, 0]
&& IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/
w], x]))

Rubi steps

\begin {align*} \int \frac {\sinh ^3\left (a+b x^2\right )}{x^2} \, dx &=-\frac {\sinh ^3\left (a+b x^2\right )}{x}+(6 b) \int \cosh \left (a+b x^2\right ) \sinh ^2\left (a+b x^2\right ) \, dx\\ &=-\frac {\sinh ^3\left (a+b x^2\right )}{x}+(6 b) \int \left (-\frac {1}{4} \cosh \left (a+b x^2\right )+\frac {1}{4} \cosh \left (3 a+3 b x^2\right )\right ) \, dx\\ &=-\frac {\sinh ^3\left (a+b x^2\right )}{x}-\frac {1}{2} (3 b) \int \cosh \left (a+b x^2\right ) \, dx+\frac {1}{2} (3 b) \int \cosh \left (3 a+3 b x^2\right ) \, dx\\ &=-\frac {\sinh ^3\left (a+b x^2\right )}{x}+\frac {1}{4} (3 b) \int e^{-3 a-3 b x^2} \, dx-\frac {1}{4} (3 b) \int e^{-a-b x^2} \, dx-\frac {1}{4} (3 b) \int e^{a+b x^2} \, dx+\frac {1}{4} (3 b) \int e^{3 a+3 b x^2} \, dx\\ &=-\frac {3}{8} \sqrt {b} e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} x\right )+\frac {1}{8} \sqrt {b} e^{-3 a} \sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {b} x\right )-\frac {3}{8} \sqrt {b} e^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x\right )+\frac {1}{8} \sqrt {b} e^{3 a} \sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )-\frac {\sinh ^3\left (a+b x^2\right )}{x}\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 204, normalized size = 1.50 \begin {gather*} \frac {-3 \sqrt {b} \sqrt {\pi } x \cosh (a) \text {Erfi}\left (\sqrt {b} x\right )+\sqrt {b} \sqrt {3 \pi } x \cosh (3 a) \text {Erfi}\left (\sqrt {3} \sqrt {b} x\right )-3 \sqrt {b} \sqrt {\pi } x \text {Erfi}\left (\sqrt {b} x\right ) \sinh (a)+3 \sqrt {b} \sqrt {\pi } x \text {Erf}\left (\sqrt {b} x\right ) (-\cosh (a)+\sinh (a))+\sqrt {b} \sqrt {3 \pi } x \text {Erf}\left (\sqrt {3} \sqrt {b} x\right ) (\cosh (3 a)-\sinh (3 a))+\sqrt {b} \sqrt {3 \pi } x \text {Erfi}\left (\sqrt {3} \sqrt {b} x\right ) \sinh (3 a)+6 \sinh \left (a+b x^2\right )-2 \sinh \left (3 \left (a+b x^2\right )\right )}{8 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*x^2]^3/x^2,x]

[Out]

(-3*Sqrt[b]*Sqrt[Pi]*x*Cosh[a]*Erfi[Sqrt[b]*x] + Sqrt[b]*Sqrt[3*Pi]*x*Cosh[3*a]*Erfi[Sqrt[3]*Sqrt[b]*x] - 3*Sq
rt[b]*Sqrt[Pi]*x*Erfi[Sqrt[b]*x]*Sinh[a] + 3*Sqrt[b]*Sqrt[Pi]*x*Erf[Sqrt[b]*x]*(-Cosh[a] + Sinh[a]) + Sqrt[b]*
Sqrt[3*Pi]*x*Erf[Sqrt[3]*Sqrt[b]*x]*(Cosh[3*a] - Sinh[3*a]) + Sqrt[b]*Sqrt[3*Pi]*x*Erfi[Sqrt[3]*Sqrt[b]*x]*Sin
h[3*a] + 6*Sinh[a + b*x^2] - 2*Sinh[3*(a + b*x^2)])/(8*x)

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Maple [A]
time = 1.25, size = 149, normalized size = 1.10

method	result	size
risch	\(\frac {{\mathrm e}^{-3 a} {\mathrm e}^{-3 x^{2} b}}{8 x}+\frac {{\mathrm e}^{-3 a} \sqrt {b}\, \sqrt {\pi }\, \sqrt {3}\, \erf \left (x \sqrt {3}\, \sqrt {b}\right )}{8}-\frac {3 \,{\mathrm e}^{-a} {\mathrm e}^{-x^{2} b}}{8 x}-\frac {3 \,{\mathrm e}^{-a} \sqrt {b}\, \sqrt {\pi }\, \erf \left (x \sqrt {b}\right )}{8}+\frac {3 \,{\mathrm e}^{a} {\mathrm e}^{x^{2} b}}{8 x}-\frac {3 \,{\mathrm e}^{a} b \sqrt {\pi }\, \erf \left (\sqrt {-b}\, x \right )}{8 \sqrt {-b}}-\frac {{\mathrm e}^{3 a} {\mathrm e}^{3 x^{2} b}}{8 x}+\frac {3 \,{\mathrm e}^{3 a} b \sqrt {\pi }\, \erf \left (\sqrt {-3 b}\, x \right )}{8 \sqrt {-3 b}}\)	\(149\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*exp(-3*a)/x*exp(-3*x^2*b)+1/8*exp(-3*a)*b^(1/2)*Pi^(1/2)*3^(1/2)*erf(x*3^(1/2)*b^(1/2))-3/8*exp(-a)/x*exp(
-x^2*b)-3/8*exp(-a)*b^(1/2)*Pi^(1/2)*erf(x*b^(1/2))+3/8*exp(a)*exp(x^2*b)/x-3/8*exp(a)*b*Pi^(1/2)/(-b)^(1/2)*e
rf((-b)^(1/2)*x)-1/8*exp(3*a)/x*exp(3*x^2*b)+3/8*exp(3*a)*b*Pi^(1/2)/(-3*b)^(1/2)*erf((-3*b)^(1/2)*x)

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Maxima [A]
time = 0.33, size = 102, normalized size = 0.75 \begin {gather*} \frac {\sqrt {3} \sqrt {b x^{2}} e^{\left (-3 \, a\right )} \Gamma \left (-\frac {1}{2}, 3 \, b x^{2}\right )}{16 \, x} - \frac {\sqrt {3} \sqrt {-b x^{2}} e^{\left (3 \, a\right )} \Gamma \left (-\frac {1}{2}, -3 \, b x^{2}\right )}{16 \, x} - \frac {3 \, \sqrt {b x^{2}} e^{\left (-a\right )} \Gamma \left (-\frac {1}{2}, b x^{2}\right )}{16 \, x} + \frac {3 \, \sqrt {-b x^{2}} e^{a} \Gamma \left (-\frac {1}{2}, -b x^{2}\right )}{16 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*sqrt(3)*sqrt(b*x^2)*e^(-3*a)*gamma(-1/2, 3*b*x^2)/x - 1/16*sqrt(3)*sqrt(-b*x^2)*e^(3*a)*gamma(-1/2, -3*b*
x^2)/x - 3/16*sqrt(b*x^2)*e^(-a)*gamma(-1/2, b*x^2)/x + 3/16*sqrt(-b*x^2)*e^a*gamma(-1/2, -b*x^2)/x

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 892 vs. \(2 (98) = 196\).
time = 0.53, size = 892, normalized size = 6.56 \begin {gather*} -\frac {\cosh \left (b x^{2} + a\right )^{6} + 6 \, \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right )^{5} + \sinh \left (b x^{2} + a\right )^{6} + 3 \, {\left (5 \, \cosh \left (b x^{2} + a\right )^{2} - 1\right )} \sinh \left (b x^{2} + a\right )^{4} - 3 \, \cosh \left (b x^{2} + a\right )^{4} + 4 \, {\left (5 \, \cosh \left (b x^{2} + a\right )^{3} - 3 \, \cosh \left (b x^{2} + a\right )\right )} \sinh \left (b x^{2} + a\right )^{3} + \sqrt {3} \sqrt {\pi } {\left (x \cosh \left (b x^{2} + a\right )^{3} \cosh \left (3 \, a\right ) + x \cosh \left (b x^{2} + a\right )^{3} \sinh \left (3 \, a\right ) + {\left (x \cosh \left (3 \, a\right ) + x \sinh \left (3 \, a\right )\right )} \sinh \left (b x^{2} + a\right )^{3} + 3 \, {\left (x \cosh \left (b x^{2} + a\right ) \cosh \left (3 \, a\right ) + x \cosh \left (b x^{2} + a\right ) \sinh \left (3 \, a\right )\right )} \sinh \left (b x^{2} + a\right )^{2} + 3 \, {\left (x \cosh \left (b x^{2} + a\right )^{2} \cosh \left (3 \, a\right ) + x \cosh \left (b x^{2} + a\right )^{2} \sinh \left (3 \, a\right )\right )} \sinh \left (b x^{2} + a\right )\right )} \sqrt {-b} \operatorname {erf}\left (\sqrt {3} \sqrt {-b} x\right ) - \sqrt {3} \sqrt {\pi } {\left (x \cosh \left (b x^{2} + a\right )^{3} \cosh \left (3 \, a\right ) - x \cosh \left (b x^{2} + a\right )^{3} \sinh \left (3 \, a\right ) + {\left (x \cosh \left (3 \, a\right ) - x \sinh \left (3 \, a\right )\right )} \sinh \left (b x^{2} + a\right )^{3} + 3 \, {\left (x \cosh \left (b x^{2} + a\right ) \cosh \left (3 \, a\right ) - x \cosh \left (b x^{2} + a\right ) \sinh \left (3 \, a\right )\right )} \sinh \left (b x^{2} + a\right )^{2} + 3 \, {\left (x \cosh \left (b x^{2} + a\right )^{2} \cosh \left (3 \, a\right ) - x \cosh \left (b x^{2} + a\right )^{2} \sinh \left (3 \, a\right )\right )} \sinh \left (b x^{2} + a\right )\right )} \sqrt {b} \operatorname {erf}\left (\sqrt {3} \sqrt {b} x\right ) - 3 \, \sqrt {\pi } {\left (x \cosh \left (b x^{2} + a\right )^{3} \cosh \left (a\right ) + x \cosh \left (b x^{2} + a\right )^{3} \sinh \left (a\right ) + {\left (x \cosh \left (a\right ) + x \sinh \left (a\right )\right )} \sinh \left (b x^{2} + a\right )^{3} + 3 \, {\left (x \cosh \left (b x^{2} + a\right ) \cosh \left (a\right ) + x \cosh \left (b x^{2} + a\right ) \sinh \left (a\right )\right )} \sinh \left (b x^{2} + a\right )^{2} + 3 \, {\left (x \cosh \left (b x^{2} + a\right )^{2} \cosh \left (a\right ) + x \cosh \left (b x^{2} + a\right )^{2} \sinh \left (a\right )\right )} \sinh \left (b x^{2} + a\right )\right )} \sqrt {-b} \operatorname {erf}\left (\sqrt {-b} x\right ) + 3 \, \sqrt {\pi } {\left (x \cosh \left (b x^{2} + a\right )^{3} \cosh \left (a\right ) - x \cosh \left (b x^{2} + a\right )^{3} \sinh \left (a\right ) + {\left (x \cosh \left (a\right ) - x \sinh \left (a\right )\right )} \sinh \left (b x^{2} + a\right )^{3} + 3 \, {\left (x \cosh \left (b x^{2} + a\right ) \cosh \left (a\right ) - x \cosh \left (b x^{2} + a\right ) \sinh \left (a\right )\right )} \sinh \left (b x^{2} + a\right )^{2} + 3 \, {\left (x \cosh \left (b x^{2} + a\right )^{2} \cosh \left (a\right ) - x \cosh \left (b x^{2} + a\right )^{2} \sinh \left (a\right )\right )} \sinh \left (b x^{2} + a\right )\right )} \sqrt {b} \operatorname {erf}\left (\sqrt {b} x\right ) + 3 \, {\left (5 \, \cosh \left (b x^{2} + a\right )^{4} - 6 \, \cosh \left (b x^{2} + a\right )^{2} + 1\right )} \sinh \left (b x^{2} + a\right )^{2} + 3 \, \cosh \left (b x^{2} + a\right )^{2} + 6 \, {\left (\cosh \left (b x^{2} + a\right )^{5} - 2 \, \cosh \left (b x^{2} + a\right )^{3} + \cosh \left (b x^{2} + a\right )\right )} \sinh \left (b x^{2} + a\right ) - 1}{8 \, {\left (x \cosh \left (b x^{2} + a\right )^{3} + 3 \, x \cosh \left (b x^{2} + a\right )^{2} \sinh \left (b x^{2} + a\right ) + 3 \, x \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right )^{2} + x \sinh \left (b x^{2} + a\right )^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(cosh(b*x^2 + a)^6 + 6*cosh(b*x^2 + a)*sinh(b*x^2 + a)^5 + sinh(b*x^2 + a)^6 + 3*(5*cosh(b*x^2 + a)^2 - 1
)*sinh(b*x^2 + a)^4 - 3*cosh(b*x^2 + a)^4 + 4*(5*cosh(b*x^2 + a)^3 - 3*cosh(b*x^2 + a))*sinh(b*x^2 + a)^3 + sq
rt(3)*sqrt(pi)*(x*cosh(b*x^2 + a)^3*cosh(3*a) + x*cosh(b*x^2 + a)^3*sinh(3*a) + (x*cosh(3*a) + x*sinh(3*a))*si
nh(b*x^2 + a)^3 + 3*(x*cosh(b*x^2 + a)*cosh(3*a) + x*cosh(b*x^2 + a)*sinh(3*a))*sinh(b*x^2 + a)^2 + 3*(x*cosh(
b*x^2 + a)^2*cosh(3*a) + x*cosh(b*x^2 + a)^2*sinh(3*a))*sinh(b*x^2 + a))*sqrt(-b)*erf(sqrt(3)*sqrt(-b)*x) - sq
rt(3)*sqrt(pi)*(x*cosh(b*x^2 + a)^3*cosh(3*a) - x*cosh(b*x^2 + a)^3*sinh(3*a) + (x*cosh(3*a) - x*sinh(3*a))*si
nh(b*x^2 + a)^3 + 3*(x*cosh(b*x^2 + a)*cosh(3*a) - x*cosh(b*x^2 + a)*sinh(3*a))*sinh(b*x^2 + a)^2 + 3*(x*cosh(
b*x^2 + a)^2*cosh(3*a) - x*cosh(b*x^2 + a)^2*sinh(3*a))*sinh(b*x^2 + a))*sqrt(b)*erf(sqrt(3)*sqrt(b)*x) - 3*sq
rt(pi)*(x*cosh(b*x^2 + a)^3*cosh(a) + x*cosh(b*x^2 + a)^3*sinh(a) + (x*cosh(a) + x*sinh(a))*sinh(b*x^2 + a)^3
+ 3*(x*cosh(b*x^2 + a)*cosh(a) + x*cosh(b*x^2 + a)*sinh(a))*sinh(b*x^2 + a)^2 + 3*(x*cosh(b*x^2 + a)^2*cosh(a)
+ x*cosh(b*x^2 + a)^2*sinh(a))*sinh(b*x^2 + a))*sqrt(-b)*erf(sqrt(-b)*x) + 3*sqrt(pi)*(x*cosh(b*x^2 + a)^3*co
sh(a) - x*cosh(b*x^2 + a)^3*sinh(a) + (x*cosh(a) - x*sinh(a))*sinh(b*x^2 + a)^3 + 3*(x*cosh(b*x^2 + a)*cosh(a)
- x*cosh(b*x^2 + a)*sinh(a))*sinh(b*x^2 + a)^2 + 3*(x*cosh(b*x^2 + a)^2*cosh(a) - x*cosh(b*x^2 + a)^2*sinh(a)
)*sinh(b*x^2 + a))*sqrt(b)*erf(sqrt(b)*x) + 3*(5*cosh(b*x^2 + a)^4 - 6*cosh(b*x^2 + a)^2 + 1)*sinh(b*x^2 + a)^
2 + 3*cosh(b*x^2 + a)^2 + 6*(cosh(b*x^2 + a)^5 - 2*cosh(b*x^2 + a)^3 + cosh(b*x^2 + a))*sinh(b*x^2 + a) - 1)/(
x*cosh(b*x^2 + a)^3 + 3*x*cosh(b*x^2 + a)^2*sinh(b*x^2 + a) + 3*x*cosh(b*x^2 + a)*sinh(b*x^2 + a)^2 + x*sinh(b
*x^2 + a)^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sinh(a + b*x**2)**3/x**2, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sinh(b*x^2 + a)^3/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b*x^2)^3/x^2,x)

[Out]

int(sinh(a + b*x^2)^3/x^2, x)

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