3.1.53 \(\int (e x)^m \sinh ^3(a+\frac {b}{x^2}) \, dx\) [53]

Optimal. Leaf size=194 \[ \frac {1}{16} 3^{\frac {1+m}{2}} e^{3 a} \left (-\frac {b}{x^2}\right )^{\frac {1+m}{2}} x (e x)^m \Gamma \left (\frac {1}{2} (-1-m),-\frac {3 b}{x^2}\right )-\frac {3}{16} e^a \left (-\frac {b}{x^2}\right )^{\frac {1+m}{2}} x (e x)^m \Gamma \left (\frac {1}{2} (-1-m),-\frac {b}{x^2}\right )+\frac {3}{16} e^{-a} \left (\frac {b}{x^2}\right )^{\frac {1+m}{2}} x (e x)^m \Gamma \left (\frac {1}{2} (-1-m),\frac {b}{x^2}\right )-\frac {1}{16} 3^{\frac {1+m}{2}} e^{-3 a} \left (\frac {b}{x^2}\right )^{\frac {1+m}{2}} x (e x)^m \Gamma \left (\frac {1}{2} (-1-m),\frac {3 b}{x^2}\right ) \]

[Out]

1/16*3^(1/2+1/2*m)*exp(3*a)*(-b/x^2)^(1/2+1/2*m)*x*(e*x)^m*GAMMA(-1/2-1/2*m,-3*b/x^2)-3/16*exp(a)*(-b/x^2)^(1/
2+1/2*m)*x*(e*x)^m*GAMMA(-1/2-1/2*m,-b/x^2)+3/16*(b/x^2)^(1/2+1/2*m)*x*(e*x)^m*GAMMA(-1/2-1/2*m,b/x^2)/exp(a)-
1/16*3^(1/2+1/2*m)*(b/x^2)^(1/2+1/2*m)*x*(e*x)^m*GAMMA(-1/2-1/2*m,3*b/x^2)/exp(3*a)

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Rubi [A]
time = 0.15, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5458, 5448, 5436, 2250} \begin {gather*} \frac {1}{16} e^{3 a} 3^{\frac {m+1}{2}} x \left (-\frac {b}{x^2}\right )^{\frac {m+1}{2}} (e x)^m \text {Gamma}\left (\frac {1}{2} (-m-1),-\frac {3 b}{x^2}\right )-\frac {3}{16} e^a x \left (-\frac {b}{x^2}\right )^{\frac {m+1}{2}} (e x)^m \text {Gamma}\left (\frac {1}{2} (-m-1),-\frac {b}{x^2}\right )+\frac {3}{16} e^{-a} x \left (\frac {b}{x^2}\right )^{\frac {m+1}{2}} (e x)^m \text {Gamma}\left (\frac {1}{2} (-m-1),\frac {b}{x^2}\right )-\frac {1}{16} e^{-3 a} 3^{\frac {m+1}{2}} x \left (\frac {b}{x^2}\right )^{\frac {m+1}{2}} (e x)^m \text {Gamma}\left (\frac {1}{2} (-m-1),\frac {3 b}{x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*Sinh[a + b/x^2]^3,x]

[Out]

(3^((1 + m)/2)*E^(3*a)*(-(b/x^2))^((1 + m)/2)*x*(e*x)^m*Gamma[(-1 - m)/2, (-3*b)/x^2])/16 - (3*E^a*(-(b/x^2))^
((1 + m)/2)*x*(e*x)^m*Gamma[(-1 - m)/2, -(b/x^2)])/16 + (3*(b/x^2)^((1 + m)/2)*x*(e*x)^m*Gamma[(-1 - m)/2, b/x
^2])/(16*E^a) - (3^((1 + m)/2)*(b/x^2)^((1 + m)/2)*x*(e*x)^m*Gamma[(-1 - m)/2, (3*b)/x^2])/(16*E^(3*a))

Rule 2250

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-F^a)*((e +
f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; Fre
eQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rule 5436

Int[((e_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[1/2, Int[(e*x)^m*E^(c + d*x^n), x], x]
- Dist[1/2, Int[(e*x)^m*E^(-c - d*x^n), x], x] /; FreeQ[{c, d, e, m}, x] && IGtQ[n, 0]

Rule 5448

Int[((e_.)*(x_))^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_), x_Symbol] :> Int[ExpandTrigReduce[(
e*x)^m, (a + b*Sinh[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 1] && IGtQ[n, 0]

Rule 5458

Int[((e_.)*(x_))^(m_)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[(-(e*x)^m)*(x^(-1
))^m, Subst[Int[(a + b*Sinh[c + d/x^n])^p/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, m}, x] && Intege
rQ[p] && ILtQ[n, 0] &&  !RationalQ[m]

Rubi steps

\begin {align*} \int (e x)^m \sinh ^3\left (a+\frac {b}{x^2}\right ) \, dx &=-\left (\left (\left (\frac {1}{x}\right )^m (e x)^m\right ) \text {Subst}\left (\int x^{-2-m} \sinh ^3\left (a+b x^2\right ) \, dx,x,\frac {1}{x}\right )\right )\\ &=-\left (\left (\left (\frac {1}{x}\right )^m (e x)^m\right ) \text {Subst}\left (\int \left (-\frac {3}{4} x^{-2-m} \sinh \left (a+b x^2\right )+\frac {1}{4} x^{-2-m} \sinh \left (3 a+3 b x^2\right )\right ) \, dx,x,\frac {1}{x}\right )\right )\\ &=-\left (\frac {1}{4} \left (\left (\frac {1}{x}\right )^m (e x)^m\right ) \text {Subst}\left (\int x^{-2-m} \sinh \left (3 a+3 b x^2\right ) \, dx,x,\frac {1}{x}\right )\right )+\frac {1}{4} \left (3 \left (\frac {1}{x}\right )^m (e x)^m\right ) \text {Subst}\left (\int x^{-2-m} \sinh \left (a+b x^2\right ) \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{8} \left (\left (\frac {1}{x}\right )^m (e x)^m\right ) \text {Subst}\left (\int e^{-3 a-3 b x^2} x^{-2-m} \, dx,x,\frac {1}{x}\right )-\frac {1}{8} \left (\left (\frac {1}{x}\right )^m (e x)^m\right ) \text {Subst}\left (\int e^{3 a+3 b x^2} x^{-2-m} \, dx,x,\frac {1}{x}\right )-\frac {1}{8} \left (3 \left (\frac {1}{x}\right )^m (e x)^m\right ) \text {Subst}\left (\int e^{-a-b x^2} x^{-2-m} \, dx,x,\frac {1}{x}\right )+\frac {1}{8} \left (3 \left (\frac {1}{x}\right )^m (e x)^m\right ) \text {Subst}\left (\int e^{a+b x^2} x^{-2-m} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{16} 3^{\frac {1+m}{2}} e^{3 a} \left (-\frac {b}{x^2}\right )^{\frac {1+m}{2}} x (e x)^m \Gamma \left (\frac {1}{2} (-1-m),-\frac {3 b}{x^2}\right )-\frac {3}{16} e^a \left (-\frac {b}{x^2}\right )^{\frac {1+m}{2}} x (e x)^m \Gamma \left (\frac {1}{2} (-1-m),-\frac {b}{x^2}\right )+\frac {3}{16} e^{-a} \left (\frac {b}{x^2}\right )^{\frac {1+m}{2}} x (e x)^m \Gamma \left (\frac {1}{2} (-1-m),\frac {b}{x^2}\right )-\frac {1}{16} 3^{\frac {1+m}{2}} e^{-3 a} \left (\frac {b}{x^2}\right )^{\frac {1+m}{2}} x (e x)^m \Gamma \left (\frac {1}{2} (-1-m),\frac {3 b}{x^2}\right )\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1039\) vs. \(2(194)=388\).
time = 18.55, size = 1039, normalized size = 5.36 \begin {gather*} x^{-m} (e x)^m \cosh ^3(a) \left (-\frac {3}{8} \left (\frac {1}{2} \left (-\frac {b}{x^2}\right )^{\frac {1+m}{2}} x^{1+m} \Gamma \left (\frac {1}{2} (-1-m),-\frac {b}{x^2}\right )-\frac {1}{2} \left (\frac {b}{x^2}\right )^{\frac {1+m}{2}} x^{1+m} \Gamma \left (\frac {1}{2} (-1-m),\frac {b}{x^2}\right )\right )+\frac {1}{8} \left (\frac {1}{2} 3^{\frac {1+m}{2}} \left (-\frac {b}{x^2}\right )^{\frac {1+m}{2}} x^{1+m} \Gamma \left (\frac {1}{2} (-1-m),-\frac {3 b}{x^2}\right )-\frac {1}{2} 3^{\frac {1+m}{2}} \left (\frac {b}{x^2}\right )^{\frac {1+m}{2}} x^{1+m} \Gamma \left (\frac {1}{2} (-1-m),\frac {3 b}{x^2}\right )\right )\right )+\frac {3}{16} x (e x)^m \cosh ^2(a) \left (-4 \cosh \left (\frac {b}{x^2}\right )+4 \cosh \left (\frac {3 b}{x^2}\right )-3^{\frac {1+m}{2}} m \left (-\frac {b}{x^2}\right )^{\frac {1+m}{2}} \Gamma \left (\frac {1}{2} (-1-m),-\frac {3 b}{x^2}\right )+m \left (-\frac {b}{x^2}\right )^{\frac {1+m}{2}} \Gamma \left (\frac {1}{2} (-1-m),-\frac {b}{x^2}\right )+m \left (\frac {b}{x^2}\right )^{\frac {1+m}{2}} \Gamma \left (\frac {1}{2} (-1-m),\frac {b}{x^2}\right )-3^{\frac {1+m}{2}} m \left (\frac {b}{x^2}\right )^{\frac {1+m}{2}} \Gamma \left (\frac {1}{2} (-1-m),\frac {3 b}{x^2}\right )-2\ 3^{\frac {1+m}{2}} \left (-\frac {b}{x^2}\right )^{\frac {1+m}{2}} \Gamma \left (\frac {1-m}{2},-\frac {3 b}{x^2}\right )+2 \left (-\frac {b}{x^2}\right )^{\frac {1+m}{2}} \Gamma \left (\frac {1-m}{2},-\frac {b}{x^2}\right )+2 \left (\frac {b}{x^2}\right )^{\frac {1+m}{2}} \Gamma \left (\frac {1-m}{2},\frac {b}{x^2}\right )-2\ 3^{\frac {1+m}{2}} \left (\frac {b}{x^2}\right )^{\frac {1+m}{2}} \Gamma \left (\frac {1-m}{2},\frac {3 b}{x^2}\right )\right ) \sinh (a)+x^{-m} (e x)^m \left (\frac {3}{8} \left (\frac {1}{2} \left (-\frac {b}{x^2}\right )^{\frac {1+m}{2}} x^{1+m} \Gamma \left (\frac {1}{2} (-1-m),-\frac {b}{x^2}\right )+\frac {1}{2} \left (\frac {b}{x^2}\right )^{\frac {1+m}{2}} x^{1+m} \Gamma \left (\frac {1}{2} (-1-m),\frac {b}{x^2}\right )\right )+\frac {1}{8} \left (\frac {1}{2} 3^{\frac {1+m}{2}} \left (-\frac {b}{x^2}\right )^{\frac {1+m}{2}} x^{1+m} \Gamma \left (\frac {1}{2} (-1-m),-\frac {3 b}{x^2}\right )+\frac {1}{2} 3^{\frac {1+m}{2}} \left (\frac {b}{x^2}\right )^{\frac {1+m}{2}} x^{1+m} \Gamma \left (\frac {1}{2} (-1-m),\frac {3 b}{x^2}\right )\right )\right ) \sinh ^3(a)+\frac {3}{16} x (e x)^m \cosh (a) \sinh ^2(a) \left (-3^{\frac {1+m}{2}} m \left (-\frac {b}{x^2}\right )^{\frac {1+m}{2}} \Gamma \left (\frac {1}{2} (-1-m),-\frac {3 b}{x^2}\right )-m \left (-\frac {b}{x^2}\right )^{\frac {1+m}{2}} \Gamma \left (\frac {1}{2} (-1-m),-\frac {b}{x^2}\right )+m \left (\frac {b}{x^2}\right )^{\frac {1+m}{2}} \Gamma \left (\frac {1}{2} (-1-m),\frac {b}{x^2}\right )+3^{\frac {1+m}{2}} m \left (\frac {b}{x^2}\right )^{\frac {1+m}{2}} \Gamma \left (\frac {1}{2} (-1-m),\frac {3 b}{x^2}\right )-2\ 3^{\frac {1+m}{2}} \left (-\frac {b}{x^2}\right )^{\frac {1+m}{2}} \Gamma \left (\frac {1-m}{2},-\frac {3 b}{x^2}\right )-2 \left (-\frac {b}{x^2}\right )^{\frac {1+m}{2}} \Gamma \left (\frac {1-m}{2},-\frac {b}{x^2}\right )+2 \left (\frac {b}{x^2}\right )^{\frac {1+m}{2}} \Gamma \left (\frac {1-m}{2},\frac {b}{x^2}\right )+2\ 3^{\frac {1+m}{2}} \left (\frac {b}{x^2}\right )^{\frac {1+m}{2}} \Gamma \left (\frac {1-m}{2},\frac {3 b}{x^2}\right )+4 \sinh \left (\frac {b}{x^2}\right )+4 \sinh \left (\frac {3 b}{x^2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*Sinh[a + b/x^2]^3,x]

[Out]

((e*x)^m*Cosh[a]^3*((-3*(((-(b/x^2))^((1 + m)/2)*x^(1 + m)*Gamma[(-1 - m)/2, -(b/x^2)])/2 - ((b/x^2)^((1 + m)/
2)*x^(1 + m)*Gamma[(-1 - m)/2, b/x^2])/2))/8 + ((3^((1 + m)/2)*(-(b/x^2))^((1 + m)/2)*x^(1 + m)*Gamma[(-1 - m)
/2, (-3*b)/x^2])/2 - (3^((1 + m)/2)*(b/x^2)^((1 + m)/2)*x^(1 + m)*Gamma[(-1 - m)/2, (3*b)/x^2])/2)/8))/x^m + (
3*x*(e*x)^m*Cosh[a]^2*(-4*Cosh[b/x^2] + 4*Cosh[(3*b)/x^2] - 3^((1 + m)/2)*m*(-(b/x^2))^((1 + m)/2)*Gamma[(-1 -
m)/2, (-3*b)/x^2] + m*(-(b/x^2))^((1 + m)/2)*Gamma[(-1 - m)/2, -(b/x^2)] + m*(b/x^2)^((1 + m)/2)*Gamma[(-1 -
m)/2, b/x^2] - 3^((1 + m)/2)*m*(b/x^2)^((1 + m)/2)*Gamma[(-1 - m)/2, (3*b)/x^2] - 2*3^((1 + m)/2)*(-(b/x^2))^(
(1 + m)/2)*Gamma[(1 - m)/2, (-3*b)/x^2] + 2*(-(b/x^2))^((1 + m)/2)*Gamma[(1 - m)/2, -(b/x^2)] + 2*(b/x^2)^((1
+ m)/2)*Gamma[(1 - m)/2, b/x^2] - 2*3^((1 + m)/2)*(b/x^2)^((1 + m)/2)*Gamma[(1 - m)/2, (3*b)/x^2])*Sinh[a])/16
+ ((e*x)^m*((3*(((-(b/x^2))^((1 + m)/2)*x^(1 + m)*Gamma[(-1 - m)/2, -(b/x^2)])/2 + ((b/x^2)^((1 + m)/2)*x^(1
+ m)*Gamma[(-1 - m)/2, b/x^2])/2))/8 + ((3^((1 + m)/2)*(-(b/x^2))^((1 + m)/2)*x^(1 + m)*Gamma[(-1 - m)/2, (-3*
b)/x^2])/2 + (3^((1 + m)/2)*(b/x^2)^((1 + m)/2)*x^(1 + m)*Gamma[(-1 - m)/2, (3*b)/x^2])/2)/8)*Sinh[a]^3)/x^m +
(3*x*(e*x)^m*Cosh[a]*Sinh[a]^2*(-(3^((1 + m)/2)*m*(-(b/x^2))^((1 + m)/2)*Gamma[(-1 - m)/2, (-3*b)/x^2]) - m*(
-(b/x^2))^((1 + m)/2)*Gamma[(-1 - m)/2, -(b/x^2)] + m*(b/x^2)^((1 + m)/2)*Gamma[(-1 - m)/2, b/x^2] + 3^((1 + m
)/2)*m*(b/x^2)^((1 + m)/2)*Gamma[(-1 - m)/2, (3*b)/x^2] - 2*3^((1 + m)/2)*(-(b/x^2))^((1 + m)/2)*Gamma[(1 - m)
/2, (-3*b)/x^2] - 2*(-(b/x^2))^((1 + m)/2)*Gamma[(1 - m)/2, -(b/x^2)] + 2*(b/x^2)^((1 + m)/2)*Gamma[(1 - m)/2,
b/x^2] + 2*3^((1 + m)/2)*(b/x^2)^((1 + m)/2)*Gamma[(1 - m)/2, (3*b)/x^2] + 4*Sinh[b/x^2] + 4*Sinh[(3*b)/x^2])
)/16

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Maple [F]
time = 1.81, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{m} \left (\sinh ^{3}\left (a +\frac {b}{x^{2}}\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*sinh(a+b/x^2)^3,x)

[Out]

int((e*x)^m*sinh(a+b/x^2)^3,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [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((e*x)^m*sinh(a+b/x^2)^3,x, algorithm="fricas")

[Out]

integral((x*e)^m*sinh((a*x^2 + b)/x^2)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b/x^2)^3*(e*x)^m,x)

[Out]

int(sinh(a + b/x^2)^3*(e*x)^m, x)

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