3.49 \(\int x^m \cosh ^3(a+b x^n) \, dx\)

Optimal. Leaf size=200 \[ -\frac {e^{3 a} 3^{-\frac {m+1}{n}} x^{m+1} \left (-b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-3 b x^n\right )}{8 n}-\frac {3 e^a x^{m+1} \left (-b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-b x^n\right )}{8 n}-\frac {3 e^{-a} x^{m+1} \left (b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},b x^n\right )}{8 n}-\frac {e^{-3 a} 3^{-\frac {m+1}{n}} x^{m+1} \left (b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},3 b x^n\right )}{8 n} \]

[Out]

-1/8*exp(3*a)*x^(1+m)*GAMMA((1+m)/n,-3*b*x^n)/(3^((1+m)/n))/n/((-b*x^n)^((1+m)/n))-3/8*exp(a)*x^(1+m)*GAMMA((1
+m)/n,-b*x^n)/n/((-b*x^n)^((1+m)/n))-3/8*x^(1+m)*GAMMA((1+m)/n,b*x^n)/exp(a)/n/((b*x^n)^((1+m)/n))-1/8*x^(1+m)
*GAMMA((1+m)/n,3*b*x^n)/(3^((1+m)/n))/exp(3*a)/n/((b*x^n)^((1+m)/n))

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Rubi [A]  time = 0.21, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5363, 5361, 2218} \[ -\frac {e^{3 a} 3^{-\frac {m+1}{n}} x^{m+1} \left (-b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},-3 b x^n\right )}{8 n}-\frac {3 e^a x^{m+1} \left (-b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},-b x^n\right )}{8 n}-\frac {3 e^{-a} x^{m+1} \left (b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},b x^n\right )}{8 n}-\frac {e^{-3 a} 3^{-\frac {m+1}{n}} x^{m+1} \left (b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},3 b x^n\right )}{8 n} \]

Antiderivative was successfully verified.

[In]

Int[x^m*Cosh[a + b*x^n]^3,x]

[Out]

-(E^(3*a)*x^(1 + m)*Gamma[(1 + m)/n, -3*b*x^n])/(8*3^((1 + m)/n)*n*(-(b*x^n))^((1 + m)/n)) - (3*E^a*x^(1 + m)*
Gamma[(1 + m)/n, -(b*x^n)])/(8*n*(-(b*x^n))^((1 + m)/n)) - (3*x^(1 + m)*Gamma[(1 + m)/n, b*x^n])/(8*E^a*n*(b*x
^n)^((1 + m)/n)) - (x^(1 + m)*Gamma[(1 + m)/n, 3*b*x^n])/(8*3^((1 + m)/n)*E^(3*a)*n*(b*x^n)^((1 + m)/n))

Rule 2218

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

Rule 5361

Int[Cosh[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), 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, n}, x]

Rule 5363

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

Rubi steps

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

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Mathematica [A]  time = 1.08, size = 182, normalized size = 0.91 \[ -\frac {e^{-3 a} 3^{-\frac {m+1}{n}} x^{m+1} \left (-b^2 x^{2 n}\right )^{-\frac {m+1}{n}} \left (\left (-b x^n\right )^{\frac {m+1}{n}} \left (e^{2 a} 3^{\frac {m+n+1}{n}} \Gamma \left (\frac {m+1}{n},b x^n\right )+\Gamma \left (\frac {m+1}{n},3 b x^n\right )\right )+e^{6 a} \left (b x^n\right )^{\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-3 b x^n\right )+e^{4 a} 3^{\frac {m+n+1}{n}} \left (b x^n\right )^{\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-b x^n\right )\right )}{8 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*Cosh[a + b*x^n]^3,x]

[Out]

-1/8*(x^(1 + m)*(E^(6*a)*(b*x^n)^((1 + m)/n)*Gamma[(1 + m)/n, -3*b*x^n] + 3^((1 + m + n)/n)*E^(4*a)*(b*x^n)^((
1 + m)/n)*Gamma[(1 + m)/n, -(b*x^n)] + (-(b*x^n))^((1 + m)/n)*(3^((1 + m + n)/n)*E^(2*a)*Gamma[(1 + m)/n, b*x^
n] + Gamma[(1 + m)/n, 3*b*x^n])))/(3^((1 + m)/n)*E^(3*a)*n*(-(b^2*x^(2*n)))^((1 + m)/n))

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fricas [F]  time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \cosh \left (b x^{n} + a\right )^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^m*cosh(b*x^n + a)^3, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \cosh \left (b x^{n} + a\right )^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m*cosh(b*x^n + a)^3, x)

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maple [F]  time = 0.34, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\cosh ^{3}\left (a +b \,x^{n}\right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.51, size = 173, normalized size = 0.86 \[ -\frac {x^{m + 1} e^{\left (-3 \, a\right )} \Gamma \left (\frac {m + 1}{n}, 3 \, b x^{n}\right )}{8 \, \left (3 \, b x^{n}\right )^{\frac {m + 1}{n}} n} - \frac {3 \, x^{m + 1} e^{\left (-a\right )} \Gamma \left (\frac {m + 1}{n}, b x^{n}\right )}{8 \, \left (b x^{n}\right )^{\frac {m + 1}{n}} n} - \frac {3 \, x^{m + 1} e^{a} \Gamma \left (\frac {m + 1}{n}, -b x^{n}\right )}{8 \, \left (-b x^{n}\right )^{\frac {m + 1}{n}} n} - \frac {x^{m + 1} e^{\left (3 \, a\right )} \Gamma \left (\frac {m + 1}{n}, -3 \, b x^{n}\right )}{8 \, \left (-3 \, b x^{n}\right )^{\frac {m + 1}{n}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*x^(m + 1)*e^(-3*a)*gamma((m + 1)/n, 3*b*x^n)/((3*b*x^n)^((m + 1)/n)*n) - 3/8*x^(m + 1)*e^(-a)*gamma((m +
1)/n, b*x^n)/((b*x^n)^((m + 1)/n)*n) - 3/8*x^(m + 1)*e^a*gamma((m + 1)/n, -b*x^n)/((-b*x^n)^((m + 1)/n)*n) - 1
/8*x^(m + 1)*e^(3*a)*gamma((m + 1)/n, -3*b*x^n)/((-3*b*x^n)^((m + 1)/n)*n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*cosh(a + b*x^n)^3,x)

[Out]

int(x^m*cosh(a + b*x^n)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*cosh(a+b*x**n)**3,x)

[Out]

Integral(x**m*cosh(a + b*x**n)**3, x)

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