3.83 \(\int x^{1+m} \cosh (a+b x) \, dx\)

Optimal. Leaf size=59 \[ -\frac {e^a x^m (-b x)^{-m} \Gamma (m+2,-b x)}{2 b^2}-\frac {e^{-a} x^m (b x)^{-m} \Gamma (m+2,b x)}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3307, 2181} \[ -\frac {e^a x^m (-b x)^{-m} \text {Gamma}(m+2,-b x)}{2 b^2}-\frac {e^{-a} x^m (b x)^{-m} \text {Gamma}(m+2,b x)}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^(1 + m)*Cosh[a + b*x],x]

[Out]

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

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 3307

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d
, e, f, m}, x] && IntegerQ[2*k]

Rubi steps

\begin {align*} \int x^{1+m} \cosh (a+b x) \, dx &=\frac {1}{2} \int e^{-i (i a+i b x)} x^{1+m} \, dx+\frac {1}{2} \int e^{i (i a+i b x)} x^{1+m} \, dx\\ &=-\frac {e^a x^m (-b x)^{-m} \Gamma (2+m,-b x)}{2 b^2}-\frac {e^{-a} x^m (b x)^{-m} \Gamma (2+m,b x)}{2 b^2}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 54, normalized size = 0.92 \[ -\frac {e^a x^m (-b x)^{-m} \Gamma (m+2,-b x)+e^{-a} x^m (b x)^{-m} \Gamma (m+2,b x)}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(1 + m)*Cosh[a + b*x],x]

[Out]

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

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fricas [A]  time = 0.44, size = 86, normalized size = 1.46 \[ -\frac {\cosh \left ({\left (m + 1\right )} \log \relax (b) + a\right ) \Gamma \left (m + 2, b x\right ) - \cosh \left ({\left (m + 1\right )} \log \left (-b\right ) - a\right ) \Gamma \left (m + 2, -b x\right ) + \Gamma \left (m + 2, -b x\right ) \sinh \left ({\left (m + 1\right )} \log \left (-b\right ) - a\right ) - \Gamma \left (m + 2, b x\right ) \sinh \left ({\left (m + 1\right )} \log \relax (b) + a\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1+m)*cosh(b*x+a),x, algorithm="fricas")

[Out]

-1/2*(cosh((m + 1)*log(b) + a)*gamma(m + 2, b*x) - cosh((m + 1)*log(-b) - a)*gamma(m + 2, -b*x) + gamma(m + 2,
-b*x)*sinh((m + 1)*log(-b) - a) - gamma(m + 2, b*x)*sinh((m + 1)*log(b) + a))/b

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m + 1} \cosh \left (b x + a\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1+m)*cosh(b*x+a),x, algorithm="giac")

[Out]

integrate(x^(m + 1)*cosh(b*x + a), x)

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maple [C]  time = 0.12, size = 73, normalized size = 1.24 \[ \frac {x^{2+m} \hypergeom \left (\left [1+\frac {m}{2}\right ], \left [\frac {1}{2}, 2+\frac {m}{2}\right ], \frac {x^{2} b^{2}}{4}\right ) \cosh \relax (a )}{2+m}+\frac {b \,x^{3+m} \hypergeom \left (\left [\frac {3}{2}+\frac {m}{2}\right ], \left [\frac {3}{2}, \frac {5}{2}+\frac {m}{2}\right ], \frac {x^{2} b^{2}}{4}\right ) \sinh \relax (a )}{3+m} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1+m)*cosh(b*x+a),x)

[Out]

1/(2+m)*x^(2+m)*hypergeom([1+1/2*m],[1/2,2+1/2*m],1/4*x^2*b^2)*cosh(a)+b/(3+m)*x^(3+m)*hypergeom([3/2+1/2*m],[
3/2,5/2+1/2*m],1/4*x^2*b^2)*sinh(a)

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maxima [A]  time = 0.74, size = 55, normalized size = 0.93 \[ -\frac {1}{2} \, \left (b x\right )^{-m - 2} x^{m + 2} e^{\left (-a\right )} \Gamma \left (m + 2, b x\right ) - \frac {1}{2} \, \left (-b x\right )^{-m - 2} x^{m + 2} e^{a} \Gamma \left (m + 2, -b x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1+m)*cosh(b*x+a),x, algorithm="maxima")

[Out]

-1/2*(b*x)^(-m - 2)*x^(m + 2)*e^(-a)*gamma(m + 2, b*x) - 1/2*(-b*x)^(-m - 2)*x^(m + 2)*e^a*gamma(m + 2, -b*x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int x^{m+1}\,\mathrm {cosh}\left (a+b\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(m + 1)*cosh(a + b*x),x)

[Out]

int(x^(m + 1)*cosh(a + b*x), x)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(1+m)*cosh(b*x+a),x)

[Out]

Exception raised: TypeError

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