3.1.87 \(\int x^{-3+m} \cosh (a+b x) \, dx\) [87]

Optimal. Leaf size=59 \[ -\frac {1}{2} b^2 e^a x^m (-b x)^{-m} \Gamma (-2+m,-b x)-\frac {1}{2} b^2 e^{-a} x^m (b x)^{-m} \Gamma (-2+m,b x) \]

[Out]

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

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Rubi [A]
time = 0.05, 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 = {3388, 2212} \begin {gather*} -\frac {1}{2} e^a b^2 x^m (-b x)^{-m} \text {Gamma}(m-2,-b x)-\frac {1}{2} e^{-a} b^2 x^m (b x)^{-m} \text {Gamma}(m-2,b x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(-3 + m)*Cosh[a + b*x],x]

[Out]

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

Rule 2212

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

Rule 3388

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^{-3+m} \cosh (a+b x) \, dx &=\frac {1}{2} \int e^{-i (i a+i b x)} x^{-3+m} \, dx+\frac {1}{2} \int e^{i (i a+i b x)} x^{-3+m} \, dx\\ &=-\frac {1}{2} b^2 e^a x^m (-b x)^{-m} \Gamma (-2+m,-b x)-\frac {1}{2} b^2 e^{-a} x^m (b x)^{-m} \Gamma (-2+m,b x)\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 55, normalized size = 0.93 \begin {gather*} \frac {1}{2} b^2 e^{-a} x^m \left (-e^{2 a} (-b x)^{-m} \Gamma (-2+m,-b x)-(b x)^{-m} \Gamma (-2+m,b x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(-3 + m)*Cosh[a + b*x],x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 4.
time = 0.33, size = 71, normalized size = 1.20

method	result	size
meijerg	\(\frac {x^{-2+m} \hypergeom \left (\left [-1+\frac {m}{2}\right ], \left [\frac {1}{2}, \frac {m}{2}\right ], \frac {b^{2} x^{2}}{4}\right ) \cosh \left (a \right )}{-2+m}+\frac {b \,x^{-1+m} \hypergeom \left (\left [-\frac {1}{2}+\frac {m}{2}\right ], \left [\frac {3}{2}, \frac {1}{2}+\frac {m}{2}\right ], \frac {b^{2} x^{2}}{4}\right ) \sinh \left (a \right )}{-1+m}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-3+m)*cosh(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.09, size = 55, normalized size = 0.93 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-3+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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Fricas [A]
time = 0.11, size = 86, normalized size = 1.46 \begin {gather*} -\frac {\cosh \left ({\left (m - 3\right )} \log \left (b\right ) + a\right ) \Gamma \left (m - 2, b x\right ) - \cosh \left ({\left (m - 3\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 - 3\right )} \log \left (-b\right ) - a\right ) - \Gamma \left (m - 2, b x\right ) \sinh \left ({\left (m - 3\right )} \log \left (b\right ) + a\right )}{2 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(cosh((m - 3)*log(b) + a)*gamma(m - 2, b*x) - cosh((m - 3)*log(-b) - a)*gamma(m - 2, -b*x) + gamma(m - 2,
-b*x)*sinh((m - 3)*log(-b) - a) - gamma(m - 2, b*x)*sinh((m - 3)*log(b) + a))/b

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> cannot determine truth value of Relational

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

[Out]

integrate(x^(m - 3)*cosh(b*x + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x^{m-3}\,\mathrm {cosh}\left (a+b\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(m - 3)*cosh(a + b*x),x)

[Out]

int(x^(m - 3)*cosh(a + b*x), x)

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