3.86 \(\int x^{-2+m} \cosh (a+b x) \, dx\)

Optimal. Leaf size=55 \[ \frac{1}{2} e^a b x^m (-b x)^{-m} \text{Gamma}(m-1,-b x)-\frac{1}{2} e^{-a} b x^m (b x)^{-m} \text{Gamma}(m-1,b x) \]

[Out]

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

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Rubi [A]  time = 0.0691807, antiderivative size = 55, normalized size of antiderivative = 1., 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} \[ \frac{1}{2} e^a b x^m (-b x)^{-m} \text{Gamma}(m-1,-b x)-\frac{1}{2} e^{-a} b x^m (b x)^{-m} \text{Gamma}(m-1,b x) \]

Antiderivative was successfully verified.

[In]

Int[x^(-2 + m)*Cosh[a + b*x],x]

[Out]

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

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]

Rubi steps

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

Mathematica [A]  time = 0.0186507, size = 52, normalized size = 0.95 \[ \frac{1}{2} e^{-a} b x^m \left (e^{2 a} (-b x)^{-m} \text{Gamma}(m-1,-b x)-(b x)^{-m} \text{Gamma}(m-1,b x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-2 + m)*Cosh[a + b*x],x]

[Out]

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

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Maple [C]  time = 0.055, size = 67, normalized size = 1.2 \begin{align*}{\frac{{x}^{-1+m}\cosh \left ( a \right ) }{-1+m}{\mbox{$_1$F$_2$}(-{\frac{1}{2}}+{\frac{m}{2}};\,{\frac{1}{2}},{\frac{1}{2}}+{\frac{m}{2}};\,{\frac{{x}^{2}{b}^{2}}{4}})}}+{\frac{b{x}^{m}\sinh \left ( a \right ) }{m}{\mbox{$_1$F$_2$}({\frac{m}{2}};\,{\frac{3}{2}},1+{\frac{m}{2}};\,{\frac{{x}^{2}{b}^{2}}{4}})}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-2+m)*cosh(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.15493, size = 74, normalized size = 1.35 \begin{align*} -\frac{1}{2} \, \left (b x\right )^{-m + 1} x^{m - 1} e^{\left (-a\right )} \Gamma \left (m - 1, b x\right ) - \frac{1}{2} \, \left (-b x\right )^{-m + 1} x^{m - 1} e^{a} \Gamma \left (m - 1, -b x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-2+m)*cosh(b*x+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.84898, size = 259, normalized size = 4.71 \begin{align*} -\frac{\cosh \left ({\left (m - 2\right )} \log \left (b\right ) + a\right ) \Gamma \left (m - 1, b x\right ) - \cosh \left ({\left (m - 2\right )} \log \left (-b\right ) - a\right ) \Gamma \left (m - 1, -b x\right ) + \Gamma \left (m - 1, -b x\right ) \sinh \left ({\left (m - 2\right )} \log \left (-b\right ) - a\right ) - \Gamma \left (m - 1, b x\right ) \sinh \left ({\left (m - 2\right )} \log \left (b\right ) + a\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-2+m)*cosh(b*x+a),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-2+m)*cosh(b*x+a),x)

[Out]

Exception raised: TypeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m - 2} \cosh \left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-2+m)*cosh(b*x+a),x, algorithm="giac")

[Out]

integrate(x^(m - 2)*cosh(b*x + a), x)