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

Optimal. Leaf size=49 \[ -\frac{1}{2} e^a x^m (-b x)^{-m} \text{Gamma}(m,-b x)-\frac{1}{2} e^{-a} x^m (b x)^{-m} \text{Gamma}(m,b x) \]

[Out]

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

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Rubi [A]  time = 0.0690003, antiderivative size = 49, 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} \[ -\frac{1}{2} e^a x^m (-b x)^{-m} \text{Gamma}(m,-b x)-\frac{1}{2} e^{-a} x^m (b x)^{-m} \text{Gamma}(m,b x) \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + m)*Cosh[a + b*x],x]

[Out]

-(E^a*x^m*Gamma[m, -(b*x)])/(2*(-(b*x))^m) - (x^m*Gamma[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^{-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{1}{2} e^a x^m (-b x)^{-m} \Gamma (m,-b x)-\frac{1}{2} e^{-a} x^m (b x)^{-m} \Gamma (m,b x)\\ \end{align*}

Mathematica [A]  time = 0.0198166, size = 49, normalized size = 1. \[ -\frac{1}{2} e^a x^m (-b x)^{-m} \text{Gamma}(m,-b x)-\frac{1}{2} e^{-a} x^m (b x)^{-m} \text{Gamma}(m,b x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + m)*Cosh[a + b*x],x]

[Out]

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

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Maple [C]  time = 0.046, size = 67, normalized size = 1.4 \begin{align*}{\frac{{x}^{m}\cosh \left ( a \right ) }{m}{\mbox{$_1$F$_2$}({\frac{m}{2}};\,{\frac{1}{2}},1+{\frac{m}{2}};\,{\frac{{x}^{2}{b}^{2}}{4}})}}+{\frac{b{x}^{1+m}\sinh \left ( a \right ) }{1+m}{\mbox{$_1$F$_2$}({\frac{1}{2}}+{\frac{m}{2}};\,{\frac{3}{2}},{\frac{3}{2}}+{\frac{m}{2}};\,{\frac{{x}^{2}{b}^{2}}{4}})}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.17037, size = 58, normalized size = 1.18 \begin{align*} -\frac{x^{m} e^{\left (-a\right )} \Gamma \left (m, b x\right )}{2 \, \left (b x\right )^{m}} - \frac{x^{m} e^{a} \Gamma \left (m, -b x\right )}{2 \, \left (-b x\right )^{m}} \end{align*}

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

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Fricas [A]  time = 1.97397, size = 238, normalized size = 4.86 \begin{align*} -\frac{\cosh \left ({\left (m - 1\right )} \log \left (b\right ) + a\right ) \Gamma \left (m, b x\right ) - \cosh \left ({\left (m - 1\right )} \log \left (-b\right ) - a\right ) \Gamma \left (m, -b x\right ) + \Gamma \left (m, -b x\right ) \sinh \left ({\left (m - 1\right )} \log \left (-b\right ) - a\right ) - \Gamma \left (m, b x\right ) \sinh \left ({\left (m - 1\right )} \log \left (b\right ) + a\right )}{2 \, b} \end{align*}

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, b*x) - cosh((m - 1)*log(-b) - a)*gamma(m, -b*x) + gamma(m, -b*x)*sinh(
(m - 1)*log(-b) - a) - gamma(m, b*x)*sinh((m - 1)*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**(-1+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 - 1} \cosh \left (b x + a\right )\,{d x} \end{align*}

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)