3.1.82 \(\int x^{-1+m} \sinh (a+b x) \, dx\) [82]

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

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 49, 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 = {3389, 2212} \begin {gather*} \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) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(-1 + m)*Sinh[a + b*x],x]

[Out]

-1/2*(E^a*x^m*Gamma[m, -(b*x)])/(-(b*x))^m + (x^m*Gamma[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 3389

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

Rubi steps

\begin {align*} \int x^{-1+m} \sinh (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*}

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Mathematica [A]
time = 0.02, size = 49, normalized size = 1.00 \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + m)*Sinh[a + b*x],x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 4.
time = 0.18, size = 67, normalized size = 1.37

method	result	size
meijerg	\(\frac {x^{m} \hypergeom \left (\left [\frac {m}{2}\right ], \left [\frac {1}{2}, 1+\frac {m}{2}\right ], \frac {b^{2} x^{2}}{4}\right ) \sinh \left (a \right )}{m}+\frac {b \,x^{1+m} \hypergeom \left (\left [\frac {1}{2}+\frac {m}{2}\right ], \left [\frac {3}{2}, \frac {3}{2}+\frac {m}{2}\right ], \frac {b^{2} x^{2}}{4}\right ) \cosh \left (a \right )}{1+m}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+m)*sinh(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.08, size = 43, normalized size = 0.88 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+m)*sinh(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 = 0.08, size = 78, normalized size = 1.59 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+m)*sinh(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 = 0.00, 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**(-1+m)*sinh(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^(-1+m)*sinh(b*x+a),x, algorithm="giac")

[Out]

integrate(x^(m - 1)*sinh(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(m - 1)*sinh(a + b*x),x)

[Out]

int(x^(m - 1)*sinh(a + b*x), x)

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