3.250 $$\int x^m \cosh (a+b x) \sinh (a+b x) \, dx$$

Optimal. Leaf size=70 $\frac{e^{2 a} 2^{-m-3} x^m (-b x)^{-m} \text{Gamma}(m+1,-2 b x)}{b}+\frac{e^{-2 a} 2^{-m-3} x^m (b x)^{-m} \text{Gamma}(m+1,2 b x)}{b}$

[Out]

(2^(-3 - m)*E^(2*a)*x^m*Gamma[1 + m, -2*b*x])/(b*(-(b*x))^m) + (2^(-3 - m)*x^m*Gamma[1 + m, 2*b*x])/(b*E^(2*a)
*(b*x)^m)

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Rubi [A]  time = 0.117687, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {5448, 12, 3308, 2181} $\frac{e^{2 a} 2^{-m-3} x^m (-b x)^{-m} \text{Gamma}(m+1,-2 b x)}{b}+\frac{e^{-2 a} 2^{-m-3} x^m (b x)^{-m} \text{Gamma}(m+1,2 b x)}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^m*Cosh[a + b*x]*Sinh[a + b*x],x]

[Out]

(2^(-3 - m)*E^(2*a)*x^m*Gamma[1 + m, -2*b*x])/(b*(-(b*x))^m) + (2^(-3 - m)*x^m*Gamma[1 + m, 2*b*x])/(b*E^(2*a)
*(b*x)^m)

Rule 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3308

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]

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^m \cosh (a+b x) \sinh (a+b x) \, dx &=\int \frac{1}{2} x^m \sinh (2 a+2 b x) \, dx\\ &=\frac{1}{2} \int x^m \sinh (2 a+2 b x) \, dx\\ &=\frac{1}{4} \int e^{-i (2 i a+2 i b x)} x^m \, dx-\frac{1}{4} \int e^{i (2 i a+2 i b x)} x^m \, dx\\ &=\frac{2^{-3-m} e^{2 a} x^m (-b x)^{-m} \Gamma (1+m,-2 b x)}{b}+\frac{2^{-3-m} e^{-2 a} x^m (b x)^{-m} \Gamma (1+m,2 b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.0459349, size = 66, normalized size = 0.94 $\frac{e^{-2 a} 2^{-m-3} x^m \left (-b^2 x^2\right )^{-m} \left (e^{4 a} (b x)^m \text{Gamma}(m+1,-2 b x)+(-b x)^m \text{Gamma}(m+1,2 b x)\right )}{b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^m*Cosh[a + b*x]*Sinh[a + b*x],x]

[Out]

(2^(-3 - m)*x^m*(E^(4*a)*(b*x)^m*Gamma[1 + m, -2*b*x] + (-(b*x))^m*Gamma[1 + m, 2*b*x]))/(b*E^(2*a)*(-(b^2*x^2
))^m)

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) \, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.26501, size = 80, normalized size = 1.14 \begin{align*} \frac{1}{4} \, \left (2 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (-2 \, a\right )} \Gamma \left (m + 1, 2 \, b x\right ) - \frac{1}{4} \, \left (-2 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (2 \, a\right )} \Gamma \left (m + 1, -2 \, b x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*b*x)^(-m - 1)*x^(m + 1)*e^(-2*a)*gamma(m + 1, 2*b*x) - 1/4*(-2*b*x)^(-m - 1)*x^(m + 1)*e^(2*a)*gamma(m
+ 1, -2*b*x)

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Fricas [A]  time = 2.20564, size = 258, normalized size = 3.69 \begin{align*} \frac{\cosh \left (m \log \left (2 \, b\right ) + 2 \, a\right ) \Gamma \left (m + 1, 2 \, b x\right ) + \cosh \left (m \log \left (-2 \, b\right ) - 2 \, a\right ) \Gamma \left (m + 1, -2 \, b x\right ) - \Gamma \left (m + 1, 2 \, b x\right ) \sinh \left (m \log \left (2 \, b\right ) + 2 \, a\right ) - \Gamma \left (m + 1, -2 \, b x\right ) \sinh \left (m \log \left (-2 \, b\right ) - 2 \, a\right )}{8 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(cosh(m*log(2*b) + 2*a)*gamma(m + 1, 2*b*x) + cosh(m*log(-2*b) - 2*a)*gamma(m + 1, -2*b*x) - gamma(m + 1,
2*b*x)*sinh(m*log(2*b) + 2*a) - gamma(m + 1, -2*b*x)*sinh(m*log(-2*b) - 2*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**m*cosh(b*x+a)*sinh(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} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m*cosh(b*x + a)*sinh(b*x + a), x)