### 3.325 $$\int x^m \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx$$

Optimal. Leaf size=155 $\frac{e^{6 a} 2^{-m-7} 3^{-m-1} x^m (-b x)^{-m} \text{Gamma}(m+1,-6 b x)}{b}-\frac{3 e^{2 a} 2^{-m-7} x^m (-b x)^{-m} \text{Gamma}(m+1,-2 b x)}{b}-\frac{3 e^{-2 a} 2^{-m-7} x^m (b x)^{-m} \text{Gamma}(m+1,2 b x)}{b}+\frac{e^{-6 a} 2^{-m-7} 3^{-m-1} x^m (b x)^{-m} \text{Gamma}(m+1,6 b x)}{b}$

[Out]

(2^(-7 - m)*3^(-1 - m)*E^(6*a)*x^m*Gamma[1 + m, -6*b*x])/(b*(-(b*x))^m) - (3*2^(-7 - m)*E^(2*a)*x^m*Gamma[1 +
m, -2*b*x])/(b*(-(b*x))^m) - (3*2^(-7 - m)*x^m*Gamma[1 + m, 2*b*x])/(b*E^(2*a)*(b*x)^m) + (2^(-7 - m)*3^(-1 -
m)*x^m*Gamma[1 + m, 6*b*x])/(b*E^(6*a)*(b*x)^m)

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Rubi [A]  time = 0.243486, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.15, Rules used = {5448, 3308, 2181} $\frac{e^{6 a} 2^{-m-7} 3^{-m-1} x^m (-b x)^{-m} \text{Gamma}(m+1,-6 b x)}{b}-\frac{3 e^{2 a} 2^{-m-7} x^m (-b x)^{-m} \text{Gamma}(m+1,-2 b x)}{b}-\frac{3 e^{-2 a} 2^{-m-7} x^m (b x)^{-m} \text{Gamma}(m+1,2 b x)}{b}+\frac{e^{-6 a} 2^{-m-7} 3^{-m-1} x^m (b x)^{-m} \text{Gamma}(m+1,6 b x)}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^m*Cosh[a + b*x]^3*Sinh[a + b*x]^3,x]

[Out]

(2^(-7 - m)*3^(-1 - m)*E^(6*a)*x^m*Gamma[1 + m, -6*b*x])/(b*(-(b*x))^m) - (3*2^(-7 - m)*E^(2*a)*x^m*Gamma[1 +
m, -2*b*x])/(b*(-(b*x))^m) - (3*2^(-7 - m)*x^m*Gamma[1 + m, 2*b*x])/(b*E^(2*a)*(b*x)^m) + (2^(-7 - m)*3^(-1 -
m)*x^m*Gamma[1 + m, 6*b*x])/(b*E^(6*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 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 ^3(a+b x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac{3}{32} x^m \sinh (2 a+2 b x)+\frac{1}{32} x^m \sinh (6 a+6 b x)\right ) \, dx\\ &=\frac{1}{32} \int x^m \sinh (6 a+6 b x) \, dx-\frac{3}{32} \int x^m \sinh (2 a+2 b x) \, dx\\ &=\frac{1}{64} \int e^{-i (6 i a+6 i b x)} x^m \, dx-\frac{1}{64} \int e^{i (6 i a+6 i b x)} x^m \, dx-\frac{3}{64} \int e^{-i (2 i a+2 i b x)} x^m \, dx+\frac{3}{64} \int e^{i (2 i a+2 i b x)} x^m \, dx\\ &=\frac{2^{-7-m} 3^{-1-m} e^{6 a} x^m (-b x)^{-m} \Gamma (1+m,-6 b x)}{b}-\frac{3\ 2^{-7-m} e^{2 a} x^m (-b x)^{-m} \Gamma (1+m,-2 b x)}{b}-\frac{3\ 2^{-7-m} e^{-2 a} x^m (b x)^{-m} \Gamma (1+m,2 b x)}{b}+\frac{2^{-7-m} 3^{-1-m} e^{-6 a} x^m (b x)^{-m} \Gamma (1+m,6 b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.152997, size = 119, normalized size = 0.77 $\frac{e^{-6 a} 2^{-m-7} 3^{-m-1} x^m \left (-b^2 x^2\right )^{-m} \left ((-b x)^m \left (\text{Gamma}(m+1,6 b x)-e^{4 a} 3^{m+2} \text{Gamma}(m+1,2 b x)\right )+e^{12 a} (b x)^m \text{Gamma}(m+1,-6 b x)-e^{8 a} 3^{m+2} (b x)^m \text{Gamma}(m+1,-2 b x)\right )}{b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^m*Cosh[a + b*x]^3*Sinh[a + b*x]^3,x]

[Out]

(2^(-7 - m)*3^(-1 - m)*x^m*(E^(12*a)*(b*x)^m*Gamma[1 + m, -6*b*x] - 3^(2 + m)*E^(8*a)*(b*x)^m*Gamma[1 + m, -2*
b*x] + (-(b*x))^m*(-(3^(2 + m)*E^(4*a)*Gamma[1 + m, 2*b*x]) + Gamma[1 + m, 6*b*x])))/(b*E^(6*a)*(-(b^2*x^2))^m
)

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( \cosh \left ( bx+a \right ) \right ) ^{3} \left ( \sinh \left ( bx+a \right ) \right ) ^{3}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*cosh(b*x+a)^3*sinh(b*x+a)^3,x)

[Out]

int(x^m*cosh(b*x+a)^3*sinh(b*x+a)^3,x)

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Maxima [A]  time = 1.19729, size = 158, normalized size = 1.02 \begin{align*} \frac{1}{64} \, \left (6 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (-6 \, a\right )} \Gamma \left (m + 1, 6 \, b x\right ) - \frac{3}{64} \, \left (2 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (-2 \, a\right )} \Gamma \left (m + 1, 2 \, b x\right ) + \frac{3}{64} \, \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}{64} \, \left (-6 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (6 \, a\right )} \Gamma \left (m + 1, -6 \, b x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(6*b*x)^(-m - 1)*x^(m + 1)*e^(-6*a)*gamma(m + 1, 6*b*x) - 3/64*(2*b*x)^(-m - 1)*x^(m + 1)*e^(-2*a)*gamma(
m + 1, 2*b*x) + 3/64*(-2*b*x)^(-m - 1)*x^(m + 1)*e^(2*a)*gamma(m + 1, -2*b*x) - 1/64*(-6*b*x)^(-m - 1)*x^(m +
1)*e^(6*a)*gamma(m + 1, -6*b*x)

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Fricas [A]  time = 1.8499, size = 520, normalized size = 3.35 \begin{align*} \frac{\cosh \left (m \log \left (6 \, b\right ) + 6 \, a\right ) \Gamma \left (m + 1, 6 \, b x\right ) - 9 \, \cosh \left (m \log \left (2 \, b\right ) + 2 \, a\right ) \Gamma \left (m + 1, 2 \, b x\right ) - 9 \, \cosh \left (m \log \left (-2 \, b\right ) - 2 \, a\right ) \Gamma \left (m + 1, -2 \, b x\right ) + \cosh \left (m \log \left (-6 \, b\right ) - 6 \, a\right ) \Gamma \left (m + 1, -6 \, b x\right ) - \Gamma \left (m + 1, 6 \, b x\right ) \sinh \left (m \log \left (6 \, b\right ) + 6 \, a\right ) + 9 \, \Gamma \left (m + 1, 2 \, b x\right ) \sinh \left (m \log \left (2 \, b\right ) + 2 \, a\right ) + 9 \, \Gamma \left (m + 1, -2 \, b x\right ) \sinh \left (m \log \left (-2 \, b\right ) - 2 \, a\right ) - \Gamma \left (m + 1, -6 \, b x\right ) \sinh \left (m \log \left (-6 \, b\right ) - 6 \, a\right )}{384 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(cosh(m*log(6*b) + 6*a)*gamma(m + 1, 6*b*x) - 9*cosh(m*log(2*b) + 2*a)*gamma(m + 1, 2*b*x) - 9*cosh(m*lo
g(-2*b) - 2*a)*gamma(m + 1, -2*b*x) + cosh(m*log(-6*b) - 6*a)*gamma(m + 1, -6*b*x) - gamma(m + 1, 6*b*x)*sinh(
m*log(6*b) + 6*a) + 9*gamma(m + 1, 2*b*x)*sinh(m*log(2*b) + 2*a) + 9*gamma(m + 1, -2*b*x)*sinh(m*log(-2*b) - 2
*a) - gamma(m + 1, -6*b*x)*sinh(m*log(-6*b) - 6*a))/b

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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