3.316 $$\int x^m \cosh ^2(a+b x) \sinh ^3(a+b x) \, dx$$

Optimal. Leaf size=209 $\frac{e^{5 a} 5^{-m-1} x^m (-b x)^{-m} \text{Gamma}(m+1,-5 b x)}{32 b}-\frac{e^{3 a} 3^{-m-1} x^m (-b x)^{-m} \text{Gamma}(m+1,-3 b x)}{32 b}-\frac{e^a x^m (-b x)^{-m} \text{Gamma}(m+1,-b x)}{16 b}-\frac{e^{-a} x^m (b x)^{-m} \text{Gamma}(m+1,b x)}{16 b}-\frac{e^{-3 a} 3^{-m-1} x^m (b x)^{-m} \text{Gamma}(m+1,3 b x)}{32 b}+\frac{e^{-5 a} 5^{-m-1} x^m (b x)^{-m} \text{Gamma}(m+1,5 b x)}{32 b}$

[Out]

(5^(-1 - m)*E^(5*a)*x^m*Gamma[1 + m, -5*b*x])/(32*b*(-(b*x))^m) - (3^(-1 - m)*E^(3*a)*x^m*Gamma[1 + m, -3*b*x]
)/(32*b*(-(b*x))^m) - (E^a*x^m*Gamma[1 + m, -(b*x)])/(16*b*(-(b*x))^m) - (x^m*Gamma[1 + m, b*x])/(16*b*E^a*(b*
x)^m) - (3^(-1 - m)*x^m*Gamma[1 + m, 3*b*x])/(32*b*E^(3*a)*(b*x)^m) + (5^(-1 - m)*x^m*Gamma[1 + m, 5*b*x])/(32
*b*E^(5*a)*(b*x)^m)

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Rubi [A]  time = 0.280038, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 3, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.15, Rules used = {5448, 3308, 2181} $\frac{e^{5 a} 5^{-m-1} x^m (-b x)^{-m} \text{Gamma}(m+1,-5 b x)}{32 b}-\frac{e^{3 a} 3^{-m-1} x^m (-b x)^{-m} \text{Gamma}(m+1,-3 b x)}{32 b}-\frac{e^a x^m (-b x)^{-m} \text{Gamma}(m+1,-b x)}{16 b}-\frac{e^{-a} x^m (b x)^{-m} \text{Gamma}(m+1,b x)}{16 b}-\frac{e^{-3 a} 3^{-m-1} x^m (b x)^{-m} \text{Gamma}(m+1,3 b x)}{32 b}+\frac{e^{-5 a} 5^{-m-1} x^m (b x)^{-m} \text{Gamma}(m+1,5 b x)}{32 b}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^m*Cosh[a + b*x]^2*Sinh[a + b*x]^3,x]

[Out]

(5^(-1 - m)*E^(5*a)*x^m*Gamma[1 + m, -5*b*x])/(32*b*(-(b*x))^m) - (3^(-1 - m)*E^(3*a)*x^m*Gamma[1 + m, -3*b*x]
)/(32*b*(-(b*x))^m) - (E^a*x^m*Gamma[1 + m, -(b*x)])/(16*b*(-(b*x))^m) - (x^m*Gamma[1 + m, b*x])/(16*b*E^a*(b*
x)^m) - (3^(-1 - m)*x^m*Gamma[1 + m, 3*b*x])/(32*b*E^(3*a)*(b*x)^m) + (5^(-1 - m)*x^m*Gamma[1 + m, 5*b*x])/(32
*b*E^(5*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 ^2(a+b x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac{1}{8} x^m \sinh (a+b x)-\frac{1}{16} x^m \sinh (3 a+3 b x)+\frac{1}{16} x^m \sinh (5 a+5 b x)\right ) \, dx\\ &=-\left (\frac{1}{16} \int x^m \sinh (3 a+3 b x) \, dx\right )+\frac{1}{16} \int x^m \sinh (5 a+5 b x) \, dx-\frac{1}{8} \int x^m \sinh (a+b x) \, dx\\ &=-\left (\frac{1}{32} \int e^{-i (3 i a+3 i b x)} x^m \, dx\right )+\frac{1}{32} \int e^{i (3 i a+3 i b x)} x^m \, dx+\frac{1}{32} \int e^{-i (5 i a+5 i b x)} x^m \, dx-\frac{1}{32} \int e^{i (5 i a+5 i b x)} x^m \, dx-\frac{1}{16} \int e^{-i (i a+i b x)} x^m \, dx+\frac{1}{16} \int e^{i (i a+i b x)} x^m \, dx\\ &=\frac{5^{-1-m} e^{5 a} x^m (-b x)^{-m} \Gamma (1+m,-5 b x)}{32 b}-\frac{3^{-1-m} e^{3 a} x^m (-b x)^{-m} \Gamma (1+m,-3 b x)}{32 b}-\frac{e^a x^m (-b x)^{-m} \Gamma (1+m,-b x)}{16 b}-\frac{e^{-a} x^m (b x)^{-m} \Gamma (1+m,b x)}{16 b}-\frac{3^{-1-m} e^{-3 a} x^m (b x)^{-m} \Gamma (1+m,3 b x)}{32 b}+\frac{5^{-1-m} e^{-5 a} x^m (b x)^{-m} \Gamma (1+m,5 b x)}{32 b}\\ \end{align*}

Mathematica [A]  time = 0.26055, size = 174, normalized size = 0.83 $\frac{e^{-5 a} x^m \left (-5 e^{2 a} 3^{-m} \left (-b^2 x^2\right )^{-m} \left (e^{6 a} (b x)^m \text{Gamma}(m+1,-3 b x)+(-b x)^m \text{Gamma}(m+1,3 b x)\right )+3\ 5^{-m} \left (-b^2 x^2\right )^{-m} \left (e^{10 a} (b x)^m \text{Gamma}(m+1,-5 b x)+(-b x)^m \text{Gamma}(m+1,5 b x)\right )-30 e^{4 a} \left (e^{2 a} (-b x)^{-m} \text{Gamma}(m+1,-b x)+(b x)^{-m} \text{Gamma}(m+1,b x)\right )\right )}{480 b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^m*Cosh[a + b*x]^2*Sinh[a + b*x]^3,x]

[Out]

(x^m*(-30*E^(4*a)*((E^(2*a)*Gamma[1 + m, -(b*x)])/(-(b*x))^m + Gamma[1 + m, b*x]/(b*x)^m) - (5*E^(2*a)*(E^(6*a
)*(b*x)^m*Gamma[1 + m, -3*b*x] + (-(b*x))^m*Gamma[1 + m, 3*b*x]))/(3^m*(-(b^2*x^2))^m) + (3*(E^(10*a)*(b*x)^m*
Gamma[1 + m, -5*b*x] + (-(b*x))^m*Gamma[1 + m, 5*b*x]))/(5^m*(-(b^2*x^2))^m)))/(480*b*E^(5*a))

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Maple [F]  time = 0.062, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( \cosh \left ( bx+a \right ) \right ) ^{2} \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)^2*sinh(b*x+a)^3,x)

[Out]

int(x^m*cosh(b*x+a)^2*sinh(b*x+a)^3,x)

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Maxima [A]  time = 1.34418, size = 231, normalized size = 1.11 \begin{align*} \frac{1}{32} \, \left (5 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (-5 \, a\right )} \Gamma \left (m + 1, 5 \, b x\right ) - \frac{1}{32} \, \left (3 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (-3 \, a\right )} \Gamma \left (m + 1, 3 \, b x\right ) - \frac{1}{16} \, \left (b x\right )^{-m - 1} x^{m + 1} e^{\left (-a\right )} \Gamma \left (m + 1, b x\right ) + \frac{1}{16} \, \left (-b x\right )^{-m - 1} x^{m + 1} e^{a} \Gamma \left (m + 1, -b x\right ) + \frac{1}{32} \, \left (-3 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (3 \, a\right )} \Gamma \left (m + 1, -3 \, b x\right ) - \frac{1}{32} \, \left (-5 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (5 \, a\right )} \Gamma \left (m + 1, -5 \, b x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*cosh(b*x+a)^2*sinh(b*x+a)^3,x, algorithm="maxima")

[Out]

1/32*(5*b*x)^(-m - 1)*x^(m + 1)*e^(-5*a)*gamma(m + 1, 5*b*x) - 1/32*(3*b*x)^(-m - 1)*x^(m + 1)*e^(-3*a)*gamma(
m + 1, 3*b*x) - 1/16*(b*x)^(-m - 1)*x^(m + 1)*e^(-a)*gamma(m + 1, b*x) + 1/16*(-b*x)^(-m - 1)*x^(m + 1)*e^a*ga
mma(m + 1, -b*x) + 1/32*(-3*b*x)^(-m - 1)*x^(m + 1)*e^(3*a)*gamma(m + 1, -3*b*x) - 1/32*(-5*b*x)^(-m - 1)*x^(m
+ 1)*e^(5*a)*gamma(m + 1, -5*b*x)

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Fricas [A]  time = 1.95095, size = 763, normalized size = 3.65 \begin{align*} \frac{3 \, \cosh \left (m \log \left (5 \, b\right ) + 5 \, a\right ) \Gamma \left (m + 1, 5 \, b x\right ) - 5 \, \cosh \left (m \log \left (3 \, b\right ) + 3 \, a\right ) \Gamma \left (m + 1, 3 \, b x\right ) - 30 \, \cosh \left (m \log \left (b\right ) + a\right ) \Gamma \left (m + 1, b x\right ) - 30 \, \cosh \left (m \log \left (-b\right ) - a\right ) \Gamma \left (m + 1, -b x\right ) - 5 \, \cosh \left (m \log \left (-3 \, b\right ) - 3 \, a\right ) \Gamma \left (m + 1, -3 \, b x\right ) + 3 \, \cosh \left (m \log \left (-5 \, b\right ) - 5 \, a\right ) \Gamma \left (m + 1, -5 \, b x\right ) - 3 \, \Gamma \left (m + 1, 5 \, b x\right ) \sinh \left (m \log \left (5 \, b\right ) + 5 \, a\right ) + 5 \, \Gamma \left (m + 1, 3 \, b x\right ) \sinh \left (m \log \left (3 \, b\right ) + 3 \, a\right ) + 30 \, \Gamma \left (m + 1, -b x\right ) \sinh \left (m \log \left (-b\right ) - a\right ) + 5 \, \Gamma \left (m + 1, -3 \, b x\right ) \sinh \left (m \log \left (-3 \, b\right ) - 3 \, a\right ) - 3 \, \Gamma \left (m + 1, -5 \, b x\right ) \sinh \left (m \log \left (-5 \, b\right ) - 5 \, a\right ) + 30 \, \Gamma \left (m + 1, b x\right ) \sinh \left (m \log \left (b\right ) + a\right )}{480 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*cosh(b*x+a)^2*sinh(b*x+a)^3,x, algorithm="fricas")

[Out]

1/480*(3*cosh(m*log(5*b) + 5*a)*gamma(m + 1, 5*b*x) - 5*cosh(m*log(3*b) + 3*a)*gamma(m + 1, 3*b*x) - 30*cosh(m
*log(b) + a)*gamma(m + 1, b*x) - 30*cosh(m*log(-b) - a)*gamma(m + 1, -b*x) - 5*cosh(m*log(-3*b) - 3*a)*gamma(m
+ 1, -3*b*x) + 3*cosh(m*log(-5*b) - 5*a)*gamma(m + 1, -5*b*x) - 3*gamma(m + 1, 5*b*x)*sinh(m*log(5*b) + 5*a)
+ 5*gamma(m + 1, 3*b*x)*sinh(m*log(3*b) + 3*a) + 30*gamma(m + 1, -b*x)*sinh(m*log(-b) - a) + 5*gamma(m + 1, -3
*b*x)*sinh(m*log(-3*b) - 3*a) - 3*gamma(m + 1, -5*b*x)*sinh(m*log(-5*b) - 5*a) + 30*gamma(m + 1, b*x)*sinh(m*l
og(b) + a))/b

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*cosh(b*x+a)**2*sinh(b*x+a)**3,x)

[Out]

Integral(x**m*sinh(a + b*x)**3*cosh(a + b*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \cosh \left (b x + a\right )^{2} \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)^2*sinh(b*x+a)^3,x, algorithm="giac")

[Out]

integrate(x^m*cosh(b*x + a)^2*sinh(b*x + a)^3, x)