### 3.289 $$\int x^m \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx$$

Optimal. Leaf size=85 $\frac{e^{4 a} 2^{-2 (m+3)} x^m (-b x)^{-m} \text{Gamma}(m+1,-4 b x)}{b}-\frac{e^{-4 a} 2^{-2 (m+3)} x^m (b x)^{-m} \text{Gamma}(m+1,4 b x)}{b}-\frac{x^{m+1}}{8 (m+1)}$

[Out]

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

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Rubi [A]  time = 0.12789, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.15, Rules used = {5448, 3307, 2181} $\frac{e^{4 a} 2^{-2 (m+3)} x^m (-b x)^{-m} \text{Gamma}(m+1,-4 b x)}{b}-\frac{e^{-4 a} 2^{-2 (m+3)} x^m (b x)^{-m} \text{Gamma}(m+1,4 b x)}{b}-\frac{x^{m+1}}{8 (m+1)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-x^(1 + m)/(8*(1 + m)) + (E^(4*a)*x^m*Gamma[1 + m, -4*b*x])/(2^(2*(3 + m))*b*(-(b*x))^m) - (x^m*Gamma[1 + m, 4
*b*x])/(2^(2*(3 + m))*b*E^(4*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 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^m \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx &=\int \left (-\frac{x^m}{8}+\frac{1}{8} x^m \cosh (4 a+4 b x)\right ) \, dx\\ &=-\frac{x^{1+m}}{8 (1+m)}+\frac{1}{8} \int x^m \cosh (4 a+4 b x) \, dx\\ &=-\frac{x^{1+m}}{8 (1+m)}+\frac{1}{16} \int e^{-i (4 i a+4 i b x)} x^m \, dx+\frac{1}{16} \int e^{i (4 i a+4 i b x)} x^m \, dx\\ &=-\frac{x^{1+m}}{8 (1+m)}+\frac{4^{-3-m} e^{4 a} x^m (-b x)^{-m} \Gamma (1+m,-4 b x)}{b}-\frac{4^{-3-m} e^{-4 a} x^m (b x)^{-m} \Gamma (1+m,4 b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.205554, size = 76, normalized size = 0.89 $\frac{1}{64} x^m \left (\frac{e^{4 a} 4^{-m} (-b x)^{-m} \text{Gamma}(m+1,-4 b x)}{b}-\frac{e^{-4 a} 4^{-m} (b x)^{-m} \text{Gamma}(m+1,4 b x)}{b}-\frac{8 x}{m+1}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x^m*((-8*x)/(1 + m) + (E^(4*a)*Gamma[1 + m, -4*b*x])/(4^m*b*(-(b*x))^m) - Gamma[1 + m, 4*b*x]/(4^m*b*E^(4*a)*
(b*x)^m)))/64

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Maple [F]  time = 0.053, 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 ) ^{2}\, 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)^2,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.88802, size = 377, normalized size = 4.44 \begin{align*} -\frac{8 \, b x \cosh \left (m \log \left (x\right )\right ) +{\left (m + 1\right )} \cosh \left (m \log \left (4 \, b\right ) + 4 \, a\right ) \Gamma \left (m + 1, 4 \, b x\right ) -{\left (m + 1\right )} \cosh \left (m \log \left (-4 \, b\right ) - 4 \, a\right ) \Gamma \left (m + 1, -4 \, b x\right ) -{\left (m + 1\right )} \Gamma \left (m + 1, 4 \, b x\right ) \sinh \left (m \log \left (4 \, b\right ) + 4 \, a\right ) +{\left (m + 1\right )} \Gamma \left (m + 1, -4 \, b x\right ) \sinh \left (m \log \left (-4 \, b\right ) - 4 \, a\right ) + 8 \, b x \sinh \left (m \log \left (x\right )\right )}{64 \,{\left (b m + b\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)^2,x, algorithm="fricas")

[Out]

-1/64*(8*b*x*cosh(m*log(x)) + (m + 1)*cosh(m*log(4*b) + 4*a)*gamma(m + 1, 4*b*x) - (m + 1)*cosh(m*log(-4*b) -
4*a)*gamma(m + 1, -4*b*x) - (m + 1)*gamma(m + 1, 4*b*x)*sinh(m*log(4*b) + 4*a) + (m + 1)*gamma(m + 1, -4*b*x)*
sinh(m*log(-4*b) - 4*a) + 8*b*x*sinh(m*log(x)))/(b*m + b)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \sinh ^{2}{\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)**2,x)

[Out]

Integral(x**m*sinh(a + b*x)**2*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 )^{2}\,{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)^2,x, algorithm="giac")

[Out]

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