### 3.194 $$\int \sinh (x) \sinh (4 x) \, dx$$

Optimal. Leaf size=17 $\frac{1}{10} \sinh (5 x)-\frac{1}{6} \sinh (3 x)$

[Out]

-Sinh[3*x]/6 + Sinh[5*x]/10

________________________________________________________________________________________

Rubi [A]  time = 0.0101462, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.143, Rules used = {4282} $\frac{1}{10} \sinh (5 x)-\frac{1}{6} \sinh (3 x)$

Antiderivative was successfully veriﬁed.

[In]

Int[Sinh[x]*Sinh[4*x],x]

[Out]

-Sinh[3*x]/6 + Sinh[5*x]/10

Rule 4282

Int[sin[(a_.) + (b_.)*(x_)]*sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[a - c + (b - d)*x]/(2*(b - d)), x]
- Simp[Sin[a + c + (b + d)*x]/(2*(b + d)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]

Rubi steps

\begin{align*} \int \sinh (x) \sinh (4 x) \, dx &=-\frac{1}{6} \sinh (3 x)+\frac{1}{10} \sinh (5 x)\\ \end{align*}

Mathematica [A]  time = 0.0066184, size = 17, normalized size = 1. $\frac{1}{10} \sinh (5 x)-\frac{1}{6} \sinh (3 x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sinh[x]*Sinh[4*x],x]

[Out]

-Sinh[3*x]/6 + Sinh[5*x]/10

________________________________________________________________________________________

Maple [A]  time = 0.02, size = 14, normalized size = 0.8 \begin{align*} -{\frac{\sinh \left ( 3\,x \right ) }{6}}+{\frac{\sinh \left ( 5\,x \right ) }{10}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)*sinh(4*x),x)

[Out]

-1/6*sinh(3*x)+1/10*sinh(5*x)

________________________________________________________________________________________

Maxima [A]  time = 1.0524, size = 36, normalized size = 2.12 \begin{align*} -\frac{1}{60} \,{\left (5 \, e^{\left (-2 \, x\right )} - 3\right )} e^{\left (5 \, x\right )} + \frac{1}{12} \, e^{\left (-3 \, x\right )} - \frac{1}{20} \, e^{\left (-5 \, x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)*sinh(4*x),x, algorithm="maxima")

[Out]

-1/60*(5*e^(-2*x) - 3)*e^(5*x) + 1/12*e^(-3*x) - 1/20*e^(-5*x)

________________________________________________________________________________________

Fricas [B]  time = 2.03812, size = 119, normalized size = 7. \begin{align*} \frac{1}{10} \, \sinh \left (x\right )^{5} + \frac{1}{6} \,{\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{3} + \frac{1}{2} \,{\left (\cosh \left (x\right )^{4} - \cosh \left (x\right )^{2}\right )} \sinh \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)*sinh(4*x),x, algorithm="fricas")

[Out]

1/10*sinh(x)^5 + 1/6*(6*cosh(x)^2 - 1)*sinh(x)^3 + 1/2*(cosh(x)^4 - cosh(x)^2)*sinh(x)

________________________________________________________________________________________

Sympy [A]  time = 0.608297, size = 20, normalized size = 1.18 \begin{align*} \frac{4 \sinh{\left (x \right )} \cosh{\left (4 x \right )}}{15} - \frac{\sinh{\left (4 x \right )} \cosh{\left (x \right )}}{15} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)*sinh(4*x),x)

[Out]

4*sinh(x)*cosh(4*x)/15 - sinh(4*x)*cosh(x)/15

________________________________________________________________________________________

Giac [A]  time = 1.1564, size = 36, normalized size = 2.12 \begin{align*} \frac{1}{60} \,{\left (5 \, e^{\left (2 \, x\right )} - 3\right )} e^{\left (-5 \, x\right )} + \frac{1}{20} \, e^{\left (5 \, x\right )} - \frac{1}{12} \, e^{\left (3 \, x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)*sinh(4*x),x, algorithm="giac")

[Out]

1/60*(5*e^(2*x) - 3)*e^(-5*x) + 1/20*e^(5*x) - 1/12*e^(3*x)