### 3.535 $$\int \frac{x \sinh (a+b x)}{\cosh ^{\frac{9}{2}}(a+b x)} \, dx$$

Optimal. Leaf size=87 $\frac{12 i E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{35 b^2}+\frac{4 \sinh (a+b x)}{35 b^2 \cosh ^{\frac{5}{2}}(a+b x)}+\frac{12 \sinh (a+b x)}{35 b^2 \sqrt{\cosh (a+b x)}}-\frac{2 x}{7 b \cosh ^{\frac{7}{2}}(a+b x)}$

[Out]

(-2*x)/(7*b*Cosh[a + b*x]^(7/2)) + (((12*I)/35)*EllipticE[(I/2)*(a + b*x), 2])/b^2 + (4*Sinh[a + b*x])/(35*b^2
*Cosh[a + b*x]^(5/2)) + (12*Sinh[a + b*x])/(35*b^2*Sqrt[Cosh[a + b*x]])

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Rubi [A]  time = 0.0534982, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.167, Rules used = {5373, 2636, 2639} $\frac{12 i E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{35 b^2}+\frac{4 \sinh (a+b x)}{35 b^2 \cosh ^{\frac{5}{2}}(a+b x)}+\frac{12 \sinh (a+b x)}{35 b^2 \sqrt{\cosh (a+b x)}}-\frac{2 x}{7 b \cosh ^{\frac{7}{2}}(a+b x)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(x*Sinh[a + b*x])/Cosh[a + b*x]^(9/2),x]

[Out]

(-2*x)/(7*b*Cosh[a + b*x]^(7/2)) + (((12*I)/35)*EllipticE[(I/2)*(a + b*x), 2])/b^2 + (4*Sinh[a + b*x])/(35*b^2
*Cosh[a + b*x]^(5/2)) + (12*Sinh[a + b*x])/(35*b^2*Sqrt[Cosh[a + b*x]])

Rule 5373

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp[(x^(m -
n + 1)*Cosh[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Cosh[a + b*x
^n]^(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]

Rule 2636

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1))/(b*d*(n +
1)), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*n]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

\begin{align*} \int \frac{x \sinh (a+b x)}{\cosh ^{\frac{9}{2}}(a+b x)} \, dx &=-\frac{2 x}{7 b \cosh ^{\frac{7}{2}}(a+b x)}+\frac{2 \int \frac{1}{\cosh ^{\frac{7}{2}}(a+b x)} \, dx}{7 b}\\ &=-\frac{2 x}{7 b \cosh ^{\frac{7}{2}}(a+b x)}+\frac{4 \sinh (a+b x)}{35 b^2 \cosh ^{\frac{5}{2}}(a+b x)}+\frac{6 \int \frac{1}{\cosh ^{\frac{3}{2}}(a+b x)} \, dx}{35 b}\\ &=-\frac{2 x}{7 b \cosh ^{\frac{7}{2}}(a+b x)}+\frac{4 \sinh (a+b x)}{35 b^2 \cosh ^{\frac{5}{2}}(a+b x)}+\frac{12 \sinh (a+b x)}{35 b^2 \sqrt{\cosh (a+b x)}}-\frac{6 \int \sqrt{\cosh (a+b x)} \, dx}{35 b}\\ &=-\frac{2 x}{7 b \cosh ^{\frac{7}{2}}(a+b x)}+\frac{12 i E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{35 b^2}+\frac{4 \sinh (a+b x)}{35 b^2 \cosh ^{\frac{5}{2}}(a+b x)}+\frac{12 \sinh (a+b x)}{35 b^2 \sqrt{\cosh (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.279004, size = 69, normalized size = 0.79 $\frac{10 \sinh (2 (a+b x))+3 \sinh (4 (a+b x))+24 i \cosh ^{\frac{7}{2}}(a+b x) E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )-20 b x}{70 b^2 \cosh ^{\frac{7}{2}}(a+b x)}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(x*Sinh[a + b*x])/Cosh[a + b*x]^(9/2),x]

[Out]

(-20*b*x + (24*I)*Cosh[a + b*x]^(7/2)*EllipticE[(I/2)*(a + b*x), 2] + 10*Sinh[2*(a + b*x)] + 3*Sinh[4*(a + b*x
)])/(70*b^2*Cosh[a + b*x]^(7/2))

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{x\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{-{\frac{9}{2}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x*sinh(b*x+a)/cosh(b*x+a)^(9/2),x)

[Out]

int(x*sinh(b*x+a)/cosh(b*x+a)^(9/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{\frac{9}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sinh(b*x+a)/cosh(b*x+a)^(9/2),x, algorithm="maxima")

[Out]

integrate(x*sinh(b*x + a)/cosh(b*x + a)^(9/2), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sinh(b*x+a)/cosh(b*x+a)^(9/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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*sinh(b*x+a)/cosh(b*x+a)**(9/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{\frac{9}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sinh(b*x+a)/cosh(b*x+a)^(9/2),x, algorithm="giac")

[Out]

integrate(x*sinh(b*x + a)/cosh(b*x + a)^(9/2), x)