3.1.70 \(\int (\frac {x}{\sinh ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\sinh (x)}) \, dx\) [70]

Optimal. Leaf size=47 \[ -\frac {2 x \cosh (x)}{5 \sinh ^{\frac {5}{2}}(x)}-\frac {4}{15 \sinh ^{\frac {3}{2}}(x)}+\frac {6 x \cosh (x)}{5 \sqrt {\sinh (x)}}-\frac {12 \sqrt {\sinh (x)}}{5} \]

[Out]

-2/5*x*cosh(x)/sinh(x)^(5/2)-4/15/sinh(x)^(3/2)+6/5*x*cosh(x)/sinh(x)^(1/2)-12/5*sinh(x)^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3396} \begin {gather*} -\frac {4}{15 \sinh ^{\frac {3}{2}}(x)}-\frac {12 \sqrt {\sinh (x)}}{5}-\frac {2 x \cosh (x)}{5 \sinh ^{\frac {5}{2}}(x)}+\frac {6 x \cosh (x)}{5 \sqrt {\sinh (x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sinh[x]^(7/2) + (3*x*Sqrt[Sinh[x]])/5,x]

[Out]

(-2*x*Cosh[x])/(5*Sinh[x]^(5/2)) - 4/(15*Sinh[x]^(3/2)) + (6*x*Cosh[x])/(5*Sqrt[Sinh[x]]) - (12*Sqrt[Sinh[x]])
/5

Rule 3396

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

Rubi steps

\begin {align*} \int \left (\frac {x}{\sinh ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\sinh (x)}\right ) \, dx &=\frac {3}{5} \int x \sqrt {\sinh (x)} \, dx+\int \frac {x}{\sinh ^{\frac {7}{2}}(x)} \, dx\\ &=-\frac {2 x \cosh (x)}{5 \sinh ^{\frac {5}{2}}(x)}-\frac {4}{15 \sinh ^{\frac {3}{2}}(x)}-\frac {3}{5} \int \frac {x}{\sinh ^{\frac {3}{2}}(x)} \, dx+\frac {3}{5} \int x \sqrt {\sinh (x)} \, dx\\ &=-\frac {2 x \cosh (x)}{5 \sinh ^{\frac {5}{2}}(x)}-\frac {4}{15 \sinh ^{\frac {3}{2}}(x)}+\frac {6 x \cosh (x)}{5 \sqrt {\sinh (x)}}-\frac {12 \sqrt {\sinh (x)}}{5}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 33, normalized size = 0.70 \begin {gather*} \frac {-21 x \cosh (x)+9 x \cosh (3 x)+46 \sinh (x)-18 \sinh (3 x)}{30 \sinh ^{\frac {5}{2}}(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sinh[x]^(7/2) + (3*x*Sqrt[Sinh[x]])/5,x]

[Out]

(-21*x*Cosh[x] + 9*x*Cosh[3*x] + 46*Sinh[x] - 18*Sinh[3*x])/(30*Sinh[x]^(5/2))

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Maple [F]
time = 1.81, size = 0, normalized size = 0.00 \[\int \frac {x}{\sinh \left (x \right )^{\frac {7}{2}}}+\frac {3 x \left (\sqrt {\sinh }\left (x \right )\right )}{5}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/sinh(x)^(7/2)+3/5*x*sinh(x)^(1/2),x)

[Out]

int(x/sinh(x)^(7/2)+3/5*x*sinh(x)^(1/2),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/sinh(x)^(7/2)+3/5*x*sinh(x)^(1/2),x, algorithm="maxima")

[Out]

integrate(3/5*x*sqrt(sinh(x)) + x/sinh(x)^(7/2), x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/sinh(x)^(7/2)+3/5*x*sinh(x)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {5 x}{\sinh ^{\frac {7}{2}}{\left (x \right )}}\, dx + \int 3 x \sqrt {\sinh {\left (x \right )}}\, dx}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/sinh(x)**(7/2)+3/5*x*sinh(x)**(1/2),x)

[Out]

(Integral(5*x/sinh(x)**(7/2), x) + Integral(3*x*sqrt(sinh(x)), x))/5

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/sinh(x)^(7/2)+3/5*x*sinh(x)^(1/2),x, algorithm="giac")

[Out]

integrate(3/5*x*sqrt(sinh(x)) + x/sinh(x)^(7/2), x)

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Mupad [B]
time = 0.29, size = 111, normalized size = 2.36 \begin {gather*} \frac {12\,x\,{\mathrm {e}}^{2\,x}\,\sqrt {\frac {{\mathrm {e}}^x}{2}-\frac {{\mathrm {e}}^{-x}}{2}}}{5\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {{\mathrm {e}}^{2\,x}\,\left (\frac {8\,x}{5}+\frac {16}{15}\right )\,\sqrt {\frac {{\mathrm {e}}^x}{2}-\frac {{\mathrm {e}}^{-x}}{2}}}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^2}-\left (\frac {6\,x}{5}+\frac {12}{5}\right )\,\sqrt {\frac {{\mathrm {e}}^x}{2}-\frac {{\mathrm {e}}^{-x}}{2}}-\frac {16\,x\,{\mathrm {e}}^{2\,x}\,\sqrt {\frac {{\mathrm {e}}^x}{2}-\frac {{\mathrm {e}}^{-x}}{2}}}{5\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x*sinh(x)^(1/2))/5 + x/sinh(x)^(7/2),x)

[Out]

(12*x*exp(2*x)*(exp(x)/2 - exp(-x)/2)^(1/2))/(5*(exp(2*x) - 1)) - (exp(2*x)*((8*x)/5 + 16/15)*(exp(x)/2 - exp(
-x)/2)^(1/2))/(exp(2*x) - 1)^2 - ((6*x)/5 + 12/5)*(exp(x)/2 - exp(-x)/2)^(1/2) - (16*x*exp(2*x)*(exp(x)/2 - ex
p(-x)/2)^(1/2))/(5*(exp(2*x) - 1)^3)

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