3.582 \(\int \text{sech}^{\frac{23}{4}}(x) \sinh ^5(x) \, dx\)

Optimal. Leaf size=31 \[ -\frac{4}{19} \text{sech}^{\frac{19}{4}}(x)+\frac{8}{11} \text{sech}^{\frac{11}{4}}(x)-\frac{4}{3} \text{sech}^{\frac{3}{4}}(x) \]

[Out]

(-4*Sech[x]^(3/4))/3 + (8*Sech[x]^(11/4))/11 - (4*Sech[x]^(19/4))/19

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Rubi [A]  time = 0.0311796, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2622, 270} \[ -\frac{4}{19} \text{sech}^{\frac{19}{4}}(x)+\frac{8}{11} \text{sech}^{\frac{11}{4}}(x)-\frac{4}{3} \text{sech}^{\frac{3}{4}}(x) \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]^(23/4)*Sinh[x]^5,x]

[Out]

(-4*Sech[x]^(3/4))/3 + (8*Sech[x]^(11/4))/11 - (4*Sech[x]^(19/4))/19

Rule 2622

Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \text{sech}^{\frac{23}{4}}(x) \sinh ^5(x) \, dx &=-\operatorname{Subst}\left (\int \frac{\left (-1+x^2\right )^2}{\sqrt [4]{x}} \, dx,x,\text{sech}(x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{1}{\sqrt [4]{x}}-2 x^{7/4}+x^{15/4}\right ) \, dx,x,\text{sech}(x)\right )\\ &=-\frac{4}{3} \text{sech}^{\frac{3}{4}}(x)+\frac{8}{11} \text{sech}^{\frac{11}{4}}(x)-\frac{4}{19} \text{sech}^{\frac{19}{4}}(x)\\ \end{align*}

Mathematica [A]  time = 0.0530409, size = 27, normalized size = 0.87 \[ \text{sech}^{\frac{3}{4}}(x) \left (-\frac{4}{19} \text{sech}^4(x)+\frac{8 \text{sech}^2(x)}{11}-\frac{4}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]^(23/4)*Sinh[x]^5,x]

[Out]

Sech[x]^(3/4)*(-4/3 + (8*Sech[x]^2)/11 - (4*Sech[x]^4)/19)

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm sech} \left (x\right ) \right ) ^{{\frac{3}{4}}} \left ( \tanh \left ( x \right ) \right ) ^{5}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^(3/4)*tanh(x)^5,x)

[Out]

int(sech(x)^(3/4)*tanh(x)^5,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{sech}\left (x\right )^{\frac{3}{4}} \tanh \left (x\right )^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^(3/4)*tanh(x)^5,x, algorithm="maxima")

[Out]

integrate(sech(x)^(3/4)*tanh(x)^5, x)

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Fricas [B]  time = 2.11994, size = 1253, normalized size = 40.42 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^(3/4)*tanh(x)^5,x, algorithm="fricas")

[Out]

-4/627*2^(3/4)*(209*cosh(x)^8 + 1672*cosh(x)*sinh(x)^7 + 209*sinh(x)^8 + 76*(77*cosh(x)^2 + 5)*sinh(x)^6 + 380
*cosh(x)^6 + 152*(77*cosh(x)^3 + 15*cosh(x))*sinh(x)^5 + 10*(1463*cosh(x)^4 + 570*cosh(x)^2 + 87)*sinh(x)^4 +
870*cosh(x)^4 + 8*(1463*cosh(x)^5 + 950*cosh(x)^3 + 435*cosh(x))*sinh(x)^3 + 4*(1463*cosh(x)^6 + 1425*cosh(x)^
4 + 1305*cosh(x)^2 + 95)*sinh(x)^2 + 380*cosh(x)^2 + 8*(209*cosh(x)^7 + 285*cosh(x)^5 + 435*cosh(x)^3 + 95*cos
h(x))*sinh(x) + 209)*((cosh(x) + sinh(x))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1))^(3/4)/(cosh(x)^8 +
8*cosh(x)*sinh(x)^7 + sinh(x)^8 + 4*(7*cosh(x)^2 + 1)*sinh(x)^6 + 4*cosh(x)^6 + 8*(7*cosh(x)^3 + 3*cosh(x))*si
nh(x)^5 + 2*(35*cosh(x)^4 + 30*cosh(x)^2 + 3)*sinh(x)^4 + 6*cosh(x)^4 + 8*(7*cosh(x)^5 + 10*cosh(x)^3 + 3*cosh
(x))*sinh(x)^3 + 4*(7*cosh(x)^6 + 15*cosh(x)^4 + 9*cosh(x)^2 + 1)*sinh(x)^2 + 4*cosh(x)^2 + 8*(cosh(x)^7 + 3*c
osh(x)^5 + 3*cosh(x)^3 + cosh(x))*sinh(x) + 1)

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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(sech(x)**(3/4)*tanh(x)**5,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{sech}\left (x\right )^{\frac{3}{4}} \tanh \left (x\right )^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^(3/4)*tanh(x)^5,x, algorithm="giac")

[Out]

integrate(sech(x)^(3/4)*tanh(x)^5, x)