\(\int \text {sech}^{\frac {23}{4}}(x) \sinh ^5(x) \, dx\) [582]

Optimal result

Integrand size = 11, antiderivative size = 31 \[ \int \text {sech}^{\frac {23}{4}}(x) \sinh ^5(x) \, dx=-\frac {4}{3} \text {sech}^{\frac {3}{4}}(x)+\frac {8}{11} \text {sech}^{\frac {11}{4}}(x)-\frac {4}{19} \text {sech}^{\frac {19}{4}}(x) \]

[Out]

-4/3*sech(x)^(3/4)+8/11*sech(x)^(11/4)-4/19*sech(x)^(19/4)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2702, 276} \[ \int \text {sech}^{\frac {23}{4}}(x) \sinh ^5(x) \, dx=-\frac {4}{19} \text {sech}^{\frac {19}{4}}(x)+\frac {8}{11} \text {sech}^{\frac {11}{4}}(x)-\frac {4}{3} \text {sech}^{\frac {3}{4}}(x) \]

[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 276

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]

Rule 2702

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])

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\left (-1+x^2\right )^2}{\sqrt [4]{x}} \, dx,x,\text {sech}(x)\right ) \\ & = -\text {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] (verified)

Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \text {sech}^{\frac {23}{4}}(x) \sinh ^5(x) \, dx=\text {sech}^{\frac {3}{4}}(x) \left (-\frac {4}{3}+\frac {8 \text {sech}^2(x)}{11}-\frac {4 \text {sech}^4(x)}{19}\right ) \]

[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)

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65

method	result	size
derivativedivides	\(-\frac {4 \operatorname {sech}\left (x \right )^{\frac {3}{4}}}{3}+\frac {8 \operatorname {sech}\left (x \right )^{\frac {11}{4}}}{11}-\frac {4 \operatorname {sech}\left (x \right )^{\frac {19}{4}}}{19}\)	\(20\)
default	\(-\frac {4 \operatorname {sech}\left (x \right )^{\frac {3}{4}}}{3}+\frac {8 \operatorname {sech}\left (x \right )^{\frac {11}{4}}}{11}-\frac {4 \operatorname {sech}\left (x \right )^{\frac {19}{4}}}{19}\)	\(20\)

[In]

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

[Out]

-4/3*sech(x)^(3/4)+8/11*sech(x)^(11/4)-4/19*sech(x)^(19/4)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (19) = 38\).

Time = 0.24 (sec) , antiderivative size = 359, normalized size of antiderivative = 11.58 \[ \int \text {sech}^{\frac {23}{4}}(x) \sinh ^5(x) \, dx=-\frac {4 \cdot 2^{\frac {3}{4}} {\left (209 \, \cosh \left (x\right )^{8} + 1672 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + 209 \, \sinh \left (x\right )^{8} + 76 \, {\left (77 \, \cosh \left (x\right )^{2} + 5\right )} \sinh \left (x\right )^{6} + 380 \, \cosh \left (x\right )^{6} + 152 \, {\left (77 \, \cosh \left (x\right )^{3} + 15 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 10 \, {\left (1463 \, \cosh \left (x\right )^{4} + 570 \, \cosh \left (x\right )^{2} + 87\right )} \sinh \left (x\right )^{4} + 870 \, \cosh \left (x\right )^{4} + 8 \, {\left (1463 \, \cosh \left (x\right )^{5} + 950 \, \cosh \left (x\right )^{3} + 435 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (1463 \, \cosh \left (x\right )^{6} + 1425 \, \cosh \left (x\right )^{4} + 1305 \, \cosh \left (x\right )^{2} + 95\right )} \sinh \left (x\right )^{2} + 380 \, \cosh \left (x\right )^{2} + 8 \, {\left (209 \, \cosh \left (x\right )^{7} + 285 \, \cosh \left (x\right )^{5} + 435 \, \cosh \left (x\right )^{3} + 95 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 209\right )} \left (\frac {\cosh \left (x\right ) + \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}\right )^{\frac {3}{4}}}{627 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{6} + 4 \, \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} + 30 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} + 10 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} + 15 \, \cosh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} + 3 \, \cosh \left (x\right )^{5} + 3 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \]

[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)

Sympy [A] (verification not implemented)

Time = 39.54 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \text {sech}^{\frac {23}{4}}(x) \sinh ^5(x) \, dx=- \frac {4 \tanh ^{4}{\left (x \right )} \operatorname {sech}^{\frac {3}{4}}{\left (x \right )}}{19} - \frac {64 \tanh ^{2}{\left (x \right )} \operatorname {sech}^{\frac {3}{4}}{\left (x \right )}}{209} - \frac {512 \operatorname {sech}^{\frac {3}{4}}{\left (x \right )}}{627} \]

[In]

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

[Out]

-4*tanh(x)**4*sech(x)**(3/4)/19 - 64*tanh(x)**2*sech(x)**(3/4)/209 - 512*sech(x)**(3/4)/627

Maxima [F]

\[ \int \text {sech}^{\frac {23}{4}}(x) \sinh ^5(x) \, dx=\int { \operatorname {sech}\left (x\right )^{\frac {3}{4}} \tanh \left (x\right )^{5} \,d x } \]

[In]

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

[Out]

integrate(sech(x)^(3/4)*tanh(x)^5, x)

Giac [F]

\[ \int \text {sech}^{\frac {23}{4}}(x) \sinh ^5(x) \, dx=\int { \operatorname {sech}\left (x\right )^{\frac {3}{4}} \tanh \left (x\right )^{5} \,d x } \]

[In]

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

[Out]

integrate(sech(x)^(3/4)*tanh(x)^5, x)

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 120, normalized size of antiderivative = 3.87 \[ \int \text {sech}^{\frac {23}{4}}(x) \sinh ^5(x) \, dx=\frac {32\,{\left (\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}\right )}^{3/4}}{11\,\left ({\mathrm {e}}^{2\,x}+1\right )}-\frac {1312\,{\left (\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}\right )}^{3/4}}{209\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^2}+\frac {128\,{\left (\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}\right )}^{3/4}}{19\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^3}-\frac {64\,{\left (\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}\right )}^{3/4}}{19\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^4}-\frac {4\,{\left (\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}\right )}^{3/4}}{3} \]

[In]

int(tanh(x)^5*(1/cosh(x))^(3/4),x)

[Out]

(32*(1/(exp(-x)/2 + exp(x)/2))^(3/4))/(11*(exp(2*x) + 1)) - (1312*(1/(exp(-x)/2 + exp(x)/2))^(3/4))/(209*(exp(
2*x) + 1)^2) + (128*(1/(exp(-x)/2 + exp(x)/2))^(3/4))/(19*(exp(2*x) + 1)^3) - (64*(1/(exp(-x)/2 + exp(x)/2))^(
3/4))/(19*(exp(2*x) + 1)^4) - (4*(1/(exp(-x)/2 + exp(x)/2))^(3/4))/3