\(\int (\frac {x}{\cosh ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cosh (x)}) \, dx\) [331]

Optimal result

Integrand size = 20, antiderivative size = 47 \[ \int \left (\frac {x}{\cosh ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cosh (x)}\right ) \, dx=\frac {4}{15 \cosh ^{\frac {3}{2}}(x)}-\frac {12 \sqrt {\cosh (x)}}{5}+\frac {2 x \sinh (x)}{5 \cosh ^{\frac {5}{2}}(x)}+\frac {6 x \sinh (x)}{5 \sqrt {\cosh (x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3396} \[ \int \left (\frac {x}{\cosh ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cosh (x)}\right ) \, dx=\frac {4}{15 \cosh ^{\frac {3}{2}}(x)}-\frac {12 \sqrt {\cosh (x)}}{5}+\frac {2 x \sinh (x)}{5 \cosh ^{\frac {5}{2}}(x)}+\frac {6 x \sinh (x)}{5 \sqrt {\cosh (x)}} \]

[In]

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

[Out]

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

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

Mathematica [A] (warning: unable to verify)

Time = 0.41 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.36 \[ \int \left (\frac {x}{\cosh ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cosh (x)}\right ) \, dx=\frac {1}{5} \sqrt {\cosh (x)} \left (-\frac {12 \sinh ^2(x)}{\sqrt {-1+\cosh (x)} (1+\cosh (x))^{3/2} \sqrt {\tanh ^2\left (\frac {x}{2}\right )}}+6 x \tanh (x)+\text {sech}^2(x) \left (\frac {4}{3}+2 x \tanh (x)\right )\right ) \]

[In]

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

[Out]

(Sqrt[Cosh[x]]*((-12*Sinh[x]^2)/(Sqrt[-1 + Cosh[x]]*(1 + Cosh[x])^(3/2)*Sqrt[Tanh[x/2]^2]) + 6*x*Tanh[x] + Sec
h[x]^2*(4/3 + 2*x*Tanh[x])))/5

Maple [F]

\[\int \left (\frac {x}{\cosh \left (x \right )^{\frac {7}{2}}}+\frac {3 x \sqrt {\cosh \left (x \right )}}{5}\right )d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \left (\frac {x}{\cosh ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cosh (x)}\right ) \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \left (\frac {x}{\cosh ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cosh (x)}\right ) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \left (\frac {x}{\cosh ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cosh (x)}\right ) \, dx=\int { \frac {3}{5} \, x \sqrt {\cosh \left (x\right )} + \frac {x}{\cosh \left (x\right )^{\frac {7}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \left (\frac {x}{\cosh ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cosh (x)}\right ) \, dx=\int { \frac {3}{5} \, x \sqrt {\cosh \left (x\right )} + \frac {x}{\cosh \left (x\right )^{\frac {7}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.91 (sec) , antiderivative size = 110, normalized size of antiderivative = 2.34 \[ \int \left (\frac {x}{\cosh ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cosh (x)}\right ) \, dx=\frac {{\mathrm {e}}^{2\,x}\,\left (\frac {8\,x}{5}+\frac {16}{15}\right )\,\sqrt {\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}}{{\left ({\mathrm {e}}^{2\,x}+1\right )}^2}-\left (\frac {6\,x}{5}+\frac {12}{5}\right )\,\sqrt {\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}+\frac {12\,x\,{\mathrm {e}}^{2\,x}\,\sqrt {\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}}{5\,\left ({\mathrm {e}}^{2\,x}+1\right )}-\frac {16\,x\,{\mathrm {e}}^{2\,x}\,\sqrt {\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}}{5\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^3} \]

[In]

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

[Out]

(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) + (12*x*exp(2*x)*(exp(-x)/2 + exp(x)/2)^(1/2))/(5*(exp(2*x) + 1)) - (16*x*exp(2*x)*(exp(-x)/2 + e
xp(x)/2)^(1/2))/(5*(exp(2*x) + 1)^3)