\(\int \frac {(a+a \cosh (x))^{3/2}}{x^3} \, dx\) [138]

Optimal result

Integrand size = 14, antiderivative size = 109 \[ \int \frac {(a+a \cosh (x))^{3/2}}{x^3} \, dx=-\frac {a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}}{x^2}+\frac {3}{16} a \sqrt {a+a \cosh (x)} \text {Chi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )+\frac {9}{16} a \sqrt {a+a \cosh (x)} \text {Chi}\left (\frac {3 x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )-\frac {3 a \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )}{2 x} \]

[Out]

-a*cosh(1/2*x)^2*(a+a*cosh(x))^(1/2)/x^2+3/16*a*Chi(1/2*x)*sech(1/2*x)*(a+a*cosh(x))^(1/2)+9/16*a*Chi(3/2*x)*s
ech(1/2*x)*(a+a*cosh(x))^(1/2)-3/2*a*cosh(1/2*x)*sinh(1/2*x)*(a+a*cosh(x))^(1/2)/x

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3400, 3395, 3382, 3393} \[ \int \frac {(a+a \cosh (x))^{3/2}}{x^3} \, dx=\frac {3}{16} a \text {Chi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}+\frac {9}{16} a \text {Chi}\left (\frac {3 x}{2}\right ) \text {sech}\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}-\frac {a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}}{x^2}-\frac {3 a \sinh \left (\frac {x}{2}\right ) \cosh \left (\frac {x}{2}\right ) \sqrt {a \cosh (x)+a}}{2 x} \]

[In]

Int[(a + a*Cosh[x])^(3/2)/x^3,x]

[Out]

-((a*Cosh[x/2]^2*Sqrt[a + a*Cosh[x]])/x^2) + (3*a*Sqrt[a + a*Cosh[x]]*CoshIntegral[x/2]*Sech[x/2])/16 + (9*a*S
qrt[a + a*Cosh[x]]*CoshIntegral[(3*x)/2]*Sech[x/2])/16 - (3*a*Cosh[x/2]*Sqrt[a + a*Cosh[x]]*Sinh[x/2])/(2*x)

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3395

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

Rule 3400

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(2*a)^IntPart[n]
*((a + b*Sin[e + f*x])^FracPart[n]/Sin[e/2 + a*(Pi/(4*b)) + f*(x/2)]^(2*FracPart[n])), Int[(c + d*x)^m*Sin[e/2
+ a*(Pi/(4*b)) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[n
+ 1/2] && (GtQ[n, 0] || IGtQ[m, 0])

Rubi steps \begin{align*} \text {integral}& = \left (2 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \frac {\cosh ^3\left (\frac {x}{2}\right )}{x^3} \, dx \\ & = -\frac {a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}}{x^2}-\frac {3 a \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )}{2 x}-\frac {1}{2} \left (3 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \frac {\cosh \left (\frac {x}{2}\right )}{x} \, dx+\frac {1}{4} \left (9 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \frac {\cosh ^3\left (\frac {x}{2}\right )}{x} \, dx \\ & = -\frac {a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}}{x^2}-\frac {3}{2} a \sqrt {a+a \cosh (x)} \text {Chi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )-\frac {3 a \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )}{2 x}+\frac {1}{4} \left (9 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \left (\frac {3 \cosh \left (\frac {x}{2}\right )}{4 x}+\frac {\cosh \left (\frac {3 x}{2}\right )}{4 x}\right ) \, dx \\ & = -\frac {a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}}{x^2}-\frac {3}{2} a \sqrt {a+a \cosh (x)} \text {Chi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )-\frac {3 a \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )}{2 x}+\frac {1}{16} \left (9 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \frac {\cosh \left (\frac {3 x}{2}\right )}{x} \, dx+\frac {1}{16} \left (27 a \sqrt {a+a \cosh (x)} \text {sech}\left (\frac {x}{2}\right )\right ) \int \frac {\cosh \left (\frac {x}{2}\right )}{x} \, dx \\ & = -\frac {a \cosh ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)}}{x^2}+\frac {3}{16} a \sqrt {a+a \cosh (x)} \text {Chi}\left (\frac {x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )+\frac {9}{16} a \sqrt {a+a \cosh (x)} \text {Chi}\left (\frac {3 x}{2}\right ) \text {sech}\left (\frac {x}{2}\right )-\frac {3 a \cosh \left (\frac {x}{2}\right ) \sqrt {a+a \cosh (x)} \sinh \left (\frac {x}{2}\right )}{2 x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.63 \[ \int \frac {(a+a \cosh (x))^{3/2}}{x^3} \, dx=\frac {(a (1+\cosh (x)))^{3/2} \left (3 x^2 \text {Chi}\left (\frac {x}{2}\right ) \text {sech}^3\left (\frac {x}{2}\right )+9 x^2 \text {Chi}\left (\frac {3 x}{2}\right ) \text {sech}^3\left (\frac {x}{2}\right )-8 \left (2+3 x \tanh \left (\frac {x}{2}\right )\right )\right )}{32 x^2} \]

[In]

Integrate[(a + a*Cosh[x])^(3/2)/x^3,x]

[Out]

((a*(1 + Cosh[x]))^(3/2)*(3*x^2*CoshIntegral[x/2]*Sech[x/2]^3 + 9*x^2*CoshIntegral[(3*x)/2]*Sech[x/2]^3 - 8*(2
+ 3*x*Tanh[x/2])))/(32*x^2)

Maple [F]

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

[In]

int((a+a*cosh(x))^(3/2)/x^3,x)

[Out]

int((a+a*cosh(x))^(3/2)/x^3,x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {(a+a \cosh (x))^{3/2}}{x^3} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {(a+a \cosh (x))^{3/2}}{x^3} \, dx=\int \frac {\left (a \left (\cosh {\left (x \right )} + 1\right )\right )^{\frac {3}{2}}}{x^{3}}\, dx \]

[In]

integrate((a+a*cosh(x))**(3/2)/x**3,x)

[Out]

Integral((a*(cosh(x) + 1))**(3/2)/x**3, x)

Maxima [F]

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

[In]

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

[Out]

integrate((a*cosh(x) + a)^(3/2)/x^3, x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (81) = 162\).

Time = 0.29 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.56 \[ \int \frac {(a+a \cosh (x))^{3/2}}{x^3} \, dx=\frac {1}{32} \, \sqrt {2} {\left (\frac {9 \, a^{\frac {3}{2}} x^{2} {\rm Ei}\left (\frac {3}{2} \, x\right ) + 3 \, a^{\frac {3}{2}} x^{2} {\rm Ei}\left (\frac {1}{2} \, x\right ) + a^{\frac {3}{2}} x^{2} {\rm Ei}\left (-\frac {1}{2} \, x\right ) - 6 \, a^{\frac {3}{2}} x e^{\left (\frac {3}{2} \, x\right )} - 6 \, a^{\frac {3}{2}} x e^{\left (\frac {1}{2} \, x\right )} + 2 \, a^{\frac {3}{2}} x e^{\left (-\frac {1}{2} \, x\right )} - 4 \, a^{\frac {3}{2}} e^{\left (\frac {3}{2} \, x\right )} - 12 \, a^{\frac {3}{2}} e^{\left (\frac {1}{2} \, x\right )} - 4 \, a^{\frac {3}{2}} e^{\left (-\frac {1}{2} \, x\right )}}{x^{2}} + \frac {2 \, a^{\frac {3}{2}} x^{2} {\rm Ei}\left (-\frac {1}{2} \, x\right ) + 9 \, a^{\frac {3}{2}} x^{2} {\rm Ei}\left (-\frac {3}{2} \, x\right ) + 4 \, a^{\frac {3}{2}} x e^{\left (-\frac {1}{2} \, x\right )} + 6 \, a^{\frac {3}{2}} x e^{\left (-\frac {3}{2} \, x\right )} - 8 \, a^{\frac {3}{2}} e^{\left (-\frac {1}{2} \, x\right )} - 4 \, a^{\frac {3}{2}} e^{\left (-\frac {3}{2} \, x\right )}}{x^{2}}\right )} \]

[In]

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

[Out]

1/32*sqrt(2)*((9*a^(3/2)*x^2*Ei(3/2*x) + 3*a^(3/2)*x^2*Ei(1/2*x) + a^(3/2)*x^2*Ei(-1/2*x) - 6*a^(3/2)*x*e^(3/2
*x) - 6*a^(3/2)*x*e^(1/2*x) + 2*a^(3/2)*x*e^(-1/2*x) - 4*a^(3/2)*e^(3/2*x) - 12*a^(3/2)*e^(1/2*x) - 4*a^(3/2)*
e^(-1/2*x))/x^2 + (2*a^(3/2)*x^2*Ei(-1/2*x) + 9*a^(3/2)*x^2*Ei(-3/2*x) + 4*a^(3/2)*x*e^(-1/2*x) + 6*a^(3/2)*x*
e^(-3/2*x) - 8*a^(3/2)*e^(-1/2*x) - 4*a^(3/2)*e^(-3/2*x))/x^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+a \cosh (x))^{3/2}}{x^3} \, dx=\int \frac {{\left (a+a\,\mathrm {cosh}\left (x\right )\right )}^{3/2}}{x^3} \,d x \]

[In]

int((a + a*cosh(x))^(3/2)/x^3,x)

[Out]

int((a + a*cosh(x))^(3/2)/x^3, x)