3.1.54 \(\int \frac {\cosh ^{\frac {3}{2}}(a+b x)}{\sinh ^{\frac {3}{2}}(a+b x)} \, dx\) [54]

Optimal. Leaf size=79 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )}{b}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )}{b}-\frac {2 \sqrt {\cosh (a+b x)}}{b \sqrt {\sinh (a+b x)}} \]

[Out]

-arctan(sinh(b*x+a)^(1/2)/cosh(b*x+a)^(1/2))/b+arctanh(sinh(b*x+a)^(1/2)/cosh(b*x+a)^(1/2))/b-2*cosh(b*x+a)^(1
/2)/b/sinh(b*x+a)^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2647, 2654, 304, 209, 212} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )}{b}-\frac {2 \sqrt {\cosh (a+b x)}}{b \sqrt {\sinh (a+b x)}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]^(3/2)/Sinh[a + b*x]^(3/2),x]

[Out]

-(ArcTan[Sqrt[Sinh[a + b*x]]/Sqrt[Cosh[a + b*x]]]/b) + ArcTanh[Sqrt[Sinh[a + b*x]]/Sqrt[Cosh[a + b*x]]]/b - (2
*Sqrt[Cosh[a + b*x]])/(b*Sqrt[Sinh[a + b*x]])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 2647

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(a*Cos[e +
f*x])^(m - 1)*((b*Sin[e + f*x])^(n + 1)/(b*f*(n + 1))), x] + Dist[a^2*((m - 1)/(b^2*(n + 1))), Int[(a*Cos[e +
f*x])^(m - 2)*(b*Sin[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && (Intege
rsQ[2*m, 2*n] || EqQ[m + n, 0])

Rule 2654

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> With[{k = Denomina
tor[m]}, Dist[k*a*(b/f), Subst[Int[x^(k*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Sin[e + f*x])^(1/k)/(b*Cos
[e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]

Rubi steps

\begin {align*} \int \frac {\cosh ^{\frac {3}{2}}(a+b x)}{\sinh ^{\frac {3}{2}}(a+b x)} \, dx &=-\frac {2 \sqrt {\cosh (a+b x)}}{b \sqrt {\sinh (a+b x)}}+\int \frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}} \, dx\\ &=-\frac {2 \sqrt {\cosh (a+b x)}}{b \sqrt {\sinh (a+b x)}}-\frac {2 \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )}{b}\\ &=-\frac {2 \sqrt {\cosh (a+b x)}}{b \sqrt {\sinh (a+b x)}}+\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )}{b}-\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )}{b}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )}{b}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )}{b}-\frac {2 \sqrt {\cosh (a+b x)}}{b \sqrt {\sinh (a+b x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.02, size = 57, normalized size = 0.72 \begin {gather*} -\frac {2 \cosh ^2(a+b x)^{3/4} \, _2F_1\left (-\frac {1}{4},-\frac {1}{4};\frac {3}{4};-\sinh ^2(a+b x)\right )}{b \cosh ^{\frac {3}{2}}(a+b x) \sqrt {\sinh (a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(Cosh[a + b*x]^2)^(3/4)*Hypergeometric2F1[-1/4, -1/4, 3/4, -Sinh[a + b*x]^2])/(b*Cosh[a + b*x]^(3/2)*Sqrt[
Sinh[a + b*x]])

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Maple [F]
time = 0.64, size = 0, normalized size = 0.00 \[\int \frac {\cosh ^{\frac {3}{2}}\left (b x +a \right )}{\sinh \left (b x +a \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (67) = 134\).
time = 0.37, size = 311, normalized size = 3.94 \begin {gather*} -\frac {2 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \arctan \left (-\cosh \left (b x + a\right )^{2} + 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \sqrt {\cosh \left (b x + a\right )} \sqrt {\sinh \left (b x + a\right )} - 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - \sinh \left (b x + a\right )^{2}\right ) + 4 \, \cosh \left (b x + a\right )^{2} + {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \log \left (-\cosh \left (b x + a\right )^{2} + 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \sqrt {\cosh \left (b x + a\right )} \sqrt {\sinh \left (b x + a\right )} - 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - \sinh \left (b x + a\right )^{2}\right ) + 8 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \sqrt {\cosh \left (b x + a\right )} \sqrt {\sinh \left (b x + a\right )} + 8 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + 4 \, \sinh \left (b x + a\right )^{2} - 4}{2 \, {\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} - b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 - 1)*arctan(-cosh(b*x + a)^2 + 2*(c
osh(b*x + a) + sinh(b*x + a))*sqrt(cosh(b*x + a))*sqrt(sinh(b*x + a)) - 2*cosh(b*x + a)*sinh(b*x + a) - sinh(b
*x + a)^2) + 4*cosh(b*x + a)^2 + (cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 - 1)*log(-
cosh(b*x + a)^2 + 2*(cosh(b*x + a) + sinh(b*x + a))*sqrt(cosh(b*x + a))*sqrt(sinh(b*x + a)) - 2*cosh(b*x + a)*
sinh(b*x + a) - sinh(b*x + a)^2) + 8*(cosh(b*x + a) + sinh(b*x + a))*sqrt(cosh(b*x + a))*sqrt(sinh(b*x + a)) +
8*cosh(b*x + a)*sinh(b*x + a) + 4*sinh(b*x + a)^2 - 4)/(b*cosh(b*x + a)^2 + 2*b*cosh(b*x + a)*sinh(b*x + a) +
b*sinh(b*x + a)^2 - b)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^{3/2}}{{\mathrm {sinh}\left (a+b\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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