3.63 \(\int \frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)} \, dx\)

Optimal. Leaf size=218 \[ -\frac {\log \left (\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1\right )}{4 b}+\frac {\log \left (\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1\right )}{4 b}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}}{\sqrt {3}}\right )}{2 b}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}{\sqrt {3}}\right )}{2 b}+\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{b} \]

[Out]

arctanh(cosh(b*x+a)^(1/3)/sinh(b*x+a)^(1/3))/b-1/4*ln(1+cosh(b*x+a)^(2/3)/sinh(b*x+a)^(2/3)-cosh(b*x+a)^(1/3)/
sinh(b*x+a)^(1/3))/b+1/4*ln(1+cosh(b*x+a)^(2/3)/sinh(b*x+a)^(2/3)+cosh(b*x+a)^(1/3)/sinh(b*x+a)^(1/3))/b+1/2*a
rctan(1/3*(1-2*cosh(b*x+a)^(1/3)/sinh(b*x+a)^(1/3))*3^(1/2))*3^(1/2)/b-1/2*arctan(1/3*(1+2*cosh(b*x+a)^(1/3)/s
inh(b*x+a)^(1/3))*3^(1/2))*3^(1/2)/b

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Rubi [A]  time = 0.20, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2575, 296, 634, 618, 204, 628, 206} \[ -\frac {\log \left (\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1\right )}{4 b}+\frac {\log \left (\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1\right )}{4 b}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}}{\sqrt {3}}\right )}{2 b}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}+1}{\sqrt {3}}\right )}{2 b}+\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[3]*ArcTan[(1 - (2*Cosh[a + b*x]^(1/3))/Sinh[a + b*x]^(1/3))/Sqrt[3]])/(2*b) - (Sqrt[3]*ArcTan[(1 + (2*Co
sh[a + b*x]^(1/3))/Sinh[a + b*x]^(1/3))/Sqrt[3]])/(2*b) + ArcTanh[Cosh[a + b*x]^(1/3)/Sinh[a + b*x]^(1/3)]/b -
Log[1 + Cosh[a + b*x]^(2/3)/Sinh[a + b*x]^(2/3) - Cosh[a + b*x]^(1/3)/Sinh[a + b*x]^(1/3)]/(4*b) + Log[1 + Co
sh[a + b*x]^(2/3)/Sinh[a + b*x]^(2/3) + Cosh[a + b*x]^(1/3)/Sinh[a + b*x]^(1/3)]/(4*b)

Rule 204

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

Rule 206

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

Rule 296

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[-(a/b), n]], s = Denominator[Rt
[-(a/b), n]], k, u}, Simp[u = Int[(r*Cos[(2*k*m*Pi)/n] - s*Cos[(2*k*(m + 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi
)/n]*x + s^2*x^2), x] + Int[(r*Cos[(2*k*m*Pi)/n] + s*Cos[(2*k*(m + 1)*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x
+ s^2*x^2), x]; (2*r^(m + 2)*Int[1/(r^2 - s^2*x^2), x])/(a*n*s^m) + Dist[(2*r^(m + 1))/(a*n*s^m), Sum[u, {k,
1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && NegQ[a/b]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 2575

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), 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*Cos[e + f*x])^(1/k)/(b*Si
n[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 {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)} \, dx &=\frac {3 \operatorname {Subst}\left (\int \frac {x^4}{1-x^6} \, dx,x,\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{b}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{2}-\frac {x}{2}}{1-x+x^2} \, dx,x,\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{b}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{2}+\frac {x}{2}}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{b}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{b}-\frac {\operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{4 b}+\frac {\operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{4 b}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{4 b}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{4 b}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{b}-\frac {\log \left (1+\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{4 b}+\frac {\log \left (1+\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{4 b}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{2 b}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{2 b}\\ &=\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}}{\sqrt {3}}\right )}{2 b}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}}{\sqrt {3}}\right )}{2 b}+\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{b}-\frac {\log \left (1+\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}-\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{4 b}+\frac {\log \left (1+\frac {\cosh ^{\frac {2}{3}}(a+b x)}{\sinh ^{\frac {2}{3}}(a+b x)}+\frac {\sqrt [3]{\cosh (a+b x)}}{\sqrt [3]{\sinh (a+b x)}}\right )}{4 b}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 57, normalized size = 0.26 \[ \frac {3 \sqrt [3]{\sinh (a+b x)} \sqrt [6]{\cosh ^2(a+b x)} \, _2F_1\left (\frac {1}{6},\frac {1}{6};\frac {7}{6};-\sinh ^2(a+b x)\right )}{b \sqrt [3]{\cosh (a+b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(Cosh[a + b*x]^2)^(1/6)*Hypergeometric2F1[1/6, 1/6, 7/6, -Sinh[a + b*x]^2]*Sinh[a + b*x]^(1/3))/(b*Cosh[a +
b*x]^(1/3))

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fricas [B]  time = 0.47, size = 723, normalized size = 3.32 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*sqrt(3)*arctan(1/3*(sqrt(3)*cosh(b*x + a)^2 + 2*sqrt(3)*cosh(b*x + a)*sinh(b*x + a) + sqrt(3)*sinh(b*x
+ a)^2 + 4*(sqrt(3)*cosh(b*x + a) + sqrt(3)*sinh(b*x + a))*cosh(b*x + a)^(2/3)*sinh(b*x + a)^(1/3) + sqrt(3))/
(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 + 1)) + 2*sqrt(3)*arctan(-1/3*(sqrt(3)*cosh
(b*x + a)^2 + 2*sqrt(3)*cosh(b*x + a)*sinh(b*x + a) + sqrt(3)*sinh(b*x + a)^2 - 4*(sqrt(3)*cosh(b*x + a) + sqr
t(3)*sinh(b*x + a))*cosh(b*x + a)^(2/3)*sinh(b*x + a)^(1/3) + sqrt(3))/(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))*cosh(b*x + a)^(2/
3)*sinh(b*x + a)^(1/3) + 2*(cosh(b*x + a) + sinh(b*x + a))*cosh(b*x + a)^(1/3)*sinh(b*x + a)^(2/3) + 2*cosh(b*
x + a)*sinh(b*x + a) + sinh(b*x + a)^2 + 1)/(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2
+ 1)) + 2*log((cosh(b*x + a)^2 + 2*(cosh(b*x + a) + sinh(b*x + a))*cosh(b*x + a)^(2/3)*sinh(b*x + a)^(1/3) +
2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 + 1)/(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))*cosh(b*x + a)^(2/3)*sinh(b*x + a)^(
1/3) + 2*(cosh(b*x + a) + sinh(b*x + a))*cosh(b*x + a)^(1/3)*sinh(b*x + a)^(2/3) + 2*cosh(b*x + a)*sinh(b*x +
a) + sinh(b*x + a)^2 + 1)/(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 + 1)) - 2*log(-(c
osh(b*x + a)^2 - 2*(cosh(b*x + a) + sinh(b*x + a))*cosh(b*x + a)^(2/3)*sinh(b*x + a)^(1/3) + 2*cosh(b*x + a)*s
inh(b*x + a) + sinh(b*x + a)^2 + 1)/(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 + 1)))/
b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{\frac {2}{3}}\left (b x +a \right )}{\sinh \left (b x +a \right )^{\frac {2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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