3.436 \(\int \coth ^6(e+f x) \sqrt{a+a \sinh ^2(e+f x)} \, dx\)
Optimal. Leaf size=124 \[ \frac{\tanh (e+f x) \sqrt{a \cosh ^2(e+f x)}}{f}-\frac{\text{csch}^5(e+f x) \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)}}{5 f}-\frac{\text{csch}^3(e+f x) \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)}}{f}-\frac{3 \text{csch}(e+f x) \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)}}{f} \]
[Out]
(-3*Sqrt[a*Cosh[e + f*x]^2]*Csch[e + f*x]*Sech[e + f*x])/f - (Sqrt[a*Cosh[e + f*x]^2]*Csch[e + f*x]^3*Sech[e +
f*x])/f - (Sqrt[a*Cosh[e + f*x]^2]*Csch[e + f*x]^5*Sech[e + f*x])/(5*f) + (Sqrt[a*Cosh[e + f*x]^2]*Tanh[e + f
*x])/f
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Rubi [A] time = 0.124526, antiderivative size = 124, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used =
{3176, 3207, 2590, 270} \[ \frac{\tanh (e+f x) \sqrt{a \cosh ^2(e+f x)}}{f}-\frac{\text{csch}^5(e+f x) \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)}}{5 f}-\frac{\text{csch}^3(e+f x) \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)}}{f}-\frac{3 \text{csch}(e+f x) \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)}}{f} \]
Antiderivative was successfully verified.
[In]
Int[Coth[e + f*x]^6*Sqrt[a + a*Sinh[e + f*x]^2],x]
[Out]
(-3*Sqrt[a*Cosh[e + f*x]^2]*Csch[e + f*x]*Sech[e + f*x])/f - (Sqrt[a*Cosh[e + f*x]^2]*Csch[e + f*x]^3*Sech[e +
f*x])/f - (Sqrt[a*Cosh[e + f*x]^2]*Csch[e + f*x]^5*Sech[e + f*x])/(5*f) + (Sqrt[a*Cosh[e + f*x]^2]*Tanh[e + f
*x])/f
Rule 3176
Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*cos[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]
Rule 3207
Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Sin[e + f*x]^n)^FracPart[p])/(Sin[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])
Rule 2590
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[f^(-1), Subst[Int[(1 - x^2
)^((m + n - 1)/2)/x^n, x], x, Cos[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]
Rule 270
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]
Rubi steps
\begin{align*} \int \coth ^6(e+f x) \sqrt{a+a \sinh ^2(e+f x)} \, dx &=\int \sqrt{a \cosh ^2(e+f x)} \coth ^6(e+f x) \, dx\\ &=\left (\sqrt{a \cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \int \cosh (e+f x) \coth ^6(e+f x) \, dx\\ &=-\frac{\left (i \sqrt{a \cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^3}{x^6} \, dx,x,-i \sinh (e+f x)\right )}{f}\\ &=-\frac{\left (i \sqrt{a \cosh ^2(e+f x)} \text{sech}(e+f x)\right ) \operatorname{Subst}\left (\int \left (-1+\frac{1}{x^6}-\frac{3}{x^4}+\frac{3}{x^2}\right ) \, dx,x,-i \sinh (e+f x)\right )}{f}\\ &=-\frac{3 \sqrt{a \cosh ^2(e+f x)} \text{csch}(e+f x) \text{sech}(e+f x)}{f}-\frac{\sqrt{a \cosh ^2(e+f x)} \text{csch}^3(e+f x) \text{sech}(e+f x)}{f}-\frac{\sqrt{a \cosh ^2(e+f x)} \text{csch}^5(e+f x) \text{sech}(e+f x)}{5 f}+\frac{\sqrt{a \cosh ^2(e+f x)} \tanh (e+f x)}{f}\\ \end{align*}
Mathematica [A] time = 0.193013, size = 67, normalized size = 0.54 \[ \frac{(235 \cosh (2 (e+f x))-90 \cosh (4 (e+f x))+5 \cosh (6 (e+f x))-182) \text{csch}^5(e+f x) \text{sech}(e+f x) \sqrt{a \cosh ^2(e+f x)}}{160 f} \]
Antiderivative was successfully verified.
[In]
Integrate[Coth[e + f*x]^6*Sqrt[a + a*Sinh[e + f*x]^2],x]
[Out]
(Sqrt[a*Cosh[e + f*x]^2]*(-182 + 235*Cosh[2*(e + f*x)] - 90*Cosh[4*(e + f*x)] + 5*Cosh[6*(e + f*x)])*Csch[e +
f*x]^5*Sech[e + f*x])/(160*f)
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Maple [A] time = 0.092, size = 65, normalized size = 0.5 \begin{align*}{\frac{\cosh \left ( fx+e \right ) a \left ( 5\, \left ( \sinh \left ( fx+e \right ) \right ) ^{6}-15\, \left ( \sinh \left ( fx+e \right ) \right ) ^{4}-5\, \left ( \sinh \left ( fx+e \right ) \right ) ^{2}-1 \right ) }{5\, \left ( \sinh \left ( fx+e \right ) \right ) ^{5}f}{\frac{1}{\sqrt{a \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(f*x+e)^6*(a+a*sinh(f*x+e)^2)^(1/2),x)
[Out]
1/5*cosh(f*x+e)*a*(5*sinh(f*x+e)^6-15*sinh(f*x+e)^4-5*sinh(f*x+e)^2-1)/sinh(f*x+e)^5/(a*cosh(f*x+e)^2)^(1/2)/f
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Maxima [B] time = 1.78392, size = 1418, normalized size = 11.44 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(f*x+e)^6*(a+a*sinh(f*x+e)^2)^(1/2),x, algorithm="maxima")
[Out]
-1/320*(105*sqrt(a)*log(e^(-f*x - e) + 1) - 105*sqrt(a)*log(e^(-f*x - e) - 1) - 2*(375*sqrt(a)*e^(-f*x - e) -
790*sqrt(a)*e^(-3*f*x - 3*e) + 896*sqrt(a)*e^(-5*f*x - 5*e) - 490*sqrt(a)*e^(-7*f*x - 7*e) + 105*sqrt(a)*e^(-9
*f*x - 9*e))/(5*e^(-2*f*x - 2*e) - 10*e^(-4*f*x - 4*e) + 10*e^(-6*f*x - 6*e) - 5*e^(-8*f*x - 8*e) + e^(-10*f*x
- 10*e) - 1))/f + 1/320*(105*sqrt(a)*log(e^(-f*x - e) + 1) - 105*sqrt(a)*log(e^(-f*x - e) - 1) + 2*(105*sqrt(
a)*e^(-f*x - e) - 490*sqrt(a)*e^(-3*f*x - 3*e) + 896*sqrt(a)*e^(-5*f*x - 5*e) - 790*sqrt(a)*e^(-7*f*x - 7*e) +
375*sqrt(a)*e^(-9*f*x - 9*e))/(5*e^(-2*f*x - 2*e) - 10*e^(-4*f*x - 4*e) + 10*e^(-6*f*x - 6*e) - 5*e^(-8*f*x -
8*e) + e^(-10*f*x - 10*e) - 1))/f + 1/256*(15*sqrt(a)*log(e^(-f*x - e) + 1) - 15*sqrt(a)*log(e^(-f*x - e) - 1
) + 2*(15*sqrt(a)*e^(-f*x - e) + 250*sqrt(a)*e^(-3*f*x - 3*e) - 128*sqrt(a)*e^(-5*f*x - 5*e) + 70*sqrt(a)*e^(-
7*f*x - 7*e) - 15*sqrt(a)*e^(-9*f*x - 9*e))/(5*e^(-2*f*x - 2*e) - 10*e^(-4*f*x - 4*e) + 10*e^(-6*f*x - 6*e) -
5*e^(-8*f*x - 8*e) + e^(-10*f*x - 10*e) - 1))/f - 1/256*(15*sqrt(a)*log(e^(-f*x - e) + 1) - 15*sqrt(a)*log(e^(
-f*x - e) - 1) + 2*(15*sqrt(a)*e^(-f*x - e) - 70*sqrt(a)*e^(-3*f*x - 3*e) + 128*sqrt(a)*e^(-5*f*x - 5*e) - 250
*sqrt(a)*e^(-7*f*x - 7*e) - 15*sqrt(a)*e^(-9*f*x - 9*e))/(5*e^(-2*f*x - 2*e) - 10*e^(-4*f*x - 4*e) + 10*e^(-6*
f*x - 6*e) - 5*e^(-8*f*x - 8*e) + e^(-10*f*x - 10*e) - 1))/f + 2*sqrt(a)*e^(-5*f*x - 5*e)/(f*(5*e^(-2*f*x - 2*
e) - 10*e^(-4*f*x - 4*e) + 10*e^(-6*f*x - 6*e) - 5*e^(-8*f*x - 8*e) + e^(-10*f*x - 10*e) - 1)) - 1/640*(2895*s
qrt(a)*e^(-2*f*x - 2*e) - 7110*sqrt(a)*e^(-4*f*x - 4*e) + 8064*sqrt(a)*e^(-6*f*x - 6*e) - 4410*sqrt(a)*e^(-8*f
*x - 8*e) + 945*sqrt(a)*e^(-10*f*x - 10*e) - 320*sqrt(a))/(f*(e^(-f*x - e) - 5*e^(-3*f*x - 3*e) + 10*e^(-5*f*x
- 5*e) - 10*e^(-7*f*x - 7*e) + 5*e^(-9*f*x - 9*e) - e^(-11*f*x - 11*e))) + 1/640*(945*sqrt(a)*e^(-f*x - e) -
4410*sqrt(a)*e^(-3*f*x - 3*e) + 8064*sqrt(a)*e^(-5*f*x - 5*e) - 7110*sqrt(a)*e^(-7*f*x - 7*e) + 2895*sqrt(a)*e
^(-9*f*x - 9*e) - 320*sqrt(a)*e^(-11*f*x - 11*e))/(f*(5*e^(-2*f*x - 2*e) - 10*e^(-4*f*x - 4*e) + 10*e^(-6*f*x
- 6*e) - 5*e^(-8*f*x - 8*e) + e^(-10*f*x - 10*e) - 1))
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Fricas [B] time = 2.11362, size = 4725, normalized size = 38.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(f*x+e)^6*(a+a*sinh(f*x+e)^2)^(1/2),x, algorithm="fricas")
[Out]
1/10*(60*cosh(f*x + e)*e^(f*x + e)*sinh(f*x + e)^11 + 5*e^(f*x + e)*sinh(f*x + e)^12 + 30*(11*cosh(f*x + e)^2
- 3)*e^(f*x + e)*sinh(f*x + e)^10 + 100*(11*cosh(f*x + e)^3 - 9*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^9 + 5
*(495*cosh(f*x + e)^4 - 810*cosh(f*x + e)^2 + 47)*e^(f*x + e)*sinh(f*x + e)^8 + 40*(99*cosh(f*x + e)^5 - 270*c
osh(f*x + e)^3 + 47*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^7 + 28*(165*cosh(f*x + e)^6 - 675*cosh(f*x + e)^4
+ 235*cosh(f*x + e)^2 - 13)*e^(f*x + e)*sinh(f*x + e)^6 + 8*(495*cosh(f*x + e)^7 - 2835*cosh(f*x + e)^5 + 164
5*cosh(f*x + e)^3 - 273*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^5 + 5*(495*cosh(f*x + e)^8 - 3780*cosh(f*x +
e)^6 + 3290*cosh(f*x + e)^4 - 1092*cosh(f*x + e)^2 + 47)*e^(f*x + e)*sinh(f*x + e)^4 + 20*(55*cosh(f*x + e)^9
- 540*cosh(f*x + e)^7 + 658*cosh(f*x + e)^5 - 364*cosh(f*x + e)^3 + 47*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e
)^3 + 10*(33*cosh(f*x + e)^10 - 405*cosh(f*x + e)^8 + 658*cosh(f*x + e)^6 - 546*cosh(f*x + e)^4 + 141*cosh(f*x
+ e)^2 - 9)*e^(f*x + e)*sinh(f*x + e)^2 + 4*(15*cosh(f*x + e)^11 - 225*cosh(f*x + e)^9 + 470*cosh(f*x + e)^7
- 546*cosh(f*x + e)^5 + 235*cosh(f*x + e)^3 - 45*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e) + (5*cosh(f*x + e)^1
2 - 90*cosh(f*x + e)^10 + 235*cosh(f*x + e)^8 - 364*cosh(f*x + e)^6 + 235*cosh(f*x + e)^4 - 90*cosh(f*x + e)^2
+ 5)*e^(f*x + e))*sqrt(a*e^(4*f*x + 4*e) + 2*a*e^(2*f*x + 2*e) + a)*e^(-f*x - e)/(f*cosh(f*x + e)^11 + (f*e^(
2*f*x + 2*e) + f)*sinh(f*x + e)^11 + 11*(f*cosh(f*x + e)*e^(2*f*x + 2*e) + f*cosh(f*x + e))*sinh(f*x + e)^10 -
5*f*cosh(f*x + e)^9 + 5*(11*f*cosh(f*x + e)^2 + (11*f*cosh(f*x + e)^2 - f)*e^(2*f*x + 2*e) - f)*sinh(f*x + e)
^9 + 15*(11*f*cosh(f*x + e)^3 - 3*f*cosh(f*x + e) + (11*f*cosh(f*x + e)^3 - 3*f*cosh(f*x + e))*e^(2*f*x + 2*e)
)*sinh(f*x + e)^8 + 10*f*cosh(f*x + e)^7 + 10*(33*f*cosh(f*x + e)^4 - 18*f*cosh(f*x + e)^2 + (33*f*cosh(f*x +
e)^4 - 18*f*cosh(f*x + e)^2 + f)*e^(2*f*x + 2*e) + f)*sinh(f*x + e)^7 + 14*(33*f*cosh(f*x + e)^5 - 30*f*cosh(f
*x + e)^3 + 5*f*cosh(f*x + e) + (33*f*cosh(f*x + e)^5 - 30*f*cosh(f*x + e)^3 + 5*f*cosh(f*x + e))*e^(2*f*x + 2
*e))*sinh(f*x + e)^6 - 10*f*cosh(f*x + e)^5 + 2*(231*f*cosh(f*x + e)^6 - 315*f*cosh(f*x + e)^4 + 105*f*cosh(f*
x + e)^2 + (231*f*cosh(f*x + e)^6 - 315*f*cosh(f*x + e)^4 + 105*f*cosh(f*x + e)^2 - 5*f)*e^(2*f*x + 2*e) - 5*f
)*sinh(f*x + e)^5 + 10*(33*f*cosh(f*x + e)^7 - 63*f*cosh(f*x + e)^5 + 35*f*cosh(f*x + e)^3 - 5*f*cosh(f*x + e)
+ (33*f*cosh(f*x + e)^7 - 63*f*cosh(f*x + e)^5 + 35*f*cosh(f*x + e)^3 - 5*f*cosh(f*x + e))*e^(2*f*x + 2*e))*s
inh(f*x + e)^4 + 5*f*cosh(f*x + e)^3 + 5*(33*f*cosh(f*x + e)^8 - 84*f*cosh(f*x + e)^6 + 70*f*cosh(f*x + e)^4 -
20*f*cosh(f*x + e)^2 + (33*f*cosh(f*x + e)^8 - 84*f*cosh(f*x + e)^6 + 70*f*cosh(f*x + e)^4 - 20*f*cosh(f*x +
e)^2 + f)*e^(2*f*x + 2*e) + f)*sinh(f*x + e)^3 + 5*(11*f*cosh(f*x + e)^9 - 36*f*cosh(f*x + e)^7 + 42*f*cosh(f*
x + e)^5 - 20*f*cosh(f*x + e)^3 + 3*f*cosh(f*x + e) + (11*f*cosh(f*x + e)^9 - 36*f*cosh(f*x + e)^7 + 42*f*cosh
(f*x + e)^5 - 20*f*cosh(f*x + e)^3 + 3*f*cosh(f*x + e))*e^(2*f*x + 2*e))*sinh(f*x + e)^2 - f*cosh(f*x + e) + (
f*cosh(f*x + e)^11 - 5*f*cosh(f*x + e)^9 + 10*f*cosh(f*x + e)^7 - 10*f*cosh(f*x + e)^5 + 5*f*cosh(f*x + e)^3 -
f*cosh(f*x + e))*e^(2*f*x + 2*e) + (11*f*cosh(f*x + e)^10 - 45*f*cosh(f*x + e)^8 + 70*f*cosh(f*x + e)^6 - 50*
f*cosh(f*x + e)^4 + 15*f*cosh(f*x + e)^2 + (11*f*cosh(f*x + e)^10 - 45*f*cosh(f*x + e)^8 + 70*f*cosh(f*x + e)^
6 - 50*f*cosh(f*x + e)^4 + 15*f*cosh(f*x + e)^2 - f)*e^(2*f*x + 2*e) - f)*sinh(f*x + e))
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(f*x+e)**6*(a+a*sinh(f*x+e)**2)**(1/2),x)
[Out]
Timed out
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Giac [A] time = 1.29864, size = 140, normalized size = 1.13 \begin{align*} -\frac{\sqrt{a}{\left (\frac{4 \,{\left (15 \, e^{\left (9 \, f x + 9 \, e\right )} - 40 \, e^{\left (7 \, f x + 7 \, e\right )} + 66 \, e^{\left (5 \, f x + 5 \, e\right )} - 40 \, e^{\left (3 \, f x + 3 \, e\right )} + 15 \, e^{\left (f x + e\right )}\right )}}{{\left (e^{\left (2 \, f x + 2 \, e\right )} - 1\right )}^{5}} - 5 \, e^{\left (f x + e\right )} + 5 \, e^{\left (-f x - e\right )}\right )}}{10 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(f*x+e)^6*(a+a*sinh(f*x+e)^2)^(1/2),x, algorithm="giac")
[Out]
-1/10*sqrt(a)*(4*(15*e^(9*f*x + 9*e) - 40*e^(7*f*x + 7*e) + 66*e^(5*f*x + 5*e) - 40*e^(3*f*x + 3*e) + 15*e^(f*
x + e))/(e^(2*f*x + 2*e) - 1)^5 - 5*e^(f*x + e) + 5*e^(-f*x - e))/f