3.447 \(\int \frac{\tanh ^5(e+f x)}{(a+a \sinh ^2(e+f x))^{3/2}} \, dx\)
Optimal. Leaf size=68 \[ -\frac{a^2}{7 f \left (a \cosh ^2(e+f x)\right )^{7/2}}+\frac{2 a}{5 f \left (a \cosh ^2(e+f x)\right )^{5/2}}-\frac{1}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \]
[Out]
-a^2/(7*f*(a*Cosh[e + f*x]^2)^(7/2)) + (2*a)/(5*f*(a*Cosh[e + f*x]^2)^(5/2)) - 1/(3*f*(a*Cosh[e + f*x]^2)^(3/2
))
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Rubi [A] time = 0.141147, antiderivative size = 68, 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, 3205, 16, 43} \[ -\frac{a^2}{7 f \left (a \cosh ^2(e+f x)\right )^{7/2}}+\frac{2 a}{5 f \left (a \cosh ^2(e+f x)\right )^{5/2}}-\frac{1}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[Tanh[e + f*x]^5/(a + a*Sinh[e + f*x]^2)^(3/2),x]
[Out]
-a^2/(7*f*(a*Cosh[e + f*x]^2)^(7/2)) + (2*a)/(5*f*(a*Cosh[e + f*x]^2)^(5/2)) - 1/(3*f*(a*Cosh[e + f*x]^2)^(3/2
))
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 3205
Int[((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = FreeFact
ors[Sin[e + f*x]^2, x]}, Dist[ff^((m + 1)/2)/(2*f), Subst[Int[(x^((m - 1)/2)*(b*ff^(n/2)*x^(n/2))^p)/(1 - ff*x
)^((m + 1)/2), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2
]
Rule 16
Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]
Rule 43
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Rubi steps
\begin{align*} \int \frac{\tanh ^5(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx &=\int \frac{\tanh ^5(e+f x)}{\left (a \cosh ^2(e+f x)\right )^{3/2}} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{(1-x)^2}{x^3 (a x)^{3/2}} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \frac{(1-x)^2}{(a x)^{9/2}} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \left (\frac{1}{(a x)^{9/2}}-\frac{2}{a (a x)^{7/2}}+\frac{1}{a^2 (a x)^{5/2}}\right ) \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=-\frac{a^2}{7 f \left (a \cosh ^2(e+f x)\right )^{7/2}}+\frac{2 a}{5 f \left (a \cosh ^2(e+f x)\right )^{5/2}}-\frac{1}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.112383, size = 51, normalized size = 0.75 \[ \frac{\left (-35 \cosh ^4(e+f x)+42 \cosh ^2(e+f x)-15\right ) \text{sech}^4(e+f x)}{105 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[Tanh[e + f*x]^5/(a + a*Sinh[e + f*x]^2)^(3/2),x]
[Out]
((-15 + 42*Cosh[e + f*x]^2 - 35*Cosh[e + f*x]^4)*Sech[e + f*x]^4)/(105*f*(a*Cosh[e + f*x]^2)^(3/2))
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Maple [C] time = 0.107, size = 44, normalized size = 0.7 \begin{align*}{\frac{1}{f}\mbox{{\tt ` int/indef0`}} \left ({\frac{ \left ( \sinh \left ( fx+e \right ) \right ) ^{5}}{ \left ( \cosh \left ( fx+e \right ) \right ) ^{8}a}{\frac{1}{\sqrt{a \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}}}},\sinh \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tanh(f*x+e)^5/(a+a*sinh(f*x+e)^2)^(3/2),x)
[Out]
`int/indef0`(sinh(f*x+e)^5/cosh(f*x+e)^8/a/(a*cosh(f*x+e)^2)^(1/2),sinh(f*x+e))/f
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Maxima [B] time = 2.21789, size = 791, normalized size = 11.63 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(f*x+e)^5/(a+a*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")
[Out]
-8/3*e^(-3*f*x - 3*e)/((7*a^(3/2)*e^(-2*f*x - 2*e) + 21*a^(3/2)*e^(-4*f*x - 4*e) + 35*a^(3/2)*e^(-6*f*x - 6*e)
+ 35*a^(3/2)*e^(-8*f*x - 8*e) + 21*a^(3/2)*e^(-10*f*x - 10*e) + 7*a^(3/2)*e^(-12*f*x - 12*e) + a^(3/2)*e^(-14
*f*x - 14*e) + a^(3/2))*f) + 32/15*e^(-5*f*x - 5*e)/((7*a^(3/2)*e^(-2*f*x - 2*e) + 21*a^(3/2)*e^(-4*f*x - 4*e)
+ 35*a^(3/2)*e^(-6*f*x - 6*e) + 35*a^(3/2)*e^(-8*f*x - 8*e) + 21*a^(3/2)*e^(-10*f*x - 10*e) + 7*a^(3/2)*e^(-1
2*f*x - 12*e) + a^(3/2)*e^(-14*f*x - 14*e) + a^(3/2))*f) - 304/35*e^(-7*f*x - 7*e)/((7*a^(3/2)*e^(-2*f*x - 2*e
) + 21*a^(3/2)*e^(-4*f*x - 4*e) + 35*a^(3/2)*e^(-6*f*x - 6*e) + 35*a^(3/2)*e^(-8*f*x - 8*e) + 21*a^(3/2)*e^(-1
0*f*x - 10*e) + 7*a^(3/2)*e^(-12*f*x - 12*e) + a^(3/2)*e^(-14*f*x - 14*e) + a^(3/2))*f) + 32/15*e^(-9*f*x - 9*
e)/((7*a^(3/2)*e^(-2*f*x - 2*e) + 21*a^(3/2)*e^(-4*f*x - 4*e) + 35*a^(3/2)*e^(-6*f*x - 6*e) + 35*a^(3/2)*e^(-8
*f*x - 8*e) + 21*a^(3/2)*e^(-10*f*x - 10*e) + 7*a^(3/2)*e^(-12*f*x - 12*e) + a^(3/2)*e^(-14*f*x - 14*e) + a^(3
/2))*f) - 8/3*e^(-11*f*x - 11*e)/((7*a^(3/2)*e^(-2*f*x - 2*e) + 21*a^(3/2)*e^(-4*f*x - 4*e) + 35*a^(3/2)*e^(-6
*f*x - 6*e) + 35*a^(3/2)*e^(-8*f*x - 8*e) + 21*a^(3/2)*e^(-10*f*x - 10*e) + 7*a^(3/2)*e^(-12*f*x - 12*e) + a^(
3/2)*e^(-14*f*x - 14*e) + a^(3/2))*f)
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Fricas [B] time = 2.14094, size = 6649, normalized size = 97.78 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(f*x+e)^5/(a+a*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")
[Out]
-8/105*(385*cosh(f*x + e)*e^(f*x + e)*sinh(f*x + e)^10 + 35*e^(f*x + e)*sinh(f*x + e)^11 + 7*(275*cosh(f*x + e
)^2 - 4)*e^(f*x + e)*sinh(f*x + e)^9 + 21*(275*cosh(f*x + e)^3 - 12*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^8
+ 6*(1925*cosh(f*x + e)^4 - 168*cosh(f*x + e)^2 + 19)*e^(f*x + e)*sinh(f*x + e)^7 + 42*(385*cosh(f*x + e)^5 -
56*cosh(f*x + e)^3 + 19*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^6 + 14*(1155*cosh(f*x + e)^6 - 252*cosh(f*x
+ e)^4 + 171*cosh(f*x + e)^2 - 2)*e^(f*x + e)*sinh(f*x + e)^5 + 14*(825*cosh(f*x + e)^7 - 252*cosh(f*x + e)^5
+ 285*cosh(f*x + e)^3 - 10*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^4 + 7*(825*cosh(f*x + e)^8 - 336*cosh(f*x
+ e)^6 + 570*cosh(f*x + e)^4 - 40*cosh(f*x + e)^2 + 5)*e^(f*x + e)*sinh(f*x + e)^3 + 7*(275*cosh(f*x + e)^9 -
144*cosh(f*x + e)^7 + 342*cosh(f*x + e)^5 - 40*cosh(f*x + e)^3 + 15*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^2
+ 7*(55*cosh(f*x + e)^10 - 36*cosh(f*x + e)^8 + 114*cosh(f*x + e)^6 - 20*cosh(f*x + e)^4 + 15*cosh(f*x + e)^2
)*e^(f*x + e)*sinh(f*x + e) + (35*cosh(f*x + e)^11 - 28*cosh(f*x + e)^9 + 114*cosh(f*x + e)^7 - 28*cosh(f*x +
e)^5 + 35*cosh(f*x + e)^3)*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)/(a^2*f*
cosh(f*x + e)^14 + 7*a^2*f*cosh(f*x + e)^12 + (a^2*f*e^(2*f*x + 2*e) + a^2*f)*sinh(f*x + e)^14 + 14*(a^2*f*cos
h(f*x + e)*e^(2*f*x + 2*e) + a^2*f*cosh(f*x + e))*sinh(f*x + e)^13 + 21*a^2*f*cosh(f*x + e)^10 + 7*(13*a^2*f*c
osh(f*x + e)^2 + a^2*f + (13*a^2*f*cosh(f*x + e)^2 + a^2*f)*e^(2*f*x + 2*e))*sinh(f*x + e)^12 + 28*(13*a^2*f*c
osh(f*x + e)^3 + 3*a^2*f*cosh(f*x + e) + (13*a^2*f*cosh(f*x + e)^3 + 3*a^2*f*cosh(f*x + e))*e^(2*f*x + 2*e))*s
inh(f*x + e)^11 + 35*a^2*f*cosh(f*x + e)^8 + 7*(143*a^2*f*cosh(f*x + e)^4 + 66*a^2*f*cosh(f*x + e)^2 + 3*a^2*f
+ (143*a^2*f*cosh(f*x + e)^4 + 66*a^2*f*cosh(f*x + e)^2 + 3*a^2*f)*e^(2*f*x + 2*e))*sinh(f*x + e)^10 + 14*(14
3*a^2*f*cosh(f*x + e)^5 + 110*a^2*f*cosh(f*x + e)^3 + 15*a^2*f*cosh(f*x + e) + (143*a^2*f*cosh(f*x + e)^5 + 11
0*a^2*f*cosh(f*x + e)^3 + 15*a^2*f*cosh(f*x + e))*e^(2*f*x + 2*e))*sinh(f*x + e)^9 + 35*a^2*f*cosh(f*x + e)^6
+ 7*(429*a^2*f*cosh(f*x + e)^6 + 495*a^2*f*cosh(f*x + e)^4 + 135*a^2*f*cosh(f*x + e)^2 + 5*a^2*f + (429*a^2*f*
cosh(f*x + e)^6 + 495*a^2*f*cosh(f*x + e)^4 + 135*a^2*f*cosh(f*x + e)^2 + 5*a^2*f)*e^(2*f*x + 2*e))*sinh(f*x +
e)^8 + 8*(429*a^2*f*cosh(f*x + e)^7 + 693*a^2*f*cosh(f*x + e)^5 + 315*a^2*f*cosh(f*x + e)^3 + 35*a^2*f*cosh(f
*x + e) + (429*a^2*f*cosh(f*x + e)^7 + 693*a^2*f*cosh(f*x + e)^5 + 315*a^2*f*cosh(f*x + e)^3 + 35*a^2*f*cosh(f
*x + e))*e^(2*f*x + 2*e))*sinh(f*x + e)^7 + 21*a^2*f*cosh(f*x + e)^4 + 7*(429*a^2*f*cosh(f*x + e)^8 + 924*a^2*
f*cosh(f*x + e)^6 + 630*a^2*f*cosh(f*x + e)^4 + 140*a^2*f*cosh(f*x + e)^2 + 5*a^2*f + (429*a^2*f*cosh(f*x + e)
^8 + 924*a^2*f*cosh(f*x + e)^6 + 630*a^2*f*cosh(f*x + e)^4 + 140*a^2*f*cosh(f*x + e)^2 + 5*a^2*f)*e^(2*f*x + 2
*e))*sinh(f*x + e)^6 + 14*(143*a^2*f*cosh(f*x + e)^9 + 396*a^2*f*cosh(f*x + e)^7 + 378*a^2*f*cosh(f*x + e)^5 +
140*a^2*f*cosh(f*x + e)^3 + 15*a^2*f*cosh(f*x + e) + (143*a^2*f*cosh(f*x + e)^9 + 396*a^2*f*cosh(f*x + e)^7 +
378*a^2*f*cosh(f*x + e)^5 + 140*a^2*f*cosh(f*x + e)^3 + 15*a^2*f*cosh(f*x + e))*e^(2*f*x + 2*e))*sinh(f*x + e
)^5 + 7*a^2*f*cosh(f*x + e)^2 + 7*(143*a^2*f*cosh(f*x + e)^10 + 495*a^2*f*cosh(f*x + e)^8 + 630*a^2*f*cosh(f*x
+ e)^6 + 350*a^2*f*cosh(f*x + e)^4 + 75*a^2*f*cosh(f*x + e)^2 + 3*a^2*f + (143*a^2*f*cosh(f*x + e)^10 + 495*a
^2*f*cosh(f*x + e)^8 + 630*a^2*f*cosh(f*x + e)^6 + 350*a^2*f*cosh(f*x + e)^4 + 75*a^2*f*cosh(f*x + e)^2 + 3*a^
2*f)*e^(2*f*x + 2*e))*sinh(f*x + e)^4 + 28*(13*a^2*f*cosh(f*x + e)^11 + 55*a^2*f*cosh(f*x + e)^9 + 90*a^2*f*co
sh(f*x + e)^7 + 70*a^2*f*cosh(f*x + e)^5 + 25*a^2*f*cosh(f*x + e)^3 + 3*a^2*f*cosh(f*x + e) + (13*a^2*f*cosh(f
*x + e)^11 + 55*a^2*f*cosh(f*x + e)^9 + 90*a^2*f*cosh(f*x + e)^7 + 70*a^2*f*cosh(f*x + e)^5 + 25*a^2*f*cosh(f*
x + e)^3 + 3*a^2*f*cosh(f*x + e))*e^(2*f*x + 2*e))*sinh(f*x + e)^3 + a^2*f + 7*(13*a^2*f*cosh(f*x + e)^12 + 66
*a^2*f*cosh(f*x + e)^10 + 135*a^2*f*cosh(f*x + e)^8 + 140*a^2*f*cosh(f*x + e)^6 + 75*a^2*f*cosh(f*x + e)^4 + 1
8*a^2*f*cosh(f*x + e)^2 + a^2*f + (13*a^2*f*cosh(f*x + e)^12 + 66*a^2*f*cosh(f*x + e)^10 + 135*a^2*f*cosh(f*x
+ e)^8 + 140*a^2*f*cosh(f*x + e)^6 + 75*a^2*f*cosh(f*x + e)^4 + 18*a^2*f*cosh(f*x + e)^2 + a^2*f)*e^(2*f*x + 2
*e))*sinh(f*x + e)^2 + (a^2*f*cosh(f*x + e)^14 + 7*a^2*f*cosh(f*x + e)^12 + 21*a^2*f*cosh(f*x + e)^10 + 35*a^2
*f*cosh(f*x + e)^8 + 35*a^2*f*cosh(f*x + e)^6 + 21*a^2*f*cosh(f*x + e)^4 + 7*a^2*f*cosh(f*x + e)^2 + a^2*f)*e^
(2*f*x + 2*e) + 14*(a^2*f*cosh(f*x + e)^13 + 6*a^2*f*cosh(f*x + e)^11 + 15*a^2*f*cosh(f*x + e)^9 + 20*a^2*f*co
sh(f*x + e)^7 + 15*a^2*f*cosh(f*x + e)^5 + 6*a^2*f*cosh(f*x + e)^3 + a^2*f*cosh(f*x + e) + (a^2*f*cosh(f*x + e
)^13 + 6*a^2*f*cosh(f*x + e)^11 + 15*a^2*f*cosh(f*x + e)^9 + 20*a^2*f*cosh(f*x + e)^7 + 15*a^2*f*cosh(f*x + e)
^5 + 6*a^2*f*cosh(f*x + e)^3 + a^2*f*cosh(f*x + e))*e^(2*f*x + 2*e))*sinh(f*x + e))
________________________________________________________________________________________
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(tanh(f*x+e)**5/(a+a*sinh(f*x+e)**2)**(3/2),x)
[Out]
Timed out
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Giac [A] time = 1.73257, size = 132, normalized size = 1.94 \begin{align*} -\frac{8 \,{\left (35 \, \sqrt{a} e^{\left (11 \, f x + 11 \, e\right )} - 28 \, \sqrt{a} e^{\left (9 \, f x + 9 \, e\right )} + 114 \, \sqrt{a} e^{\left (7 \, f x + 7 \, e\right )} - 28 \, \sqrt{a} e^{\left (5 \, f x + 5 \, e\right )} + 35 \, \sqrt{a} e^{\left (3 \, f x + 3 \, e\right )}\right )}}{105 \, a^{2} f{\left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tanh(f*x+e)^5/(a+a*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")
[Out]
-8/105*(35*sqrt(a)*e^(11*f*x + 11*e) - 28*sqrt(a)*e^(9*f*x + 9*e) + 114*sqrt(a)*e^(7*f*x + 7*e) - 28*sqrt(a)*e
^(5*f*x + 5*e) + 35*sqrt(a)*e^(3*f*x + 3*e))/(a^2*f*(e^(2*f*x + 2*e) + 1)^7)