3.5.47 \(\int \frac {\tanh ^5(e+f x)}{(a+a \sinh ^2(e+f x))^{3/2}} \, dx\) [447]
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]
-1/7*a^2/f/(a*cosh(f*x+e)^2)^(7/2)+2/5*a/f/(a*cosh(f*x+e)^2)^(5/2)-1/3/f/(a*cosh(f*x+e)^2)^(3/2)
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Rubi [A]
time = 0.10, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3255, 3284, 16,
45} \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
[In]
Int[Tanh[e + f*x]^5/(a + a*Sinh[e + f*x]^2)^(3/2),x]
[Out]
-1/7*a^2/(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 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 45
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])
Rule 3255
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 3284
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
]
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 {\text {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 \text {Subst}\left (\int \frac {(1-x)^2}{(a x)^{9/2}} \, dx,x,\cosh ^2(e+f x)\right )}{2 f}\\ &=\frac {a^3 \text {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*}
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Mathematica [A]
time = 0.08, size = 51, normalized size = 0.75 \begin {gather*} \frac {\left (-15+42 \cosh ^2(e+f x)-35 \cosh ^4(e+f x)\right ) \text {sech}^4(e+f x)}{105 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \end {gather*}
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] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.25, size = 44, normalized size = 0.65
| | |
method |
result |
size |
| | |
default |
\(\frac {\mathit {`\,int/indef0`\,}\left (\frac {\sinh ^{5}\left (f x +e \right )}{\cosh \left (f x +e \right )^{8} a \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}}, \sinh \left (f x +e \right )\right )}{f}\) |
\(44\) |
risch |
\(-\frac {8 \left (35 \,{\mathrm e}^{8 f x +8 e}-28 \,{\mathrm e}^{6 f x +6 e}+114 \,{\mathrm e}^{4 f x +4 e}-28 \,{\mathrm e}^{2 f x +2 e}+35\right ) {\mathrm e}^{2 f x +2 e}}{105 f \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, \left ({\mathrm e}^{2 f x +2 e}+1\right )^{6} a}\) |
\(103\) |
| | |
|
|
|
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,method=_RETURNVERBOSE)
[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] Leaf count of result is larger than twice the leaf count of optimal. 626 vs.
\(2 (59) = 118\).
time = 0.57, size = 626, normalized size = 9.21 \begin {gather*} -\frac {8 \, e^{\left (-3 \, f x - 3 \, e\right )}}{3 \, {\left (7 \, a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 21 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} + 35 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} + 35 \, a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + 21 \, a^{\frac {3}{2}} e^{\left (-10 \, f x - 10 \, e\right )} + 7 \, a^{\frac {3}{2}} e^{\left (-12 \, f x - 12 \, e\right )} + a^{\frac {3}{2}} e^{\left (-14 \, f x - 14 \, e\right )} + a^{\frac {3}{2}}\right )} f} + \frac {32 \, e^{\left (-5 \, f x - 5 \, e\right )}}{15 \, {\left (7 \, a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 21 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} + 35 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} + 35 \, a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + 21 \, a^{\frac {3}{2}} e^{\left (-10 \, f x - 10 \, e\right )} + 7 \, a^{\frac {3}{2}} e^{\left (-12 \, f x - 12 \, e\right )} + a^{\frac {3}{2}} e^{\left (-14 \, f x - 14 \, e\right )} + a^{\frac {3}{2}}\right )} f} - \frac {304 \, e^{\left (-7 \, f x - 7 \, e\right )}}{35 \, {\left (7 \, a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 21 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} + 35 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} + 35 \, a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + 21 \, a^{\frac {3}{2}} e^{\left (-10 \, f x - 10 \, e\right )} + 7 \, a^{\frac {3}{2}} e^{\left (-12 \, f x - 12 \, e\right )} + a^{\frac {3}{2}} e^{\left (-14 \, f x - 14 \, e\right )} + a^{\frac {3}{2}}\right )} f} + \frac {32 \, e^{\left (-9 \, f x - 9 \, e\right )}}{15 \, {\left (7 \, a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 21 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} + 35 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} + 35 \, a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + 21 \, a^{\frac {3}{2}} e^{\left (-10 \, f x - 10 \, e\right )} + 7 \, a^{\frac {3}{2}} e^{\left (-12 \, f x - 12 \, e\right )} + a^{\frac {3}{2}} e^{\left (-14 \, f x - 14 \, e\right )} + a^{\frac {3}{2}}\right )} f} - \frac {8 \, e^{\left (-11 \, f x - 11 \, e\right )}}{3 \, {\left (7 \, a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 21 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} + 35 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} + 35 \, a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + 21 \, a^{\frac {3}{2}} e^{\left (-10 \, f x - 10 \, e\right )} + 7 \, a^{\frac {3}{2}} e^{\left (-12 \, f x - 12 \, e\right )} + a^{\frac {3}{2}} e^{\left (-14 \, f x - 14 \, e\right )} + a^{\frac {3}{2}}\right )} f} \end {gather*}
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] Leaf count of result is larger than twice the leaf count of optimal. 2507 vs.
\(2 (56) = 112\).
time = 0.45, size = 2507, normalized size = 36.87 \begin {gather*} \text {Too large to display} \end {gather*}
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))
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh ^{5}{\left (e + f x \right )}}{\left (a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
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]
Integral(tanh(e + f*x)**5/(a*(sinh(e + f*x)**2 + 1))**(3/2), x)
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
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Mupad [B]
time = 0.16, size = 457, normalized size = 6.72 \begin {gather*} \frac {464\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{15\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^3\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {16\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{3\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^2\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {3072\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{35\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^4\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}+\frac {4736\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{35\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^5\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {768\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{7\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^6\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}+\frac {256\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{7\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^7\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tanh(e + f*x)^5/(a + a*sinh(e + f*x)^2)^(3/2),x)
[Out]
(464*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(15*a^2*f*(exp(2*e + 2*f*x) + 1)^3*
(exp(e + f*x) + exp(3*e + 3*f*x))) - (16*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))
/(3*a^2*f*(exp(2*e + 2*f*x) + 1)^2*(exp(e + f*x) + exp(3*e + 3*f*x))) - (3072*exp(3*e + 3*f*x)*(a + a*(exp(e +
f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(35*a^2*f*(exp(2*e + 2*f*x) + 1)^4*(exp(e + f*x) + exp(3*e + 3*f*x))) +
(4736*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(35*a^2*f*(exp(2*e + 2*f*x) + 1)^5
*(exp(e + f*x) + exp(3*e + 3*f*x))) - (768*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2
))/(7*a^2*f*(exp(2*e + 2*f*x) + 1)^6*(exp(e + f*x) + exp(3*e + 3*f*x))) + (256*exp(3*e + 3*f*x)*(a + a*(exp(e
+ f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(7*a^2*f*(exp(2*e + 2*f*x) + 1)^7*(exp(e + f*x) + exp(3*e + 3*f*x)))
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