3.7.23 \(\int \frac {1}{(a \text {sech}(x)+b \tanh (x))^5} \, dx\) [623]

Optimal. Leaf size=95 \[ \frac {\log (a+b \sinh (x))}{b^5}-\frac {\left (a^2+b^2\right )^2}{4 b^5 (a+b \sinh (x))^4}+\frac {4 a \left (a^2+b^2\right )}{3 b^5 (a+b \sinh (x))^3}-\frac {3 a^2+b^2}{b^5 (a+b \sinh (x))^2}+\frac {4 a}{b^5 (a+b \sinh (x))} \]

[Out]

ln(a+b*sinh(x))/b^5-1/4*(a^2+b^2)^2/b^5/(a+b*sinh(x))^4+4/3*a*(a^2+b^2)/b^5/(a+b*sinh(x))^3+(-3*a^2-b^2)/b^5/(
a+b*sinh(x))^2+4*a/b^5/(a+b*sinh(x))

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4476, 2747, 711} \begin {gather*} -\frac {\left (a^2+b^2\right )^2}{4 b^5 (a+b \sinh (x))^4}+\frac {4 a \left (a^2+b^2\right )}{3 b^5 (a+b \sinh (x))^3}-\frac {3 a^2+b^2}{b^5 (a+b \sinh (x))^2}+\frac {4 a}{b^5 (a+b \sinh (x))}+\frac {\log (a+b \sinh (x))}{b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*Sech[x] + b*Tanh[x])^(-5),x]

[Out]

Log[a + b*Sinh[x]]/b^5 - (a^2 + b^2)^2/(4*b^5*(a + b*Sinh[x])^4) + (4*a*(a^2 + b^2))/(3*b^5*(a + b*Sinh[x])^3)
- (3*a^2 + b^2)/(b^5*(a + b*Sinh[x])^2) + (4*a)/(b^5*(a + b*Sinh[x]))

Rule 711

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rule 2747

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 4476

Int[(u_.)*((b_.)*sec[(c_.) + (d_.)*(x_)]^(n_.) + (a_.)*tan[(c_.) + (d_.)*(x_)]^(n_.))^(p_), x_Symbol] :> Int[A
ctivateTrig[u]*Sec[c + d*x]^(n*p)*(b + a*Sin[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]

Rubi steps

\begin {align*} \int \frac {1}{(a \text {sech}(x)+b \tanh (x))^5} \, dx &=\int \frac {\cosh ^5(x)}{(a+b \sinh (x))^5} \, dx\\ &=\frac {\text {Subst}\left (\int \frac {\left (-b^2-x^2\right )^2}{(a+x)^5} \, dx,x,b \sinh (x)\right )}{b^5}\\ &=\frac {\text {Subst}\left (\int \left (\frac {\left (a^2+b^2\right )^2}{(a+x)^5}-\frac {4 a \left (a^2+b^2\right )}{(a+x)^4}+\frac {2 \left (3 a^2+b^2\right )}{(a+x)^3}-\frac {4 a}{(a+x)^2}+\frac {1}{a+x}\right ) \, dx,x,b \sinh (x)\right )}{b^5}\\ &=\frac {\log (a+b \sinh (x))}{b^5}-\frac {\left (a^2+b^2\right )^2}{4 b^5 (a+b \sinh (x))^4}+\frac {4 a \left (a^2+b^2\right )}{3 b^5 (a+b \sinh (x))^3}-\frac {3 a^2+b^2}{b^5 (a+b \sinh (x))^2}+\frac {4 a}{b^5 (a+b \sinh (x))}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.23, size = 140, normalized size = 1.47 \begin {gather*} \frac {25 a^4-2 a^2 b^2-3 b^4+12 a^4 \log (a+b \sinh (x))+8 a b \left (11 a^2-b^2+6 a^2 \log (a+b \sinh (x))\right ) \sinh (x)-12 b^2 \left (-9 a^2+b^2-6 a^2 \log (a+b \sinh (x))\right ) \sinh ^2(x)+48 a b^3 (1+\log (a+b \sinh (x))) \sinh ^3(x)+12 b^4 \log (a+b \sinh (x)) \sinh ^4(x)}{12 b^5 (a+b \sinh (x))^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*Sech[x] + b*Tanh[x])^(-5),x]

[Out]

(25*a^4 - 2*a^2*b^2 - 3*b^4 + 12*a^4*Log[a + b*Sinh[x]] + 8*a*b*(11*a^2 - b^2 + 6*a^2*Log[a + b*Sinh[x]])*Sinh
[x] - 12*b^2*(-9*a^2 + b^2 - 6*a^2*Log[a + b*Sinh[x]])*Sinh[x]^2 + 48*a*b^3*(1 + Log[a + b*Sinh[x]])*Sinh[x]^3
+ 12*b^4*Log[a + b*Sinh[x]]*Sinh[x]^4)/(12*b^5*(a + b*Sinh[x])^4)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(284\) vs. \(2(92)=184\).
time = 1.26, size = 285, normalized size = 3.00

method	result	size
risch	\(-\frac {x}{b^{5}}+\frac {4 \left (6 a \,b^{3} {\mathrm e}^{6 x}+27 a^{2} b^{2} {\mathrm e}^{5 x}-3 b^{4} {\mathrm e}^{5 x}+44 \,{\mathrm e}^{4 x} a^{3} b -22 a \,b^{3} {\mathrm e}^{4 x}+25 a^{4} {\mathrm e}^{3 x}-56 a^{2} b^{2} {\mathrm e}^{3 x}+3 b^{4} {\mathrm e}^{3 x}-44 \,{\mathrm e}^{2 x} a^{3} b +22 a \,b^{3} {\mathrm e}^{2 x}+27 a^{2} b^{2} {\mathrm e}^{x}-3 b^{4} {\mathrm e}^{x}-6 a \,b^{3}\right ) {\mathrm e}^{x}}{3 b^{5} \left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}-b \right )^{4}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right )}{b^{5}}\)	\(176\)
default	\(\frac {\frac {2 \left (\frac {\left (a^{4}-b^{4}\right ) b \left (\tanh ^{7}\left (\frac {x}{2}\right )\right )}{a}-\frac {b^{2} \left (7 a^{4}-3 b^{4}\right ) \left (\tanh ^{6}\left (\frac {x}{2}\right )\right )}{a^{2}}-\frac {b \left (9 a^{6}-52 a^{4} b^{2}-a^{2} b^{4}+12 b^{6}\right ) \left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{3 a^{3}}+\frac {2 b^{2} \left (21 a^{6}-25 a^{4} b^{2}-7 a^{2} b^{4}+3 b^{6}\right ) \left (\tanh ^{4}\left (\frac {x}{2}\right )\right )}{3 a^{4}}+\frac {b \left (9 a^{6}-52 a^{4} b^{2}-a^{2} b^{4}+12 b^{6}\right ) \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3 a^{3}}-\frac {b^{2} \left (7 a^{4}-3 b^{4}\right ) \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{a^{2}}-\frac {\left (a^{4}-b^{4}\right ) b \tanh \left (\frac {x}{2}\right )}{a}\right )}{\left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 b \tanh \left (\frac {x}{2}\right )-a \right )^{4}}+\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 b \tanh \left (\frac {x}{2}\right )-a \right )}{b^{5}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b^{5}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b^{5}}\)	\(285\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*sech(x)+b*tanh(x))^5,x,method=_RETURNVERBOSE)

[Out]

2/b^5*(((a^4-b^4)*b/a*tanh(1/2*x)^7-b^2*(7*a^4-3*b^4)/a^2*tanh(1/2*x)^6-1/3/a^3*b*(9*a^6-52*a^4*b^2-a^2*b^4+12
*b^6)*tanh(1/2*x)^5+2/3/a^4*b^2*(21*a^6-25*a^4*b^2-7*a^2*b^4+3*b^6)*tanh(1/2*x)^4+1/3/a^3*b*(9*a^6-52*a^4*b^2-
a^2*b^4+12*b^6)*tanh(1/2*x)^3-b^2*(7*a^4-3*b^4)/a^2*tanh(1/2*x)^2-(a^4-b^4)*b/a*tanh(1/2*x))/(a*tanh(1/2*x)^2-
2*b*tanh(1/2*x)-a)^4+1/2*ln(a*tanh(1/2*x)^2-2*b*tanh(1/2*x)-a))-1/b^5*ln(tanh(1/2*x)-1)-1/b^5*ln(tanh(1/2*x)+1
)

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 297 vs. \(2 (91) = 182\).
time = 0.30, size = 297, normalized size = 3.13 \begin {gather*} \frac {4 \, {\left (6 \, a b^{3} e^{\left (-x\right )} - 6 \, a b^{3} e^{\left (-7 \, x\right )} + 3 \, {\left (9 \, a^{2} b^{2} - b^{4}\right )} e^{\left (-2 \, x\right )} + 22 \, {\left (2 \, a^{3} b - a b^{3}\right )} e^{\left (-3 \, x\right )} + {\left (25 \, a^{4} - 56 \, a^{2} b^{2} + 3 \, b^{4}\right )} e^{\left (-4 \, x\right )} - 22 \, {\left (2 \, a^{3} b - a b^{3}\right )} e^{\left (-5 \, x\right )} + 3 \, {\left (9 \, a^{2} b^{2} - b^{4}\right )} e^{\left (-6 \, x\right )}\right )}}{3 \, {\left (8 \, a b^{8} e^{\left (-x\right )} - 8 \, a b^{8} e^{\left (-7 \, x\right )} + b^{9} e^{\left (-8 \, x\right )} + b^{9} + 4 \, {\left (6 \, a^{2} b^{7} - b^{9}\right )} e^{\left (-2 \, x\right )} + 8 \, {\left (4 \, a^{3} b^{6} - 3 \, a b^{8}\right )} e^{\left (-3 \, x\right )} + 2 \, {\left (8 \, a^{4} b^{5} - 24 \, a^{2} b^{7} + 3 \, b^{9}\right )} e^{\left (-4 \, x\right )} - 8 \, {\left (4 \, a^{3} b^{6} - 3 \, a b^{8}\right )} e^{\left (-5 \, x\right )} + 4 \, {\left (6 \, a^{2} b^{7} - b^{9}\right )} e^{\left (-6 \, x\right )}\right )}} + \frac {x}{b^{5}} + \frac {\log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)+b*tanh(x))^5,x, algorithm="maxima")

[Out]

4/3*(6*a*b^3*e^(-x) - 6*a*b^3*e^(-7*x) + 3*(9*a^2*b^2 - b^4)*e^(-2*x) + 22*(2*a^3*b - a*b^3)*e^(-3*x) + (25*a^
4 - 56*a^2*b^2 + 3*b^4)*e^(-4*x) - 22*(2*a^3*b - a*b^3)*e^(-5*x) + 3*(9*a^2*b^2 - b^4)*e^(-6*x))/(8*a*b^8*e^(-
x) - 8*a*b^8*e^(-7*x) + b^9*e^(-8*x) + b^9 + 4*(6*a^2*b^7 - b^9)*e^(-2*x) + 8*(4*a^3*b^6 - 3*a*b^8)*e^(-3*x) +
2*(8*a^4*b^5 - 24*a^2*b^7 + 3*b^9)*e^(-4*x) - 8*(4*a^3*b^6 - 3*a*b^8)*e^(-5*x) + 4*(6*a^2*b^7 - b^9)*e^(-6*x)
) + x/b^5 + log(-2*a*e^(-x) + b*e^(-2*x) - b)/b^5

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2640 vs. \(2 (91) = 182\).
time = 0.40, size = 2640, normalized size = 27.79 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)+b*tanh(x))^5,x, algorithm="fricas")

[Out]

-1/3*(3*b^4*x*cosh(x)^8 + 3*b^4*x*sinh(x)^8 + 24*(a*b^3*x - a*b^3)*cosh(x)^7 + 24*(b^4*x*cosh(x) + a*b^3*x - a
*b^3)*sinh(x)^7 - 12*(9*a^2*b^2 - b^4 - (6*a^2*b^2 - b^4)*x)*cosh(x)^6 + 12*(7*b^4*x*cosh(x)^2 - 9*a^2*b^2 + b
^4 + (6*a^2*b^2 - b^4)*x + 14*(a*b^3*x - a*b^3)*cosh(x))*sinh(x)^6 - 8*(22*a^3*b - 11*a*b^3 - 3*(4*a^3*b - 3*a
*b^3)*x)*cosh(x)^5 + 8*(21*b^4*x*cosh(x)^3 - 22*a^3*b + 11*a*b^3 + 63*(a*b^3*x - a*b^3)*cosh(x)^2 + 3*(4*a^3*b
- 3*a*b^3)*x - 9*(9*a^2*b^2 - b^4 - (6*a^2*b^2 - b^4)*x)*cosh(x))*sinh(x)^5 + 3*b^4*x - 2*(50*a^4 - 112*a^2*b
^2 + 6*b^4 - 3*(8*a^4 - 24*a^2*b^2 + 3*b^4)*x)*cosh(x)^4 + 2*(105*b^4*x*cosh(x)^4 - 50*a^4 + 112*a^2*b^2 - 6*b
^4 + 420*(a*b^3*x - a*b^3)*cosh(x)^3 - 90*(9*a^2*b^2 - b^4 - (6*a^2*b^2 - b^4)*x)*cosh(x)^2 + 3*(8*a^4 - 24*a^
2*b^2 + 3*b^4)*x - 20*(22*a^3*b - 11*a*b^3 - 3*(4*a^3*b - 3*a*b^3)*x)*cosh(x))*sinh(x)^4 + 8*(22*a^3*b - 11*a*
b^3 - 3*(4*a^3*b - 3*a*b^3)*x)*cosh(x)^3 + 8*(21*b^4*x*cosh(x)^5 + 105*(a*b^3*x - a*b^3)*cosh(x)^4 + 22*a^3*b
- 11*a*b^3 - 30*(9*a^2*b^2 - b^4 - (6*a^2*b^2 - b^4)*x)*cosh(x)^3 - 10*(22*a^3*b - 11*a*b^3 - 3*(4*a^3*b - 3*a
*b^3)*x)*cosh(x)^2 - 3*(4*a^3*b - 3*a*b^3)*x - (50*a^4 - 112*a^2*b^2 + 6*b^4 - 3*(8*a^4 - 24*a^2*b^2 + 3*b^4)*
x)*cosh(x))*sinh(x)^3 - 12*(9*a^2*b^2 - b^4 - (6*a^2*b^2 - b^4)*x)*cosh(x)^2 + 4*(21*b^4*x*cosh(x)^6 + 126*(a*
b^3*x - a*b^3)*cosh(x)^5 - 45*(9*a^2*b^2 - b^4 - (6*a^2*b^2 - b^4)*x)*cosh(x)^4 - 27*a^2*b^2 + 3*b^4 - 20*(22*
a^3*b - 11*a*b^3 - 3*(4*a^3*b - 3*a*b^3)*x)*cosh(x)^3 - 3*(50*a^4 - 112*a^2*b^2 + 6*b^4 - 3*(8*a^4 - 24*a^2*b^
2 + 3*b^4)*x)*cosh(x)^2 + 3*(6*a^2*b^2 - b^4)*x + 6*(22*a^3*b - 11*a*b^3 - 3*(4*a^3*b - 3*a*b^3)*x)*cosh(x))*s
inh(x)^2 - 24*(a*b^3*x - a*b^3)*cosh(x) - 3*(b^4*cosh(x)^8 + b^4*sinh(x)^8 + 8*a*b^3*cosh(x)^7 + 8*(b^4*cosh(x
) + a*b^3)*sinh(x)^7 + 4*(6*a^2*b^2 - b^4)*cosh(x)^6 + 4*(7*b^4*cosh(x)^2 + 14*a*b^3*cosh(x) + 6*a^2*b^2 - b^4
)*sinh(x)^6 + 8*(4*a^3*b - 3*a*b^3)*cosh(x)^5 + 8*(7*b^4*cosh(x)^3 + 21*a*b^3*cosh(x)^2 + 4*a^3*b - 3*a*b^3 +
3*(6*a^2*b^2 - b^4)*cosh(x))*sinh(x)^5 - 8*a*b^3*cosh(x) + 2*(8*a^4 - 24*a^2*b^2 + 3*b^4)*cosh(x)^4 + 2*(35*b^
4*cosh(x)^4 + 140*a*b^3*cosh(x)^3 + 8*a^4 - 24*a^2*b^2 + 3*b^4 + 30*(6*a^2*b^2 - b^4)*cosh(x)^2 + 20*(4*a^3*b
- 3*a*b^3)*cosh(x))*sinh(x)^4 + b^4 - 8*(4*a^3*b - 3*a*b^3)*cosh(x)^3 + 8*(7*b^4*cosh(x)^5 + 35*a*b^3*cosh(x)^
4 - 4*a^3*b + 3*a*b^3 + 10*(6*a^2*b^2 - b^4)*cosh(x)^3 + 10*(4*a^3*b - 3*a*b^3)*cosh(x)^2 + (8*a^4 - 24*a^2*b^
2 + 3*b^4)*cosh(x))*sinh(x)^3 + 4*(6*a^2*b^2 - b^4)*cosh(x)^2 + 4*(7*b^4*cosh(x)^6 + 42*a*b^3*cosh(x)^5 + 15*(
6*a^2*b^2 - b^4)*cosh(x)^4 + 6*a^2*b^2 - b^4 + 20*(4*a^3*b - 3*a*b^3)*cosh(x)^3 + 3*(8*a^4 - 24*a^2*b^2 + 3*b^
4)*cosh(x)^2 - 6*(4*a^3*b - 3*a*b^3)*cosh(x))*sinh(x)^2 + 8*(b^4*cosh(x)^7 + 7*a*b^3*cosh(x)^6 + 3*(6*a^2*b^2
- b^4)*cosh(x)^5 + 5*(4*a^3*b - 3*a*b^3)*cosh(x)^4 - a*b^3 + (8*a^4 - 24*a^2*b^2 + 3*b^4)*cosh(x)^3 - 3*(4*a^3
*b - 3*a*b^3)*cosh(x)^2 + (6*a^2*b^2 - b^4)*cosh(x))*sinh(x))*log(2*(b*sinh(x) + a)/(cosh(x) - sinh(x))) + 8*(
3*b^4*x*cosh(x)^7 + 21*(a*b^3*x - a*b^3)*cosh(x)^6 - 9*(9*a^2*b^2 - b^4 - (6*a^2*b^2 - b^4)*x)*cosh(x)^5 - 3*a
*b^3*x - 5*(22*a^3*b - 11*a*b^3 - 3*(4*a^3*b - 3*a*b^3)*x)*cosh(x)^4 + 3*a*b^3 - (50*a^4 - 112*a^2*b^2 + 6*b^4
- 3*(8*a^4 - 24*a^2*b^2 + 3*b^4)*x)*cosh(x)^3 + 3*(22*a^3*b - 11*a*b^3 - 3*(4*a^3*b - 3*a*b^3)*x)*cosh(x)^2 -
3*(9*a^2*b^2 - b^4 - (6*a^2*b^2 - b^4)*x)*cosh(x))*sinh(x))/(b^9*cosh(x)^8 + b^9*sinh(x)^8 + 8*a*b^8*cosh(x)^
7 - 8*a*b^8*cosh(x) + b^9 + 8*(b^9*cosh(x) + a*b^8)*sinh(x)^7 + 4*(6*a^2*b^7 - b^9)*cosh(x)^6 + 4*(7*b^9*cosh(
x)^2 + 14*a*b^8*cosh(x) + 6*a^2*b^7 - b^9)*sinh(x)^6 + 8*(4*a^3*b^6 - 3*a*b^8)*cosh(x)^5 + 8*(7*b^9*cosh(x)^3
+ 21*a*b^8*cosh(x)^2 + 4*a^3*b^6 - 3*a*b^8 + 3*(6*a^2*b^7 - b^9)*cosh(x))*sinh(x)^5 + 2*(8*a^4*b^5 - 24*a^2*b^
7 + 3*b^9)*cosh(x)^4 + 2*(35*b^9*cosh(x)^4 + 140*a*b^8*cosh(x)^3 + 8*a^4*b^5 - 24*a^2*b^7 + 3*b^9 + 30*(6*a^2*
b^7 - b^9)*cosh(x)^2 + 20*(4*a^3*b^6 - 3*a*b^8)*cosh(x))*sinh(x)^4 - 8*(4*a^3*b^6 - 3*a*b^8)*cosh(x)^3 + 8*(7*
b^9*cosh(x)^5 + 35*a*b^8*cosh(x)^4 - 4*a^3*b^6 + 3*a*b^8 + 10*(6*a^2*b^7 - b^9)*cosh(x)^3 + 10*(4*a^3*b^6 - 3*
a*b^8)*cosh(x)^2 + (8*a^4*b^5 - 24*a^2*b^7 + 3*b^9)*cosh(x))*sinh(x)^3 + 4*(6*a^2*b^7 - b^9)*cosh(x)^2 + 4*(7*
b^9*cosh(x)^6 + 42*a*b^8*cosh(x)^5 + 6*a^2*b^7 - b^9 + 15*(6*a^2*b^7 - b^9)*cosh(x)^4 + 20*(4*a^3*b^6 - 3*a*b^
8)*cosh(x)^3 + 3*(8*a^4*b^5 - 24*a^2*b^7 + 3*b^9)*cosh(x)^2 - 6*(4*a^3*b^6 - 3*a*b^8)*cosh(x))*sinh(x)^2 + 8*(
b^9*cosh(x)^7 + 7*a*b^8*cosh(x)^6 - a*b^8 + 3*(6*a^2*b^7 - b^9)*cosh(x)^5 + 5*(4*a^3*b^6 - 3*a*b^8)*cosh(x)^4
+ (8*a^4*b^5 - 24*a^2*b^7 + 3*b^9)*cosh(x)^3 - 3*(4*a^3*b^6 - 3*a*b^8)*cosh(x)^2 + (6*a^2*b^7 - b^9)*cosh(x))*
sinh(x))

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2162 vs. \(2 (92) = 184\).
time = 7.26, size = 2162, normalized size = 22.76 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)+b*tanh(x))**5,x)

[Out]

Piecewise((36*a**4*x*sech(x)**4/(36*a**4*b**5*sech(x)**4 + 144*a**3*b**6*tanh(x)*sech(x)**3 + 216*a**2*b**7*ta
nh(x)**2*sech(x)**2 + 144*a*b**8*tanh(x)**3*sech(x) + 36*b**9*tanh(x)**4) + 36*a**4*log(a*sech(x)/b + tanh(x))
*sech(x)**4/(36*a**4*b**5*sech(x)**4 + 144*a**3*b**6*tanh(x)*sech(x)**3 + 216*a**2*b**7*tanh(x)**2*sech(x)**2
+ 144*a*b**8*tanh(x)**3*sech(x) + 36*b**9*tanh(x)**4) - 36*a**4*log(tanh(x) + 1)*sech(x)**4/(36*a**4*b**5*sech
(x)**4 + 144*a**3*b**6*tanh(x)*sech(x)**3 + 216*a**2*b**7*tanh(x)**2*sech(x)**2 + 144*a*b**8*tanh(x)**3*sech(x
) + 36*b**9*tanh(x)**4) + 20*a**4*sech(x)**4/(36*a**4*b**5*sech(x)**4 + 144*a**3*b**6*tanh(x)*sech(x)**3 + 216
*a**2*b**7*tanh(x)**2*sech(x)**2 + 144*a*b**8*tanh(x)**3*sech(x) + 36*b**9*tanh(x)**4) + 144*a**3*b*x*tanh(x)*
sech(x)**3/(36*a**4*b**5*sech(x)**4 + 144*a**3*b**6*tanh(x)*sech(x)**3 + 216*a**2*b**7*tanh(x)**2*sech(x)**2 +
144*a*b**8*tanh(x)**3*sech(x) + 36*b**9*tanh(x)**4) + 144*a**3*b*log(a*sech(x)/b + tanh(x))*tanh(x)*sech(x)**
3/(36*a**4*b**5*sech(x)**4 + 144*a**3*b**6*tanh(x)*sech(x)**3 + 216*a**2*b**7*tanh(x)**2*sech(x)**2 + 144*a*b*
*8*tanh(x)**3*sech(x) + 36*b**9*tanh(x)**4) - 144*a**3*b*log(tanh(x) + 1)*tanh(x)*sech(x)**3/(36*a**4*b**5*sec
h(x)**4 + 144*a**3*b**6*tanh(x)*sech(x)**3 + 216*a**2*b**7*tanh(x)**2*sech(x)**2 + 144*a*b**8*tanh(x)**3*sech(
x) + 36*b**9*tanh(x)**4) + 44*a**3*b*tanh(x)*sech(x)**3/(36*a**4*b**5*sech(x)**4 + 144*a**3*b**6*tanh(x)*sech(
x)**3 + 216*a**2*b**7*tanh(x)**2*sech(x)**2 + 144*a*b**8*tanh(x)**3*sech(x) + 36*b**9*tanh(x)**4) + 216*a**2*b
**2*x*tanh(x)**2*sech(x)**2/(36*a**4*b**5*sech(x)**4 + 144*a**3*b**6*tanh(x)*sech(x)**3 + 216*a**2*b**7*tanh(x
)**2*sech(x)**2 + 144*a*b**8*tanh(x)**3*sech(x) + 36*b**9*tanh(x)**4) + 216*a**2*b**2*log(a*sech(x)/b + tanh(x
))*tanh(x)**2*sech(x)**2/(36*a**4*b**5*sech(x)**4 + 144*a**3*b**6*tanh(x)*sech(x)**3 + 216*a**2*b**7*tanh(x)**
2*sech(x)**2 + 144*a*b**8*tanh(x)**3*sech(x) + 36*b**9*tanh(x)**4) - 216*a**2*b**2*log(tanh(x) + 1)*tanh(x)**2
*sech(x)**2/(36*a**4*b**5*sech(x)**4 + 144*a**3*b**6*tanh(x)*sech(x)**3 + 216*a**2*b**7*tanh(x)**2*sech(x)**2
+ 144*a*b**8*tanh(x)**3*sech(x) + 36*b**9*tanh(x)**4) - 6*a**2*b**2*sech(x)**2/(36*a**4*b**5*sech(x)**4 + 144*
a**3*b**6*tanh(x)*sech(x)**3 + 216*a**2*b**7*tanh(x)**2*sech(x)**2 + 144*a*b**8*tanh(x)**3*sech(x) + 36*b**9*t
anh(x)**4) + 144*a*b**3*x*tanh(x)**3*sech(x)/(36*a**4*b**5*sech(x)**4 + 144*a**3*b**6*tanh(x)*sech(x)**3 + 216
*a**2*b**7*tanh(x)**2*sech(x)**2 + 144*a*b**8*tanh(x)**3*sech(x) + 36*b**9*tanh(x)**4) + 144*a*b**3*log(a*sech
(x)/b + tanh(x))*tanh(x)**3*sech(x)/(36*a**4*b**5*sech(x)**4 + 144*a**3*b**6*tanh(x)*sech(x)**3 + 216*a**2*b**
7*tanh(x)**2*sech(x)**2 + 144*a*b**8*tanh(x)**3*sech(x) + 36*b**9*tanh(x)**4) - 144*a*b**3*log(tanh(x) + 1)*ta
nh(x)**3*sech(x)/(36*a**4*b**5*sech(x)**4 + 144*a**3*b**6*tanh(x)*sech(x)**3 + 216*a**2*b**7*tanh(x)**2*sech(x
)**2 + 144*a*b**8*tanh(x)**3*sech(x) + 36*b**9*tanh(x)**4) - 52*a*b**3*tanh(x)**3*sech(x)/(36*a**4*b**5*sech(x
)**4 + 144*a**3*b**6*tanh(x)*sech(x)**3 + 216*a**2*b**7*tanh(x)**2*sech(x)**2 + 144*a*b**8*tanh(x)**3*sech(x)
+ 36*b**9*tanh(x)**4) - 24*a*b**3*tanh(x)*sech(x)/(36*a**4*b**5*sech(x)**4 + 144*a**3*b**6*tanh(x)*sech(x)**3
+ 216*a**2*b**7*tanh(x)**2*sech(x)**2 + 144*a*b**8*tanh(x)**3*sech(x) + 36*b**9*tanh(x)**4) + 36*b**4*x*tanh(x
)**4/(36*a**4*b**5*sech(x)**4 + 144*a**3*b**6*tanh(x)*sech(x)**3 + 216*a**2*b**7*tanh(x)**2*sech(x)**2 + 144*a
*b**8*tanh(x)**3*sech(x) + 36*b**9*tanh(x)**4) + 36*b**4*log(a*sech(x)/b + tanh(x))*tanh(x)**4/(36*a**4*b**5*s
ech(x)**4 + 144*a**3*b**6*tanh(x)*sech(x)**3 + 216*a**2*b**7*tanh(x)**2*sech(x)**2 + 144*a*b**8*tanh(x)**3*sec
h(x) + 36*b**9*tanh(x)**4) - 36*b**4*log(tanh(x) + 1)*tanh(x)**4/(36*a**4*b**5*sech(x)**4 + 144*a**3*b**6*tanh
(x)*sech(x)**3 + 216*a**2*b**7*tanh(x)**2*sech(x)**2 + 144*a*b**8*tanh(x)**3*sech(x) + 36*b**9*tanh(x)**4) - 2
8*b**4*tanh(x)**4/(36*a**4*b**5*sech(x)**4 + 144*a**3*b**6*tanh(x)*sech(x)**3 + 216*a**2*b**7*tanh(x)**2*sech(
x)**2 + 144*a*b**8*tanh(x)**3*sech(x) + 36*b**9*tanh(x)**4) - 18*b**4*tanh(x)**2/(36*a**4*b**5*sech(x)**4 + 14
4*a**3*b**6*tanh(x)*sech(x)**3 + 216*a**2*b**7*tanh(x)**2*sech(x)**2 + 144*a*b**8*tanh(x)**3*sech(x) + 36*b**9
*tanh(x)**4) - 9*b**4/(36*a**4*b**5*sech(x)**4 + 144*a**3*b**6*tanh(x)*sech(x)**3 + 216*a**2*b**7*tanh(x)**2*s
ech(x)**2 + 144*a*b**8*tanh(x)**3*sech(x) + 36*b**9*tanh(x)**4), Ne(b, 0)), ((8*tanh(x)**5/(15*sech(x)**5) - 4
*tanh(x)**3/(3*sech(x)**5) + tanh(x)/sech(x)**5)/a**5, True))

________________________________________________________________________________________

Giac [A]
time = 0.43, size = 152, normalized size = 1.60 \begin {gather*} \frac {\log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{b^{5}} - \frac {25 \, b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{4} - 104 \, a b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 168 \, a^{2} b {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 48 \, b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} - 96 \, a^{3} {\left (e^{\left (-x\right )} - e^{x}\right )} - 64 \, a b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )} + 32 \, a^{2} b + 48 \, b^{3}}{12 \, {\left (b {\left (e^{\left (-x\right )} - e^{x}\right )} - 2 \, a\right )}^{4} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)+b*tanh(x))^5,x, algorithm="giac")

[Out]

log(abs(-b*(e^(-x) - e^x) + 2*a))/b^5 - 1/12*(25*b^3*(e^(-x) - e^x)^4 - 104*a*b^2*(e^(-x) - e^x)^3 + 168*a^2*b
*(e^(-x) - e^x)^2 + 48*b^3*(e^(-x) - e^x)^2 - 96*a^3*(e^(-x) - e^x) - 64*a*b^2*(e^(-x) - e^x) + 32*a^2*b + 48*
b^3)/((b*(e^(-x) - e^x) - 2*a)^4*b^4)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (b\,\mathrm {tanh}\left (x\right )+\frac {a}{\mathrm {cosh}\left (x\right )}\right )}^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*tanh(x) + a/cosh(x))^5,x)

[Out]

int(1/(b*tanh(x) + a/cosh(x))^5, x)

________________________________________________________________________________________