3.2.48 \(\int \frac {\tanh ^5(c+d x)}{(a+b \text {sech}^2(c+d x))^2} \, dx\) [148]

Optimal. Leaf size=76 \[ \frac {(a+b)^2}{2 a^2 b d \left (b+a \cosh ^2(c+d x)\right )}+\frac {\log (\cosh (c+d x))}{b^2 d}+\frac {\left (\frac {1}{a^2}-\frac {1}{b^2}\right ) \log \left (b+a \cosh ^2(c+d x)\right )}{2 d} \]

[Out]

1/2*(a+b)^2/a^2/b/d/(b+a*cosh(d*x+c)^2)+ln(cosh(d*x+c))/b^2/d+1/2*(1/a^2-1/b^2)*ln(b+a*cosh(d*x+c)^2)/d

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Rubi [A]
time = 0.08, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4223, 457, 90} \begin {gather*} \frac {\left (\frac {1}{a^2}-\frac {1}{b^2}\right ) \log \left (a \cosh ^2(c+d x)+b\right )}{2 d}+\frac {(a+b)^2}{2 a^2 b d \left (a \cosh ^2(c+d x)+b\right )}+\frac {\log (\cosh (c+d x))}{b^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tanh[c + d*x]^5/(a + b*Sech[c + d*x]^2)^2,x]

[Out]

(a + b)^2/(2*a^2*b*d*(b + a*Cosh[c + d*x]^2)) + Log[Cosh[c + d*x]]/(b^2*d) + ((a^(-2) - b^(-2))*Log[b + a*Cosh
[c + d*x]^2])/(2*d)

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 4223

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> Module[{ff =
FreeFactors[Cos[e + f*x], x]}, Dist[-(f*ff^(m + n*p - 1))^(-1), Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*
(ff*x)^n)^p/x^(m + n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[(m - 1)/2] &&
IntegerQ[n] && IntegerQ[p]

Rubi steps

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

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Mathematica [A]
time = 0.31, size = 109, normalized size = 1.43 \begin {gather*} \frac {(a+2 b+a \cosh (2 c+2 d x))^2 \left (\frac {(a+b)^2}{a^2 b \left (b+a \cosh ^2(c+d x)\right )}+\frac {2 \log (\cosh (c+d x))}{b^2}+\left (\frac {1}{a^2}-\frac {1}{b^2}\right ) \log \left (b+a \cosh ^2(c+d x)\right )\right ) \text {sech}^4(c+d x)}{8 d \left (a+b \text {sech}^2(c+d x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tanh[c + d*x]^5/(a + b*Sech[c + d*x]^2)^2,x]

[Out]

((a + 2*b + a*Cosh[2*c + 2*d*x])^2*((a + b)^2/(a^2*b*(b + a*Cosh[c + d*x]^2)) + (2*Log[Cosh[c + d*x]])/b^2 + (
a^(-2) - b^(-2))*Log[b + a*Cosh[c + d*x]^2])*Sech[c + d*x]^4)/(8*d*(a + b*Sech[c + d*x]^2)^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(208\) vs. \(2(72)=144\).
time = 2.36, size = 209, normalized size = 2.75

method	result	size
risch	\(-\frac {x}{a^{2}}-\frac {2 c}{a^{2} d}+\frac {2 \left (a^{2}+2 a b +b^{2}\right ) {\mathrm e}^{2 d x +2 c}}{a^{2} b d \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )}+\frac {\ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{b^{2} d}-\frac {\ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (2 b +a \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 b^{2} d}+\frac {\ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (2 b +a \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 a^{2} d}\)	\(184\)
derivativedivides	\(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}-\frac {\left (a +b \right ) \left (\frac {2 a b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (a -b \right ) \ln \left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}{2}\right )}{a^{2} b^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}+\frac {\ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{2}}}{d}\)	\(209\)
default	\(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}-\frac {\left (a +b \right ) \left (\frac {2 a b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (a -b \right ) \ln \left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}{2}\right )}{a^{2} b^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}+\frac {\ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{2}}}{d}\)	\(209\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(d*x+c)^5/(a+b*sech(d*x+c)^2)^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/a^2*ln(tanh(1/2*d*x+1/2*c)-1)-1/a^2/b^2*(a+b)*(2*a*b*tanh(1/2*d*x+1/2*c)^2/(a*tanh(1/2*d*x+1/2*c)^4+b*
tanh(1/2*d*x+1/2*c)^4+2*a*tanh(1/2*d*x+1/2*c)^2-2*b*tanh(1/2*d*x+1/2*c)^2+a+b)+1/2*(a-b)*ln(a*tanh(1/2*d*x+1/2
*c)^4+b*tanh(1/2*d*x+1/2*c)^4+2*a*tanh(1/2*d*x+1/2*c)^2-2*b*tanh(1/2*d*x+1/2*c)^2+a+b))-1/a^2*ln(tanh(1/2*d*x+
1/2*c)+1)+1/b^2*ln(tanh(1/2*d*x+1/2*c)^2+1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (72) = 144\).
time = 0.48, size = 154, normalized size = 2.03 \begin {gather*} \frac {2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (a^{3} b e^{\left (-4 \, d x - 4 \, c\right )} + a^{3} b + 2 \, {\left (a^{3} b + 2 \, a^{2} b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} d} + \frac {d x + c}{a^{2} d} + \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{b^{2} d} - \frac {{\left (a^{2} - b^{2}\right )} \log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{2 \, a^{2} b^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)^5/(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

2*(a^2 + 2*a*b + b^2)*e^(-2*d*x - 2*c)/((a^3*b*e^(-4*d*x - 4*c) + a^3*b + 2*(a^3*b + 2*a^2*b^2)*e^(-2*d*x - 2*
c))*d) + (d*x + c)/(a^2*d) + log(e^(-2*d*x - 2*c) + 1)/(b^2*d) - 1/2*(a^2 - b^2)*log(2*(a + 2*b)*e^(-2*d*x - 2
*c) + a*e^(-4*d*x - 4*c) + a)/(a^2*b^2*d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 853 vs. \(2 (72) = 144\).
time = 0.46, size = 853, normalized size = 11.22 \begin {gather*} -\frac {2 \, a b^{2} d x \cosh \left (d x + c\right )^{4} + 8 \, a b^{2} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 2 \, a b^{2} d x \sinh \left (d x + c\right )^{4} + 2 \, a b^{2} d x - 4 \, {\left (a^{2} b + 2 \, a b^{2} + b^{3} - {\left (a b^{2} + 2 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{2} + 4 \, {\left (3 \, a b^{2} d x \cosh \left (d x + c\right )^{2} - a^{2} b - 2 \, a b^{2} - b^{3} + {\left (a b^{2} + 2 \, b^{3}\right )} d x\right )} \sinh \left (d x + c\right )^{2} + {\left ({\left (a^{3} - a b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{3} - a b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{3} - a b^{2}\right )} \sinh \left (d x + c\right )^{4} + a^{3} - a b^{2} + 2 \, {\left (a^{3} + 2 \, a^{2} b - a b^{2} - 2 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{3} + 2 \, a^{2} b - a b^{2} - 2 \, b^{3} + 3 \, {\left (a^{3} - a b^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a^{3} - a b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (a^{3} + 2 \, a^{2} b - a b^{2} - 2 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, {\left (a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + a + 2 \, b\right )}}{\cosh \left (d x + c\right )^{2} - 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2}}\right ) - 2 \, {\left (a^{3} \cosh \left (d x + c\right )^{4} + 4 \, a^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{3} \sinh \left (d x + c\right )^{4} + a^{3} + 2 \, {\left (a^{3} + 2 \, a^{2} b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{3} \cosh \left (d x + c\right )^{2} + a^{3} + 2 \, a^{2} b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a^{3} \cosh \left (d x + c\right )^{3} + {\left (a^{3} + 2 \, a^{2} b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 8 \, {\left (a b^{2} d x \cosh \left (d x + c\right )^{3} - {\left (a^{2} b + 2 \, a b^{2} + b^{3} - {\left (a b^{2} + 2 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left (a^{3} b^{2} d \cosh \left (d x + c\right )^{4} + 4 \, a^{3} b^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{3} b^{2} d \sinh \left (d x + c\right )^{4} + a^{3} b^{2} d + 2 \, {\left (a^{3} b^{2} + 2 \, a^{2} b^{3}\right )} d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{3} b^{2} d \cosh \left (d x + c\right )^{2} + {\left (a^{3} b^{2} + 2 \, a^{2} b^{3}\right )} d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a^{3} b^{2} d \cosh \left (d x + c\right )^{3} + {\left (a^{3} b^{2} + 2 \, a^{2} b^{3}\right )} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)^5/(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

-1/2*(2*a*b^2*d*x*cosh(d*x + c)^4 + 8*a*b^2*d*x*cosh(d*x + c)*sinh(d*x + c)^3 + 2*a*b^2*d*x*sinh(d*x + c)^4 +
2*a*b^2*d*x - 4*(a^2*b + 2*a*b^2 + b^3 - (a*b^2 + 2*b^3)*d*x)*cosh(d*x + c)^2 + 4*(3*a*b^2*d*x*cosh(d*x + c)^2
- a^2*b - 2*a*b^2 - b^3 + (a*b^2 + 2*b^3)*d*x)*sinh(d*x + c)^2 + ((a^3 - a*b^2)*cosh(d*x + c)^4 + 4*(a^3 - a*
b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^3 - a*b^2)*sinh(d*x + c)^4 + a^3 - a*b^2 + 2*(a^3 + 2*a^2*b - a*b^2 -
2*b^3)*cosh(d*x + c)^2 + 2*(a^3 + 2*a^2*b - a*b^2 - 2*b^3 + 3*(a^3 - a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 +
4*((a^3 - a*b^2)*cosh(d*x + c)^3 + (a^3 + 2*a^2*b - a*b^2 - 2*b^3)*cosh(d*x + c))*sinh(d*x + c))*log(2*(a*cos
h(d*x + c)^2 + a*sinh(d*x + c)^2 + a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2
)) - 2*(a^3*cosh(d*x + c)^4 + 4*a^3*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*sinh(d*x + c)^4 + a^3 + 2*(a^3 + 2*a^2
*b)*cosh(d*x + c)^2 + 2*(3*a^3*cosh(d*x + c)^2 + a^3 + 2*a^2*b)*sinh(d*x + c)^2 + 4*(a^3*cosh(d*x + c)^3 + (a^
3 + 2*a^2*b)*cosh(d*x + c))*sinh(d*x + c))*log(2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 8*(a*b^2*d*x
*cosh(d*x + c)^3 - (a^2*b + 2*a*b^2 + b^3 - (a*b^2 + 2*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c))/(a^3*b^2*d*cosh
(d*x + c)^4 + 4*a^3*b^2*d*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*b^2*d*sinh(d*x + c)^4 + a^3*b^2*d + 2*(a^3*b^2 +
2*a^2*b^3)*d*cosh(d*x + c)^2 + 2*(3*a^3*b^2*d*cosh(d*x + c)^2 + (a^3*b^2 + 2*a^2*b^3)*d)*sinh(d*x + c)^2 + 4*
(a^3*b^2*d*cosh(d*x + c)^3 + (a^3*b^2 + 2*a^2*b^3)*d*cosh(d*x + c))*sinh(d*x + c))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh ^{5}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)**5/(a+b*sech(d*x+c)**2)**2,x)

[Out]

Integral(tanh(c + d*x)**5/(a + b*sech(c + d*x)**2)**2, x)

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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(d*x+c)^5/(a+b*sech(d*x+c)^2)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^4\,{\mathrm {tanh}\left (c+d\,x\right )}^5}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(c + d*x)^5/(a + b/cosh(c + d*x)^2)^2,x)

[Out]

int((cosh(c + d*x)^4*tanh(c + d*x)^5)/(b + a*cosh(c + d*x)^2)^2, x)

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