3.2.15 \(\int (a+b \text {sech}^2(c+d x))^2 \tanh (c+d x) \, dx\) [115]

Optimal. Leaf size=48 \[ \frac {a^2 \log (\cosh (c+d x))}{d}-\frac {a b \text {sech}^2(c+d x)}{d}-\frac {b^2 \text {sech}^4(c+d x)}{4 d} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4223, 272, 45} \begin {gather*} \frac {a^2 \log (\cosh (c+d x))}{d}-\frac {a b \text {sech}^2(c+d x)}{d}-\frac {b^2 \text {sech}^4(c+d x)}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && 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 \left (a+b \text {sech}^2(c+d x)\right )^2 \tanh (c+d x) \, dx &=\frac {\text {Subst}\left (\int \frac {\left (b+a x^2\right )^2}{x^5} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {(b+a x)^2}{x^3} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {b^2}{x^3}+\frac {2 a b}{x^2}+\frac {a^2}{x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {a^2 \log (\cosh (c+d x))}{d}-\frac {a b \text {sech}^2(c+d x)}{d}-\frac {b^2 \text {sech}^4(c+d x)}{4 d}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 81, normalized size = 1.69 \begin {gather*} \frac {\left (b+a \cosh ^2(c+d x)\right )^2 \left (-b^2-4 a b \cosh ^2(c+d x)+4 a^2 \cosh ^4(c+d x) \log (\cosh (c+d x))\right ) \text {sech}^4(c+d x)}{d (a+2 b+a \cosh (2 (c+d x)))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.67, size = 42, normalized size = 0.88

method	result	size
derivativedivides	\(-\frac {\frac {\mathrm {sech}\left (d x +c \right )^{4} b^{2}}{4}+\mathrm {sech}\left (d x +c \right )^{2} a b +a^{2} \ln \left (\mathrm {sech}\left (d x +c \right )\right )}{d}\)	\(42\)
default	\(-\frac {\frac {\mathrm {sech}\left (d x +c \right )^{4} b^{2}}{4}+\mathrm {sech}\left (d x +c \right )^{2} a b +a^{2} \ln \left (\mathrm {sech}\left (d x +c \right )\right )}{d}\)	\(42\)
risch	\(-a^{2} x -\frac {2 a^{2} c}{d}-\frac {4 b \,{\mathrm e}^{2 d x +2 c} \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+b \,{\mathrm e}^{2 d x +2 c}+a \right )}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{4}}+\frac {a^{2} \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d}\)	\(100\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 55, normalized size = 1.15 \begin {gather*} \frac {a b \tanh \left (d x + c\right )^{2}}{d} + \frac {a^{2} \log \left (\cosh \left (d x + c\right )\right )}{d} - \frac {4 \, b^{2}}{d {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1180 vs. \(2 (46) = 92\).
time = 0.37, size = 1180, normalized size = 24.58 \begin {gather*} -\frac {a^{2} d x \cosh \left (d x + c\right )^{8} + 8 \, a^{2} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + a^{2} d x \sinh \left (d x + c\right )^{8} + 4 \, {\left (a^{2} d x + a b\right )} \cosh \left (d x + c\right )^{6} + 4 \, {\left (7 \, a^{2} d x \cosh \left (d x + c\right )^{2} + a^{2} d x + a b\right )} \sinh \left (d x + c\right )^{6} + 8 \, {\left (7 \, a^{2} d x \cosh \left (d x + c\right )^{3} + 3 \, {\left (a^{2} d x + a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 2 \, {\left (3 \, a^{2} d x + 4 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 2 \, {\left (35 \, a^{2} d x \cosh \left (d x + c\right )^{4} + 3 \, a^{2} d x + 30 \, {\left (a^{2} d x + a b\right )} \cosh \left (d x + c\right )^{2} + 4 \, a b + 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{4} + a^{2} d x + 8 \, {\left (7 \, a^{2} d x \cosh \left (d x + c\right )^{5} + 10 \, {\left (a^{2} d x + a b\right )} \cosh \left (d x + c\right )^{3} + {\left (3 \, a^{2} d x + 4 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 4 \, {\left (a^{2} d x + a b\right )} \cosh \left (d x + c\right )^{2} + 4 \, {\left (7 \, a^{2} d x \cosh \left (d x + c\right )^{6} + 15 \, {\left (a^{2} d x + a b\right )} \cosh \left (d x + c\right )^{4} + a^{2} d x + 3 \, {\left (3 \, a^{2} d x + 4 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + a b\right )} \sinh \left (d x + c\right )^{2} - {\left (a^{2} \cosh \left (d x + c\right )^{8} + 8 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + a^{2} \sinh \left (d x + c\right )^{8} + 4 \, a^{2} \cosh \left (d x + c\right )^{6} + 4 \, {\left (7 \, a^{2} \cosh \left (d x + c\right )^{2} + a^{2}\right )} \sinh \left (d x + c\right )^{6} + 6 \, a^{2} \cosh \left (d x + c\right )^{4} + 8 \, {\left (7 \, a^{2} \cosh \left (d x + c\right )^{3} + 3 \, a^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 2 \, {\left (35 \, a^{2} \cosh \left (d x + c\right )^{4} + 30 \, a^{2} \cosh \left (d x + c\right )^{2} + 3 \, a^{2}\right )} \sinh \left (d x + c\right )^{4} + 4 \, a^{2} \cosh \left (d x + c\right )^{2} + 8 \, {\left (7 \, a^{2} \cosh \left (d x + c\right )^{5} + 10 \, a^{2} \cosh \left (d x + c\right )^{3} + 3 \, a^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 4 \, {\left (7 \, a^{2} \cosh \left (d x + c\right )^{6} + 15 \, a^{2} \cosh \left (d x + c\right )^{4} + 9 \, a^{2} \cosh \left (d x + c\right )^{2} + a^{2}\right )} \sinh \left (d x + c\right )^{2} + a^{2} + 8 \, {\left (a^{2} \cosh \left (d x + c\right )^{7} + 3 \, a^{2} \cosh \left (d x + c\right )^{5} + 3 \, a^{2} \cosh \left (d x + c\right )^{3} + a^{2} \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^{2} d x \cosh \left (d x + c\right )^{7} + 3 \, {\left (a^{2} d x + a b\right )} \cosh \left (d x + c\right )^{5} + {\left (3 \, a^{2} d x + 4 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (a^{2} d x + a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{d \cosh \left (d x + c\right )^{8} + 8 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + d \sinh \left (d x + c\right )^{8} + 4 \, d \cosh \left (d x + c\right )^{6} + 4 \, {\left (7 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )^{6} + 8 \, {\left (7 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 6 \, d \cosh \left (d x + c\right )^{4} + 2 \, {\left (35 \, d \cosh \left (d x + c\right )^{4} + 30 \, d \cosh \left (d x + c\right )^{2} + 3 \, d\right )} \sinh \left (d x + c\right )^{4} + 8 \, {\left (7 \, d \cosh \left (d x + c\right )^{5} + 10 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 4 \, d \cosh \left (d x + c\right )^{2} + 4 \, {\left (7 \, d \cosh \left (d x + c\right )^{6} + 15 \, d \cosh \left (d x + c\right )^{4} + 9 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )^{2} + 8 \, {\left (d \cosh \left (d x + c\right )^{7} + 3 \, d \cosh \left (d x + c\right )^{5} + 3 \, d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^2*d*x*cosh(d*x + c)^8 + 8*a^2*d*x*cosh(d*x + c)*sinh(d*x + c)^7 + a^2*d*x*sinh(d*x + c)^8 + 4*(a^2*d*x + a
*b)*cosh(d*x + c)^6 + 4*(7*a^2*d*x*cosh(d*x + c)^2 + a^2*d*x + a*b)*sinh(d*x + c)^6 + 8*(7*a^2*d*x*cosh(d*x +
c)^3 + 3*(a^2*d*x + a*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^2*d*x + 4*a*b + 2*b^2)*cosh(d*x + c)^4 + 2*(3
5*a^2*d*x*cosh(d*x + c)^4 + 3*a^2*d*x + 30*(a^2*d*x + a*b)*cosh(d*x + c)^2 + 4*a*b + 2*b^2)*sinh(d*x + c)^4 +
a^2*d*x + 8*(7*a^2*d*x*cosh(d*x + c)^5 + 10*(a^2*d*x + a*b)*cosh(d*x + c)^3 + (3*a^2*d*x + 4*a*b + 2*b^2)*cosh
(d*x + c))*sinh(d*x + c)^3 + 4*(a^2*d*x + a*b)*cosh(d*x + c)^2 + 4*(7*a^2*d*x*cosh(d*x + c)^6 + 15*(a^2*d*x +
a*b)*cosh(d*x + c)^4 + a^2*d*x + 3*(3*a^2*d*x + 4*a*b + 2*b^2)*cosh(d*x + c)^2 + a*b)*sinh(d*x + c)^2 - (a^2*c
osh(d*x + c)^8 + 8*a^2*cosh(d*x + c)*sinh(d*x + c)^7 + a^2*sinh(d*x + c)^8 + 4*a^2*cosh(d*x + c)^6 + 4*(7*a^2*
cosh(d*x + c)^2 + a^2)*sinh(d*x + c)^6 + 6*a^2*cosh(d*x + c)^4 + 8*(7*a^2*cosh(d*x + c)^3 + 3*a^2*cosh(d*x + c
))*sinh(d*x + c)^5 + 2*(35*a^2*cosh(d*x + c)^4 + 30*a^2*cosh(d*x + c)^2 + 3*a^2)*sinh(d*x + c)^4 + 4*a^2*cosh(
d*x + c)^2 + 8*(7*a^2*cosh(d*x + c)^5 + 10*a^2*cosh(d*x + c)^3 + 3*a^2*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*a
^2*cosh(d*x + c)^6 + 15*a^2*cosh(d*x + c)^4 + 9*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*x + c)^2 + a^2 + 8*(a^2*cosh
(d*x + c)^7 + 3*a^2*cosh(d*x + c)^5 + 3*a^2*cosh(d*x + c)^3 + a^2*cosh(d*x + c))*sinh(d*x + c))*log(2*cosh(d*x
+ c)/(cosh(d*x + c) - sinh(d*x + c))) + 8*(a^2*d*x*cosh(d*x + c)^7 + 3*(a^2*d*x + a*b)*cosh(d*x + c)^5 + (3*a
^2*d*x + 4*a*b + 2*b^2)*cosh(d*x + c)^3 + (a^2*d*x + a*b)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^8 + 8
*d*cosh(d*x + c)*sinh(d*x + c)^7 + d*sinh(d*x + c)^8 + 4*d*cosh(d*x + c)^6 + 4*(7*d*cosh(d*x + c)^2 + d)*sinh(
d*x + c)^6 + 8*(7*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^5 + 6*d*cosh(d*x + c)^4 + 2*(35*d*cosh(
d*x + c)^4 + 30*d*cosh(d*x + c)^2 + 3*d)*sinh(d*x + c)^4 + 8*(7*d*cosh(d*x + c)^5 + 10*d*cosh(d*x + c)^3 + 3*d
*cosh(d*x + c))*sinh(d*x + c)^3 + 4*d*cosh(d*x + c)^2 + 4*(7*d*cosh(d*x + c)^6 + 15*d*cosh(d*x + c)^4 + 9*d*co
sh(d*x + c)^2 + d)*sinh(d*x + c)^2 + 8*(d*cosh(d*x + c)^7 + 3*d*cosh(d*x + c)^5 + 3*d*cosh(d*x + c)^3 + d*cosh
(d*x + c))*sinh(d*x + c) + d)

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Sympy [A]
time = 0.42, size = 63, normalized size = 1.31 \begin {gather*} \begin {cases} a^{2} x - \frac {a^{2} \log {\left (\tanh {\left (c + d x \right )} + 1 \right )}}{d} - \frac {a b \operatorname {sech}^{2}{\left (c + d x \right )}}{d} - \frac {b^{2} \operatorname {sech}^{4}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {sech}^{2}{\left (c \right )}\right )^{2} \tanh {\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*x - a**2*log(tanh(c + d*x) + 1)/d - a*b*sech(c + d*x)**2/d - b**2*sech(c + d*x)**4/(4*d), Ne(d
, 0)), (x*(a + b*sech(c)**2)**2*tanh(c), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (46) = 92\).
time = 0.41, size = 162, normalized size = 3.38 \begin {gather*} -\frac {12 \, {\left (d x + c\right )} a^{2} - 12 \, a^{2} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) + \frac {25 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} + 100 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} + 48 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 150 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 96 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 48 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 100 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 48 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 25 \, a^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{4}}}{12 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(12*(d*x + c)*a^2 - 12*a^2*log(e^(2*d*x + 2*c) + 1) + (25*a^2*e^(8*d*x + 8*c) + 100*a^2*e^(6*d*x + 6*c)
+ 48*a*b*e^(6*d*x + 6*c) + 150*a^2*e^(4*d*x + 4*c) + 96*a*b*e^(4*d*x + 4*c) + 48*b^2*e^(4*d*x + 4*c) + 100*a^2
*e^(2*d*x + 2*c) + 48*a*b*e^(2*d*x + 2*c) + 25*a^2)/(e^(2*d*x + 2*c) + 1)^4)/d

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Mupad [B]
time = 1.69, size = 182, normalized size = 3.79 \begin {gather*} \frac {4\,\left (a\,b-b^2\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-a^2\,x+\frac {8\,b^2}{d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {4\,b^2}{d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}+\frac {a^2\,\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+1\right )}{d}-\frac {4\,a\,b}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*(a*b - b^2))/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) - a^2*x + (8*b^2)/(d*(3*exp(2*c + 2*d*x) + 3*e
xp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) - (4*b^2)/(d*(4*exp(2*c + 2*d*x) + 6*exp(4*c + 4*d*x) + 4*exp(6*c + 6
*d*x) + exp(8*c + 8*d*x) + 1)) + (a^2*log(exp(2*c)*exp(2*d*x) + 1))/d - (4*a*b)/(d*(exp(2*c + 2*d*x) + 1))

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