3.2.63 \(\int \frac {\tanh (c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\) [163]

Optimal. Leaf size=73 \[ -\frac {b^2}{4 a^3 d \left (b+a \cosh ^2(c+d x)\right )^2}+\frac {b}{a^3 d \left (b+a \cosh ^2(c+d x)\right )}+\frac {\log \left (b+a \cosh ^2(c+d x)\right )}{2 a^3 d} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 73, 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 {b^2}{4 a^3 d \left (a \cosh ^2(c+d x)+b\right )^2}+\frac {b}{a^3 d \left (a \cosh ^2(c+d x)+b\right )}+\frac {\log \left (a \cosh ^2(c+d x)+b\right )}{2 a^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tanh[c + d*x]/(a + b*Sech[c + d*x]^2)^3,x]

[Out]

-1/4*b^2/(a^3*d*(b + a*Cosh[c + d*x]^2)^2) + b/(a^3*d*(b + a*Cosh[c + d*x]^2)) + Log[b + a*Cosh[c + d*x]^2]/(2
*a^3*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 \frac {\tanh (c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {x^5}{\left (b+a x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {x^2}{(b+a x)^3} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {b^2}{a^2 (b+a x)^3}-\frac {2 b}{a^2 (b+a x)^2}+\frac {1}{a^2 (b+a x)}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {b^2}{4 a^3 d \left (b+a \cosh ^2(c+d x)\right )^2}+\frac {b}{a^3 d \left (b+a \cosh ^2(c+d x)\right )}+\frac {\log \left (b+a \cosh ^2(c+d x)\right )}{2 a^3 d}\\ \end {align*}

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Mathematica [A]
time = 1.33, size = 129, normalized size = 1.77 \begin {gather*} \frac {2 b (2 a+3 b)+(a+2 b)^2 \log (a+2 b+a \cosh (2 (c+d x)))+a^2 \cosh ^2(2 (c+d x)) \log (a+2 b+a \cosh (2 (c+d x)))+2 a \cosh (2 (c+d x)) (2 b+(a+2 b) \log (a+2 b+a \cosh (2 (c+d x))))}{2 a^3 d (a+2 b+a \cosh (2 (c+d x)))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tanh[c + d*x]/(a + b*Sech[c + d*x]^2)^3,x]

[Out]

(2*b*(2*a + 3*b) + (a + 2*b)^2*Log[a + 2*b + a*Cosh[2*(c + d*x)]] + a^2*Cosh[2*(c + d*x)]^2*Log[a + 2*b + a*Co
sh[2*(c + d*x)]] + 2*a*Cosh[2*(c + d*x)]*(2*b + (a + 2*b)*Log[a + 2*b + a*Cosh[2*(c + d*x)]]))/(2*a^3*d*(a + 2
*b + a*Cosh[2*(c + d*x)])^2)

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Maple [A]
time = 1.12, size = 84, normalized size = 1.15

method	result	size
derivativedivides	\(-\frac {-\frac {b \left (-\frac {a^{2}}{2 b \left (a +b \mathrm {sech}\left (d x +c \right )^{2}\right )^{2}}+\frac {\ln \left (a +b \mathrm {sech}\left (d x +c \right )^{2}\right )}{b}-\frac {a}{b \left (a +b \mathrm {sech}\left (d x +c \right )^{2}\right )}\right )}{2 a^{3}}+\frac {\ln \left (\mathrm {sech}\left (d x +c \right )\right )}{a^{3}}}{d}\)	\(84\)
default	\(-\frac {-\frac {b \left (-\frac {a^{2}}{2 b \left (a +b \mathrm {sech}\left (d x +c \right )^{2}\right )^{2}}+\frac {\ln \left (a +b \mathrm {sech}\left (d x +c \right )^{2}\right )}{b}-\frac {a}{b \left (a +b \mathrm {sech}\left (d x +c \right )^{2}\right )}\right )}{2 a^{3}}+\frac {\ln \left (\mathrm {sech}\left (d x +c \right )\right )}{a^{3}}}{d}\)	\(84\)
risch	\(-\frac {x}{a^{3}}-\frac {2 c}{a^{3} d}+\frac {4 \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+3 b \,{\mathrm e}^{2 d x +2 c}+a \right ) {\mathrm e}^{2 d x +2 c} b}{a^{3} \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2} d}+\frac {\ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (2 b +a \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 a^{3} d}\)	\(150\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(d*x+c)/(a+b*sech(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

-1/d*(-1/2/a^3*b*(-1/2*a^2/b/(a+b*sech(d*x+c)^2)^2+1/b*ln(a+b*sech(d*x+c)^2)-a/b/(a+b*sech(d*x+c)^2))+1/a^3*ln
(sech(d*x+c)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (69) = 138\).
time = 0.28, size = 193, normalized size = 2.64 \begin {gather*} \frac {4 \, {\left (a b e^{\left (-2 \, d x - 2 \, c\right )} + a b e^{\left (-6 \, d x - 6 \, c\right )} + {\left (2 \, a b + 3 \, b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )}}{{\left (a^{5} e^{\left (-8 \, d x - 8 \, c\right )} + a^{5} + 4 \, {\left (a^{5} + 2 \, a^{4} b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (3 \, a^{5} + 8 \, a^{4} b + 8 \, a^{3} b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, {\left (a^{5} + 2 \, a^{4} b\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )} d} + \frac {d x + c}{a^{3} d} + \frac {\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^{3} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

4*(a*b*e^(-2*d*x - 2*c) + a*b*e^(-6*d*x - 6*c) + (2*a*b + 3*b^2)*e^(-4*d*x - 4*c))/((a^5*e^(-8*d*x - 8*c) + a^
5 + 4*(a^5 + 2*a^4*b)*e^(-2*d*x - 2*c) + 2*(3*a^5 + 8*a^4*b + 8*a^3*b^2)*e^(-4*d*x - 4*c) + 4*(a^5 + 2*a^4*b)*
e^(-6*d*x - 6*c))*d) + (d*x + c)/(a^3*d) + 1/2*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/(a^3
*d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1666 vs. \(2 (69) = 138\).
time = 0.40, size = 1666, normalized size = 22.82 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

-1/2*(2*a^2*d*x*cosh(d*x + c)^8 + 16*a^2*d*x*cosh(d*x + c)*sinh(d*x + c)^7 + 2*a^2*d*x*sinh(d*x + c)^8 + 8*((a
^2 + 2*a*b)*d*x - a*b)*cosh(d*x + c)^6 + 8*(7*a^2*d*x*cosh(d*x + c)^2 + (a^2 + 2*a*b)*d*x - a*b)*sinh(d*x + c)
^6 + 16*(7*a^2*d*x*cosh(d*x + c)^3 + 3*((a^2 + 2*a*b)*d*x - a*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 4*((3*a^2 +
8*a*b + 8*b^2)*d*x - 4*a*b - 6*b^2)*cosh(d*x + c)^4 + 4*(35*a^2*d*x*cosh(d*x + c)^4 + (3*a^2 + 8*a*b + 8*b^2)*
d*x + 30*((a^2 + 2*a*b)*d*x - a*b)*cosh(d*x + c)^2 - 4*a*b - 6*b^2)*sinh(d*x + c)^4 + 2*a^2*d*x + 16*(7*a^2*d*
x*cosh(d*x + c)^5 + 10*((a^2 + 2*a*b)*d*x - a*b)*cosh(d*x + c)^3 + ((3*a^2 + 8*a*b + 8*b^2)*d*x - 4*a*b - 6*b^
2)*cosh(d*x + c))*sinh(d*x + c)^3 + 8*((a^2 + 2*a*b)*d*x - a*b)*cosh(d*x + c)^2 + 8*(7*a^2*d*x*cosh(d*x + c)^6
+ 15*((a^2 + 2*a*b)*d*x - a*b)*cosh(d*x + c)^4 + (a^2 + 2*a*b)*d*x + 3*((3*a^2 + 8*a*b + 8*b^2)*d*x - 4*a*b -
6*b^2)*cosh(d*x + c)^2 - a*b)*sinh(d*x + c)^2 - (a^2*cosh(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 + 2*a*b)*cosh(d*x + c)^6 + 4*(7*a^2*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^
6 + 8*(7*a^2*cosh(d*x + c)^3 + 3*(a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^2 + 8*a*b + 8*b^2)*cosh
(d*x + c)^4 + 2*(35*a^2*cosh(d*x + c)^4 + 30*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 3*a^2 + 8*a*b + 8*b^2)*sinh(d*x +
c)^4 + 8*(7*a^2*cosh(d*x + c)^5 + 10*(a^2 + 2*a*b)*cosh(d*x + c)^3 + (3*a^2 + 8*a*b + 8*b^2)*cosh(d*x + c))*s
inh(d*x + c)^3 + 4*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 4*(7*a^2*cosh(d*x + c)^6 + 15*(a^2 + 2*a*b)*cosh(d*x + c)^4
+ 3*(3*a^2 + 8*a*b + 8*b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 8*(a^2*cosh(d*x + c)^7 + 3
*(a^2 + 2*a*b)*cosh(d*x + c)^5 + (3*a^2 + 8*a*b + 8*b^2)*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d
*x + c))*log(2*(a*cosh(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)) + 16*(a^2*d*x*cosh(d*x + c)^7 + 3*((a^2 + 2*a*b)*d*x - a*b)*cosh(d*x + c)^5 + ((3*a^2
+ 8*a*b + 8*b^2)*d*x - 4*a*b - 6*b^2)*cosh(d*x + c)^3 + ((a^2 + 2*a*b)*d*x - a*b)*cosh(d*x + c))*sinh(d*x + c)
)/(a^5*d*cosh(d*x + c)^8 + 8*a^5*d*cosh(d*x + c)*sinh(d*x + c)^7 + a^5*d*sinh(d*x + c)^8 + 4*(a^5 + 2*a^4*b)*d
*cosh(d*x + c)^6 + 4*(7*a^5*d*cosh(d*x + c)^2 + (a^5 + 2*a^4*b)*d)*sinh(d*x + c)^6 + a^5*d + 2*(3*a^5 + 8*a^4*
b + 8*a^3*b^2)*d*cosh(d*x + c)^4 + 8*(7*a^5*d*cosh(d*x + c)^3 + 3*(a^5 + 2*a^4*b)*d*cosh(d*x + c))*sinh(d*x +
c)^5 + 2*(35*a^5*d*cosh(d*x + c)^4 + 30*(a^5 + 2*a^4*b)*d*cosh(d*x + c)^2 + (3*a^5 + 8*a^4*b + 8*a^3*b^2)*d)*s
inh(d*x + c)^4 + 4*(a^5 + 2*a^4*b)*d*cosh(d*x + c)^2 + 8*(7*a^5*d*cosh(d*x + c)^5 + 10*(a^5 + 2*a^4*b)*d*cosh(
d*x + c)^3 + (3*a^5 + 8*a^4*b + 8*a^3*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*a^5*d*cosh(d*x + c)^6 + 15*
(a^5 + 2*a^4*b)*d*cosh(d*x + c)^4 + 3*(3*a^5 + 8*a^4*b + 8*a^3*b^2)*d*cosh(d*x + c)^2 + (a^5 + 2*a^4*b)*d)*sin
h(d*x + c)^2 + 8*(a^5*d*cosh(d*x + c)^7 + 3*(a^5 + 2*a^4*b)*d*cosh(d*x + c)^5 + (3*a^5 + 8*a^4*b + 8*a^3*b^2)*
d*cosh(d*x + c)^3 + (a^5 + 2*a^4*b)*d*cosh(d*x + c))*sinh(d*x + c))

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)/(a+b*sech(d*x+c)**2)**3,x)

[Out]

Timed out

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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)/(a+b*sech(d*x+c)^2)^3,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 [B]
time = 1.60, size = 94, normalized size = 1.29 \begin {gather*} \frac {\ln \left ({\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )\right )}{2\,a^3\,d}-\frac {b^2}{4\,a^3\,d\,{\mathrm {cosh}\left (c+d\,x\right )}^4\,{\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )}^2}+\frac {b}{a^3\,d\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\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)^3,x)

[Out]

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

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