3.2.23 \(\int \coth ^7(c+d x) (a+b \text {sech}^2(c+d x))^2 \, dx\) [123]

Optimal. Leaf size=86 \[ -\frac {a (a+b) \text {csch}^2(c+d x)}{d}-\frac {(a+b)^2 \text {csch}^4(c+d x)}{4 d}-\frac {\left (b+a \cosh ^2(c+d x)\right )^3 \text {csch}^6(c+d x)}{6 (a+b) d}+\frac {a^2 \log (\sinh (c+d x))}{d} \]

[Out]

-a*(a+b)*csch(d*x+c)^2/d-1/4*(a+b)^2*csch(d*x+c)^4/d-1/6*(b+a*cosh(d*x+c)^2)^3*csch(d*x+c)^6/(a+b)/d+a^2*ln(si
nh(d*x+c))/d

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Rubi [A]
time = 0.09, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4223, 457, 79, 45} \begin {gather*} \frac {a^2 \log (\sinh (c+d x))}{d}-\frac {(a+b)^2 \text {csch}^4(c+d x)}{4 d}-\frac {a (a+b) \text {csch}^2(c+d x)}{d}-\frac {\text {csch}^6(c+d x) \left (a \cosh ^2(c+d x)+b\right )^3}{6 d (a+b)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[c + d*x]^7*(a + b*Sech[c + d*x]^2)^2,x]

[Out]

-((a*(a + b)*Csch[c + d*x]^2)/d) - ((a + b)^2*Csch[c + d*x]^4)/(4*d) - ((b + a*Cosh[c + d*x]^2)^3*Csch[c + d*x
]^6)/(6*(a + b)*d) + (a^2*Log[Sinh[c + d*x]])/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 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

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 \coth ^7(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int \frac {x^3 \left (b+a x^2\right )^2}{\left (1-x^2\right )^4} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {x (b+a x)^2}{(1-x)^4} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {\left (b+a \cosh ^2(c+d x)\right )^3 \text {csch}^6(c+d x)}{6 (a+b) d}-\frac {\text {Subst}\left (\int \frac {(b+a x)^2}{(1-x)^3} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {\left (b+a \cosh ^2(c+d x)\right )^3 \text {csch}^6(c+d x)}{6 (a+b) d}-\frac {\text {Subst}\left (\int \left (-\frac {(a+b)^2}{(-1+x)^3}-\frac {2 a (a+b)}{(-1+x)^2}-\frac {a^2}{-1+x}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=-\frac {a (a+b) \text {csch}^2(c+d x)}{d}-\frac {(a+b)^2 \text {csch}^4(c+d x)}{4 d}-\frac {\left (b+a \cosh ^2(c+d x)\right )^3 \text {csch}^6(c+d x)}{6 (a+b) d}+\frac {a^2 \log (\sinh (c+d x))}{d}\\ \end {align*}

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Mathematica [A]
time = 0.36, size = 107, normalized size = 1.24 \begin {gather*} -\frac {\left (b+a \cosh ^2(c+d x)\right )^2 \left (6 a (3 a+2 b) \text {csch}^2(c+d x)+3 \left (3 a^2+4 a b+b^2\right ) \text {csch}^4(c+d x)+2 (a+b)^2 \text {csch}^6(c+d x)-12 a^2 \log (\sinh (c+d x))\right )}{3 d (a+2 b+a \cosh (2 (c+d x)))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[c + d*x]^7*(a + b*Sech[c + d*x]^2)^2,x]

[Out]

-1/3*((b + a*Cosh[c + d*x]^2)^2*(6*a*(3*a + 2*b)*Csch[c + d*x]^2 + 3*(3*a^2 + 4*a*b + b^2)*Csch[c + d*x]^4 + 2
*(a + b)^2*Csch[c + d*x]^6 - 12*a^2*Log[Sinh[c + d*x]]))/(d*(a + 2*b + a*Cosh[2*(c + d*x)])^2)

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Maple [A]
time = 1.81, size = 132, normalized size = 1.53

method	result	size
derivativedivides	\(\frac {a^{2} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}-\frac {\left (\coth ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\coth ^{6}\left (d x +c \right )\right )}{6}\right )+2 a b \left (-\frac {\cosh ^{4}\left (d x +c \right )}{2 \sinh \left (d x +c \right )^{6}}+\frac {\cosh ^{2}\left (d x +c \right )}{2 \sinh \left (d x +c \right )^{6}}-\frac {1}{6 \sinh \left (d x +c \right )^{6}}\right )+b^{2} \left (-\frac {\cosh ^{2}\left (d x +c \right )}{4 \sinh \left (d x +c \right )^{6}}+\frac {1}{12 \sinh \left (d x +c \right )^{6}}\right )}{d}\)	\(132\)
default	\(\frac {a^{2} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\left (\coth ^{2}\left (d x +c \right )\right )}{2}-\frac {\left (\coth ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\coth ^{6}\left (d x +c \right )\right )}{6}\right )+2 a b \left (-\frac {\cosh ^{4}\left (d x +c \right )}{2 \sinh \left (d x +c \right )^{6}}+\frac {\cosh ^{2}\left (d x +c \right )}{2 \sinh \left (d x +c \right )^{6}}-\frac {1}{6 \sinh \left (d x +c \right )^{6}}\right )+b^{2} \left (-\frac {\cosh ^{2}\left (d x +c \right )}{4 \sinh \left (d x +c \right )^{6}}+\frac {1}{12 \sinh \left (d x +c \right )^{6}}\right )}{d}\)	\(132\)
risch	\(-a^{2} x -\frac {2 a^{2} c}{d}-\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (9 a^{2} {\mathrm e}^{8 d x +8 c}+6 a b \,{\mathrm e}^{8 d x +8 c}-18 a^{2} {\mathrm e}^{6 d x +6 c}+6 b^{2} {\mathrm e}^{6 d x +6 c}+34 a^{2} {\mathrm e}^{4 d x +4 c}+20 a b \,{\mathrm e}^{4 d x +4 c}+4 b^{2} {\mathrm e}^{4 d x +4 c}-18 a^{2} {\mathrm e}^{2 d x +2 c}+6 b^{2} {\mathrm e}^{2 d x +2 c}+9 a^{2}+6 a b \right )}{3 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{6}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a^{2}}{d}\)	\(197\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*x+c)^7*(a+b*sech(d*x+c)^2)^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(a^2*(ln(sinh(d*x+c))-1/2*coth(d*x+c)^2-1/4*coth(d*x+c)^4-1/6*coth(d*x+c)^6)+2*a*b*(-1/2/sinh(d*x+c)^6*cos
h(d*x+c)^4+1/2/sinh(d*x+c)^6*cosh(d*x+c)^2-1/6/sinh(d*x+c)^6)+b^2*(-1/4/sinh(d*x+c)^6*cosh(d*x+c)^2+1/12/sinh(
d*x+c)^6))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 696 vs. \(2 (82) = 164\).
time = 0.28, size = 696, normalized size = 8.09 \begin {gather*} \frac {1}{3} \, a^{2} {\left (3 \, x + \frac {3 \, c}{d} + \frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 18 \, e^{\left (-4 \, d x - 4 \, c\right )} + 34 \, e^{\left (-6 \, d x - 6 \, c\right )} - 18 \, e^{\left (-8 \, d x - 8 \, c\right )} + 9 \, e^{\left (-10 \, d x - 10 \, c\right )}\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}}\right )} + \frac {4}{3} \, a b {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}} + \frac {10 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}} + \frac {3 \, e^{\left (-10 \, d x - 10 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}}\right )} + \frac {4}{3} \, b^{2} {\left (\frac {3 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}} + \frac {2 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}} + \frac {3 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^7*(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

1/3*a^2*(3*x + 3*c/d + 3*log(e^(-d*x - c) + 1)/d + 3*log(e^(-d*x - c) - 1)/d + 2*(9*e^(-2*d*x - 2*c) - 18*e^(-
4*d*x - 4*c) + 34*e^(-6*d*x - 6*c) - 18*e^(-8*d*x - 8*c) + 9*e^(-10*d*x - 10*c))/(d*(6*e^(-2*d*x - 2*c) - 15*e
^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) - 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) - e^(-12*d*x - 12*c) - 1)))
+ 4/3*a*b*(3*e^(-2*d*x - 2*c)/(d*(6*e^(-2*d*x - 2*c) - 15*e^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) - 15*e^(-8*d
*x - 8*c) + 6*e^(-10*d*x - 10*c) - e^(-12*d*x - 12*c) - 1)) + 10*e^(-6*d*x - 6*c)/(d*(6*e^(-2*d*x - 2*c) - 15*
e^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) - 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) - e^(-12*d*x - 12*c) - 1))
+ 3*e^(-10*d*x - 10*c)/(d*(6*e^(-2*d*x - 2*c) - 15*e^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) - 15*e^(-8*d*x - 8*
c) + 6*e^(-10*d*x - 10*c) - e^(-12*d*x - 12*c) - 1))) + 4/3*b^2*(3*e^(-4*d*x - 4*c)/(d*(6*e^(-2*d*x - 2*c) - 1
5*e^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) - 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) - e^(-12*d*x - 12*c) - 1
)) + 2*e^(-6*d*x - 6*c)/(d*(6*e^(-2*d*x - 2*c) - 15*e^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) - 15*e^(-8*d*x - 8*
c) + 6*e^(-10*d*x - 10*c) - e^(-12*d*x - 12*c) - 1)) + 3*e^(-8*d*x - 8*c)/(d*(6*e^(-2*d*x - 2*c) - 15*e^(-4*d*
x - 4*c) + 20*e^(-6*d*x - 6*c) - 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) - e^(-12*d*x - 12*c) - 1)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2548 vs. \(2 (82) = 164\).
time = 0.48, size = 2548, normalized size = 29.63 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^7*(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

-1/3*(3*a^2*d*x*cosh(d*x + c)^12 + 36*a^2*d*x*cosh(d*x + c)*sinh(d*x + c)^11 + 3*a^2*d*x*sinh(d*x + c)^12 - 6*
(3*a^2*d*x - 3*a^2 - 2*a*b)*cosh(d*x + c)^10 + 6*(33*a^2*d*x*cosh(d*x + c)^2 - 3*a^2*d*x + 3*a^2 + 2*a*b)*sinh
(d*x + c)^10 + 60*(11*a^2*d*x*cosh(d*x + c)^3 - (3*a^2*d*x - 3*a^2 - 2*a*b)*cosh(d*x + c))*sinh(d*x + c)^9 + 3
*(15*a^2*d*x - 12*a^2 + 4*b^2)*cosh(d*x + c)^8 + 3*(495*a^2*d*x*cosh(d*x + c)^4 + 15*a^2*d*x - 90*(3*a^2*d*x -
3*a^2 - 2*a*b)*cosh(d*x + c)^2 - 12*a^2 + 4*b^2)*sinh(d*x + c)^8 + 24*(99*a^2*d*x*cosh(d*x + c)^5 - 30*(3*a^2
*d*x - 3*a^2 - 2*a*b)*cosh(d*x + c)^3 + (15*a^2*d*x - 12*a^2 + 4*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 4*(15*a
^2*d*x - 17*a^2 - 10*a*b - 2*b^2)*cosh(d*x + c)^6 + 4*(693*a^2*d*x*cosh(d*x + c)^6 - 315*(3*a^2*d*x - 3*a^2 -
2*a*b)*cosh(d*x + c)^4 - 15*a^2*d*x + 21*(15*a^2*d*x - 12*a^2 + 4*b^2)*cosh(d*x + c)^2 + 17*a^2 + 10*a*b + 2*b
^2)*sinh(d*x + c)^6 + 24*(99*a^2*d*x*cosh(d*x + c)^7 - 63*(3*a^2*d*x - 3*a^2 - 2*a*b)*cosh(d*x + c)^5 + 7*(15*
a^2*d*x - 12*a^2 + 4*b^2)*cosh(d*x + c)^3 - (15*a^2*d*x - 17*a^2 - 10*a*b - 2*b^2)*cosh(d*x + c))*sinh(d*x + c
)^5 + 3*(15*a^2*d*x - 12*a^2 + 4*b^2)*cosh(d*x + c)^4 + 3*(495*a^2*d*x*cosh(d*x + c)^8 - 420*(3*a^2*d*x - 3*a^
2 - 2*a*b)*cosh(d*x + c)^6 + 70*(15*a^2*d*x - 12*a^2 + 4*b^2)*cosh(d*x + c)^4 + 15*a^2*d*x - 20*(15*a^2*d*x -
17*a^2 - 10*a*b - 2*b^2)*cosh(d*x + c)^2 - 12*a^2 + 4*b^2)*sinh(d*x + c)^4 + 3*a^2*d*x + 4*(165*a^2*d*x*cosh(d
*x + c)^9 - 180*(3*a^2*d*x - 3*a^2 - 2*a*b)*cosh(d*x + c)^7 + 42*(15*a^2*d*x - 12*a^2 + 4*b^2)*cosh(d*x + c)^5
- 20*(15*a^2*d*x - 17*a^2 - 10*a*b - 2*b^2)*cosh(d*x + c)^3 + 3*(15*a^2*d*x - 12*a^2 + 4*b^2)*cosh(d*x + c))*
sinh(d*x + c)^3 - 6*(3*a^2*d*x - 3*a^2 - 2*a*b)*cosh(d*x + c)^2 + 6*(33*a^2*d*x*cosh(d*x + c)^10 - 45*(3*a^2*d
*x - 3*a^2 - 2*a*b)*cosh(d*x + c)^8 + 14*(15*a^2*d*x - 12*a^2 + 4*b^2)*cosh(d*x + c)^6 - 10*(15*a^2*d*x - 17*a
^2 - 10*a*b - 2*b^2)*cosh(d*x + c)^4 - 3*a^2*d*x + 3*(15*a^2*d*x - 12*a^2 + 4*b^2)*cosh(d*x + c)^2 + 3*a^2 + 2
*a*b)*sinh(d*x + c)^2 - 3*(a^2*cosh(d*x + c)^12 + 12*a^2*cosh(d*x + c)*sinh(d*x + c)^11 + a^2*sinh(d*x + c)^12
- 6*a^2*cosh(d*x + c)^10 + 6*(11*a^2*cosh(d*x + c)^2 - a^2)*sinh(d*x + c)^10 + 15*a^2*cosh(d*x + c)^8 + 20*(1
1*a^2*cosh(d*x + c)^3 - 3*a^2*cosh(d*x + c))*sinh(d*x + c)^9 + 15*(33*a^2*cosh(d*x + c)^4 - 18*a^2*cosh(d*x +
c)^2 + a^2)*sinh(d*x + c)^8 - 20*a^2*cosh(d*x + c)^6 + 24*(33*a^2*cosh(d*x + c)^5 - 30*a^2*cosh(d*x + c)^3 + 5
*a^2*cosh(d*x + c))*sinh(d*x + c)^7 + 4*(231*a^2*cosh(d*x + c)^6 - 315*a^2*cosh(d*x + c)^4 + 105*a^2*cosh(d*x
+ c)^2 - 5*a^2)*sinh(d*x + c)^6 + 15*a^2*cosh(d*x + c)^4 + 24*(33*a^2*cosh(d*x + c)^7 - 63*a^2*cosh(d*x + c)^5
+ 35*a^2*cosh(d*x + c)^3 - 5*a^2*cosh(d*x + c))*sinh(d*x + c)^5 + 15*(33*a^2*cosh(d*x + c)^8 - 84*a^2*cosh(d*
x + c)^6 + 70*a^2*cosh(d*x + c)^4 - 20*a^2*cosh(d*x + c)^2 + a^2)*sinh(d*x + c)^4 - 6*a^2*cosh(d*x + c)^2 + 20
*(11*a^2*cosh(d*x + c)^9 - 36*a^2*cosh(d*x + c)^7 + 42*a^2*cosh(d*x + c)^5 - 20*a^2*cosh(d*x + c)^3 + 3*a^2*co
sh(d*x + c))*sinh(d*x + c)^3 + 6*(11*a^2*cosh(d*x + c)^10 - 45*a^2*cosh(d*x + c)^8 + 70*a^2*cosh(d*x + c)^6 -
50*a^2*cosh(d*x + c)^4 + 15*a^2*cosh(d*x + c)^2 - a^2)*sinh(d*x + c)^2 + a^2 + 12*(a^2*cosh(d*x + c)^11 - 5*a^
2*cosh(d*x + c)^9 + 10*a^2*cosh(d*x + c)^7 - 10*a^2*cosh(d*x + c)^5 + 5*a^2*cosh(d*x + c)^3 - a^2*cosh(d*x + c
))*sinh(d*x + c))*log(2*sinh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 12*(3*a^2*d*x*cosh(d*x + c)^11 - 5*(3
*a^2*d*x - 3*a^2 - 2*a*b)*cosh(d*x + c)^9 + 2*(15*a^2*d*x - 12*a^2 + 4*b^2)*cosh(d*x + c)^7 - 2*(15*a^2*d*x -
17*a^2 - 10*a*b - 2*b^2)*cosh(d*x + c)^5 + (15*a^2*d*x - 12*a^2 + 4*b^2)*cosh(d*x + c)^3 - (3*a^2*d*x - 3*a^2
- 2*a*b)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^12 + 12*d*cosh(d*x + c)*sinh(d*x + c)^11 + d*sinh(d*x
+ c)^12 - 6*d*cosh(d*x + c)^10 + 6*(11*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^10 + 20*(11*d*cosh(d*x + c)^3 - 3*
d*cosh(d*x + c))*sinh(d*x + c)^9 + 15*d*cosh(d*x + c)^8 + 15*(33*d*cosh(d*x + c)^4 - 18*d*cosh(d*x + c)^2 + d)
*sinh(d*x + c)^8 + 24*(33*d*cosh(d*x + c)^5 - 30*d*cosh(d*x + c)^3 + 5*d*cosh(d*x + c))*sinh(d*x + c)^7 - 20*d
*cosh(d*x + c)^6 + 4*(231*d*cosh(d*x + c)^6 - 315*d*cosh(d*x + c)^4 + 105*d*cosh(d*x + c)^2 - 5*d)*sinh(d*x +
c)^6 + 24*(33*d*cosh(d*x + c)^7 - 63*d*cosh(d*x + c)^5 + 35*d*cosh(d*x + c)^3 - 5*d*cosh(d*x + c))*sinh(d*x +
c)^5 + 15*d*cosh(d*x + c)^4 + 15*(33*d*cosh(d*x + c)^8 - 84*d*cosh(d*x + c)^6 + 70*d*cosh(d*x + c)^4 - 20*d*co
sh(d*x + c)^2 + d)*sinh(d*x + c)^4 + 20*(11*d*cosh(d*x + c)^9 - 36*d*cosh(d*x + c)^7 + 42*d*cosh(d*x + c)^5 -
20*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^3 - 6*d*cosh(d*x + c)^2 + 6*(11*d*cosh(d*x + c)^10 - 4
5*d*cosh(d*x + c)^8 + 70*d*cosh(d*x + c)^6 - 50*d*cosh(d*x + c)^4 + 15*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^2
+ 12*(d*cosh(d*x + c)^11 - 5*d*cosh(d*x + c)^9 + 10*d*cosh(d*x + c)^7 - 10*d*cosh(d*x + c)^5 + 5*d*cosh(d*x +
c)^3 - d*cosh(d*x + c))*sinh(d*x + c) + d)

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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(coth(d*x+c)**7*(a+b*sech(d*x+c)**2)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (82) = 164\).
time = 0.49, size = 219, normalized size = 2.55 \begin {gather*} -\frac {60 \, {\left (d x + c\right )} a^{2} - 60 \, a^{2} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac {147 \, a^{2} e^{\left (12 \, d x + 12 \, c\right )} - 522 \, a^{2} e^{\left (10 \, d x + 10 \, c\right )} + 240 \, a b e^{\left (10 \, d x + 10 \, c\right )} + 1485 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} + 240 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 1580 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} + 800 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 160 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 1485 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 240 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 522 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 240 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 147 \, a^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{6}}}{60 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^7*(a+b*sech(d*x+c)^2)^2,x, algorithm="giac")

[Out]

-1/60*(60*(d*x + c)*a^2 - 60*a^2*log(abs(e^(2*d*x + 2*c) - 1)) + (147*a^2*e^(12*d*x + 12*c) - 522*a^2*e^(10*d*
x + 10*c) + 240*a*b*e^(10*d*x + 10*c) + 1485*a^2*e^(8*d*x + 8*c) + 240*b^2*e^(8*d*x + 8*c) - 1580*a^2*e^(6*d*x
+ 6*c) + 800*a*b*e^(6*d*x + 6*c) + 160*b^2*e^(6*d*x + 6*c) + 1485*a^2*e^(4*d*x + 4*c) + 240*b^2*e^(4*d*x + 4*
c) - 522*a^2*e^(2*d*x + 2*c) + 240*a*b*e^(2*d*x + 2*c) + 147*a^2)/(e^(2*d*x + 2*c) - 1)^6)/d

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Mupad [B]
time = 1.53, size = 377, normalized size = 4.38 \begin {gather*} \frac {a^2\,\ln \left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-1\right )}{d}-\frac {32\,\left (a^2+2\,a\,b+b^2\right )}{d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1\right )}-\frac {2\,\left (3\,a^2+2\,b\,a\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {32\,\left (a^2+2\,a\,b+b^2\right )}{3\,d\,\left (15\,{\mathrm {e}}^{4\,c+4\,d\,x}-6\,{\mathrm {e}}^{2\,c+2\,d\,x}-20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}-6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1\right )}-\frac {2\,\left (9\,a^2+10\,a\,b+2\,b^2\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {8\,\left (13\,a^2+20\,a\,b+7\,b^2\right )}{3\,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\,\left (11\,a^2+20\,a\,b+9\,b^2\right )}{d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-a^2\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(c + d*x)^7*(a + b/cosh(c + d*x)^2)^2,x)

[Out]

(a^2*log(exp(2*c)*exp(2*d*x) - 1))/d - (32*(2*a*b + a^2 + b^2))/(d*(5*exp(2*c + 2*d*x) - 10*exp(4*c + 4*d*x) +
10*exp(6*c + 6*d*x) - 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) - 1)) - (2*(2*a*b + 3*a^2))/(d*(exp(2*c + 2*d*x
) - 1)) - (32*(2*a*b + a^2 + b^2))/(3*d*(15*exp(4*c + 4*d*x) - 6*exp(2*c + 2*d*x) - 20*exp(6*c + 6*d*x) + 15*e
xp(8*c + 8*d*x) - 6*exp(10*c + 10*d*x) + exp(12*c + 12*d*x) + 1)) - (2*(10*a*b + 9*a^2 + 2*b^2))/(d*(exp(4*c +
4*d*x) - 2*exp(2*c + 2*d*x) + 1)) - (8*(20*a*b + 13*a^2 + 7*b^2))/(3*d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*
x) + exp(6*c + 6*d*x) - 1)) - (4*(20*a*b + 11*a^2 + 9*b^2))/(d*(6*exp(4*c + 4*d*x) - 4*exp(2*c + 2*d*x) - 4*ex
p(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1)) - a^2*x

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