3.15 \(\int \text{csch}^3(c+d x) (a+b \text{sech}^2(c+d x))^2 \, dx\)
Optimal. Leaf size=104 \[ -\frac{\left (3 a^2+6 a b+5 b^2\right ) \coth (c+d x) \text{csch}(c+d x)}{6 d}-\frac{b (6 a+5 b) \text{sech}(c+d x)}{3 d}+\frac{(a+b) (a+5 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}+\frac{b^2 \text{csch}^2(c+d x) \text{sech}^3(c+d x)}{3 d} \]
[Out]
((a + b)*(a + 5*b)*ArcTanh[Cosh[c + d*x]])/(2*d) - ((3*a^2 + 6*a*b + 5*b^2)*Coth[c + d*x]*Csch[c + d*x])/(6*d)
- (b*(6*a + 5*b)*Sech[c + d*x])/(3*d) + (b^2*Csch[c + d*x]^2*Sech[c + d*x]^3)/(3*d)
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Rubi [A] time = 0.135715, antiderivative size = 104, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used =
{4133, 462, 456, 453, 206} \[ -\frac{\left (3 a^2+6 a b+5 b^2\right ) \coth (c+d x) \text{csch}(c+d x)}{6 d}-\frac{b (6 a+5 b) \text{sech}(c+d x)}{3 d}+\frac{(a+b) (a+5 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}+\frac{b^2 \text{csch}^2(c+d x) \text{sech}^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
[In]
Int[Csch[c + d*x]^3*(a + b*Sech[c + d*x]^2)^2,x]
[Out]
((a + b)*(a + 5*b)*ArcTanh[Cosh[c + d*x]])/(2*d) - ((3*a^2 + 6*a*b + 5*b^2)*Coth[c + d*x]*Csch[c + d*x])/(6*d)
- (b*(6*a + 5*b)*Sech[c + d*x])/(3*d) + (b^2*Csch[c + d*x]^2*Sech[c + d*x]^3)/(3*d)
Rule 4133
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = F
reeFactors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[((1 - ff^2*x^2)^((m - 1)/2)*(b + a*(ff*x)^n)^p)/(ff*x)^(n*
p), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p
]
Rule 462
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^2, x_Symbol] :> Simp[(c^2*(e*x)^(
m + 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*x^n)^p*Simp[b
*c^2*n*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*(m + 1)*d^2*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && Ne
Q[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && GtQ[n, 0]
Rule 456
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[x^m*(a + b*x^2)^(p +
1)*ExpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)]
- ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &
& ILtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
Rule 453
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \text{csch}^3(c+d x) \left (a+b \text{sech}^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (b+a x^2\right )^2}{x^4 \left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{b^2 \text{csch}^2(c+d x) \text{sech}^3(c+d x)}{3 d}+\frac{\operatorname{Subst}\left (\int \frac{b (6 a+5 b)+3 a^2 x^2}{x^2 \left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{3 d}\\ &=-\frac{\left (3 a^2+6 a b+5 b^2\right ) \coth (c+d x) \text{csch}(c+d x)}{6 d}+\frac{b^2 \text{csch}^2(c+d x) \text{sech}^3(c+d x)}{3 d}-\frac{\operatorname{Subst}\left (\int \frac{-2 b (6 a+5 b)-\left (3 a^2+6 a b+5 b^2\right ) x^2}{x^2 \left (1-x^2\right )} \, dx,x,\cosh (c+d x)\right )}{6 d}\\ &=-\frac{\left (3 a^2+6 a b+5 b^2\right ) \coth (c+d x) \text{csch}(c+d x)}{6 d}-\frac{b (6 a+5 b) \text{sech}(c+d x)}{3 d}+\frac{b^2 \text{csch}^2(c+d x) \text{sech}^3(c+d x)}{3 d}+\frac{((a+b) (a+5 b)) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{2 d}\\ &=\frac{(a+b) (a+5 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{\left (3 a^2+6 a b+5 b^2\right ) \coth (c+d x) \text{csch}(c+d x)}{6 d}-\frac{b (6 a+5 b) \text{sech}(c+d x)}{3 d}+\frac{b^2 \text{csch}^2(c+d x) \text{sech}^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 1.53897, size = 144, normalized size = 1.38 \[ -\frac{\text{sech}^3(c+d x) \left (a \cosh ^2(c+d x)+b\right )^2 \left (48 b (a+b) \cosh ^2(c+d x)+3 (a+b) \cosh ^3(c+d x) \left ((a+b) \text{csch}^2\left (\frac{1}{2} (c+d x)\right )+(a+b) \text{sech}^2\left (\frac{1}{2} (c+d x)\right )-4 (a+5 b) \left (\log \left (\cosh \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sinh \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )+8 b^2\right )}{6 d (a \cosh (2 (c+d x))+a+2 b)^2} \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[c + d*x]^3*(a + b*Sech[c + d*x]^2)^2,x]
[Out]
-((b + a*Cosh[c + d*x]^2)^2*(8*b^2 + 48*b*(a + b)*Cosh[c + d*x]^2 + 3*(a + b)*Cosh[c + d*x]^3*((a + b)*Csch[(c
+ d*x)/2]^2 - 4*(a + 5*b)*(Log[Cosh[(c + d*x)/2]] - Log[Sinh[(c + d*x)/2]]) + (a + b)*Sech[(c + d*x)/2]^2))*S
ech[c + d*x]^3)/(6*d*(a + 2*b + a*Cosh[2*(c + d*x)])^2)
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Maple [A] time = 0.046, size = 126, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( -{\frac{{\rm csch} \left (dx+c\right ){\rm coth} \left (dx+c\right )}{2}}+{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \right ) +2\,ab \left ( -1/2\,{\frac{1}{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}\cosh \left ( dx+c \right ) }}-3/2\, \left ( \cosh \left ( dx+c \right ) \right ) ^{-1}+3\,{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \right ) +{b}^{2} \left ( -{\frac{1}{2\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2} \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}-{\frac{5}{6\, \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}-{\frac{5}{2\,\cosh \left ( dx+c \right ) }}+5\,{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(d*x+c)^3*(a+b*sech(d*x+c)^2)^2,x)
[Out]
1/d*(a^2*(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp(d*x+c)))+2*a*b*(-1/2/sinh(d*x+c)^2/cosh(d*x+c)-3/2/cosh(d*x
+c)+3*arctanh(exp(d*x+c)))+b^2*(-1/2/sinh(d*x+c)^2/cosh(d*x+c)^3-5/6/cosh(d*x+c)^3-5/2/cosh(d*x+c)+5*arctanh(e
xp(d*x+c))))
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Maxima [B] time = 1.09309, size = 478, normalized size = 4.6 \begin{align*} \frac{1}{6} \, b^{2}{\left (\frac{15 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac{15 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac{2 \,{\left (15 \, e^{\left (-d x - c\right )} + 20 \, e^{\left (-3 \, d x - 3 \, c\right )} - 22 \, e^{\left (-5 \, d x - 5 \, c\right )} + 20 \, e^{\left (-7 \, d x - 7 \, c\right )} + 15 \, e^{\left (-9 \, d x - 9 \, c\right )}\right )}}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-4 \, d x - 4 \, c\right )} - 2 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} + a b{\left (\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 (3 \, e^{\left (-d x - c\right )} - 2 \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-5 \, d x - 5 \, c\right )}\right )}}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} - e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + \frac{1}{2} \, a^{2}{\left (\frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac{2 \,{\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^3*(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")
[Out]
1/6*b^2*(15*log(e^(-d*x - c) + 1)/d - 15*log(e^(-d*x - c) - 1)/d - 2*(15*e^(-d*x - c) + 20*e^(-3*d*x - 3*c) -
22*e^(-5*d*x - 5*c) + 20*e^(-7*d*x - 7*c) + 15*e^(-9*d*x - 9*c))/(d*(e^(-2*d*x - 2*c) - 2*e^(-4*d*x - 4*c) - 2
*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1))) + a*b*(3*log(e^(-d*x - c) + 1)/d - 3*log(e^(-
d*x - c) - 1)/d + 2*(3*e^(-d*x - c) - 2*e^(-3*d*x - 3*c) + 3*e^(-5*d*x - 5*c))/(d*(e^(-2*d*x - 2*c) + e^(-4*d*
x - 4*c) - e^(-6*d*x - 6*c) - 1))) + 1/2*a^2*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d + 2*(e^(-d*x -
c) + e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c) - 1)))
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Fricas [B] time = 2.97845, size = 7422, normalized size = 71.37 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^3*(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")
[Out]
-1/6*(6*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^9 + 54*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)*sinh(d*x + c)^8 + 6*(a^
2 + 6*a*b + 5*b^2)*sinh(d*x + c)^9 + 8*(3*a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^7 + 8*(27*(a^2 + 6*a*b + 5*b^2)*c
osh(d*x + c)^2 + 3*a^2 + 6*a*b + 5*b^2)*sinh(d*x + c)^7 + 56*(9*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^3 + (3*a^2
+ 6*a*b + 5*b^2)*cosh(d*x + c))*sinh(d*x + c)^6 + 4*(9*a^2 + 6*a*b - 11*b^2)*cosh(d*x + c)^5 + 4*(189*(a^2 +
6*a*b + 5*b^2)*cosh(d*x + c)^4 + 42*(3*a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^2 + 9*a^2 + 6*a*b - 11*b^2)*sinh(d*x
+ c)^5 + 4*(189*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^5 + 70*(3*a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^3 + 5*(9*a^2
+ 6*a*b - 11*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 + 8*(3*a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^3 + 8*(63*(a^2 + 6
*a*b + 5*b^2)*cosh(d*x + c)^6 + 35*(3*a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^4 + 5*(9*a^2 + 6*a*b - 11*b^2)*cosh(d
*x + c)^2 + 3*a^2 + 6*a*b + 5*b^2)*sinh(d*x + c)^3 + 8*(27*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^7 + 21*(3*a^2 +
6*a*b + 5*b^2)*cosh(d*x + c)^5 + 5*(9*a^2 + 6*a*b - 11*b^2)*cosh(d*x + c)^3 + 3*(3*a^2 + 6*a*b + 5*b^2)*cosh(
d*x + c))*sinh(d*x + c)^2 + 6*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c) - 3*((a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^10
+ 10*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)*sinh(d*x + c)^9 + (a^2 + 6*a*b + 5*b^2)*sinh(d*x + c)^10 + (a^2 + 6*a
*b + 5*b^2)*cosh(d*x + c)^8 + (45*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^2 + a^2 + 6*a*b + 5*b^2)*sinh(d*x + c)^8
+ 8*(15*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^3 + (a^2 + 6*a*b + 5*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 2*(a^2
+ 6*a*b + 5*b^2)*cosh(d*x + c)^6 + 2*(105*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^4 + 14*(a^2 + 6*a*b + 5*b^2)*co
sh(d*x + c)^2 - a^2 - 6*a*b - 5*b^2)*sinh(d*x + c)^6 + 4*(63*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^5 + 14*(a^2 +
6*a*b + 5*b^2)*cosh(d*x + c)^3 - 3*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(a^2 + 6*a*b + 5*
b^2)*cosh(d*x + c)^4 + 2*(105*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^6 + 35*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^4
- 15*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^2 - a^2 - 6*a*b - 5*b^2)*sinh(d*x + c)^4 + 8*(15*(a^2 + 6*a*b + 5*b^
2)*cosh(d*x + c)^7 + 7*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^5 - 5*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^3 - (a^2
+ 6*a*b + 5*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + (a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^2 + (45*(a^2 + 6*a*b + 5
*b^2)*cosh(d*x + c)^8 + 28*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^6 - 30*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^4 -
12*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^2 + a^2 + 6*a*b + 5*b^2)*sinh(d*x + c)^2 + a^2 + 6*a*b + 5*b^2 + 2*(5*(
a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^9 + 4*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^7 - 6*(a^2 + 6*a*b + 5*b^2)*cosh(
d*x + c)^5 - 4*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^3 + (a^2 + 6*a*b + 5*b^2)*cosh(d*x + c))*sinh(d*x + c))*log
(cosh(d*x + c) + sinh(d*x + c) + 1) + 3*((a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^10 + 10*(a^2 + 6*a*b + 5*b^2)*cos
h(d*x + c)*sinh(d*x + c)^9 + (a^2 + 6*a*b + 5*b^2)*sinh(d*x + c)^10 + (a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^8 +
(45*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^2 + a^2 + 6*a*b + 5*b^2)*sinh(d*x + c)^8 + 8*(15*(a^2 + 6*a*b + 5*b^2)
*cosh(d*x + c)^3 + (a^2 + 6*a*b + 5*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 2*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c
)^6 + 2*(105*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^4 + 14*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^2 - a^2 - 6*a*b -
5*b^2)*sinh(d*x + c)^6 + 4*(63*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^5 + 14*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^
3 - 3*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^4 + 2*(105*
(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^6 + 35*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^4 - 15*(a^2 + 6*a*b + 5*b^2)*co
sh(d*x + c)^2 - a^2 - 6*a*b - 5*b^2)*sinh(d*x + c)^4 + 8*(15*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^7 + 7*(a^2 +
6*a*b + 5*b^2)*cosh(d*x + c)^5 - 5*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^3 - (a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)
)*sinh(d*x + c)^3 + (a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^2 + (45*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^8 + 28*(a^
2 + 6*a*b + 5*b^2)*cosh(d*x + c)^6 - 30*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^4 - 12*(a^2 + 6*a*b + 5*b^2)*cosh(
d*x + c)^2 + a^2 + 6*a*b + 5*b^2)*sinh(d*x + c)^2 + a^2 + 6*a*b + 5*b^2 + 2*(5*(a^2 + 6*a*b + 5*b^2)*cosh(d*x
+ c)^9 + 4*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^7 - 6*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^5 - 4*(a^2 + 6*a*b +
5*b^2)*cosh(d*x + c)^3 + (a^2 + 6*a*b + 5*b^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c)
- 1) + 2*(27*(a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^8 + 28*(3*a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^6 + 10*(9*a^2 +
6*a*b - 11*b^2)*cosh(d*x + c)^4 + 12*(3*a^2 + 6*a*b + 5*b^2)*cosh(d*x + c)^2 + 3*a^2 + 18*a*b + 15*b^2)*sinh(
d*x + c))/(d*cosh(d*x + c)^10 + 10*d*cosh(d*x + c)*sinh(d*x + c)^9 + d*sinh(d*x + c)^10 + d*cosh(d*x + c)^8 +
(45*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^8 + 8*(15*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c)^7 - 2*d*
cosh(d*x + c)^6 + 2*(105*d*cosh(d*x + c)^4 + 14*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^6 + 4*(63*d*cosh(d*x + c)
^5 + 14*d*cosh(d*x + c)^3 - 3*d*cosh(d*x + c))*sinh(d*x + c)^5 - 2*d*cosh(d*x + c)^4 + 2*(105*d*cosh(d*x + c)^
6 + 35*d*cosh(d*x + c)^4 - 15*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^4 + 8*(15*d*cosh(d*x + c)^7 + 7*d*cosh(d*x
+ c)^5 - 5*d*cosh(d*x + c)^3 - d*cosh(d*x + c))*sinh(d*x + c)^3 + d*cosh(d*x + c)^2 + (45*d*cosh(d*x + c)^8 +
28*d*cosh(d*x + c)^6 - 30*d*cosh(d*x + c)^4 - 12*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^2 + 2*(5*d*cosh(d*x + c)
^9 + 4*d*cosh(d*x + c)^7 - 6*d*cosh(d*x + c)^5 - 4*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c) + d)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{sech}^{2}{\left (c + d x \right )}\right )^{2} \operatorname{csch}^{3}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)**3*(a+b*sech(d*x+c)**2)**2,x)
[Out]
Integral((a + b*sech(c + d*x)**2)**2*csch(c + d*x)**3, x)
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Giac [B] time = 1.22353, size = 317, normalized size = 3.05 \begin{align*} \frac{{\left (a^{2} + 6 \, a b + 5 \, b^{2}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right )}{4 \, d} - \frac{{\left (a^{2} + 6 \, a b + 5 \, b^{2}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right )}{4 \, d} - \frac{a^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 2 \, a b{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + b^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4\right )} d} - \frac{4 \,{\left (3 \, a b{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 3 \, b^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 2 \, b^{2}\right )}}{3 \, d{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(d*x+c)^3*(a+b*sech(d*x+c)^2)^2,x, algorithm="giac")
[Out]
1/4*(a^2 + 6*a*b + 5*b^2)*log(e^(d*x + c) + e^(-d*x - c) + 2)/d - 1/4*(a^2 + 6*a*b + 5*b^2)*log(e^(d*x + c) +
e^(-d*x - c) - 2)/d - (a^2*(e^(d*x + c) + e^(-d*x - c)) + 2*a*b*(e^(d*x + c) + e^(-d*x - c)) + b^2*(e^(d*x + c
) + e^(-d*x - c)))/(((e^(d*x + c) + e^(-d*x - c))^2 - 4)*d) - 4/3*(3*a*b*(e^(d*x + c) + e^(-d*x - c))^2 + 3*b^
2*(e^(d*x + c) + e^(-d*x - c))^2 + 2*b^2)/(d*(e^(d*x + c) + e^(-d*x - c))^3)