3.21 \(\int \text{csch}(c+d x) (a+b \text{sech}^2(c+d x))^3 \, dx\)

Optimal. Leaf size=83 \[ \frac{b \left (3 a^2+3 a b+b^2\right ) \text{sech}(c+d x)}{d}+\frac{b^2 (3 a+b) \text{sech}^3(c+d x)}{3 d}-\frac{(a+b)^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b^3 \text{sech}^5(c+d x)}{5 d} \]

[Out]

-(((a + b)^3*ArcTanh[Cosh[c + d*x]])/d) + (b*(3*a^2 + 3*a*b + b^2)*Sech[c + d*x])/d + (b^2*(3*a + b)*Sech[c +
d*x]^3)/(3*d) + (b^3*Sech[c + d*x]^5)/(5*d)

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Rubi [A]  time = 0.10033, antiderivative size = 83, normalized size of antiderivative = 1., 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 = {4133, 461, 207} \[ \frac{b \left (3 a^2+3 a b+b^2\right ) \text{sech}(c+d x)}{d}+\frac{b^2 (3 a+b) \text{sech}^3(c+d x)}{3 d}-\frac{(a+b)^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b^3 \text{sech}^5(c+d x)}{5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((a + b)^3*ArcTanh[Cosh[c + d*x]])/d) + (b*(3*a^2 + 3*a*b + b^2)*Sech[c + d*x])/d + (b^2*(3*a + b)*Sech[c +
d*x]^3)/(3*d) + (b^3*Sech[c + d*x]^5)/(5*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 461

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[((e*x)^m*(a + b*x^n)^p)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \text{csch}(c+d x) \left (a+b \text{sech}^2(c+d x)\right )^3 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (b+a x^2\right )^3}{x^6 \left (1-x^2\right )} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{b^3}{x^6}+\frac{b^2 (3 a+b)}{x^4}+\frac{b \left (3 a^2+3 a b+b^2\right )}{x^2}-\frac{(a+b)^3}{-1+x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{b \left (3 a^2+3 a b+b^2\right ) \text{sech}(c+d x)}{d}+\frac{b^2 (3 a+b) \text{sech}^3(c+d x)}{3 d}+\frac{b^3 \text{sech}^5(c+d x)}{5 d}+\frac{(a+b)^3 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{(a+b)^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b \left (3 a^2+3 a b+b^2\right ) \text{sech}(c+d x)}{d}+\frac{b^2 (3 a+b) \text{sech}^3(c+d x)}{3 d}+\frac{b^3 \text{sech}^5(c+d x)}{5 d}\\ \end{align*}

Mathematica [A]  time = 1.21971, size = 134, normalized size = 1.61 \[ -\frac{8 \text{sech}^5(c+d x) \left (a \cosh ^2(c+d x)+b\right )^3 \left (-15 b \left (3 a^2+3 a b+b^2\right ) \cosh ^4(c+d x)-5 b^2 (3 a+b) \cosh ^2(c+d x)+15 (a+b)^3 \cosh ^5(c+d x) \left (\log \left (\cosh \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sinh \left (\frac{1}{2} (c+d x)\right )\right )\right )-3 b^3\right )}{15 d (a \cosh (2 (c+d x))+a+2 b)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*(b + a*Cosh[c + d*x]^2)^3*(-3*b^3 - 5*b^2*(3*a + b)*Cosh[c + d*x]^2 - 15*b*(3*a^2 + 3*a*b + b^2)*Cosh[c +
d*x]^4 + 15*(a + b)^3*Cosh[c + d*x]^5*(Log[Cosh[(c + d*x)/2]] - Log[Sinh[(c + d*x)/2]]))*Sech[c + d*x]^5)/(15*
d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)

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Maple [A]  time = 0.04, size = 118, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ( -2\,{a}^{3}{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) +3\,{a}^{2}b \left ( \left ( \cosh \left ( dx+c \right ) \right ) ^{-1}-2\,{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \right ) +3\,a{b}^{2} \left ( 1/3\, \left ( \cosh \left ( dx+c \right ) \right ) ^{-3}+ \left ( \cosh \left ( dx+c \right ) \right ) ^{-1}-2\,{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \right ) +{b}^{3} \left ({\frac{1}{5\, \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}+{\frac{1}{3\, \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}+ \left ( \cosh \left ( dx+c \right ) \right ) ^{-1}-2\,{\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)*(a+b*sech(d*x+c)^2)^3,x)

[Out]

1/d*(-2*a^3*arctanh(exp(d*x+c))+3*a^2*b*(1/cosh(d*x+c)-2*arctanh(exp(d*x+c)))+3*a*b^2*(1/3/cosh(d*x+c)^3+1/cos
h(d*x+c)-2*arctanh(exp(d*x+c)))+b^3*(1/5/cosh(d*x+c)^5+1/3/cosh(d*x+c)^3+1/cosh(d*x+c)-2*arctanh(exp(d*x+c))))

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Maxima [B]  time = 1.10028, size = 483, normalized size = 5.82 \begin{align*} -\frac{1}{15} \, b^{3}{\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 )} + 80 \, e^{\left (-3 \, d x - 3 \, c\right )} + 178 \, e^{\left (-5 \, d x - 5 \, c\right )} + 80 \, e^{\left (-7 \, d x - 7 \, c\right )} + 15 \, e^{\left (-9 \, d x - 9 \, c\right )}\right )}}{d{\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} - a b^{2}{\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 )} + 10 \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-5 \, d x - 5 \, c\right )}\right )}}{d{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} - 3 \, a^{2} b{\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 \, e^{\left (-d x - c\right )}}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} + \frac{a^{3} \log \left (\tanh \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*b^3*(15*log(e^(-d*x - c) + 1)/d - 15*log(e^(-d*x - c) - 1)/d - 2*(15*e^(-d*x - c) + 80*e^(-3*d*x - 3*c)
+ 178*e^(-5*d*x - 5*c) + 80*e^(-7*d*x - 7*c) + 15*e^(-9*d*x - 9*c))/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*
c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1))) - a*b^2*(3*log(e^(-d*x - c) + 1)/d -
3*log(e^(-d*x - c) - 1)/d - 2*(3*e^(-d*x - c) + 10*e^(-3*d*x - 3*c) + 3*e^(-5*d*x - 5*c))/(d*(3*e^(-2*d*x - 2
*c) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) + 1))) - 3*a^2*b*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/
d - 2*e^(-d*x - c)/(d*(e^(-2*d*x - 2*c) + 1))) + a^3*log(tanh(1/2*d*x + 1/2*c))/d

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Fricas [B]  time = 3.09183, size = 8645, normalized size = 104.16 \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)*(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

1/15*(30*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^9 + 270*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)*sinh(d*x + c)
^8 + 30*(3*a^2*b + 3*a*b^2 + b^3)*sinh(d*x + c)^9 + 40*(9*a^2*b + 12*a*b^2 + 4*b^3)*cosh(d*x + c)^7 + 40*(9*a^
2*b + 12*a*b^2 + 4*b^3 + 27*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 280*(9*(3*a^2*b + 3*a
*b^2 + b^3)*cosh(d*x + c)^3 + (9*a^2*b + 12*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 + 4*(135*a^2*b + 195
*a*b^2 + 89*b^3)*cosh(d*x + c)^5 + 4*(945*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 + 135*a^2*b + 195*a*b^2 +
89*b^3 + 210*(9*a^2*b + 12*a*b^2 + 4*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 20*(189*(3*a^2*b + 3*a*b^2 + b^3)
*cosh(d*x + c)^5 + 70*(9*a^2*b + 12*a*b^2 + 4*b^3)*cosh(d*x + c)^3 + (135*a^2*b + 195*a*b^2 + 89*b^3)*cosh(d*x
+ c))*sinh(d*x + c)^4 + 40*(9*a^2*b + 12*a*b^2 + 4*b^3)*cosh(d*x + c)^3 + 40*(63*(3*a^2*b + 3*a*b^2 + b^3)*co
sh(d*x + c)^6 + 35*(9*a^2*b + 12*a*b^2 + 4*b^3)*cosh(d*x + c)^4 + 9*a^2*b + 12*a*b^2 + 4*b^3 + (135*a^2*b + 19
5*a*b^2 + 89*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 40*(27*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^7 + 21*(9*
a^2*b + 12*a*b^2 + 4*b^3)*cosh(d*x + c)^5 + (135*a^2*b + 195*a*b^2 + 89*b^3)*cosh(d*x + c)^3 + 3*(9*a^2*b + 12
*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + 30*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c) - 15*((a^3 + 3*a^2
*b + 3*a*b^2 + b^3)*cosh(d*x + c)^10 + 10*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)*sinh(d*x + c)^9 + (a^3
+ 3*a^2*b + 3*a*b^2 + b^3)*sinh(d*x + c)^10 + 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^8 + 5*(a^3 + 3*
a^2*b + 3*a*b^2 + b^3 + 9*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 40*(3*(a^3 + 3*a^
2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^3 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^7 + 10*(a^
3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^6 + 10*(21*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 + a^3 +
3*a^2*b + 3*a*b^2 + b^3 + 14*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*(a^3 + 3
*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^5 + 70*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^3 + 15*(a^3 + 3*a^2
*b + 3*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 10*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 + 10*(
21*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^6 + 35*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 + a^3
+ 3*a^2*b + 3*a*b^2 + b^3 + 15*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 40*(3*(a^3 +
3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^7 + 7*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^5 + 5*(a^3 + 3*a^2
*b + 3*a*b^2 + b^3)*cosh(d*x + c)^3 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + a^3 + 3
*a^2*b + 3*a*b^2 + b^3 + 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2 + 5*(9*(a^3 + 3*a^2*b + 3*a*b^2 + b
^3)*cosh(d*x + c)^8 + 28*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^6 + 30*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*
cosh(d*x + c)^4 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 + 12*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x
+ c)^2 + 10*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^9 + 4*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c
)^7 + 6*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^5 + 4*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^3 +
(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) + 15*((a^
3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^10 + 10*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)*sinh(d*x + c)
^9 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sinh(d*x + c)^10 + 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^8 + 5*
(a^3 + 3*a^2*b + 3*a*b^2 + b^3 + 9*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 40*(3*(a
^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^3 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^7
+ 10*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^6 + 10*(21*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4
+ a^3 + 3*a^2*b + 3*a*b^2 + b^3 + 14*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63
*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^5 + 70*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^3 + 15*(a^
3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 10*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)
^4 + 10*(21*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^6 + 35*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)
^4 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 + 15*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 40*
(3*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^7 + 7*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^5 + 5*(a^
3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^3 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^3
+ a^3 + 3*a^2*b + 3*a*b^2 + b^3 + 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2 + 5*(9*(a^3 + 3*a^2*b + 3*
a*b^2 + b^3)*cosh(d*x + c)^8 + 28*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^6 + 30*(a^3 + 3*a^2*b + 3*a*b^
2 + b^3)*cosh(d*x + c)^4 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 + 12*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^2)
*sinh(d*x + c)^2 + 10*((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^9 + 4*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos
h(d*x + c)^7 + 6*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^5 + 4*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x
+ c)^3 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1)
+ 10*(27*(3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^8 + 28*(9*a^2*b + 12*a*b^2 + 4*b^3)*cosh(d*x + c)^6 + 2*(135*
a^2*b + 195*a*b^2 + 89*b^3)*cosh(d*x + c)^4 + 9*a^2*b + 9*a*b^2 + 3*b^3 + 12*(9*a^2*b + 12*a*b^2 + 4*b^3)*cosh
(d*x + c)^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 + 5*
d*cosh(d*x + c)^8 + 5*(9*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^8 + 40*(3*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*s
inh(d*x + c)^7 + 10*d*cosh(d*x + c)^6 + 10*(21*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 + 70*d*cosh(d*x + c)^3 + 15*d*cosh(d*x + c))*sinh(d*x + c)^5 + 10*d*cosh(d*x + c)^4 +
10*(21*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 + 40*(3*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 + 5*d*cosh(d*x + c)^2
+ 5*(9*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 + 10*(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 )^{3} \operatorname{csch}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*sech(c + d*x)**2)**3*csch(c + d*x), x)

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Giac [B]  time = 1.18934, size = 313, normalized size = 3.77 \begin{align*} -\frac{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right )}{2 \, d} + \frac{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right )}{2 \, d} + \frac{2 \,{\left (45 \, a^{2} b{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} + 45 \, a b^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} + 15 \, b^{3}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} + 60 \, a b^{2}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 20 \, b^{3}{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 48 \, b^{3}\right )}}{15 \, d{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)*(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")

[Out]

-1/2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*log(e^(d*x + c) + e^(-d*x - c) + 2)/d + 1/2*(a^3 + 3*a^2*b + 3*a*b^2 + b^
3)*log(e^(d*x + c) + e^(-d*x - c) - 2)/d + 2/15*(45*a^2*b*(e^(d*x + c) + e^(-d*x - c))^4 + 45*a*b^2*(e^(d*x +
c) + e^(-d*x - c))^4 + 15*b^3*(e^(d*x + c) + e^(-d*x - c))^4 + 60*a*b^2*(e^(d*x + c) + e^(-d*x - c))^2 + 20*b^
3*(e^(d*x + c) + e^(-d*x - c))^2 + 48*b^3)/(d*(e^(d*x + c) + e^(-d*x - c))^5)