### 3.23 $$\int \text{csch}^3(c+d x) (a+b \tanh ^2(c+d x))^3 \, dx$$

Optimal. Leaf size=152 $\frac{b \left (81 a^2-28 a b-4 b^2\right ) \text{sech}(c+d x)}{30 d}+\frac{a^2 (a-6 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}+\frac{7 b \text{sech}(c+d x) \left (a-b \text{sech}^2(c+d x)+b\right )^2}{10 d}+\frac{b (33 a-2 b) \text{sech}(c+d x) \left (a-b \text{sech}^2(c+d x)+b\right )}{30 d}-\frac{\coth (c+d x) \text{csch}(c+d x) \left (a-b \text{sech}^2(c+d x)+b\right )^3}{2 d}$

[Out]

(a^2*(a - 6*b)*ArcTanh[Cosh[c + d*x]])/(2*d) + (b*(81*a^2 - 28*a*b - 4*b^2)*Sech[c + d*x])/(30*d) + ((33*a - 2
*b)*b*Sech[c + d*x]*(a + b - b*Sech[c + d*x]^2))/(30*d) + (7*b*Sech[c + d*x]*(a + b - b*Sech[c + d*x]^2)^2)/(1
0*d) - (Coth[c + d*x]*Csch[c + d*x]*(a + b - b*Sech[c + d*x]^2)^3)/(2*d)

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Rubi [A]  time = 0.225033, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.217, Rules used = {3664, 467, 528, 388, 207} $\frac{b \left (81 a^2-28 a b-4 b^2\right ) \text{sech}(c+d x)}{30 d}+\frac{a^2 (a-6 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}+\frac{7 b \text{sech}(c+d x) \left (a-b \text{sech}^2(c+d x)+b\right )^2}{10 d}+\frac{b (33 a-2 b) \text{sech}(c+d x) \left (a-b \text{sech}^2(c+d x)+b\right )}{30 d}-\frac{\coth (c+d x) \text{csch}(c+d x) \left (a-b \text{sech}^2(c+d x)+b\right )^3}{2 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(a^2*(a - 6*b)*ArcTanh[Cosh[c + d*x]])/(2*d) + (b*(81*a^2 - 28*a*b - 4*b^2)*Sech[c + d*x])/(30*d) + ((33*a - 2
*b)*b*Sech[c + d*x]*(a + b - b*Sech[c + d*x]^2))/(30*d) + (7*b*Sech[c + d*x]*(a + b - b*Sech[c + d*x]^2)^2)/(1
0*d) - (Coth[c + d*x]*Csch[c + d*x]*(a + b - b*Sech[c + d*x]^2)^3)/(2*d)

Rule 3664

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sec[e + f*x], x]}, Dist[1/(f*ff^m), Subst[Int[((-1 + ff^2*x^2)^((m - 1)/2)*(a - b + b*ff^2*x^2)^p)/x^(
m + 1), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 467

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(n -
1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*n*(p + 1)), x] - Dist[e^n/(b*n*(p + 1)), Int[(e*x)^
(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m - n + 1) + d*(m + n*(q - 1) + 1)*x^n, x], x], x] /;
FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 0] && GtQ[m - n + 1, 0] &
& IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 528

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[
(f*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*(n*(p + q + 1) + 1)), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a +
b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*
d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1
, 0]

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

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}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (a+b-b x^2\right )^3}{\left (-1+x^2\right )^2} \, dx,x,\text{sech}(c+d x)\right )}{d}\\ &=-\frac{\coth (c+d x) \text{csch}(c+d x) \left (a+b-b \text{sech}^2(c+d x)\right )^3}{2 d}-\frac{\operatorname{Subst}\left (\int \frac{\left (a+b-7 b x^2\right ) \left (a+b-b x^2\right )^2}{-1+x^2} \, dx,x,\text{sech}(c+d x)\right )}{2 d}\\ &=\frac{7 b \text{sech}(c+d x) \left (a+b-b \text{sech}^2(c+d x)\right )^2}{10 d}-\frac{\coth (c+d x) \text{csch}(c+d x) \left (a+b-b \text{sech}^2(c+d x)\right )^3}{2 d}-\frac{\operatorname{Subst}\left (\int \frac{\left (a+b-b x^2\right ) \left ((5 a-2 b) (a+b)-(33 a-2 b) b x^2\right )}{-1+x^2} \, dx,x,\text{sech}(c+d x)\right )}{10 d}\\ &=\frac{(33 a-2 b) b \text{sech}(c+d x) \left (a+b-b \text{sech}^2(c+d x)\right )}{30 d}+\frac{7 b \text{sech}(c+d x) \left (a+b-b \text{sech}^2(c+d x)\right )^2}{10 d}-\frac{\coth (c+d x) \text{csch}(c+d x) \left (a+b-b \text{sech}^2(c+d x)\right )^3}{2 d}-\frac{\operatorname{Subst}\left (\int \frac{(a+b) \left (15 a^2-24 a b-4 b^2\right )-b \left (81 a^2-28 a b-4 b^2\right ) x^2}{-1+x^2} \, dx,x,\text{sech}(c+d x)\right )}{30 d}\\ &=\frac{b \left (81 a^2-28 a b-4 b^2\right ) \text{sech}(c+d x)}{30 d}+\frac{(33 a-2 b) b \text{sech}(c+d x) \left (a+b-b \text{sech}^2(c+d x)\right )}{30 d}+\frac{7 b \text{sech}(c+d x) \left (a+b-b \text{sech}^2(c+d x)\right )^2}{10 d}-\frac{\coth (c+d x) \text{csch}(c+d x) \left (a+b-b \text{sech}^2(c+d x)\right )^3}{2 d}-\frac{\left (a^2 (a-6 b)\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\text{sech}(c+d x)\right )}{2 d}\\ &=\frac{a^2 (a-6 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}+\frac{b \left (81 a^2-28 a b-4 b^2\right ) \text{sech}(c+d x)}{30 d}+\frac{(33 a-2 b) b \text{sech}(c+d x) \left (a+b-b \text{sech}^2(c+d x)\right )}{30 d}+\frac{7 b \text{sech}(c+d x) \left (a+b-b \text{sech}^2(c+d x)\right )^2}{10 d}-\frac{\coth (c+d x) \text{csch}(c+d x) \left (a+b-b \text{sech}^2(c+d x)\right )^3}{2 d}\\ \end{align*}

Mathematica [A]  time = 6.17533, size = 127, normalized size = 0.84 $\frac{3 a^2 b \text{sech}(c+d x)}{d}-\frac{a^2 (a-6 b) \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}-\frac{a^3 \text{csch}^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{a^3 \text{sech}^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{b^2 (3 a+b) \text{sech}^3(c+d x)}{3 d}+\frac{b^3 \text{sech}^5(c+d x)}{5 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(a^3*Csch[(c + d*x)/2]^2)/(8*d) - (a^2*(a - 6*b)*Log[Tanh[(c + d*x)/2]])/(2*d) - (a^3*Sech[(c + d*x)/2]^2)/(8
*d) + (3*a^2*b*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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Maple [A]  time = 0.063, size = 174, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( -{\frac{{\rm csch} \left (dx+c\right ){\rm coth} \left (dx+c\right )}{2}}+{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \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\,{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{ \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}+1/3\,{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{\cosh \left ( dx+c \right ) }}-1/3\,\cosh \left ( dx+c \right ) \right ) +{b}^{3} \left ( -{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{5\, \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}+{\frac{2\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{15\, \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}+{\frac{2\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{15\,\cosh \left ( dx+c \right ) }}-{\frac{2\,\cosh \left ( dx+c \right ) }{15}} \right ) \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^3*(a+b*tanh(d*x+c)^2)^3,x)

[Out]

1/d*(a^3*(-1/2*csch(d*x+c)*coth(d*x+c)+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*sinh(d*x+c)^2/cosh(d*x+c)^3+1/3*sinh(d*x+c)^2/cosh(d*x+c)-1/3*cosh(d*x+c))+b^3*(-1/5*sinh(d*x+c)^2/co
sh(d*x+c)^5+2/15*sinh(d*x+c)^2/cosh(d*x+c)^3+2/15*sinh(d*x+c)^2/cosh(d*x+c)-2/15*cosh(d*x+c)))

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Maxima [B]  time = 1.0929, size = 544, normalized size = 3.58 \begin{align*} \frac{1}{2} \, a^{3}{\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 )} - 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{8}{15} \, b^{3}{\left (\frac{5 \, e^{\left (-3 \, d x - 3 \, c\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 )}} - \frac{2 \, e^{\left (-5 \, d x - 5 \, c\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 )}} + \frac{5 \, e^{\left (-7 \, d x - 7 \, c\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 )} - \frac{8 \, a b^{2}}{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*tanh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

1/2*a^3*(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))) - 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))) - 8/15*b^3*(5*e^(-3*d*x - 3*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)) - 2*e^(-5*d*x - 5*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)) + 5*e^(-7*d*x -
7*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))) - 8*a*b^2/(d*(e^(d*x + c) + e^(-d*x - c))^3)

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Fricas [B]  time = 2.47568, size = 12984, normalized size = 85.42 \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*tanh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

-1/30*(30*(a^3 - 6*a^2*b)*cosh(d*x + c)^13 + 390*(a^3 - 6*a^2*b)*cosh(d*x + c)*sinh(d*x + c)^12 + 30*(a^3 - 6*
a^2*b)*sinh(d*x + c)^13 + 20*(9*a^3 - 18*a^2*b + 12*a*b^2 + 4*b^3)*cosh(d*x + c)^11 + 20*(9*a^3 - 18*a^2*b + 1
2*a*b^2 + 4*b^3 + 117*(a^3 - 6*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^11 + 220*(39*(a^3 - 6*a^2*b)*cosh(d*x + c
)^3 + (9*a^3 - 18*a^2*b + 12*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c)^10 + 6*(75*a^3 + 30*a^2*b - 32*b^3)*c
osh(d*x + c)^9 + 2*(10725*(a^3 - 6*a^2*b)*cosh(d*x + c)^4 + 225*a^3 + 90*a^2*b - 96*b^3 + 550*(9*a^3 - 18*a^2*
b + 12*a*b^2 + 4*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^9 + 6*(6435*(a^3 - 6*a^2*b)*cosh(d*x + c)^5 + 550*(9*a^3
- 18*a^2*b + 12*a*b^2 + 4*b^3)*cosh(d*x + c)^3 + 9*(75*a^3 + 30*a^2*b - 32*b^3)*cosh(d*x + c))*sinh(d*x + c)^8
+ 8*(75*a^3 + 90*a^2*b - 60*a*b^2 + 28*b^3)*cosh(d*x + c)^7 + 8*(6435*(a^3 - 6*a^2*b)*cosh(d*x + c)^6 + 825*(
9*a^3 - 18*a^2*b + 12*a*b^2 + 4*b^3)*cosh(d*x + c)^4 + 75*a^3 + 90*a^2*b - 60*a*b^2 + 28*b^3 + 27*(75*a^3 + 30
*a^2*b - 32*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 8*(6435*(a^3 - 6*a^2*b)*cosh(d*x + c)^7 + 1155*(9*a^3 - 18
*a^2*b + 12*a*b^2 + 4*b^3)*cosh(d*x + c)^5 + 63*(75*a^3 + 30*a^2*b - 32*b^3)*cosh(d*x + c)^3 + 7*(75*a^3 + 90*
a^2*b - 60*a*b^2 + 28*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 + 6*(75*a^3 + 30*a^2*b - 32*b^3)*cosh(d*x + c)^5 + 6
*(6435*(a^3 - 6*a^2*b)*cosh(d*x + c)^8 + 1540*(9*a^3 - 18*a^2*b + 12*a*b^2 + 4*b^3)*cosh(d*x + c)^6 + 126*(75*
a^3 + 30*a^2*b - 32*b^3)*cosh(d*x + c)^4 + 75*a^3 + 30*a^2*b - 32*b^3 + 28*(75*a^3 + 90*a^2*b - 60*a*b^2 + 28*
b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 2*(10725*(a^3 - 6*a^2*b)*cosh(d*x + c)^9 + 3300*(9*a^3 - 18*a^2*b + 12
*a*b^2 + 4*b^3)*cosh(d*x + c)^7 + 378*(75*a^3 + 30*a^2*b - 32*b^3)*cosh(d*x + c)^5 + 140*(75*a^3 + 90*a^2*b -
60*a*b^2 + 28*b^3)*cosh(d*x + c)^3 + 15*(75*a^3 + 30*a^2*b - 32*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 20*(9*a^
3 - 18*a^2*b + 12*a*b^2 + 4*b^3)*cosh(d*x + c)^3 + 4*(2145*(a^3 - 6*a^2*b)*cosh(d*x + c)^10 + 825*(9*a^3 - 18*
a^2*b + 12*a*b^2 + 4*b^3)*cosh(d*x + c)^8 + 126*(75*a^3 + 30*a^2*b - 32*b^3)*cosh(d*x + c)^6 + 70*(75*a^3 + 90
*a^2*b - 60*a*b^2 + 28*b^3)*cosh(d*x + c)^4 + 45*a^3 - 90*a^2*b + 60*a*b^2 + 20*b^3 + 15*(75*a^3 + 30*a^2*b -
32*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 4*(585*(a^3 - 6*a^2*b)*cosh(d*x + c)^11 + 275*(9*a^3 - 18*a^2*b + 1
2*a*b^2 + 4*b^3)*cosh(d*x + c)^9 + 54*(75*a^3 + 30*a^2*b - 32*b^3)*cosh(d*x + c)^7 + 42*(75*a^3 + 90*a^2*b - 6
0*a*b^2 + 28*b^3)*cosh(d*x + c)^5 + 15*(75*a^3 + 30*a^2*b - 32*b^3)*cosh(d*x + c)^3 + 15*(9*a^3 - 18*a^2*b + 1
2*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + 30*(a^3 - 6*a^2*b)*cosh(d*x + c) - 15*((a^3 - 6*a^2*b)*cosh(
d*x + c)^14 + 14*(a^3 - 6*a^2*b)*cosh(d*x + c)*sinh(d*x + c)^13 + (a^3 - 6*a^2*b)*sinh(d*x + c)^14 + 3*(a^3 -
6*a^2*b)*cosh(d*x + c)^12 + (3*a^3 - 18*a^2*b + 91*(a^3 - 6*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^12 + 4*(91*(
a^3 - 6*a^2*b)*cosh(d*x + c)^3 + 9*(a^3 - 6*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^11 + (a^3 - 6*a^2*b)*cosh(d*x
+ c)^10 + (1001*(a^3 - 6*a^2*b)*cosh(d*x + c)^4 + a^3 - 6*a^2*b + 198*(a^3 - 6*a^2*b)*cosh(d*x + c)^2)*sinh(d*
x + c)^10 + 2*(1001*(a^3 - 6*a^2*b)*cosh(d*x + c)^5 + 330*(a^3 - 6*a^2*b)*cosh(d*x + c)^3 + 5*(a^3 - 6*a^2*b)*
cosh(d*x + c))*sinh(d*x + c)^9 - 5*(a^3 - 6*a^2*b)*cosh(d*x + c)^8 + (3003*(a^3 - 6*a^2*b)*cosh(d*x + c)^6 + 1
485*(a^3 - 6*a^2*b)*cosh(d*x + c)^4 - 5*a^3 + 30*a^2*b + 45*(a^3 - 6*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^8 +
8*(429*(a^3 - 6*a^2*b)*cosh(d*x + c)^7 + 297*(a^3 - 6*a^2*b)*cosh(d*x + c)^5 + 15*(a^3 - 6*a^2*b)*cosh(d*x +
c)^3 - 5*(a^3 - 6*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^7 - 5*(a^3 - 6*a^2*b)*cosh(d*x + c)^6 + (3003*(a^3 - 6*a
^2*b)*cosh(d*x + c)^8 + 2772*(a^3 - 6*a^2*b)*cosh(d*x + c)^6 + 210*(a^3 - 6*a^2*b)*cosh(d*x + c)^4 - 5*a^3 + 3
0*a^2*b - 140*(a^3 - 6*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 2*(1001*(a^3 - 6*a^2*b)*cosh(d*x + c)^9 + 118
8*(a^3 - 6*a^2*b)*cosh(d*x + c)^7 + 126*(a^3 - 6*a^2*b)*cosh(d*x + c)^5 - 140*(a^3 - 6*a^2*b)*cosh(d*x + c)^3
- 15*(a^3 - 6*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^5 + (a^3 - 6*a^2*b)*cosh(d*x + c)^4 + (1001*(a^3 - 6*a^2*b)*
cosh(d*x + c)^10 + 1485*(a^3 - 6*a^2*b)*cosh(d*x + c)^8 + 210*(a^3 - 6*a^2*b)*cosh(d*x + c)^6 - 350*(a^3 - 6*a
^2*b)*cosh(d*x + c)^4 + a^3 - 6*a^2*b - 75*(a^3 - 6*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(91*(a^3 - 6*a
^2*b)*cosh(d*x + c)^11 + 165*(a^3 - 6*a^2*b)*cosh(d*x + c)^9 + 30*(a^3 - 6*a^2*b)*cosh(d*x + c)^7 - 70*(a^3 -
6*a^2*b)*cosh(d*x + c)^5 - 25*(a^3 - 6*a^2*b)*cosh(d*x + c)^3 + (a^3 - 6*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^3
+ a^3 - 6*a^2*b + 3*(a^3 - 6*a^2*b)*cosh(d*x + c)^2 + (91*(a^3 - 6*a^2*b)*cosh(d*x + c)^12 + 198*(a^3 - 6*a^2
*b)*cosh(d*x + c)^10 + 45*(a^3 - 6*a^2*b)*cosh(d*x + c)^8 - 140*(a^3 - 6*a^2*b)*cosh(d*x + c)^6 - 75*(a^3 - 6*
a^2*b)*cosh(d*x + c)^4 + 3*a^3 - 18*a^2*b + 6*(a^3 - 6*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(7*(a^3 - 6
*a^2*b)*cosh(d*x + c)^13 + 18*(a^3 - 6*a^2*b)*cosh(d*x + c)^11 + 5*(a^3 - 6*a^2*b)*cosh(d*x + c)^9 - 20*(a^3 -
6*a^2*b)*cosh(d*x + c)^7 - 15*(a^3 - 6*a^2*b)*cosh(d*x + c)^5 + 2*(a^3 - 6*a^2*b)*cosh(d*x + c)^3 + 3*(a^3 -
6*a^2*b)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) + 15*((a^3 - 6*a^2*b)*cosh(d*x +
c)^14 + 14*(a^3 - 6*a^2*b)*cosh(d*x + c)*sinh(d*x + c)^13 + (a^3 - 6*a^2*b)*sinh(d*x + c)^14 + 3*(a^3 - 6*a^2
*b)*cosh(d*x + c)^12 + (3*a^3 - 18*a^2*b + 91*(a^3 - 6*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^12 + 4*(91*(a^3 -
6*a^2*b)*cosh(d*x + c)^3 + 9*(a^3 - 6*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^11 + (a^3 - 6*a^2*b)*cosh(d*x + c)^
10 + (1001*(a^3 - 6*a^2*b)*cosh(d*x + c)^4 + a^3 - 6*a^2*b + 198*(a^3 - 6*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c
)^10 + 2*(1001*(a^3 - 6*a^2*b)*cosh(d*x + c)^5 + 330*(a^3 - 6*a^2*b)*cosh(d*x + c)^3 + 5*(a^3 - 6*a^2*b)*cosh(
d*x + c))*sinh(d*x + c)^9 - 5*(a^3 - 6*a^2*b)*cosh(d*x + c)^8 + (3003*(a^3 - 6*a^2*b)*cosh(d*x + c)^6 + 1485*(
a^3 - 6*a^2*b)*cosh(d*x + c)^4 - 5*a^3 + 30*a^2*b + 45*(a^3 - 6*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 8*(4
29*(a^3 - 6*a^2*b)*cosh(d*x + c)^7 + 297*(a^3 - 6*a^2*b)*cosh(d*x + c)^5 + 15*(a^3 - 6*a^2*b)*cosh(d*x + c)^3
- 5*(a^3 - 6*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^7 - 5*(a^3 - 6*a^2*b)*cosh(d*x + c)^6 + (3003*(a^3 - 6*a^2*b)
*cosh(d*x + c)^8 + 2772*(a^3 - 6*a^2*b)*cosh(d*x + c)^6 + 210*(a^3 - 6*a^2*b)*cosh(d*x + c)^4 - 5*a^3 + 30*a^2
*b - 140*(a^3 - 6*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 2*(1001*(a^3 - 6*a^2*b)*cosh(d*x + c)^9 + 1188*(a^
3 - 6*a^2*b)*cosh(d*x + c)^7 + 126*(a^3 - 6*a^2*b)*cosh(d*x + c)^5 - 140*(a^3 - 6*a^2*b)*cosh(d*x + c)^3 - 15*
(a^3 - 6*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^5 + (a^3 - 6*a^2*b)*cosh(d*x + c)^4 + (1001*(a^3 - 6*a^2*b)*cosh(
d*x + c)^10 + 1485*(a^3 - 6*a^2*b)*cosh(d*x + c)^8 + 210*(a^3 - 6*a^2*b)*cosh(d*x + c)^6 - 350*(a^3 - 6*a^2*b)
*cosh(d*x + c)^4 + a^3 - 6*a^2*b - 75*(a^3 - 6*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(91*(a^3 - 6*a^2*b)
*cosh(d*x + c)^11 + 165*(a^3 - 6*a^2*b)*cosh(d*x + c)^9 + 30*(a^3 - 6*a^2*b)*cosh(d*x + c)^7 - 70*(a^3 - 6*a^2
*b)*cosh(d*x + c)^5 - 25*(a^3 - 6*a^2*b)*cosh(d*x + c)^3 + (a^3 - 6*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^3 + a^
3 - 6*a^2*b + 3*(a^3 - 6*a^2*b)*cosh(d*x + c)^2 + (91*(a^3 - 6*a^2*b)*cosh(d*x + c)^12 + 198*(a^3 - 6*a^2*b)*c
osh(d*x + c)^10 + 45*(a^3 - 6*a^2*b)*cosh(d*x + c)^8 - 140*(a^3 - 6*a^2*b)*cosh(d*x + c)^6 - 75*(a^3 - 6*a^2*b
)*cosh(d*x + c)^4 + 3*a^3 - 18*a^2*b + 6*(a^3 - 6*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(7*(a^3 - 6*a^2*
b)*cosh(d*x + c)^13 + 18*(a^3 - 6*a^2*b)*cosh(d*x + c)^11 + 5*(a^3 - 6*a^2*b)*cosh(d*x + c)^9 - 20*(a^3 - 6*a^
2*b)*cosh(d*x + c)^7 - 15*(a^3 - 6*a^2*b)*cosh(d*x + c)^5 + 2*(a^3 - 6*a^2*b)*cosh(d*x + c)^3 + 3*(a^3 - 6*a^2
*b)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*(195*(a^3 - 6*a^2*b)*cosh(d*x + c
)^12 + 110*(9*a^3 - 18*a^2*b + 12*a*b^2 + 4*b^3)*cosh(d*x + c)^10 + 27*(75*a^3 + 30*a^2*b - 32*b^3)*cosh(d*x +
c)^8 + 28*(75*a^3 + 90*a^2*b - 60*a*b^2 + 28*b^3)*cosh(d*x + c)^6 + 15*(75*a^3 + 30*a^2*b - 32*b^3)*cosh(d*x
+ c)^4 + 15*a^3 - 90*a^2*b + 30*(9*a^3 - 18*a^2*b + 12*a*b^2 + 4*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/(d*cosh(
d*x + c)^14 + 14*d*cosh(d*x + c)*sinh(d*x + c)^13 + d*sinh(d*x + c)^14 + 3*d*cosh(d*x + c)^12 + (91*d*cosh(d*x
+ c)^2 + 3*d)*sinh(d*x + c)^12 + 4*(91*d*cosh(d*x + c)^3 + 9*d*cosh(d*x + c))*sinh(d*x + c)^11 + d*cosh(d*x +
c)^10 + (1001*d*cosh(d*x + c)^4 + 198*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^10 + 2*(1001*d*cosh(d*x + c)^5 + 3
30*d*cosh(d*x + c)^3 + 5*d*cosh(d*x + c))*sinh(d*x + c)^9 - 5*d*cosh(d*x + c)^8 + (3003*d*cosh(d*x + c)^6 + 14
85*d*cosh(d*x + c)^4 + 45*d*cosh(d*x + c)^2 - 5*d)*sinh(d*x + c)^8 + 8*(429*d*cosh(d*x + c)^7 + 297*d*cosh(d*x
+ c)^5 + 15*d*cosh(d*x + c)^3 - 5*d*cosh(d*x + c))*sinh(d*x + c)^7 - 5*d*cosh(d*x + c)^6 + (3003*d*cosh(d*x +
c)^8 + 2772*d*cosh(d*x + c)^6 + 210*d*cosh(d*x + c)^4 - 140*d*cosh(d*x + c)^2 - 5*d)*sinh(d*x + c)^6 + 2*(100
1*d*cosh(d*x + c)^9 + 1188*d*cosh(d*x + c)^7 + 126*d*cosh(d*x + c)^5 - 140*d*cosh(d*x + c)^3 - 15*d*cosh(d*x +
c))*sinh(d*x + c)^5 + d*cosh(d*x + c)^4 + (1001*d*cosh(d*x + c)^10 + 1485*d*cosh(d*x + c)^8 + 210*d*cosh(d*x
+ c)^6 - 350*d*cosh(d*x + c)^4 - 75*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^4 + 4*(91*d*cosh(d*x + c)^11 + 165*d*
cosh(d*x + c)^9 + 30*d*cosh(d*x + c)^7 - 70*d*cosh(d*x + c)^5 - 25*d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d
*x + c)^3 + 3*d*cosh(d*x + c)^2 + (91*d*cosh(d*x + c)^12 + 198*d*cosh(d*x + c)^10 + 45*d*cosh(d*x + c)^8 - 140
*d*cosh(d*x + c)^6 - 75*d*cosh(d*x + c)^4 + 6*d*cosh(d*x + c)^2 + 3*d)*sinh(d*x + c)^2 + 2*(7*d*cosh(d*x + c)^
13 + 18*d*cosh(d*x + c)^11 + 5*d*cosh(d*x + c)^9 - 20*d*cosh(d*x + c)^7 - 15*d*cosh(d*x + c)^5 + 2*d*cosh(d*x
+ c)^3 + 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 \tanh ^{2}{\left (c + d x \right )}\right )^{3} \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*tanh(d*x+c)**2)**3,x)

[Out]

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

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Giac [A]  time = 1.76989, size = 379, normalized size = 2.49 \begin{align*} \frac{15 \,{\left (a^{3} e^{c} - 6 \, a^{2} b e^{c}\right )} e^{\left (-c\right )} \log \left (e^{\left (d x + c\right )} + 1\right ) - 15 \,{\left (a^{3} e^{c} - 6 \, a^{2} b e^{c}\right )} e^{\left (-c\right )} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac{30 \,{\left (a^{3} e^{\left (3 \, d x + 3 \, c\right )} + a^{3} e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}} + \frac{4 \,{\left (45 \, a^{2} b e^{\left (9 \, d x + 9 \, c\right )} + 180 \, a^{2} b e^{\left (7 \, d x + 7 \, c\right )} - 60 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} - 20 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} + 270 \, a^{2} b e^{\left (5 \, d x + 5 \, c\right )} - 120 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 8 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 180 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} - 60 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 20 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 45 \, a^{2} b e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}}{30 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30*(15*(a^3*e^c - 6*a^2*b*e^c)*e^(-c)*log(e^(d*x + c) + 1) - 15*(a^3*e^c - 6*a^2*b*e^c)*e^(-c)*log(abs(e^(d*
x + c) - 1)) - 30*(a^3*e^(3*d*x + 3*c) + a^3*e^(d*x + c))/(e^(2*d*x + 2*c) - 1)^2 + 4*(45*a^2*b*e^(9*d*x + 9*c
) + 180*a^2*b*e^(7*d*x + 7*c) - 60*a*b^2*e^(7*d*x + 7*c) - 20*b^3*e^(7*d*x + 7*c) + 270*a^2*b*e^(5*d*x + 5*c)
- 120*a*b^2*e^(5*d*x + 5*c) + 8*b^3*e^(5*d*x + 5*c) + 180*a^2*b*e^(3*d*x + 3*c) - 60*a*b^2*e^(3*d*x + 3*c) - 2
0*b^3*e^(3*d*x + 3*c) + 45*a^2*b*e^(d*x + c))/(e^(2*d*x + 2*c) + 1)^5)/d