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

Optimal. Leaf size=141 \[ \frac {(a+4 b) \tanh ^{-1}(\cosh (c+d x))}{2 a^3 d}-\frac {\sqrt {b} (3 a+4 b) \tanh ^{-1}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{2 a^3 d \sqrt {a+b}}-\frac {b \text {sech}(c+d x)}{a^2 d \left (a-b \text {sech}^2(c+d x)+b\right )}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d \left (a-b \text {sech}^2(c+d x)+b\right )} \]

[Out]

1/2*(a+4*b)*arctanh(cosh(d*x+c))/a^3/d-1/2*coth(d*x+c)*csch(d*x+c)/a/d/(a+b-b*sech(d*x+c)^2)-b*sech(d*x+c)/a^2
/d/(a+b-b*sech(d*x+c)^2)-1/2*(3*a+4*b)*arctanh(sech(d*x+c)*b^(1/2)/(a+b)^(1/2))*b^(1/2)/a^3/d/(a+b)^(1/2)

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Rubi [A]  time = 0.21, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3664, 471, 527, 522, 207, 208} \[ -\frac {b \text {sech}(c+d x)}{a^2 d \left (a-b \text {sech}^2(c+d x)+b\right )}+\frac {(a+4 b) \tanh ^{-1}(\cosh (c+d x))}{2 a^3 d}-\frac {\sqrt {b} (3 a+4 b) \tanh ^{-1}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{2 a^3 d \sqrt {a+b}}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d \left (a-b \text {sech}^2(c+d x)+b\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + 4*b)*ArcTanh[Cosh[c + d*x]])/(2*a^3*d) - (Sqrt[b]*(3*a + 4*b)*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b
]])/(2*a^3*Sqrt[a + b]*d) - (Coth[c + d*x]*Csch[c + d*x])/(2*a*d*(a + b - b*Sech[c + d*x]^2)) - (b*Sech[c + d*
x])/(a^2*d*(a + b - b*Sech[c + d*x]^2))

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])

Rule 208

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

Rule 471

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 + 1))/(n*(b*c - a*d)*(p + 1)), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 522

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

Rule 527

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

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]

Rubi steps

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

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Mathematica [C]  time = 4.35, size = 203, normalized size = 1.44 \[ -\frac {\frac {8 a b \cosh (c+d x)}{(a+b) \cosh (2 (c+d x))+a-b}+4 (a+4 b) \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+\frac {4 i \sqrt {b} (3 a+4 b) \tan ^{-1}\left (\frac {-\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )-i \sqrt {a+b}}{\sqrt {b}}\right )}{\sqrt {a+b}}+\frac {4 i \sqrt {b} (3 a+4 b) \tan ^{-1}\left (\frac {\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )-i \sqrt {a+b}}{\sqrt {b}}\right )}{\sqrt {a+b}}+a \text {csch}^2\left (\frac {1}{2} (c+d x)\right )+a \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(((4*I)*Sqrt[b]*(3*a + 4*b)*ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]])/Sqrt[a + b] +
((4*I)*Sqrt[b]*(3*a + 4*b)*ArcTan[((-I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]])/Sqrt[a + b] + (8*a
*b*Cosh[c + d*x])/(a - b + (a + b)*Cosh[2*(c + d*x)]) + a*Csch[(c + d*x)/2]^2 + 4*(a + 4*b)*Log[Tanh[(c + d*x)
/2]] + a*Sech[(c + d*x)/2]^2)/(a^3*d)

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fricas [B]  time = 0.73, size = 6335, normalized size = 44.93 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*(a^2 + 2*a*b)*cosh(d*x + c)^7 + 28*(a^2 + 2*a*b)*cosh(d*x + c)*sinh(d*x + c)^6 + 4*(a^2 + 2*a*b)*sinh(
d*x + c)^7 + 4*(3*a^2 - 2*a*b)*cosh(d*x + c)^5 + 4*(21*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 3*a^2 - 2*a*b)*sinh(d*x
+ c)^5 + 20*(7*(a^2 + 2*a*b)*cosh(d*x + c)^3 + (3*a^2 - 2*a*b)*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(3*a^2 - 2*
a*b)*cosh(d*x + c)^3 + 4*(35*(a^2 + 2*a*b)*cosh(d*x + c)^4 + 10*(3*a^2 - 2*a*b)*cosh(d*x + c)^2 + 3*a^2 - 2*a*
b)*sinh(d*x + c)^3 + 4*(21*(a^2 + 2*a*b)*cosh(d*x + c)^5 + 10*(3*a^2 - 2*a*b)*cosh(d*x + c)^3 + 3*(3*a^2 - 2*a
*b)*cosh(d*x + c))*sinh(d*x + c)^2 - ((3*a^2 + 7*a*b + 4*b^2)*cosh(d*x + c)^8 + 8*(3*a^2 + 7*a*b + 4*b^2)*cosh
(d*x + c)*sinh(d*x + c)^7 + (3*a^2 + 7*a*b + 4*b^2)*sinh(d*x + c)^8 - 4*(3*a*b + 4*b^2)*cosh(d*x + c)^6 + 4*(7
*(3*a^2 + 7*a*b + 4*b^2)*cosh(d*x + c)^2 - 3*a*b - 4*b^2)*sinh(d*x + c)^6 + 8*(7*(3*a^2 + 7*a*b + 4*b^2)*cosh(
d*x + c)^3 - 3*(3*a*b + 4*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(3*a^2 - 5*a*b - 12*b^2)*cosh(d*x + c)^4 + 2
*(35*(3*a^2 + 7*a*b + 4*b^2)*cosh(d*x + c)^4 - 30*(3*a*b + 4*b^2)*cosh(d*x + c)^2 - 3*a^2 + 5*a*b + 12*b^2)*si
nh(d*x + c)^4 + 8*(7*(3*a^2 + 7*a*b + 4*b^2)*cosh(d*x + c)^5 - 10*(3*a*b + 4*b^2)*cosh(d*x + c)^3 - (3*a^2 - 5
*a*b - 12*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(3*a*b + 4*b^2)*cosh(d*x + c)^2 + 4*(7*(3*a^2 + 7*a*b + 4*b^
2)*cosh(d*x + c)^6 - 15*(3*a*b + 4*b^2)*cosh(d*x + c)^4 - 3*(3*a^2 - 5*a*b - 12*b^2)*cosh(d*x + c)^2 - 3*a*b -
4*b^2)*sinh(d*x + c)^2 + 3*a^2 + 7*a*b + 4*b^2 + 8*((3*a^2 + 7*a*b + 4*b^2)*cosh(d*x + c)^7 - 3*(3*a*b + 4*b^
2)*cosh(d*x + c)^5 - (3*a^2 - 5*a*b - 12*b^2)*cosh(d*x + c)^3 - (3*a*b + 4*b^2)*cosh(d*x + c))*sinh(d*x + c))*
sqrt(b/(a + b))*log(((a + b)*cosh(d*x + c)^4 + 4*(a + b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + b)*sinh(d*x + c)
^4 + 2*(a + 3*b)*cosh(d*x + c)^2 + 2*(3*(a + b)*cosh(d*x + c)^2 + a + 3*b)*sinh(d*x + c)^2 + 4*((a + b)*cosh(d
*x + c)^3 + (a + 3*b)*cosh(d*x + c))*sinh(d*x + c) - 4*((a + b)*cosh(d*x + c)^3 + 3*(a + b)*cosh(d*x + c)*sinh
(d*x + c)^2 + (a + b)*sinh(d*x + c)^3 + (a + b)*cosh(d*x + c) + (3*(a + b)*cosh(d*x + c)^2 + a + b)*sinh(d*x +
c))*sqrt(b/(a + b)) + a + b)/((a + b)*cosh(d*x + c)^4 + 4*(a + b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + b)*sin
h(d*x + c)^4 + 2*(a - b)*cosh(d*x + c)^2 + 2*(3*(a + b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c)^2 + 4*((a + b)*
cosh(d*x + c)^3 + (a - b)*cosh(d*x + c))*sinh(d*x + c) + a + b)) + 4*(a^2 + 2*a*b)*cosh(d*x + c) - 2*((a^2 + 5
*a*b + 4*b^2)*cosh(d*x + c)^8 + 8*(a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2 + 5*a*b + 4*b^2)*
sinh(d*x + c)^8 - 4*(a*b + 4*b^2)*cosh(d*x + c)^6 + 4*(7*(a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)^2 - a*b - 4*b^2)*
sinh(d*x + c)^6 + 8*(7*(a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)^3 - 3*(a*b + 4*b^2)*cosh(d*x + c))*sinh(d*x + c)^5
- 2*(a^2 + a*b - 12*b^2)*cosh(d*x + c)^4 + 2*(35*(a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)^4 - 30*(a*b + 4*b^2)*cosh
(d*x + c)^2 - a^2 - a*b + 12*b^2)*sinh(d*x + c)^4 + 8*(7*(a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)^5 - 10*(a*b + 4*b
^2)*cosh(d*x + c)^3 - (a^2 + a*b - 12*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(a*b + 4*b^2)*cosh(d*x + c)^2 +
4*(7*(a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)^6 - 15*(a*b + 4*b^2)*cosh(d*x + c)^4 - 3*(a^2 + a*b - 12*b^2)*cosh(d*
x + c)^2 - a*b - 4*b^2)*sinh(d*x + c)^2 + a^2 + 5*a*b + 4*b^2 + 8*((a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)^7 - 3*(
a*b + 4*b^2)*cosh(d*x + c)^5 - (a^2 + a*b - 12*b^2)*cosh(d*x + c)^3 - (a*b + 4*b^2)*cosh(d*x + c))*sinh(d*x +
c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) + 2*((a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)^8 + 8*(a^2 + 5*a*b + 4*b^2
)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2 + 5*a*b + 4*b^2)*sinh(d*x + c)^8 - 4*(a*b + 4*b^2)*cosh(d*x + c)^6 + 4*
(7*(a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)^2 - a*b - 4*b^2)*sinh(d*x + c)^6 + 8*(7*(a^2 + 5*a*b + 4*b^2)*cosh(d*x
+ c)^3 - 3*(a*b + 4*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(a^2 + a*b - 12*b^2)*cosh(d*x + c)^4 + 2*(35*(a^2
+ 5*a*b + 4*b^2)*cosh(d*x + c)^4 - 30*(a*b + 4*b^2)*cosh(d*x + c)^2 - a^2 - a*b + 12*b^2)*sinh(d*x + c)^4 + 8*
(7*(a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)^5 - 10*(a*b + 4*b^2)*cosh(d*x + c)^3 - (a^2 + a*b - 12*b^2)*cosh(d*x +
c))*sinh(d*x + c)^3 - 4*(a*b + 4*b^2)*cosh(d*x + c)^2 + 4*(7*(a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)^6 - 15*(a*b +
4*b^2)*cosh(d*x + c)^4 - 3*(a^2 + a*b - 12*b^2)*cosh(d*x + c)^2 - a*b - 4*b^2)*sinh(d*x + c)^2 + a^2 + 5*a*b
+ 4*b^2 + 8*((a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)^7 - 3*(a*b + 4*b^2)*cosh(d*x + c)^5 - (a^2 + a*b - 12*b^2)*co
sh(d*x + c)^3 - (a*b + 4*b^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 4*(7*(a^2
+ 2*a*b)*cosh(d*x + c)^6 + 5*(3*a^2 - 2*a*b)*cosh(d*x + c)^4 + 3*(3*a^2 - 2*a*b)*cosh(d*x + c)^2 + a^2 + 2*a*
b)*sinh(d*x + c))/(4*a^3*b*d*cosh(d*x + c)^6 - (a^4 + a^3*b)*d*cosh(d*x + c)^8 - 8*(a^4 + a^3*b)*d*cosh(d*x +
c)*sinh(d*x + c)^7 - (a^4 + a^3*b)*d*sinh(d*x + c)^8 + 4*a^3*b*d*cosh(d*x + c)^2 + 4*(a^3*b*d - 7*(a^4 + a^3*b
)*d*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 2*(a^4 - 3*a^3*b)*d*cosh(d*x + c)^4 + 8*(3*a^3*b*d*cosh(d*x + c) - 7*(a
^4 + a^3*b)*d*cosh(d*x + c)^3)*sinh(d*x + c)^5 + 2*(30*a^3*b*d*cosh(d*x + c)^2 - 35*(a^4 + a^3*b)*d*cosh(d*x +
c)^4 + (a^4 - 3*a^3*b)*d)*sinh(d*x + c)^4 + 8*(10*a^3*b*d*cosh(d*x + c)^3 - 7*(a^4 + a^3*b)*d*cosh(d*x + c)^5
+ (a^4 - 3*a^3*b)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(15*a^3*b*d*cosh(d*x + c)^4 - 7*(a^4 + a^3*b)*d*cosh(d
*x + c)^6 + a^3*b*d + 3*(a^4 - 3*a^3*b)*d*cosh(d*x + c)^2)*sinh(d*x + c)^2 - (a^4 + a^3*b)*d + 8*(3*a^3*b*d*co
sh(d*x + c)^5 - (a^4 + a^3*b)*d*cosh(d*x + c)^7 + a^3*b*d*cosh(d*x + c) + (a^4 - 3*a^3*b)*d*cosh(d*x + c)^3)*s
inh(d*x + c)), 1/2*(2*(a^2 + 2*a*b)*cosh(d*x + c)^7 + 14*(a^2 + 2*a*b)*cosh(d*x + c)*sinh(d*x + c)^6 + 2*(a^2
+ 2*a*b)*sinh(d*x + c)^7 + 2*(3*a^2 - 2*a*b)*cosh(d*x + c)^5 + 2*(21*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 3*a^2 - 2
*a*b)*sinh(d*x + c)^5 + 10*(7*(a^2 + 2*a*b)*cosh(d*x + c)^3 + (3*a^2 - 2*a*b)*cosh(d*x + c))*sinh(d*x + c)^4 +
2*(3*a^2 - 2*a*b)*cosh(d*x + c)^3 + 2*(35*(a^2 + 2*a*b)*cosh(d*x + c)^4 + 10*(3*a^2 - 2*a*b)*cosh(d*x + c)^2
+ 3*a^2 - 2*a*b)*sinh(d*x + c)^3 + 2*(21*(a^2 + 2*a*b)*cosh(d*x + c)^5 + 10*(3*a^2 - 2*a*b)*cosh(d*x + c)^3 +
3*(3*a^2 - 2*a*b)*cosh(d*x + c))*sinh(d*x + c)^2 + ((3*a^2 + 7*a*b + 4*b^2)*cosh(d*x + c)^8 + 8*(3*a^2 + 7*a*b
+ 4*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a^2 + 7*a*b + 4*b^2)*sinh(d*x + c)^8 - 4*(3*a*b + 4*b^2)*cosh(d*x
+ c)^6 + 4*(7*(3*a^2 + 7*a*b + 4*b^2)*cosh(d*x + c)^2 - 3*a*b - 4*b^2)*sinh(d*x + c)^6 + 8*(7*(3*a^2 + 7*a*b
+ 4*b^2)*cosh(d*x + c)^3 - 3*(3*a*b + 4*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(3*a^2 - 5*a*b - 12*b^2)*cosh(
d*x + c)^4 + 2*(35*(3*a^2 + 7*a*b + 4*b^2)*cosh(d*x + c)^4 - 30*(3*a*b + 4*b^2)*cosh(d*x + c)^2 - 3*a^2 + 5*a*
b + 12*b^2)*sinh(d*x + c)^4 + 8*(7*(3*a^2 + 7*a*b + 4*b^2)*cosh(d*x + c)^5 - 10*(3*a*b + 4*b^2)*cosh(d*x + c)^
3 - (3*a^2 - 5*a*b - 12*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(3*a*b + 4*b^2)*cosh(d*x + c)^2 + 4*(7*(3*a^2
+ 7*a*b + 4*b^2)*cosh(d*x + c)^6 - 15*(3*a*b + 4*b^2)*cosh(d*x + c)^4 - 3*(3*a^2 - 5*a*b - 12*b^2)*cosh(d*x +
c)^2 - 3*a*b - 4*b^2)*sinh(d*x + c)^2 + 3*a^2 + 7*a*b + 4*b^2 + 8*((3*a^2 + 7*a*b + 4*b^2)*cosh(d*x + c)^7 - 3
*(3*a*b + 4*b^2)*cosh(d*x + c)^5 - (3*a^2 - 5*a*b - 12*b^2)*cosh(d*x + c)^3 - (3*a*b + 4*b^2)*cosh(d*x + c))*s
inh(d*x + c))*sqrt(-b/(a + b))*arctan(1/2*((a + b)*cosh(d*x + c)^3 + 3*(a + b)*cosh(d*x + c)*sinh(d*x + c)^2 +
(a + b)*sinh(d*x + c)^3 + (a - 3*b)*cosh(d*x + c) + (3*(a + b)*cosh(d*x + c)^2 + a - 3*b)*sinh(d*x + c))*sqrt
(-b/(a + b))/b) - ((3*a^2 + 7*a*b + 4*b^2)*cosh(d*x + c)^8 + 8*(3*a^2 + 7*a*b + 4*b^2)*cosh(d*x + c)*sinh(d*x
+ c)^7 + (3*a^2 + 7*a*b + 4*b^2)*sinh(d*x + c)^8 - 4*(3*a*b + 4*b^2)*cosh(d*x + c)^6 + 4*(7*(3*a^2 + 7*a*b + 4
*b^2)*cosh(d*x + c)^2 - 3*a*b - 4*b^2)*sinh(d*x + c)^6 + 8*(7*(3*a^2 + 7*a*b + 4*b^2)*cosh(d*x + c)^3 - 3*(3*a
*b + 4*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(3*a^2 - 5*a*b - 12*b^2)*cosh(d*x + c)^4 + 2*(35*(3*a^2 + 7*a*b
+ 4*b^2)*cosh(d*x + c)^4 - 30*(3*a*b + 4*b^2)*cosh(d*x + c)^2 - 3*a^2 + 5*a*b + 12*b^2)*sinh(d*x + c)^4 + 8*(
7*(3*a^2 + 7*a*b + 4*b^2)*cosh(d*x + c)^5 - 10*(3*a*b + 4*b^2)*cosh(d*x + c)^3 - (3*a^2 - 5*a*b - 12*b^2)*cosh
(d*x + c))*sinh(d*x + c)^3 - 4*(3*a*b + 4*b^2)*cosh(d*x + c)^2 + 4*(7*(3*a^2 + 7*a*b + 4*b^2)*cosh(d*x + c)^6
- 15*(3*a*b + 4*b^2)*cosh(d*x + c)^4 - 3*(3*a^2 - 5*a*b - 12*b^2)*cosh(d*x + c)^2 - 3*a*b - 4*b^2)*sinh(d*x +
c)^2 + 3*a^2 + 7*a*b + 4*b^2 + 8*((3*a^2 + 7*a*b + 4*b^2)*cosh(d*x + c)^7 - 3*(3*a*b + 4*b^2)*cosh(d*x + c)^5
- (3*a^2 - 5*a*b - 12*b^2)*cosh(d*x + c)^3 - (3*a*b + 4*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-b/(a + b))*ar
ctan(1/2*((a + b)*cosh(d*x + c) + (a + b)*sinh(d*x + c))*sqrt(-b/(a + b))/b) + 2*(a^2 + 2*a*b)*cosh(d*x + c) -
((a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)^8 + 8*(a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2 + 5*a*b
+ 4*b^2)*sinh(d*x + c)^8 - 4*(a*b + 4*b^2)*cosh(d*x + c)^6 + 4*(7*(a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)^2 - a*b
- 4*b^2)*sinh(d*x + c)^6 + 8*(7*(a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)^3 - 3*(a*b + 4*b^2)*cosh(d*x + c))*sinh(d
*x + c)^5 - 2*(a^2 + a*b - 12*b^2)*cosh(d*x + c)^4 + 2*(35*(a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)^4 - 30*(a*b + 4
*b^2)*cosh(d*x + c)^2 - a^2 - a*b + 12*b^2)*sinh(d*x + c)^4 + 8*(7*(a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)^5 - 10*
(a*b + 4*b^2)*cosh(d*x + c)^3 - (a^2 + a*b - 12*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(a*b + 4*b^2)*cosh(d*x
+ c)^2 + 4*(7*(a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)^6 - 15*(a*b + 4*b^2)*cosh(d*x + c)^4 - 3*(a^2 + a*b - 12*b^
2)*cosh(d*x + c)^2 - a*b - 4*b^2)*sinh(d*x + c)^2 + a^2 + 5*a*b + 4*b^2 + 8*((a^2 + 5*a*b + 4*b^2)*cosh(d*x +
c)^7 - 3*(a*b + 4*b^2)*cosh(d*x + c)^5 - (a^2 + a*b - 12*b^2)*cosh(d*x + c)^3 - (a*b + 4*b^2)*cosh(d*x + c))*s
inh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) + ((a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)^8 + 8*(a^2 + 5*a*b
+ 4*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2 + 5*a*b + 4*b^2)*sinh(d*x + c)^8 - 4*(a*b + 4*b^2)*cosh(d*x + c
)^6 + 4*(7*(a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)^2 - a*b - 4*b^2)*sinh(d*x + c)^6 + 8*(7*(a^2 + 5*a*b + 4*b^2)*c
osh(d*x + c)^3 - 3*(a*b + 4*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(a^2 + a*b - 12*b^2)*cosh(d*x + c)^4 + 2*(
35*(a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)^4 - 30*(a*b + 4*b^2)*cosh(d*x + c)^2 - a^2 - a*b + 12*b^2)*sinh(d*x + c
)^4 + 8*(7*(a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)^5 - 10*(a*b + 4*b^2)*cosh(d*x + c)^3 - (a^2 + a*b - 12*b^2)*cos
h(d*x + c))*sinh(d*x + c)^3 - 4*(a*b + 4*b^2)*cosh(d*x + c)^2 + 4*(7*(a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)^6 - 1
5*(a*b + 4*b^2)*cosh(d*x + c)^4 - 3*(a^2 + a*b - 12*b^2)*cosh(d*x + c)^2 - a*b - 4*b^2)*sinh(d*x + c)^2 + a^2
+ 5*a*b + 4*b^2 + 8*((a^2 + 5*a*b + 4*b^2)*cosh(d*x + c)^7 - 3*(a*b + 4*b^2)*cosh(d*x + c)^5 - (a^2 + a*b - 12
*b^2)*cosh(d*x + c)^3 - (a*b + 4*b^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2
*(7*(a^2 + 2*a*b)*cosh(d*x + c)^6 + 5*(3*a^2 - 2*a*b)*cosh(d*x + c)^4 + 3*(3*a^2 - 2*a*b)*cosh(d*x + c)^2 + a^
2 + 2*a*b)*sinh(d*x + c))/(4*a^3*b*d*cosh(d*x + c)^6 - (a^4 + a^3*b)*d*cosh(d*x + c)^8 - 8*(a^4 + a^3*b)*d*cos
h(d*x + c)*sinh(d*x + c)^7 - (a^4 + a^3*b)*d*sinh(d*x + c)^8 + 4*a^3*b*d*cosh(d*x + c)^2 + 4*(a^3*b*d - 7*(a^4
+ a^3*b)*d*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 2*(a^4 - 3*a^3*b)*d*cosh(d*x + c)^4 + 8*(3*a^3*b*d*cosh(d*x + c
) - 7*(a^4 + a^3*b)*d*cosh(d*x + c)^3)*sinh(d*x + c)^5 + 2*(30*a^3*b*d*cosh(d*x + c)^2 - 35*(a^4 + a^3*b)*d*co
sh(d*x + c)^4 + (a^4 - 3*a^3*b)*d)*sinh(d*x + c)^4 + 8*(10*a^3*b*d*cosh(d*x + c)^3 - 7*(a^4 + a^3*b)*d*cosh(d*
x + c)^5 + (a^4 - 3*a^3*b)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(15*a^3*b*d*cosh(d*x + c)^4 - 7*(a^4 + a^3*b)*
d*cosh(d*x + c)^6 + a^3*b*d + 3*(a^4 - 3*a^3*b)*d*cosh(d*x + c)^2)*sinh(d*x + c)^2 - (a^4 + a^3*b)*d + 8*(3*a^
3*b*d*cosh(d*x + c)^5 - (a^4 + a^3*b)*d*cosh(d*x + c)^7 + a^3*b*d*cosh(d*x + c) + (a^4 - 3*a^3*b)*d*cosh(d*x +
c)^3)*sinh(d*x + c))]

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giac [B]  time = 0.71, size = 1074, normalized size = 7.62 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((2*(12*a^3*b^2 + 4*a^2*b^3 - 16*a*b^4 + (3*a^3*b - 14*a^2*b^2 - 21*a*b^3 + 4*b^4)*sqrt(-a*b))*a^2*abs(a*
e^(2*c) + b*e^(2*c)) + (3*a^5*b - 11*a^4*b^2 - 35*a^3*b^3 - 17*a^2*b^4 + 4*a*b^5 - 4*(3*a^4*b + 4*a^3*b^2 - 3*
a^2*b^3 - 4*a*b^4)*sqrt(-a*b))*abs(a*e^(2*c) + b*e^(2*c))*abs(a) + (12*a^6*b - 8*a^5*b^2 - 20*a^4*b^3 + 16*a^3
*b^4 + (3*a^6 - 17*a^5*b - 7*a^4*b^2 + 25*a^3*b^3 - 4*a^2*b^4)*sqrt(-a*b))*abs(a*e^(2*c) + b*e^(2*c)))*arctan(
e^(d*x)/sqrt((a^4*e^(2*c) - a^3*b*e^(2*c) + sqrt((a^4*e^(2*c) - a^3*b*e^(2*c))^2 - (a^4*e^(4*c) + a^3*b*e^(4*c
))*(a^4 + a^3*b)))/(a^4*e^(4*c) + a^3*b*e^(4*c))))*e^(-2*c)/((a^8 + 2*a^7*b - 2*a^5*b^3 - a^4*b^4 - 2*(a^7 + 3
*a^6*b + 3*a^5*b^2 + a^4*b^3)*sqrt(-a*b))*sqrt(a^2 - b^2 - 2*sqrt(-a*b)*(a + b))*abs(a)) + (2*(12*a^3*b^2 + 4*
a^2*b^3 - 16*a*b^4 - (3*a^3*b - 14*a^2*b^2 - 21*a*b^3 + 4*b^4)*sqrt(-a*b))*a^2*abs(a*e^(2*c) + b*e^(2*c)) + (3
*a^5*b - 11*a^4*b^2 - 35*a^3*b^3 - 17*a^2*b^4 + 4*a*b^5 + 4*(3*a^4*b + 4*a^3*b^2 - 3*a^2*b^3 - 4*a*b^4)*sqrt(-
a*b))*abs(a*e^(2*c) + b*e^(2*c))*abs(a) + (12*a^6*b - 8*a^5*b^2 - 20*a^4*b^3 + 16*a^3*b^4 - (3*a^6 - 17*a^5*b
- 7*a^4*b^2 + 25*a^3*b^3 - 4*a^2*b^4)*sqrt(-a*b))*abs(a*e^(2*c) + b*e^(2*c)))*arctan(e^(d*x)/sqrt((a^4*e^(2*c)
- a^3*b*e^(2*c) - sqrt((a^4*e^(2*c) - a^3*b*e^(2*c))^2 - (a^4*e^(4*c) + a^3*b*e^(4*c))*(a^4 + a^3*b)))/(a^4*e
^(4*c) + a^3*b*e^(4*c))))*e^(-2*c)/((a^8 + 2*a^7*b - 2*a^5*b^3 - a^4*b^4 + 2*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*
b^3)*sqrt(-a*b))*sqrt(a^2 - b^2 + 2*sqrt(-a*b)*(a + b))*abs(a)) - (a*e^c + 4*b*e^c)*e^(-c)*log(e^(d*x + c) + 1
)/a^3 + (a*e^c + 4*b*e^c)*e^(-c)*log(abs(e^(d*x + c) - 1))/a^3 + 2*(b*e^(3*d*x + 3*c) + b*e^(d*x + c))/((a*e^(
4*d*x + 4*c) + b*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) - 2*b*e^(2*d*x + 2*c) + a + b)*a^2) + 2*(e^(3*d*x + 3*c
) + e^(d*x + c))/(a^2*(e^(2*d*x + 2*c) - 1)^2))/d

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maple [B]  time = 0.46, size = 367, normalized size = 2.60 \[ \frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{2}}-\frac {b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a \right )}-\frac {2 b^{2} \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3} \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a \right )}-\frac {b}{d \,a^{2} \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a \right )}-\frac {3 b \arctanh \left (\frac {2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{2 d \,a^{2} \sqrt {a b +b^{2}}}-\frac {2 b^{2} \arctanh \left (\frac {2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{d \,a^{3} \sqrt {a b +b^{2}}}-\frac {1}{8 d \,a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{2}}-\frac {2 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b}{d \,a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/d*tanh(1/2*d*x+1/2*c)^2/a^2-1/d*b/a^2/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/
2*c)^2*b+a)*tanh(1/2*d*x+1/2*c)^2-2/d*b^2/a^3/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*
x+1/2*c)^2*b+a)*tanh(1/2*d*x+1/2*c)^2-1/d*b/a^2/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*
d*x+1/2*c)^2*b+a)-3/2/d*b/a^2/(a*b+b^2)^(1/2)*arctanh(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a+2*a+4*b)/(a*b+b^2)^(1/2))
-2/d*b^2/a^3/(a*b+b^2)^(1/2)*arctanh(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a+2*a+4*b)/(a*b+b^2)^(1/2))-1/8/d/a^2/tanh(1
/2*d*x+1/2*c)^2-1/2/d/a^2*ln(tanh(1/2*d*x+1/2*c))-2/d/a^3*ln(tanh(1/2*d*x+1/2*c))*b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (a e^{\left (7 \, c\right )} + 2 \, b e^{\left (7 \, c\right )}\right )} e^{\left (7 \, d x\right )} + {\left (3 \, a e^{\left (5 \, c\right )} - 2 \, b e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} + {\left (3 \, a e^{\left (3 \, c\right )} - 2 \, b e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + {\left (a e^{c} + 2 \, b e^{c}\right )} e^{\left (d x\right )}}{4 \, a^{2} b d e^{\left (6 \, d x + 6 \, c\right )} + 4 \, a^{2} b d e^{\left (2 \, d x + 2 \, c\right )} - a^{3} d - a^{2} b d - {\left (a^{3} d e^{\left (8 \, c\right )} + a^{2} b d e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} + 2 \, {\left (a^{3} d e^{\left (4 \, c\right )} - 3 \, a^{2} b d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )}} + \frac {{\left (a + 4 \, b\right )} \log \left ({\left (e^{\left (d x + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{2 \, a^{3} d} - \frac {{\left (a + 4 \, b\right )} \log \left ({\left (e^{\left (d x + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{2 \, a^{3} d} + 8 \, \int \frac {{\left (3 \, a b e^{\left (3 \, c\right )} + 4 \, b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} - {\left (3 \, a b e^{c} + 4 \, b^{2} e^{c}\right )} e^{\left (d x\right )}}{8 \, {\left (a^{4} + a^{3} b + {\left (a^{4} e^{\left (4 \, c\right )} + a^{3} b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \, {\left (a^{4} e^{\left (2 \, c\right )} - a^{3} b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a*e^(7*c) + 2*b*e^(7*c))*e^(7*d*x) + (3*a*e^(5*c) - 2*b*e^(5*c))*e^(5*d*x) + (3*a*e^(3*c) - 2*b*e^(3*c))*e^(
3*d*x) + (a*e^c + 2*b*e^c)*e^(d*x))/(4*a^2*b*d*e^(6*d*x + 6*c) + 4*a^2*b*d*e^(2*d*x + 2*c) - a^3*d - a^2*b*d -
(a^3*d*e^(8*c) + a^2*b*d*e^(8*c))*e^(8*d*x) + 2*(a^3*d*e^(4*c) - 3*a^2*b*d*e^(4*c))*e^(4*d*x)) + 1/2*(a + 4*b
)*log((e^(d*x + c) + 1)*e^(-c))/(a^3*d) - 1/2*(a + 4*b)*log((e^(d*x + c) - 1)*e^(-c))/(a^3*d) + 8*integrate(1/
8*((3*a*b*e^(3*c) + 4*b^2*e^(3*c))*e^(3*d*x) - (3*a*b*e^c + 4*b^2*e^c)*e^(d*x))/(a^4 + a^3*b + (a^4*e^(4*c) +
a^3*b*e^(4*c))*e^(4*d*x) + 2*(a^4*e^(2*c) - a^3*b*e^(2*c))*e^(2*d*x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)^3*(a + b*tanh(c + d*x)^2)^2),x)

[Out]

int(1/(sinh(c + d*x)^3*(a + b*tanh(c + d*x)^2)^2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}^{3}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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