### 3.37 $$\int \frac{\text{csch}(c+d x)}{(a+b \tanh ^2(c+d x))^2} \, dx$$

Optimal. Leaf size=103 $\frac{\sqrt{b} (3 a+2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right )}{2 a^2 d (a+b)^{3/2}}-\frac{\tanh ^{-1}(\cosh (c+d x))}{a^2 d}+\frac{b \text{sech}(c+d x)}{2 a d (a+b) \left (a-b \text{sech}^2(c+d x)+b\right )}$

[Out]

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

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Rubi [A]  time = 0.143505, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.238, Rules used = {3664, 414, 522, 207, 208} $\frac{\sqrt{b} (3 a+2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \text{sech}(c+d x)}{\sqrt{a+b}}\right )}{2 a^2 d (a+b)^{3/2}}-\frac{\tanh ^{-1}(\cosh (c+d x))}{a^2 d}+\frac{b \text{sech}(c+d x)}{2 a d (a+b) \left (a-b \text{sech}^2(c+d x)+b\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, 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 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]

Rubi steps

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

Mathematica [C]  time = 0.663079, size = 175, normalized size = 1.7 $\frac{\frac{2 a b \cosh (c+d x)}{(a+b) ((a+b) \cosh (2 (c+d x))+a-b)}+\frac{i \sqrt{b} (3 a+2 b) \tan ^{-1}\left (\frac{-\sqrt{a} \tanh \left (\frac{1}{2} (c+d x)\right )-i \sqrt{a+b}}{\sqrt{b}}\right )}{(a+b)^{3/2}}+\frac{i \sqrt{b} (3 a+2 b) \tan ^{-1}\left (\frac{\sqrt{a} \tanh \left (\frac{1}{2} (c+d x)\right )-i \sqrt{a+b}}{\sqrt{b}}\right )}{(a+b)^{3/2}}+2 \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{2 a^2 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((I*Sqrt[b]*(3*a + 2*b)*ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]])/(a + b)^(3/2) + (I*Sqr
t[b]*(3*a + 2*b)*ArcTan[((-I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]])/(a + b)^(3/2) + (2*a*b*Cosh[c
+ d*x])/((a + b)*(a - b + (a + b)*Cosh[2*(c + d*x)])) + 2*Log[Tanh[(c + d*x)/2]])/(2*a^2*d)

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Maple [B]  time = 0.092, size = 331, normalized size = 3.2 \begin{align*}{\frac{b}{da \left ( a+b \right ) } \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}a+2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+4\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a \right ) ^{-1}}+2\,{\frac{{b}^{2} \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{d{a}^{2} \left ( \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}a+2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+4\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a \right ) \left ( a+b \right ) }}+{\frac{b}{da \left ( a+b \right ) } \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}a+2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+4\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a \right ) ^{-1}}+{\frac{3\,b}{2\,da \left ( a+b \right ) }{\it Artanh} \left ({\frac{1}{4} \left ( 2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+2\,a+4\,b \right ){\frac{1}{\sqrt{ab+{b}^{2}}}}} \right ){\frac{1}{\sqrt{ab+{b}^{2}}}}}+{\frac{{b}^{2}}{d{a}^{2} \left ( a+b \right ) }{\it Artanh} \left ({\frac{1}{4} \left ( 2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+2\,a+4\,b \right ){\frac{1}{\sqrt{ab+{b}^{2}}}}} \right ){\frac{1}{\sqrt{ab+{b}^{2}}}}}+{\frac{1}{d{a}^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{b e^{\left (3 \, d x + 3 \, c\right )} + b e^{\left (d x + c\right )}}{a^{3} d + 2 \, a^{2} b d + a b^{2} d +{\left (a^{3} d e^{\left (4 \, c\right )} + 2 \, a^{2} b d e^{\left (4 \, c\right )} + a b^{2} d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (a^{3} d e^{\left (2 \, c\right )} - a b^{2} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}} - \frac{\log \left ({\left (e^{\left (d x + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{a^{2} d} + \frac{\log \left ({\left (e^{\left (d x + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{a^{2} d} - 2 \, \int \frac{{\left (3 \, a b e^{\left (3 \, c\right )} + 2 \, b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} -{\left (3 \, a b e^{c} + 2 \, b^{2} e^{c}\right )} e^{\left (d x\right )}}{2 \,{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2} +{\left (a^{4} e^{\left (4 \, c\right )} + 2 \, a^{3} b e^{\left (4 \, c\right )} + a^{2} b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (a^{4} e^{\left (2 \, c\right )} - a^{2} b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 3.28411, size = 6604, normalized size = 64.12 \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*tanh(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[1/4*(4*a*b*cosh(d*x + c)^3 + 12*a*b*cosh(d*x + c)*sinh(d*x + c)^2 + 4*a*b*sinh(d*x + c)^3 + 4*a*b*cosh(d*x +
c) + ((3*a^2 + 5*a*b + 2*b^2)*cosh(d*x + c)^4 + 4*(3*a^2 + 5*a*b + 2*b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (3*a
^2 + 5*a*b + 2*b^2)*sinh(d*x + c)^4 + 2*(3*a^2 - a*b - 2*b^2)*cosh(d*x + c)^2 + 2*(3*(3*a^2 + 5*a*b + 2*b^2)*c
osh(d*x + c)^2 + 3*a^2 - a*b - 2*b^2)*sinh(d*x + c)^2 + 3*a^2 + 5*a*b + 2*b^2 + 4*((3*a^2 + 5*a*b + 2*b^2)*cos
h(d*x + c)^3 + (3*a^2 - a*b - 2*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)*sinh(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))*si
nh(d*x + c) + a + b)) - 4*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 4*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x
+ c)^3 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 + 2*(a^2 - b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 + 2*a*b + b^2)*cosh(d
*x + c)^2 + a^2 - b^2)*sinh(d*x + c)^2 + a^2 + 2*a*b + b^2 + 4*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + (a^2 - b
^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) + 4*((a^2 + 2*a*b + b^2)*cosh(d*x + c
)^4 + 4*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 + 2*(a^2 - b^2
)*cosh(d*x + c)^2 + 2*(3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 - b^2)*sinh(d*x + c)^2 + a^2 + 2*a*b + b^2
+ 4*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + (a^2 - b^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(
d*x + c) - 1) + 4*(3*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c))/((a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^4 +
4*(a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 + 2*a^3*b + a^2*b^2)*d*sinh(d*x + c)^4 + 2*
(a^4 - a^2*b^2)*d*cosh(d*x + c)^2 + 2*(3*(a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^2 + (a^4 - a^2*b^2)*d)*sinh
(d*x + c)^2 + (a^4 + 2*a^3*b + a^2*b^2)*d + 4*((a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^3 + (a^4 - a^2*b^2)*d
*cosh(d*x + c))*sinh(d*x + c)), 1/2*(2*a*b*cosh(d*x + c)^3 + 6*a*b*cosh(d*x + c)*sinh(d*x + c)^2 + 2*a*b*sinh(
d*x + c)^3 + 2*a*b*cosh(d*x + c) + ((3*a^2 + 5*a*b + 2*b^2)*cosh(d*x + c)^4 + 4*(3*a^2 + 5*a*b + 2*b^2)*cosh(d
*x + c)*sinh(d*x + c)^3 + (3*a^2 + 5*a*b + 2*b^2)*sinh(d*x + c)^4 + 2*(3*a^2 - a*b - 2*b^2)*cosh(d*x + c)^2 +
2*(3*(3*a^2 + 5*a*b + 2*b^2)*cosh(d*x + c)^2 + 3*a^2 - a*b - 2*b^2)*sinh(d*x + c)^2 + 3*a^2 + 5*a*b + 2*b^2 +
4*((3*a^2 + 5*a*b + 2*b^2)*cosh(d*x + c)^3 + (3*a^2 - a*b - 2*b^2)*cosh(d*x + c))*sinh(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 +
5*a*b + 2*b^2)*cosh(d*x + c)^4 + 4*(3*a^2 + 5*a*b + 2*b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (3*a^2 + 5*a*b + 2
*b^2)*sinh(d*x + c)^4 + 2*(3*a^2 - a*b - 2*b^2)*cosh(d*x + c)^2 + 2*(3*(3*a^2 + 5*a*b + 2*b^2)*cosh(d*x + c)^2
+ 3*a^2 - a*b - 2*b^2)*sinh(d*x + c)^2 + 3*a^2 + 5*a*b + 2*b^2 + 4*((3*a^2 + 5*a*b + 2*b^2)*cosh(d*x + c)^3 +
(3*a^2 - a*b - 2*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-b/(a + b))*arctan(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 + b^2)*cosh(d*x + c)^4 + 4*(a^2 + 2*a*b + b^2)*cosh(d
*x + c)*sinh(d*x + c)^3 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^4 + 2*(a^2 - b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 + 2*
a*b + b^2)*cosh(d*x + c)^2 + a^2 - b^2)*sinh(d*x + c)^2 + a^2 + 2*a*b + b^2 + 4*((a^2 + 2*a*b + b^2)*cosh(d*x
+ c)^3 + (a^2 - b^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) + 2*((a^2 + 2*a*b +
b^2)*cosh(d*x + c)^4 + 4*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)
^4 + 2*(a^2 - b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 - b^2)*sinh(d*x + c)^2 + a
^2 + 2*a*b + b^2 + 4*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + (a^2 - b^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh
(d*x + c) + sinh(d*x + c) - 1) + 2*(3*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c))/((a^4 + 2*a^3*b + a^2*b^2)*d*c
osh(d*x + c)^4 + 4*(a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 + 2*a^3*b + a^2*b^2)*d*sin
h(d*x + c)^4 + 2*(a^4 - a^2*b^2)*d*cosh(d*x + c)^2 + 2*(3*(a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^2 + (a^4 -
a^2*b^2)*d)*sinh(d*x + c)^2 + (a^4 + 2*a^3*b + a^2*b^2)*d + 4*((a^4 + 2*a^3*b + a^2*b^2)*d*cosh(d*x + c)^3 +
(a^4 - a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [C]  time = 1.70439, size = 7329, normalized size = 71.16 \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*tanh(d*x+c)^2)^2,x, algorithm="giac")

[Out]

-1/8*(2*(3*(3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*cos(1/2*real_part(arc
cos(-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3*sin(1/2*real_part(arccos
(-a/(a + b) + b/(a + b)))) - (3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*cos
h(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3 - 9*(3
*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b
) + b/(a + b))))^2*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2*sin(1/2*real_part(arccos(-a/(a + b) +
b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 3*(3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7
*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2*sin(1/2*real_part(
arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 9*(3*a^5*b*e^(4*c) +
8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^
2*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(
1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c)
+ 2*a^2*b^4*e^(4*c))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) +
b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c)
+ 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*sin(1/2*real_pa
rt(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + (3*a^5*b*e^(4*c) +
8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))
^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 - (3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^
(4*c) + 2*a^2*b^4*e^(4*c))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(a
+ b) + b/(a + b)))) + (3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*sin(1/2*re
al_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))))*arctan((((a^4 +
2*a^3*b + a^2*b^2)/(a^4*e^(4*c) + 2*a^3*b*e^(4*c) + a^2*b^2*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b)))
+ e^(d*x))/(((a^4 + 2*a^3*b + a^2*b^2)/(a^4*e^(4*c) + 2*a^3*b*e^(4*c) + a^2*b^2*e^(4*c)))^(1/4)*sin(1/2*arccos
(-(a - b)/(a + b)))))/(2*(a^3*e^(2*c) + a^2*b*e^(2*c))^2*a*b - (a^4*e^(2*c) - a^2*b^2*e^(2*c))*sqrt(-a*b)*abs(
-a^3*e^(2*c) - a^2*b*e^(2*c))) + 2*(3*(3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(
4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))
^3*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b)))) - (3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*
c) + 2*a^2*b^4*e^(4*c))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3*sin(1/2*real_part(arccos(-a/(a +
b) + b/(a + b))))^3 - 9*(3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*cos(1/2
*real_part(arccos(-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2*sin(1/2*re
al_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 3*(3*a^5*b*e^(4
*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a +
b))))^2*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)
))) + 9*(3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*cos(1/2*real_part(arccos
(-a/(a + b) + b/(a + b))))^2*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*real_part(arccos(-a/(
a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(3*a^5*b*e^(4*c) + 8*a^4*b^2*e
^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(1/2*re
al_part(arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(3*a^5*b*
e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(
a + b))))^2*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b
))))^3 + (3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*sin(1/2*real_part(arcco
s(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 - (3*a^5*b*e^(4*c) + 8*a^4
*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin(
1/2*real_part(arccos(-a/(a + b) + b/(a + b)))) + (3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*
a^2*b^4*e^(4*c))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(
a + b)))))*arctan(-(((a^4 + 2*a^3*b + a^2*b^2)/(a^4*e^(4*c) + 2*a^3*b*e^(4*c) + a^2*b^2*e^(4*c)))^(1/4)*cos(1/
2*arccos(-(a - b)/(a + b))) - e^(d*x))/(((a^4 + 2*a^3*b + a^2*b^2)/(a^4*e^(4*c) + 2*a^3*b*e^(4*c) + a^2*b^2*e^
(4*c)))^(1/4)*sin(1/2*arccos(-(a - b)/(a + b)))))/(2*(a^3*e^(2*c) + a^2*b*e^(2*c))^2*a*b - (a^4*e^(2*c) - a^2*
b^2*e^(2*c))*sqrt(-a*b)*abs(-a^3*e^(2*c) - a^2*b*e^(2*c))) + ((3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3
*e^(4*c) + 2*a^2*b^4*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_part(arccos(-
a/(a + b) + b/(a + b))))^3 - 3*(3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*c
os(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3*sin(1/
2*real_part(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) +
2*a^2*b^4*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_part(arccos(-a/(a + b) +
b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 9*(3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) +
7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(
arccos(-a/(a + b) + b/(a + b))))^2*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arc
cos(-a/(a + b) + b/(a + b)))) + 3*(3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c)
)*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sin
h(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 9*(3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c
) + 2*a^2*b^4*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(arccos(-a/(a + b)
+ b/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/
(a + b))))^2 - (3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*cos(1/2*real_part
(arccos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + 3*(3*a^5*b*e^(4*c)
+ 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))
))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3
- (3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*cos(1/2*real_part(arccos(-a/(a
+ b) + b/(a + b))))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + (3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c
) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_pa
rt(arccos(-a/(a + b) + b/(a + b)))))*log(2*((a^4 + 2*a^3*b + a^2*b^2)/(a^4*e^(4*c) + 2*a^3*b*e^(4*c) + a^2*b^2
*e^(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b)))*e^(d*x) + sqrt((a^4 + 2*a^3*b + a^2*b^2)/(a^4*e^(4*c) + 2*a
^3*b*e^(4*c) + a^2*b^2*e^(4*c))) + e^(2*d*x))/(2*(a^3*e^(2*c) + a^2*b*e^(2*c))^2*a*b - (a^4*e^(2*c) - a^2*b^2*
e^(2*c))*sqrt(-a*b)*abs(-a^3*e^(2*c) - a^2*b*e^(2*c))) - ((3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(
4*c) + 2*a^2*b^4*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_part(arccos(-a/(a
+ b) + b/(a + b))))^3 - 3*(3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*cos(1
/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3*sin(1/2*re
al_part(arccos(-a/(a + b) + b/(a + b))))^2 - 3*(3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^
2*b^4*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(
a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + 9*(3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a
^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(arcc
os(-a/(a + b) + b/(a + b))))^2*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(
-a/(a + b) + b/(a + b)))) + 3*(3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*co
s(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^3*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/
2*imag_part(arccos(-a/(a + b) + b/(a + b))))^2 - 9*(3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) +
2*a^2*b^4*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*cosh(1/2*imag_part(arccos(-a/(a + b) + b
/(a + b))))*sin(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a +
b))))^2 - (3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*cos(1/2*real_part(arc
cos(-a/(a + b) + b/(a + b))))^3*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 + 3*(3*a^5*b*e^(4*c) + 8
*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*s
in(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))^2*sinh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b))))^3 - (3
*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) + 7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b
) + b/(a + b))))*cosh(1/2*imag_part(arccos(-a/(a + b) + b/(a + b)))) + (3*a^5*b*e^(4*c) + 8*a^4*b^2*e^(4*c) +
7*a^3*b^3*e^(4*c) + 2*a^2*b^4*e^(4*c))*cos(1/2*real_part(arccos(-a/(a + b) + b/(a + b))))*sinh(1/2*imag_part(a
rccos(-a/(a + b) + b/(a + b)))))*log(-2*((a^4 + 2*a^3*b + a^2*b^2)/(a^4*e^(4*c) + 2*a^3*b*e^(4*c) + a^2*b^2*e^
(4*c)))^(1/4)*cos(1/2*arccos(-(a - b)/(a + b)))*e^(d*x) + sqrt((a^4 + 2*a^3*b + a^2*b^2)/(a^4*e^(4*c) + 2*a^3*
b*e^(4*c) + a^2*b^2*e^(4*c))) + e^(2*d*x))/(2*(a^3*e^(2*c) + a^2*b*e^(2*c))^2*a*b - (a^4*e^(2*c) - a^2*b^2*e^(
2*c))*sqrt(-a*b)*abs(-a^3*e^(2*c) - a^2*b*e^(2*c))) - 8*(b*e^(3*d*x + 3*c) + b*e^(d*x + c))/((a^2 + a*b)*(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)) + 8*log(e^(d*x + c) +
1)/a^2 - 8*log(abs(e^(d*x + c) - 1))/a^2)/d