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

Optimal. Leaf size=85 \[ \frac {(a+2 b) \tanh ^{-1}(\cosh (c+d x))}{2 a^2 d}-\frac {\sqrt {b} \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d} \]

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 85, normalized size of antiderivative = 1.00, 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 = {3664, 471, 522, 207, 208} \[ \frac {(a+2 b) \tanh ^{-1}(\cosh (c+d x))}{2 a^2 d}-\frac {\sqrt {b} \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [C]  time = 0.70, size = 170, normalized size = 2.00 \[ -\frac {8 i \sqrt {b} \sqrt {a+b} \tan ^{-1}\left (\frac {-\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )-i \sqrt {a+b}}{\sqrt {b}}\right )+8 i \sqrt {b} \sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )-i \sqrt {a+b}}{\sqrt {b}}\right )+a \text {csch}^2\left (\frac {1}{2} (c+d x)\right )+a \text {sech}^2\left (\frac {1}{2} (c+d x)\right )+4 a \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+8 b \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.27, size = 1790, normalized size = 21.06 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(2*a*cosh(d*x + c)^3 + 6*a*cosh(d*x + c)*sinh(d*x + c)^2 + 2*a*sinh(d*x + c)^3 - (cosh(d*x + c)^4 + 4*co
sh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4 + 2*(3*cosh(d*x + c)^2 - 1)*sinh(d*x + c)^2 - 2*cosh(d*x + c)^2
+ 4*(cosh(d*x + c)^3 - cosh(d*x + c))*sinh(d*x + c) + 1)*sqrt(a*b + b^2)*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*(cosh(d*x + c)^3 + 3*cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x + c)^2 + 1)*sinh(d*x + c)
+ cosh(d*x + c))*sqrt(a*b + b^2) + 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))*sinh(d*x + c) + a + b)) + 2*a*cosh(d*x + c) - ((a + 2*b)
*cosh(d*x + c)^4 + 4*(a + 2*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + 2*b)*sinh(d*x + c)^4 - 2*(a + 2*b)*cosh(d*
x + c)^2 + 2*(3*(a + 2*b)*cosh(d*x + c)^2 - a - 2*b)*sinh(d*x + c)^2 + 4*((a + 2*b)*cosh(d*x + c)^3 - (a + 2*b
)*cosh(d*x + c))*sinh(d*x + c) + a + 2*b)*log(cosh(d*x + c) + sinh(d*x + c) + 1) + ((a + 2*b)*cosh(d*x + c)^4
+ 4*(a + 2*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + 2*b)*sinh(d*x + c)^4 - 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*(
a + 2*b)*cosh(d*x + c)^2 - a - 2*b)*sinh(d*x + c)^2 + 4*((a + 2*b)*cosh(d*x + c)^3 - (a + 2*b)*cosh(d*x + c))*
sinh(d*x + c) + a + 2*b)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*(3*a*cosh(d*x + c)^2 + a)*sinh(d*x + c))/(
a^2*d*cosh(d*x + c)^4 + 4*a^2*d*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*d*sinh(d*x + c)^4 - 2*a^2*d*cosh(d*x + c)^
2 + a^2*d + 2*(3*a^2*d*cosh(d*x + c)^2 - a^2*d)*sinh(d*x + c)^2 + 4*(a^2*d*cosh(d*x + c)^3 - a^2*d*cosh(d*x +
c))*sinh(d*x + c)), -1/2*(2*a*cosh(d*x + c)^3 + 6*a*cosh(d*x + c)*sinh(d*x + c)^2 + 2*a*sinh(d*x + c)^3 + 2*(c
osh(d*x + c)^4 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4 + 2*(3*cosh(d*x + c)^2 - 1)*sinh(d*x + c)^2
- 2*cosh(d*x + c)^2 + 4*(cosh(d*x + c)^3 - cosh(d*x + c))*sinh(d*x + c) + 1)*sqrt(-a*b - b^2)*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(-a*b - b^2)/(a*b + b^2)) - 2*(cosh(d*x + c)^4
+ 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4 + 2*(3*cosh(d*x + c)^2 - 1)*sinh(d*x + c)^2 - 2*cosh(d*x
+ c)^2 + 4*(cosh(d*x + c)^3 - cosh(d*x + c))*sinh(d*x + c) + 1)*sqrt(-a*b - b^2)*arctan(1/2*sqrt(-a*b - b^2)*(
cosh(d*x + c) + sinh(d*x + c))/b) + 2*a*cosh(d*x + c) - ((a + 2*b)*cosh(d*x + c)^4 + 4*(a + 2*b)*cosh(d*x + c)
*sinh(d*x + c)^3 + (a + 2*b)*sinh(d*x + c)^4 - 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*(a + 2*b)*cosh(d*x + c)^2 -
a - 2*b)*sinh(d*x + c)^2 + 4*((a + 2*b)*cosh(d*x + c)^3 - (a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + a + 2*b)*lo
g(cosh(d*x + c) + sinh(d*x + c) + 1) + ((a + 2*b)*cosh(d*x + c)^4 + 4*(a + 2*b)*cosh(d*x + c)*sinh(d*x + c)^3
+ (a + 2*b)*sinh(d*x + c)^4 - 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*(a + 2*b)*cosh(d*x + c)^2 - a - 2*b)*sinh(d*x
+ c)^2 + 4*((a + 2*b)*cosh(d*x + c)^3 - (a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + a + 2*b)*log(cosh(d*x + c) +
sinh(d*x + c) - 1) + 2*(3*a*cosh(d*x + c)^2 + a)*sinh(d*x + c))/(a^2*d*cosh(d*x + c)^4 + 4*a^2*d*cosh(d*x + c
)*sinh(d*x + c)^3 + a^2*d*sinh(d*x + c)^4 - 2*a^2*d*cosh(d*x + c)^2 + a^2*d + 2*(3*a^2*d*cosh(d*x + c)^2 - a^2
*d)*sinh(d*x + c)^2 + 4*(a^2*d*cosh(d*x + c)^3 - a^2*d*cosh(d*x + c))*sinh(d*x + c))]

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giac [B]  time = 0.38, size = 475, normalized size = 5.59 \[ \frac {\frac {2 \, {\left (3 \, a b - b^{2} - \sqrt {-a b} {\left (a - 3 \, b\right )}\right )} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} \arctan \left (\frac {e^{\left (d x\right )}}{\sqrt {\frac {a^{3} e^{\left (2 \, c\right )} - a^{2} b e^{\left (2 \, c\right )} + \sqrt {{\left (a^{3} e^{\left (2 \, c\right )} - a^{2} b e^{\left (2 \, c\right )}\right )}^{2} - {\left (a^{3} e^{\left (4 \, c\right )} + a^{2} b e^{\left (4 \, c\right )}\right )} {\left (a^{3} + a^{2} b\right )}}}{a^{3} e^{\left (4 \, c\right )} + a^{2} b e^{\left (4 \, c\right )}}}}\right ) e^{\left (-2 \, c\right )}}{{\left (a^{3} - a^{2} b + 2 \, \sqrt {-a b} a^{2}\right )} \sqrt {a^{2} - b^{2} + 2 \, \sqrt {-a b} {\left (a + b\right )}}} + \frac {2 \, {\left (3 \, a b - b^{2} + \sqrt {-a b} {\left (a - 3 \, b\right )}\right )} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} \arctan \left (\frac {e^{\left (d x\right )}}{\sqrt {\frac {a^{3} e^{\left (2 \, c\right )} - a^{2} b e^{\left (2 \, c\right )} - \sqrt {{\left (a^{3} e^{\left (2 \, c\right )} - a^{2} b e^{\left (2 \, c\right )}\right )}^{2} - {\left (a^{3} e^{\left (4 \, c\right )} + a^{2} b e^{\left (4 \, c\right )}\right )} {\left (a^{3} + a^{2} b\right )}}}{a^{3} e^{\left (4 \, c\right )} + a^{2} b e^{\left (4 \, c\right )}}}}\right ) e^{\left (-2 \, c\right )}}{{\left (a^{3} - a^{2} b - 2 \, \sqrt {-a b} a^{2}\right )} \sqrt {a^{2} - b^{2} - 2 \, \sqrt {-a b} {\left (a + b\right )}}} + \frac {{\left (a e^{c} + 2 \, b e^{c}\right )} e^{\left (-c\right )} \log \left (e^{\left (d x + c\right )} + 1\right )}{a^{2}} - \frac {{\left (a e^{c} + 2 \, b e^{c}\right )} e^{\left (-c\right )} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a^{2}} - \frac {2 \, {\left (e^{\left (3 \, d x + 3 \, c\right )} + e^{\left (d x + c\right )}\right )}}{a {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(3*a*b - b^2 - sqrt(-a*b)*(a - 3*b))*abs(a*e^(2*c) + b*e^(2*c))*arctan(e^(d*x)/sqrt((a^3*e^(2*c) - a^2*
b*e^(2*c) + sqrt((a^3*e^(2*c) - a^2*b*e^(2*c))^2 - (a^3*e^(4*c) + a^2*b*e^(4*c))*(a^3 + a^2*b)))/(a^3*e^(4*c)
+ a^2*b*e^(4*c))))*e^(-2*c)/((a^3 - a^2*b + 2*sqrt(-a*b)*a^2)*sqrt(a^2 - b^2 + 2*sqrt(-a*b)*(a + b))) + 2*(3*a
*b - b^2 + sqrt(-a*b)*(a - 3*b))*abs(a*e^(2*c) + b*e^(2*c))*arctan(e^(d*x)/sqrt((a^3*e^(2*c) - a^2*b*e^(2*c) -
sqrt((a^3*e^(2*c) - a^2*b*e^(2*c))^2 - (a^3*e^(4*c) + a^2*b*e^(4*c))*(a^3 + a^2*b)))/(a^3*e^(4*c) + a^2*b*e^(
4*c))))*e^(-2*c)/((a^3 - a^2*b - 2*sqrt(-a*b)*a^2)*sqrt(a^2 - b^2 - 2*sqrt(-a*b)*(a + b))) + (a*e^c + 2*b*e^c)
*e^(-c)*log(e^(d*x + c) + 1)/a^2 - (a*e^c + 2*b*e^c)*e^(-c)*log(abs(e^(d*x + c) - 1))/a^2 - 2*(e^(3*d*x + 3*c)
+ e^(d*x + c))/(a*(e^(2*d*x + 2*c) - 1)^2))/d

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maple [B]  time = 0.39, size = 181, normalized size = 2.13 \[ \frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}-\frac {b \arctanh \left (\frac {2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{d a \sqrt {a b +b^{2}}}-\frac {b^{2} \arctanh \left (\frac {2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{d \,a^{2} \sqrt {a b +b^{2}}}-\frac {1}{8 d a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}-\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.80, size = 787, normalized size = 9.26 \[ \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (18\,b^7\,\sqrt {-a^4\,d^2}+48\,a^2\,b^5\,\sqrt {-a^4\,d^2}+27\,a^3\,b^4\,\sqrt {-a^4\,d^2}+8\,a^4\,b^3\,\sqrt {-a^4\,d^2}+a^5\,b^2\,\sqrt {-a^4\,d^2}+45\,a\,b^6\,\sqrt {-a^4\,d^2}\right )}{9\,a^2\,b^6\,d\,\sqrt {a^2+4\,a\,b+4\,b^2}+18\,a^3\,b^5\,d\,\sqrt {a^2+4\,a\,b+4\,b^2}+15\,a^4\,b^4\,d\,\sqrt {a^2+4\,a\,b+4\,b^2}+6\,a^5\,b^3\,d\,\sqrt {a^2+4\,a\,b+4\,b^2}+a^6\,b^2\,d\,\sqrt {a^2+4\,a\,b+4\,b^2}}\right )\,\sqrt {a^2+4\,a\,b+4\,b^2}}{\sqrt {-a^4\,d^2}}-\frac {\left (2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a+b\right )\,\sqrt {-a^4\,d^2}}{2\,a^2\,d\,\sqrt {b\,\left (a+b\right )}}\right )+2\,\mathrm {atan}\left (\frac {\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {64\,\left (2\,a^4\,b\,d\,\sqrt {b^2+a\,b}+6\,a^2\,b^3\,d\,\sqrt {b^2+a\,b}+6\,a^3\,b^2\,d\,\sqrt {b^2+a\,b}\right )}{a^9\,d^2\,{\left (a+b\right )}^2\,\left (a^2+2\,a\,b+b^2\right )}-\frac {32\,\left (3\,b^4\,\sqrt {-a^4\,d^2}+4\,a^2\,b^2\,\sqrt {-a^4\,d^2}+6\,a\,b^3\,\sqrt {-a^4\,d^2}+a^3\,b\,\sqrt {-a^4\,d^2}\right )}{a^7\,d\,\left (a+b\right )\,\sqrt {-a^4\,d^2}\,\sqrt {b\,\left (a+b\right )}\,\left (a^2+2\,a\,b+b^2\right )}\right )-\frac {32\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\left (3\,b^4\,\sqrt {-a^4\,d^2}+4\,a^2\,b^2\,\sqrt {-a^4\,d^2}+6\,a\,b^3\,\sqrt {-a^4\,d^2}+a^3\,b\,\sqrt {-a^4\,d^2}\right )}{a^7\,d\,\left (a+b\right )\,\sqrt {-a^4\,d^2}\,\sqrt {b\,\left (a+b\right )}\,\left (a^2+2\,a\,b+b^2\right )}\right )\,\left (a^8\,\sqrt {-a^4\,d^2}+a^5\,b^3\,\sqrt {-a^4\,d^2}+3\,a^6\,b^2\,\sqrt {-a^4\,d^2}+3\,a^7\,b\,\sqrt {-a^4\,d^2}\right )}{64\,a^2\,b+192\,a\,b^2+192\,b^3}\right )\right )\,\sqrt {b^2+a\,b}}{2\,\sqrt {-a^4\,d^2}}-\frac {{\mathrm {e}}^{c+d\,x}}{a\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}}{a\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]

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)),x)

[Out]

(atan((exp(d*x)*exp(c)*(18*b^7*(-a^4*d^2)^(1/2) + 48*a^2*b^5*(-a^4*d^2)^(1/2) + 27*a^3*b^4*(-a^4*d^2)^(1/2) +
8*a^4*b^3*(-a^4*d^2)^(1/2) + a^5*b^2*(-a^4*d^2)^(1/2) + 45*a*b^6*(-a^4*d^2)^(1/2)))/(9*a^2*b^6*d*(4*a*b + a^2
+ 4*b^2)^(1/2) + 18*a^3*b^5*d*(4*a*b + a^2 + 4*b^2)^(1/2) + 15*a^4*b^4*d*(4*a*b + a^2 + 4*b^2)^(1/2) + 6*a^5*b
^3*d*(4*a*b + a^2 + 4*b^2)^(1/2) + a^6*b^2*d*(4*a*b + a^2 + 4*b^2)^(1/2)))*(4*a*b + a^2 + 4*b^2)^(1/2))/(-a^4*
d^2)^(1/2) - ((2*atan((exp(d*x)*exp(c)*(a + b)*(-a^4*d^2)^(1/2))/(2*a^2*d*(b*(a + b))^(1/2))) + 2*atan(((exp(d
*x)*exp(c)*((64*(2*a^4*b*d*(a*b + b^2)^(1/2) + 6*a^2*b^3*d*(a*b + b^2)^(1/2) + 6*a^3*b^2*d*(a*b + b^2)^(1/2)))
/(a^9*d^2*(a + b)^2*(2*a*b + a^2 + b^2)) - (32*(3*b^4*(-a^4*d^2)^(1/2) + 4*a^2*b^2*(-a^4*d^2)^(1/2) + 6*a*b^3*
(-a^4*d^2)^(1/2) + a^3*b*(-a^4*d^2)^(1/2)))/(a^7*d*(a + b)*(-a^4*d^2)^(1/2)*(b*(a + b))^(1/2)*(2*a*b + a^2 + b
^2))) - (32*exp(3*c)*exp(3*d*x)*(3*b^4*(-a^4*d^2)^(1/2) + 4*a^2*b^2*(-a^4*d^2)^(1/2) + 6*a*b^3*(-a^4*d^2)^(1/2
) + a^3*b*(-a^4*d^2)^(1/2)))/(a^7*d*(a + b)*(-a^4*d^2)^(1/2)*(b*(a + b))^(1/2)*(2*a*b + a^2 + b^2)))*(a^8*(-a^
4*d^2)^(1/2) + a^5*b^3*(-a^4*d^2)^(1/2) + 3*a^6*b^2*(-a^4*d^2)^(1/2) + 3*a^7*b*(-a^4*d^2)^(1/2)))/(192*a*b^2 +
64*a^2*b + 192*b^3)))*(a*b + b^2)^(1/2))/(2*(-a^4*d^2)^(1/2)) - exp(c + d*x)/(a*d*(exp(2*c + 2*d*x) - 1)) - (
2*exp(c + d*x))/(a*d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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