3.1.36 \(\int \frac {\sinh (c+d x)}{(a+b \tanh ^2(c+d x))^2} \, dx\) [36]
Optimal. Leaf size=92 \[ -\frac {3 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{2 (a+b)^{5/2} d}+\frac {3 \cosh (c+d x)}{2 (a+b)^2 d}-\frac {\cosh (c+d x)}{2 (a+b) d \left (a+b-b \text {sech}^2(c+d x)\right )} \]
[Out]
3/2*cosh(d*x+c)/(a+b)^2/d-1/2*cosh(d*x+c)/(a+b)/d/(a+b-b*sech(d*x+c)^2)-3/2*arctanh(sech(d*x+c)*b^(1/2)/(a+b)^
(1/2))*b^(1/2)/(a+b)^(5/2)/d
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Rubi [A]
time = 0.05, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3745, 296, 331,
214} \begin {gather*} \frac {3 \cosh (c+d x)}{2 d (a+b)^2}-\frac {\cosh (c+d x)}{2 d (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )}-\frac {3 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{2 d (a+b)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sinh[c + d*x]/(a + b*Tanh[c + d*x]^2)^2,x]
[Out]
(-3*Sqrt[b]*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(2*(a + b)^(5/2)*d) + (3*Cosh[c + d*x])/(2*(a + b)^2
*d) - Cosh[c + d*x]/(2*(a + b)*d*(a + b - b*Sech[c + d*x]^2))
Rule 214
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]
Rule 296
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]
Rule 331
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]
Rule 3745
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 {\sinh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (a+b-b x^2\right )^2} \, dx,x,\text {sech}(c+d x)\right )}{d}\\ &=-\frac {\cosh (c+d x)}{2 (a+b) d \left (a+b-b \text {sech}^2(c+d x)\right )}-\frac {3 \text {Subst}\left (\int \frac {1}{x^2 \left (a+b-b x^2\right )} \, dx,x,\text {sech}(c+d x)\right )}{2 (a+b) d}\\ &=\frac {3 \cosh (c+d x)}{2 (a+b)^2 d}-\frac {\cosh (c+d x)}{2 (a+b) d \left (a+b-b \text {sech}^2(c+d x)\right )}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\text {sech}(c+d x)\right )}{2 (a+b)^2 d}\\ &=-\frac {3 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{2 (a+b)^{5/2} d}+\frac {3 \cosh (c+d x)}{2 (a+b)^2 d}-\frac {\cosh (c+d x)}{2 (a+b) d \left (a+b-b \text {sech}^2(c+d x)\right )}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.54, size = 133, normalized size = 1.45 \begin {gather*} \frac {-\frac {3 i \sqrt {b} \left (\text {ArcTan}\left (\frac {-i \sqrt {a+b}-\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )+\text {ArcTan}\left (\frac {-i \sqrt {a+b}+\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )\right )}{(a+b)^{5/2}}+\frac {2 \cosh (c+d x) \left (1-\frac {b}{a-b+(a+b) \cosh (2 (c+d x))}\right )}{(a+b)^2}}{2 d} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sinh[c + d*x]/(a + b*Tanh[c + d*x]^2)^2,x]
[Out]
(((-3*I)*Sqrt[b]*(ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]] + ArcTan[((-I)*Sqrt[a + b] +
Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]]))/(a + b)^(5/2) + (2*Cosh[c + d*x]*(1 - b/(a - b + (a + b)*Cosh[2*(c + d*x
)])))/(a + b)^2)/(2*d)
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(166\) vs.
\(2(78)=156\).
time = 2.16, size = 167, normalized size = 1.82
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(\frac {-\frac {1}{\left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {2 b \left (\frac {-\frac {\left (2 b +a \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {1}{2}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}-\frac {3 \arctanh \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{4 \sqrt {a b +b^{2}}}\right )}{\left (a +b \right )^{2}}+\frac {1}{\left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\) |
\(167\) |
| default |
\(\frac {-\frac {1}{\left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {2 b \left (\frac {-\frac {\left (2 b +a \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {1}{2}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}-\frac {3 \arctanh \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{4 \sqrt {a b +b^{2}}}\right )}{\left (a +b \right )^{2}}+\frac {1}{\left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\) |
\(167\) |
| risch |
\(\frac {{\mathrm e}^{d x +c}}{2 \left (a^{2}+2 a b +b^{2}\right ) d}+\frac {{\mathrm e}^{-d x -c}}{2 \left (a^{2}+2 a b +b^{2}\right ) d}-\frac {b \,{\mathrm e}^{d x +c} \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d \left (a +b \right )^{2} \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )}+\frac {3 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {b \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{4 \left (a +b \right )^{3} d}-\frac {3 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {b \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{4 \left (a +b \right )^{3} d}\) |
\(230\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sinh(d*x+c)/(a+b*tanh(d*x+c)^2)^2,x,method=_RETURNVERBOSE)
[Out]
1/d*(-1/(a+b)^2/(tanh(1/2*d*x+1/2*c)-1)+2*b/(a+b)^2*((-1/2*(2*b+a)/a*tanh(1/2*d*x+1/2*c)^2-1/2)/(a*tanh(1/2*d*
x+1/2*c)^4+2*a*tanh(1/2*d*x+1/2*c)^2+4*b*tanh(1/2*d*x+1/2*c)^2+a)-3/4/(a*b+b^2)^(1/2)*arctanh(1/4*(2*a*tanh(1/
2*d*x+1/2*c)^2+2*a+4*b)/(a*b+b^2)^(1/2)))+1/(a+b)^2/(tanh(1/2*d*x+1/2*c)+1))
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sinh(d*x+c)/(a+b*tanh(d*x+c)^2)^2,x, algorithm="maxima")
[Out]
1/2*((a*e^(6*c) + b*e^(6*c))*e^(6*d*x) + 3*(a*e^(4*c) - b*e^(4*c))*e^(4*d*x) + 3*(a*e^(2*c) - b*e^(2*c))*e^(2*
d*x) + a + b)/((a^3*d*e^(5*c) + 3*a^2*b*d*e^(5*c) + 3*a*b^2*d*e^(5*c) + b^3*d*e^(5*c))*e^(5*d*x) + 2*(a^3*d*e^
(3*c) + a^2*b*d*e^(3*c) - a*b^2*d*e^(3*c) - b^3*d*e^(3*c))*e^(3*d*x) + (a^3*d*e^c + 3*a^2*b*d*e^c + 3*a*b^2*d*
e^c + b^3*d*e^c)*e^(d*x)) + 1/2*integrate(6*(b*e^(3*d*x + 3*c) - b*e^(d*x + c))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3
+ (a^3*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) + b^3*e^(4*c))*e^(4*d*x) + 2*(a^3*e^(2*c) + a^2*b*e^(2*c)
- a*b^2*e^(2*c) - b^3*e^(2*c))*e^(2*d*x)), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1114 vs.
\(2 (81) = 162\).
time = 0.39, size = 2252, normalized size = 24.48 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sinh(d*x+c)/(a+b*tanh(d*x+c)^2)^2,x, algorithm="fricas")
[Out]
[1/4*(2*(a + b)*cosh(d*x + c)^6 + 12*(a + b)*cosh(d*x + c)*sinh(d*x + c)^5 + 2*(a + b)*sinh(d*x + c)^6 + 6*(a
- b)*cosh(d*x + c)^4 + 6*(5*(a + b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c)^4 + 8*(5*(a + b)*cosh(d*x + c)^3 +
3*(a - b)*cosh(d*x + c))*sinh(d*x + c)^3 + 6*(a - b)*cosh(d*x + c)^2 + 6*(5*(a + b)*cosh(d*x + c)^4 + 6*(a - b
)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c)^2 + 3*((a + b)*cosh(d*x + c)^5 + 5*(a + b)*cosh(d*x + c)*sinh(d*x + c
)^4 + (a + b)*sinh(d*x + c)^5 + 2*(a - b)*cosh(d*x + c)^3 + 2*(5*(a + b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c
)^3 + 2*(5*(a + b)*cosh(d*x + c)^3 + 3*(a - b)*cosh(d*x + c))*sinh(d*x + c)^2 + (a + b)*cosh(d*x + c) + (5*(a
+ b)*cosh(d*x + c)^4 + 6*(a - b)*cosh(d*x + c)^2 + a + b)*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))*sinh(d*x + c) + a + b)) + 12*((a + b)*cosh(d*x + c)^5 + 2*(a - b)*cosh(d*x + c)^3 + (a - b)*cosh(d*x + c))
*sinh(d*x + c) + 2*a + 2*b)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c)^5 + 5*(a^3 + 3*a^2*b + 3*a*b^2 +
b^3)*d*cosh(d*x + c)*sinh(d*x + c)^4 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*sinh(d*x + c)^5 + 2*(a^3 + a^2*b - a*
b^2 - b^3)*d*cosh(d*x + c)^3 + 2*(5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c)^2 + (a^3 + a^2*b - a*b^2 -
b^3)*d)*sinh(d*x + c)^3 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c) + 2*(5*(a^3 + 3*a^2*b + 3*a*b^2 + b
^3)*d*cosh(d*x + c)^3 + 3*(a^3 + a^2*b - a*b^2 - b^3)*d*cosh(d*x + c))*sinh(d*x + c)^2 + (5*(a^3 + 3*a^2*b + 3
*a*b^2 + b^3)*d*cosh(d*x + c)^4 + 6*(a^3 + a^2*b - a*b^2 - b^3)*d*cosh(d*x + c)^2 + (a^3 + 3*a^2*b + 3*a*b^2 +
b^3)*d)*sinh(d*x + c)), 1/2*((a + b)*cosh(d*x + c)^6 + 6*(a + b)*cosh(d*x + c)*sinh(d*x + c)^5 + (a + b)*sinh
(d*x + c)^6 + 3*(a - b)*cosh(d*x + c)^4 + 3*(5*(a + b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c)^4 + 4*(5*(a + b)
*cosh(d*x + c)^3 + 3*(a - b)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(a - b)*cosh(d*x + c)^2 + 3*(5*(a + b)*cosh(d*
x + c)^4 + 6*(a - b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c)^2 - 3*((a + b)*cosh(d*x + c)^5 + 5*(a + b)*cosh(d*
x + c)*sinh(d*x + c)^4 + (a + b)*sinh(d*x + c)^5 + 2*(a - b)*cosh(d*x + c)^3 + 2*(5*(a + b)*cosh(d*x + c)^2 +
a - b)*sinh(d*x + c)^3 + 2*(5*(a + b)*cosh(d*x + c)^3 + 3*(a - b)*cosh(d*x + c))*sinh(d*x + c)^2 + (a + b)*cos
h(d*x + c) + (5*(a + b)*cosh(d*x + c)^4 + 6*(a - b)*cosh(d*x + c)^2 + a + b)*sinh(d*x + c))*sqrt(-b/(a + b))*a
rctan(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 + b)*cos
h(d*x + c)^5 + 5*(a + b)*cosh(d*x + c)*sinh(d*x + c)^4 + (a + b)*sinh(d*x + c)^5 + 2*(a - b)*cosh(d*x + c)^3 +
2*(5*(a + b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c)^3 + 2*(5*(a + b)*cosh(d*x + c)^3 + 3*(a - b)*cosh(d*x + c
))*sinh(d*x + c)^2 + (a + b)*cosh(d*x + c) + (5*(a + b)*cosh(d*x + c)^4 + 6*(a - b)*cosh(d*x + c)^2 + a + b)*s
inh(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)
+ 6*((a + b)*cosh(d*x + c)^5 + 2*(a - b)*cosh(d*x + c)^3 + (a - b)*cosh(d*x + c))*sinh(d*x + c) + a + b)/((a^3
+ 3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c)^5 + 5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c)*sinh(d*x + c
)^4 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*sinh(d*x + c)^5 + 2*(a^3 + a^2*b - a*b^2 - b^3)*d*cosh(d*x + c)^3 + 2*
(5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c)^2 + (a^3 + a^2*b - a*b^2 - b^3)*d)*sinh(d*x + c)^3 + (a^3 +
3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c) + 2*(5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c)^3 + 3*(a^3 +
a^2*b - a*b^2 - b^3)*d*cosh(d*x + c))*sinh(d*x + c)^2 + (5*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c)^4 +
6*(a^3 + a^2*b - a*b^2 - b^3)*d*cosh(d*x + c)^2 + (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d)*sinh(d*x + c))]
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh {\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sinh(d*x+c)/(a+b*tanh(d*x+c)**2)**2,x)
[Out]
Integral(sinh(c + d*x)/(a + b*tanh(c + d*x)**2)**2, x)
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sinh(d*x+c)/(a+b*tanh(d*x+c)^2)^2,x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {sinh}\left (c+d\,x\right )}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sinh(c + d*x)/(a + b*tanh(c + d*x)^2)^2,x)
[Out]
int(sinh(c + d*x)/(a + b*tanh(c + d*x)^2)^2, x)
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