3.2.17 \(\int \frac {\cosh (c+d x)}{(a+b \tanh ^2(c+d x))^2} \, dx\) [117]
Optimal. Leaf size=101 \[ \frac {b (4 a+b) \text {ArcTan}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^{5/2} d}+\frac {\sinh (c+d x)}{(a+b)^2 d}+\frac {b^2 \sinh (c+d x)}{2 a (a+b)^2 d \left (a+(a+b) \sinh ^2(c+d x)\right )} \]
[Out]
1/2*b*(4*a+b)*arctan(sinh(d*x+c)*(a+b)^(1/2)/a^(1/2))/a^(3/2)/(a+b)^(5/2)/d+sinh(d*x+c)/(a+b)^2/d+1/2*b^2*sinh
(d*x+c)/a/(a+b)^2/d/(a+(a+b)*sinh(d*x+c)^2)
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Rubi [A]
time = 0.10, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3757, 398, 393,
211} \begin {gather*} \frac {b (4 a+b) \text {ArcTan}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} d (a+b)^{5/2}}+\frac {b^2 \sinh (c+d x)}{2 a d (a+b)^2 \left ((a+b) \sinh ^2(c+d x)+a\right )}+\frac {\sinh (c+d x)}{d (a+b)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Cosh[c + d*x]/(a + b*Tanh[c + d*x]^2)^2,x]
[Out]
(b*(4*a + b)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^(5/2)*d) + Sinh[c + d*x]/((a + b)
^2*d) + (b^2*Sinh[c + d*x])/(2*a*(a + b)^2*d*(a + (a + b)*Sinh[c + d*x]^2))
Rule 211
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 393
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])
Rule 398
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]
Rule 3757
Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> With[{ff = F
reeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[ExpandToSum[b*(ff*x)^n + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 -
ff^2*x^2)^((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] &&
IntegerQ[n/2] && IntegerQ[p]
Rubi steps
\begin {align*} \int \frac {\cosh (c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{\left (a+(a+b) x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{(a+b)^2}+\frac {b (2 a+b)+2 b (a+b) x^2}{(a+b)^2 \left (a+(a+b) x^2\right )^2}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\sinh (c+d x)}{(a+b)^2 d}+\frac {\text {Subst}\left (\int \frac {b (2 a+b)+2 b (a+b) x^2}{\left (a+(a+b) x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{(a+b)^2 d}\\ &=\frac {\sinh (c+d x)}{(a+b)^2 d}+\frac {b^2 \sinh (c+d x)}{2 a (a+b)^2 d \left (a+(a+b) \sinh ^2(c+d x)\right )}+\frac {(b (4 a+b)) \text {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\sinh (c+d x)\right )}{2 a (a+b)^2 d}\\ &=\frac {b (4 a+b) \tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^{5/2} d}+\frac {\sinh (c+d x)}{(a+b)^2 d}+\frac {b^2 \sinh (c+d x)}{2 a (a+b)^2 d \left (a+(a+b) \sinh ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.60, size = 89, normalized size = 0.88 \begin {gather*} \frac {-\frac {b (4 a+b) \text {ArcTan}\left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {a+b}}\right )}{a^{3/2} (a+b)^{5/2}}+\frac {2 \left (1+\frac {b^2}{a (a-b+(a+b) \cosh (2 (c+d x)))}\right ) \sinh (c+d x)}{(a+b)^2}}{2 d} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Cosh[c + d*x]/(a + b*Tanh[c + d*x]^2)^2,x]
[Out]
(-((b*(4*a + b)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[a + b]])/(a^(3/2)*(a + b)^(5/2))) + (2*(1 + b^2/(a*(a - b
+ (a + b)*Cosh[2*(c + d*x)])))*Sinh[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. \(285\) vs.
\(2(89)=178\).
time = 2.88, size = 286, normalized size = 2.83
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(\frac {-\frac {1}{\left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {2 b \left (\frac {-\frac {b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\left (4 a +b \right ) \left (-\frac {\left (\sqrt {b \left (a +b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\left (\sqrt {b \left (a +b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2}\right )}{\left (a +b \right )^{2}}-\frac {1}{\left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) |
\(286\) |
| default |
\(\frac {-\frac {1}{\left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {2 b \left (\frac {-\frac {b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\left (4 a +b \right ) \left (-\frac {\left (\sqrt {b \left (a +b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\left (\sqrt {b \left (a +b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2}\right )}{\left (a +b \right )^{2}}-\frac {1}{\left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) |
\(286\) |
| risch |
\(\frac {{\mathrm e}^{d x +c}}{2 \left (a^{2}+2 a b +b^{2}\right ) d}-\frac {{\mathrm e}^{-d x -c}}{2 \left (a^{2}+2 a b +b^{2}\right ) d}+\frac {b^{2} {\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \left (a +b \right )^{2} a \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )}-\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{\sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d}-\frac {b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d a}+\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{\sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d}+\frac {b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d a}\) |
\(369\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cosh(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/a*b*tanh(1/2*d*x+1/2*c)^3+1/2/a*b*tanh(1/2*d*x+1/2*
c))/(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)+1/2*(4*a+b)*(-1/2*((b*(a+b
))^(1/2)-b)/a/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^
(1/2)-a-2*b)*a)^(1/2))+1/2*((b*(a+b))^(1/2)+b)/a/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*
tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(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(cosh(d*x+c)/(a+b*tanh(d*x+c)^2)^2,x, algorithm="maxima")
[Out]
-1/2*(a^2 + a*b - (a^2*e^(6*c) + a*b*e^(6*c))*e^(6*d*x) - (a^2*e^(4*c) - 3*a*b*e^(4*c) + 2*b^2*e^(4*c))*e^(4*d
*x) + (a^2*e^(2*c) - 3*a*b*e^(2*c) + 2*b^2*e^(2*c))*e^(2*d*x))/((a^4*d*e^(5*c) + 3*a^3*b*d*e^(5*c) + 3*a^2*b^2
*d*e^(5*c) + a*b^3*d*e^(5*c))*e^(5*d*x) + 2*(a^4*d*e^(3*c) + a^3*b*d*e^(3*c) - a^2*b^2*d*e^(3*c) - a*b^3*d*e^(
3*c))*e^(3*d*x) + (a^4*d*e^c + 3*a^3*b*d*e^c + 3*a^2*b^2*d*e^c + a*b^3*d*e^c)*e^(d*x)) + 1/2*integrate(2*((4*a
*b*e^(3*c) + b^2*e^(3*c))*e^(3*d*x) + (4*a*b*e^c + b^2*e^c)*e^(d*x))/(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3 + (a^4
*e^(4*c) + 3*a^3*b*e^(4*c) + 3*a^2*b^2*e^(4*c) + a*b^3*e^(4*c))*e^(4*d*x) + 2*(a^4*e^(2*c) + a^3*b*e^(2*c) - a
^2*b^2*e^(2*c) - a*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. 1815 vs.
\(2 (89) = 178\).
time = 0.43, size = 3502, normalized size = 34.67 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cosh(d*x+c)/(a+b*tanh(d*x+c)^2)^2,x, algorithm="fricas")
[Out]
[1/4*(2*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^6 + 12*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^5
+ 2*(a^4 + 2*a^3*b + a^2*b^2)*sinh(d*x + c)^6 + 2*(a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^4 + 2*(a^
4 - 2*a^3*b - a^2*b^2 + 2*a*b^3 + 15*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - 2*a^4 - 4*a^
3*b - 2*a^2*b^2 + 8*(5*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^3 + (a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^3)*cosh(d*
x + c))*sinh(d*x + c)^3 - 2*(a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^2 + 2*(15*(a^4 + 2*a^3*b + a^2*b
^2)*cosh(d*x + c)^4 - a^4 + 2*a^3*b + a^2*b^2 - 2*a*b^3 + 6*(a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^
2)*sinh(d*x + c)^2 - ((4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c)^5 + 5*(4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c)*si
nh(d*x + c)^4 + (4*a^2*b + 5*a*b^2 + b^3)*sinh(d*x + c)^5 + 2*(4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c)^3 + 2*(4
*a^2*b - 3*a*b^2 - b^3 + 5*(4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(5*(4*a^2*b + 5*a*b^
2 + b^3)*cosh(d*x + c)^3 + 3*(4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + (4*a^2*b + 5*a*b^2 + b
^3)*cosh(d*x + c) + (5*(4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c)^4 + 4*a^2*b + 5*a*b^2 + b^3 + 6*(4*a^2*b - 3*a*
b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c))*sqrt(-a^2 - 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*(3*a + b)*cosh(d*x + c)^2 + 2*(3*(a + b)*cosh(d*x + c)^2 -
3*a - b)*sinh(d*x + c)^2 + 4*((a + b)*cosh(d*x + c)^3 - (3*a + 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^2 - 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)*si
nh(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*(3*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x +
c)^5 + 2*(a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^3 - (a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^3)*cosh(d*x +
c))*sinh(d*x + c))/((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^5 + 5*(a^6 + 4*a^5*b + 6
*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)*sinh(d*x + c)^4 + (a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2
*b^4)*d*sinh(d*x + c)^5 + 2*(a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c)^3 + 2*(5*(a^6 + 4*a^5*b + 6*
a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^2 + (a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4)*d)*sinh(d*x + c)^3 +
(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c) + 2*(5*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^
3 + a^2*b^4)*d*cosh(d*x + c)^3 + 3*(a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^2 + (5
*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^4 + 6*(a^6 + 2*a^5*b - 2*a^3*b^3 - a^2*b^4)
*d*cosh(d*x + c)^2 + (a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d)*sinh(d*x + c)), 1/2*((a^4 + 2*a^3*b
+ a^2*b^2)*cosh(d*x + c)^6 + 6*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^5 + (a^4 + 2*a^3*b + a^2*
b^2)*sinh(d*x + c)^6 + (a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^4 + (a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^
3 + 15*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - a^4 - 2*a^3*b - a^2*b^2 + 4*(5*(a^4 + 2*a^
3*b + a^2*b^2)*cosh(d*x + c)^3 + (a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - (a^4 - 2
*a^3*b - a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^2 + (15*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^4 - a^4 + 2*a^3*b +
a^2*b^2 - 2*a*b^3 + 6*(a^4 - 2*a^3*b - a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((4*a^2*b + 5*a*b
^2 + b^3)*cosh(d*x + c)^5 + 5*(4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c)*sinh(d*x + c)^4 + (4*a^2*b + 5*a*b^2 + b
^3)*sinh(d*x + c)^5 + 2*(4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c)^3 + 2*(4*a^2*b - 3*a*b^2 - b^3 + 5*(4*a^2*b +
5*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(5*(4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c)^3 + 3*(4*a^2*b
- 3*a*b^2 - b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + (4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c) + (5*(4*a^2*b + 5*a*
b^2 + b^3)*cosh(d*x + c)^4 + 4*a^2*b + 5*a*b^2 + b^3 + 6*(4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x +
c))*sqrt(a^2 + 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)*s
inh(d*x + c)^3 + (3*a - b)*cosh(d*x + c) + (3*(a + b)*cosh(d*x + c)^2 + 3*a - b)*sinh(d*x + c))/sqrt(a^2 + a*b
)) + ((4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c)^5 + 5*(4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c)*sinh(d*x + c)^4 +
(4*a^2*b + 5*a*b^2 + b^3)*sinh(d*x + c)^5 + 2*(4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c)^3 + 2*(4*a^2*b - 3*a*b^2
- b^3 + 5*(4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(5*(4*a^2*b + 5*a*b^2 + b^3)*cosh(d*
x + c)^3 + 3*(4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + (4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c
) + (5*(4*a^2*b + 5*a*b^2 + b^3)*cosh(d*x + c)^4 + 4*a^2*b + 5*a*b^2 + b^3 + 6*(4*a^2*b - 3*a*b^2 - b^3)*cosh(
d*x + c)^2)*sinh(d*x + c))*sqrt(a^2 + a*b)*arct...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cosh {\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(cosh(d*x+c)/(a+b*tanh(d*x+c)**2)**2,x)
[Out]
Integral(cosh(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(cosh(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 {cosh}\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(cosh(c + d*x)/(a + b*tanh(c + d*x)^2)^2,x)
[Out]
int(cosh(c + d*x)/(a + b*tanh(c + d*x)^2)^2, x)
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