3.2.26 \(\int \frac {\cosh (c+d x)}{(a+b \tanh ^2(c+d x))^3} \, dx\) [126]

Optimal. Leaf size=154 \[ \frac {3 b \left (8 a^2+4 a b+b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} (a+b)^{7/2} d}+\frac {\sinh (c+d x)}{(a+b)^3 d}+\frac {b^3 \sinh (c+d x)}{4 a (a+b)^3 d \left (a+(a+b) \sinh ^2(c+d x)\right )^2}+\frac {3 b^2 (4 a+b) \sinh (c+d x)}{8 a^2 (a+b)^3 d \left (a+(a+b) \sinh ^2(c+d x)\right )} \]

[Out]

3/8*b*(8*a^2+4*a*b+b^2)*arctan(sinh(d*x+c)*(a+b)^(1/2)/a^(1/2))/a^(5/2)/(a+b)^(7/2)/d+sinh(d*x+c)/(a+b)^3/d+1/
4*b^3*sinh(d*x+c)/a/(a+b)^3/d/(a+(a+b)*sinh(d*x+c)^2)^2+3/8*b^2*(4*a+b)*sinh(d*x+c)/a^2/(a+b)^3/d/(a+(a+b)*sin
h(d*x+c)^2)

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Rubi [A]
time = 0.16, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3757, 398, 1171, 393, 211} \begin {gather*} \frac {3 b^2 (4 a+b) \sinh (c+d x)}{8 a^2 d (a+b)^3 \left ((a+b) \sinh ^2(c+d x)+a\right )}+\frac {3 b \left (8 a^2+4 a b+b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} d (a+b)^{7/2}}+\frac {b^3 \sinh (c+d x)}{4 a d (a+b)^3 \left ((a+b) \sinh ^2(c+d x)+a\right )^2}+\frac {\sinh (c+d x)}{d (a+b)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b*(8*a^2 + 4*a*b + b^2)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a + b)^(7/2)*d) + Sinh[c +
d*x]/((a + b)^3*d) + (b^3*Sinh[c + d*x])/(4*a*(a + b)^3*d*(a + (a + b)*Sinh[c + d*x]^2)^2) + (3*b^2*(4*a + b)
*Sinh[c + d*x])/(8*a^2*(a + b)^3*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 1171

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1
)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

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 )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{\left (a+(a+b) x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{(a+b)^3}+\frac {b \left (3 a^2+3 a b+b^2\right )+3 b (a+b) (2 a+b) x^2+3 b (a+b)^2 x^4}{(a+b)^3 \left (a+(a+b) x^2\right )^3}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\sinh (c+d x)}{(a+b)^3 d}+\frac {\text {Subst}\left (\int \frac {b \left (3 a^2+3 a b+b^2\right )+3 b (a+b) (2 a+b) x^2+3 b (a+b)^2 x^4}{\left (a+(a+b) x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{(a+b)^3 d}\\ &=\frac {\sinh (c+d x)}{(a+b)^3 d}+\frac {b^3 \sinh (c+d x)}{4 a (a+b)^3 d \left (a+(a+b) \sinh ^2(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-3 b (2 a+b)^2-12 a b (a+b) x^2}{\left (a+(a+b) x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{4 a (a+b)^3 d}\\ &=\frac {\sinh (c+d x)}{(a+b)^3 d}+\frac {b^3 \sinh (c+d x)}{4 a (a+b)^3 d \left (a+(a+b) \sinh ^2(c+d x)\right )^2}+\frac {3 b^2 (4 a+b) \sinh (c+d x)}{8 a^2 (a+b)^3 d \left (a+(a+b) \sinh ^2(c+d x)\right )}+\frac {\left (3 b \left (8 a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\sinh (c+d x)\right )}{8 a^2 (a+b)^3 d}\\ &=\frac {3 b \left (8 a^2+4 a b+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} (a+b)^{7/2} d}+\frac {\sinh (c+d x)}{(a+b)^3 d}+\frac {b^3 \sinh (c+d x)}{4 a (a+b)^3 d \left (a+(a+b) \sinh ^2(c+d x)\right )^2}+\frac {3 b^2 (4 a+b) \sinh (c+d x)}{8 a^2 (a+b)^3 d \left (a+(a+b) \sinh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [A]
time = 1.98, size = 145, normalized size = 0.94 \begin {gather*} \frac {-\frac {3 b \left (8 a^2+4 a b+b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {a+b}}\right )}{a^{5/2} (a+b)^{7/2}}+\frac {2 \left (4+\frac {3 b^3}{a^2 (a-b+(a+b) \cosh (2 (c+d x)))}+\frac {4 b^2 (3 a-2 b+3 (a+b) \cosh (2 (c+d x)))}{a (a-b+(a+b) \cosh (2 (c+d x)))^2}\right ) \sinh (c+d x)}{(a+b)^3}}{8 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-3*b*(8*a^2 + 4*a*b + b^2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[a + b]])/(a^(5/2)*(a + b)^(7/2)) + (2*(4 + (3
*b^3)/(a^2*(a - b + (a + b)*Cosh[2*(c + d*x)])) + (4*b^2*(3*a - 2*b + 3*(a + b)*Cosh[2*(c + d*x)]))/(a*(a - b
+ (a + b)*Cosh[2*(c + d*x)])^2))*Sinh[c + d*x])/(a + b)^3)/(8*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(374\) vs. \(2(140)=280\).
time = 3.26, size = 375, normalized size = 2.44

method	result	size
derivativedivides	\(\frac {-\frac {1}{\left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {2 b \left (\frac {-\frac {b \left (12 a +5 b \right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {3 \left (4 a^{2}+15 a b +4 b^{2}\right ) b \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}+\frac {3 \left (4 a^{2}+15 a b +4 b^{2}\right ) b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}+\frac {b \left (12 a +5 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {3 \left (8 a^{2}+4 a b +b^{2}\right ) \left (-\frac {\left (\sqrt {b \left (a +b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\left (\sqrt {b \left (a +b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{8 a}\right )}{\left (a +b \right )^{3}}-\frac {1}{\left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\)	\(375\)
default	\(\frac {-\frac {1}{\left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {2 b \left (\frac {-\frac {b \left (12 a +5 b \right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {3 \left (4 a^{2}+15 a b +4 b^{2}\right ) b \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}+\frac {3 \left (4 a^{2}+15 a b +4 b^{2}\right ) b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}+\frac {b \left (12 a +5 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {3 \left (8 a^{2}+4 a b +b^{2}\right ) \left (-\frac {\left (\sqrt {b \left (a +b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\left (\sqrt {b \left (a +b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{8 a}\right )}{\left (a +b \right )^{3}}-\frac {1}{\left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\)	\(375\)
risch	\(\frac {{\mathrm e}^{d x +c}}{2 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}-\frac {{\mathrm e}^{-d x -c}}{2 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}+\frac {\left (12 a^{2} {\mathrm e}^{6 d x +6 c}+15 a b \,{\mathrm e}^{6 d x +6 c}+3 b^{2} {\mathrm e}^{6 d x +6 c}+12 a^{2} {\mathrm e}^{4 d x +4 c}-25 a b \,{\mathrm e}^{4 d x +4 c}-9 b^{2} {\mathrm e}^{4 d x +4 c}-12 a^{2} {\mathrm e}^{2 d x +2 c}+25 a b \,{\mathrm e}^{2 d x +2 c}+9 b^{2} {\mathrm e}^{2 d x +2 c}-12 a^{2}-15 a b -3 b^{2}\right ) {\mathrm e}^{d x +c} b^{2}}{4 \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )^{2} \left (a +b \right ) \left (a^{2}+2 a b +b^{2}\right ) a^{2} d}-\frac {3 b \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3} d}-\frac {3 b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3} d a}-\frac {3 b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{16 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3} d \,a^{2}}+\frac {3 b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3} d}+\frac {3 b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3} d a}+\frac {3 b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{16 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3} d \,a^{2}}\)	\(649\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)/(a+b*tanh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/(a+b)^3/(tanh(1/2*d*x+1/2*c)-1)+2*b/(a+b)^3*((-1/8*b*(12*a+5*b)/a*tanh(1/2*d*x+1/2*c)^7-3/8*(4*a^2+15*
a*b+4*b^2)/a^2*b*tanh(1/2*d*x+1/2*c)^5+3/8*(4*a^2+15*a*b+4*b^2)/a^2*b*tanh(1/2*d*x+1/2*c)^3+1/8*b*(12*a+5*b)/a
*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)^2+3/8/a*
(8*a^2+4*a*b+b^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)*arctanh(a*ta
nh(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)^3/(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)^3,x, algorithm="maxima")

[Out]

-1/4*(2*a^4 + 4*a^3*b + 2*a^2*b^2 - 2*(a^4*e^(10*c) + 2*a^3*b*e^(10*c) + a^2*b^2*e^(10*c))*e^(10*d*x) - (6*a^4
*e^(8*c) - 4*a^3*b*e^(8*c) + 2*a^2*b^2*e^(8*c) + 15*a*b^3*e^(8*c) + 3*b^4*e^(8*c))*e^(8*d*x) - (4*a^4*e^(6*c)
- 8*a^3*b*e^(6*c) + 32*a^2*b^2*e^(6*c) - 25*a*b^3*e^(6*c) - 9*b^4*e^(6*c))*e^(6*d*x) + (4*a^4*e^(4*c) - 8*a^3*
b*e^(4*c) + 32*a^2*b^2*e^(4*c) - 25*a*b^3*e^(4*c) - 9*b^4*e^(4*c))*e^(4*d*x) + (6*a^4*e^(2*c) - 4*a^3*b*e^(2*c
) + 2*a^2*b^2*e^(2*c) + 15*a*b^3*e^(2*c) + 3*b^4*e^(2*c))*e^(2*d*x))/((a^7*d*e^(9*c) + 5*a^6*b*d*e^(9*c) + 10*
a^5*b^2*d*e^(9*c) + 10*a^4*b^3*d*e^(9*c) + 5*a^3*b^4*d*e^(9*c) + a^2*b^5*d*e^(9*c))*e^(9*d*x) + 4*(a^7*d*e^(7*
c) + 3*a^6*b*d*e^(7*c) + 2*a^5*b^2*d*e^(7*c) - 2*a^4*b^3*d*e^(7*c) - 3*a^3*b^4*d*e^(7*c) - a^2*b^5*d*e^(7*c))*
e^(7*d*x) + 2*(3*a^7*d*e^(5*c) + 7*a^6*b*d*e^(5*c) + 6*a^5*b^2*d*e^(5*c) + 6*a^4*b^3*d*e^(5*c) + 7*a^3*b^4*d*e
^(5*c) + 3*a^2*b^5*d*e^(5*c))*e^(5*d*x) + 4*(a^7*d*e^(3*c) + 3*a^6*b*d*e^(3*c) + 2*a^5*b^2*d*e^(3*c) - 2*a^4*b
^3*d*e^(3*c) - 3*a^3*b^4*d*e^(3*c) - a^2*b^5*d*e^(3*c))*e^(3*d*x) + (a^7*d*e^c + 5*a^6*b*d*e^c + 10*a^5*b^2*d*
e^c + 10*a^4*b^3*d*e^c + 5*a^3*b^4*d*e^c + a^2*b^5*d*e^c)*e^(d*x)) + 1/2*integrate(3/2*((8*a^2*b*e^(3*c) + 4*a
*b^2*e^(3*c) + b^3*e^(3*c))*e^(3*d*x) + (8*a^2*b*e^c + 4*a*b^2*e^c + b^3*e^c)*e^(d*x))/(a^6 + 4*a^5*b + 6*a^4*
b^2 + 4*a^3*b^3 + a^2*b^4 + (a^6*e^(4*c) + 4*a^5*b*e^(4*c) + 6*a^4*b^2*e^(4*c) + 4*a^3*b^3*e^(4*c) + a^2*b^4*e
^(4*c))*e^(4*d*x) + 2*(a^6*e^(2*c) + 2*a^5*b*e^(2*c) - 2*a^3*b^3*e^(2*c) - a^2*b^4*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. 6260 vs. \(2 (140) = 280\).
time = 0.50, size = 11392, normalized size = 73.97 \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)^3,x, algorithm="fricas")

[Out]

[1/16*(8*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cosh(d*x + c)^10 + 80*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cos
h(d*x + c)*sinh(d*x + c)^9 + 8*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*sinh(d*x + c)^10 + 4*(6*a^6 + 2*a^5*b - 2
*a^4*b^2 + 17*a^3*b^3 + 18*a^2*b^4 + 3*a*b^5)*cosh(d*x + c)^8 + 4*(6*a^6 + 2*a^5*b - 2*a^4*b^2 + 17*a^3*b^3 +
18*a^2*b^4 + 3*a*b^5 + 90*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 32*(30*(a^6
+ 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cosh(d*x + c)^3 + (6*a^6 + 2*a^5*b - 2*a^4*b^2 + 17*a^3*b^3 + 18*a^2*b^4 + 3
*a*b^5)*cosh(d*x + c))*sinh(d*x + c)^7 + 4*(4*a^6 - 4*a^5*b + 24*a^4*b^2 + 7*a^3*b^3 - 34*a^2*b^4 - 9*a*b^5)*c
osh(d*x + c)^6 + 4*(4*a^6 - 4*a^5*b + 24*a^4*b^2 + 7*a^3*b^3 - 34*a^2*b^4 - 9*a*b^5 + 420*(a^6 + 3*a^5*b + 3*a
^4*b^2 + a^3*b^3)*cosh(d*x + c)^4 + 28*(6*a^6 + 2*a^5*b - 2*a^4*b^2 + 17*a^3*b^3 + 18*a^2*b^4 + 3*a*b^5)*cosh(
d*x + c)^2)*sinh(d*x + c)^6 - 8*a^6 - 24*a^5*b - 24*a^4*b^2 - 8*a^3*b^3 + 8*(252*(a^6 + 3*a^5*b + 3*a^4*b^2 +
a^3*b^3)*cosh(d*x + c)^5 + 28*(6*a^6 + 2*a^5*b - 2*a^4*b^2 + 17*a^3*b^3 + 18*a^2*b^4 + 3*a*b^5)*cosh(d*x + c)^
3 + 3*(4*a^6 - 4*a^5*b + 24*a^4*b^2 + 7*a^3*b^3 - 34*a^2*b^4 - 9*a*b^5)*cosh(d*x + c))*sinh(d*x + c)^5 - 4*(4*
a^6 - 4*a^5*b + 24*a^4*b^2 + 7*a^3*b^3 - 34*a^2*b^4 - 9*a*b^5)*cosh(d*x + c)^4 + 4*(420*(a^6 + 3*a^5*b + 3*a^4
*b^2 + a^3*b^3)*cosh(d*x + c)^6 - 4*a^6 + 4*a^5*b - 24*a^4*b^2 - 7*a^3*b^3 + 34*a^2*b^4 + 9*a*b^5 + 70*(6*a^6
+ 2*a^5*b - 2*a^4*b^2 + 17*a^3*b^3 + 18*a^2*b^4 + 3*a*b^5)*cosh(d*x + c)^4 + 15*(4*a^6 - 4*a^5*b + 24*a^4*b^2
+ 7*a^3*b^3 - 34*a^2*b^4 - 9*a*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*(60*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3
*b^3)*cosh(d*x + c)^7 + 14*(6*a^6 + 2*a^5*b - 2*a^4*b^2 + 17*a^3*b^3 + 18*a^2*b^4 + 3*a*b^5)*cosh(d*x + c)^5 +
5*(4*a^6 - 4*a^5*b + 24*a^4*b^2 + 7*a^3*b^3 - 34*a^2*b^4 - 9*a*b^5)*cosh(d*x + c)^3 - (4*a^6 - 4*a^5*b + 24*a
^4*b^2 + 7*a^3*b^3 - 34*a^2*b^4 - 9*a*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(6*a^6 + 2*a^5*b - 2*a^4*b^2 + 1
7*a^3*b^3 + 18*a^2*b^4 + 3*a*b^5)*cosh(d*x + c)^2 + 4*(90*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cosh(d*x + c)^
8 + 28*(6*a^6 + 2*a^5*b - 2*a^4*b^2 + 17*a^3*b^3 + 18*a^2*b^4 + 3*a*b^5)*cosh(d*x + c)^6 - 6*a^6 - 2*a^5*b + 2
*a^4*b^2 - 17*a^3*b^3 - 18*a^2*b^4 - 3*a*b^5 + 15*(4*a^6 - 4*a^5*b + 24*a^4*b^2 + 7*a^3*b^3 - 34*a^2*b^4 - 9*a
*b^5)*cosh(d*x + c)^4 - 6*(4*a^6 - 4*a^5*b + 24*a^4*b^2 + 7*a^3*b^3 - 34*a^2*b^4 - 9*a*b^5)*cosh(d*x + c)^2)*s
inh(d*x + c)^2 - 3*((8*a^4*b + 20*a^3*b^2 + 17*a^2*b^3 + 6*a*b^4 + b^5)*cosh(d*x + c)^9 + 9*(8*a^4*b + 20*a^3*
b^2 + 17*a^2*b^3 + 6*a*b^4 + b^5)*cosh(d*x + c)*sinh(d*x + c)^8 + (8*a^4*b + 20*a^3*b^2 + 17*a^2*b^3 + 6*a*b^4
+ b^5)*sinh(d*x + c)^9 + 4*(8*a^4*b + 4*a^3*b^2 - 7*a^2*b^3 - 4*a*b^4 - b^5)*cosh(d*x + c)^7 + 4*(8*a^4*b + 4
*a^3*b^2 - 7*a^2*b^3 - 4*a*b^4 - b^5 + 9*(8*a^4*b + 20*a^3*b^2 + 17*a^2*b^3 + 6*a*b^4 + b^5)*cosh(d*x + c)^2)*
sinh(d*x + c)^7 + 28*(3*(8*a^4*b + 20*a^3*b^2 + 17*a^2*b^3 + 6*a*b^4 + b^5)*cosh(d*x + c)^3 + (8*a^4*b + 4*a^3
*b^2 - 7*a^2*b^3 - 4*a*b^4 - b^5)*cosh(d*x + c))*sinh(d*x + c)^6 + 2*(24*a^4*b - 4*a^3*b^2 + 19*a^2*b^3 + 10*a
*b^4 + 3*b^5)*cosh(d*x + c)^5 + 2*(24*a^4*b - 4*a^3*b^2 + 19*a^2*b^3 + 10*a*b^4 + 3*b^5 + 63*(8*a^4*b + 20*a^3
*b^2 + 17*a^2*b^3 + 6*a*b^4 + b^5)*cosh(d*x + c)^4 + 42*(8*a^4*b + 4*a^3*b^2 - 7*a^2*b^3 - 4*a*b^4 - b^5)*cosh
(d*x + c)^2)*sinh(d*x + c)^5 + 2*(63*(8*a^4*b + 20*a^3*b^2 + 17*a^2*b^3 + 6*a*b^4 + b^5)*cosh(d*x + c)^5 + 70*
(8*a^4*b + 4*a^3*b^2 - 7*a^2*b^3 - 4*a*b^4 - b^5)*cosh(d*x + c)^3 + 5*(24*a^4*b - 4*a^3*b^2 + 19*a^2*b^3 + 10*
a*b^4 + 3*b^5)*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(8*a^4*b + 4*a^3*b^2 - 7*a^2*b^3 - 4*a*b^4 - b^5)*cosh(d*x +
c)^3 + 4*(21*(8*a^4*b + 20*a^3*b^2 + 17*a^2*b^3 + 6*a*b^4 + b^5)*cosh(d*x + c)^6 + 8*a^4*b + 4*a^3*b^2 - 7*a^
2*b^3 - 4*a*b^4 - b^5 + 35*(8*a^4*b + 4*a^3*b^2 - 7*a^2*b^3 - 4*a*b^4 - b^5)*cosh(d*x + c)^4 + 5*(24*a^4*b - 4
*a^3*b^2 + 19*a^2*b^3 + 10*a*b^4 + 3*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 4*(9*(8*a^4*b + 20*a^3*b^2 + 17*a
^2*b^3 + 6*a*b^4 + b^5)*cosh(d*x + c)^7 + 21*(8*a^4*b + 4*a^3*b^2 - 7*a^2*b^3 - 4*a*b^4 - b^5)*cosh(d*x + c)^5
+ 5*(24*a^4*b - 4*a^3*b^2 + 19*a^2*b^3 + 10*a*b^4 + 3*b^5)*cosh(d*x + c)^3 + 3*(8*a^4*b + 4*a^3*b^2 - 7*a^2*b
^3 - 4*a*b^4 - b^5)*cosh(d*x + c))*sinh(d*x + c)^2 + (8*a^4*b + 20*a^3*b^2 + 17*a^2*b^3 + 6*a*b^4 + b^5)*cosh(
d*x + c) + (9*(8*a^4*b + 20*a^3*b^2 + 17*a^2*b^3 + 6*a*b^4 + b^5)*cosh(d*x + c)^8 + 28*(8*a^4*b + 4*a^3*b^2 -
7*a^2*b^3 - 4*a*b^4 - b^5)*cosh(d*x + c)^6 + 8*a^4*b + 20*a^3*b^2 + 17*a^2*b^3 + 6*a*b^4 + b^5 + 10*(24*a^4*b
- 4*a^3*b^2 + 19*a^2*b^3 + 10*a*b^4 + 3*b^5)*cosh(d*x + c)^4 + 12*(8*a^4*b + 4*a^3*b^2 - 7*a^2*b^3 - 4*a*b^4 -
b^5)*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 +...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**3,x)

[Out]

Timed out

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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)^3,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 )}^3} \,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)^3,x)

[Out]

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

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