3.1.42 \(\int \frac {\sinh ^3(c+d x)}{(a+b \tanh ^2(c+d x))^3} \, dx\) [42]

Optimal. Leaf size=166 \[ \frac {5 (3 a-4 b) \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{8 (a+b)^{9/2} d}-\frac {(a-2 b) \cosh (c+d x)}{(a+b)^4 d}+\frac {\cosh ^3(c+d x)}{3 (a+b)^3 d}+\frac {a b \text {sech}(c+d x)}{4 (a+b)^3 d \left (a+b-b \text {sech}^2(c+d x)\right )^2}+\frac {(7 a-4 b) b \text {sech}(c+d x)}{8 (a+b)^4 d \left (a+b-b \text {sech}^2(c+d x)\right )} \]

[Out]

-(a-2*b)*cosh(d*x+c)/(a+b)^4/d+1/3*cosh(d*x+c)^3/(a+b)^3/d+1/4*a*b*sech(d*x+c)/(a+b)^3/d/(a+b-b*sech(d*x+c)^2)
^2+1/8*(7*a-4*b)*b*sech(d*x+c)/(a+b)^4/d/(a+b-b*sech(d*x+c)^2)+5/8*(3*a-4*b)*arctanh(sech(d*x+c)*b^(1/2)/(a+b)
^(1/2))*b^(1/2)/(a+b)^(9/2)/d

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Rubi [A]
time = 0.21, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3745, 467, 1273, 1275, 214} \begin {gather*} \frac {\cosh ^3(c+d x)}{3 d (a+b)^3}-\frac {(a-2 b) \cosh (c+d x)}{d (a+b)^4}+\frac {b (7 a-4 b) \text {sech}(c+d x)}{8 d (a+b)^4 \left (a-b \text {sech}^2(c+d x)+b\right )}+\frac {a b \text {sech}(c+d x)}{4 d (a+b)^3 \left (a-b \text {sech}^2(c+d x)+b\right )^2}+\frac {5 \sqrt {b} (3 a-4 b) \tanh ^{-1}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{8 d (a+b)^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(3*a - 4*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(8*(a + b)^(9/2)*d) - ((a - 2*b)*Cosh[c +
d*x])/((a + b)^4*d) + Cosh[c + d*x]^3/(3*(a + b)^3*d) + (a*b*Sech[c + d*x])/(4*(a + b)^3*d*(a + b - b*Sech[c
+ d*x]^2)^2) + ((7*a - 4*b)*b*Sech[c + d*x])/(8*(a + b)^4*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 467

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x
*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[x^m*(a + b*x^2)^(p +
1)*ExpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)]
- ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &
& ILtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1273

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[(-d)^(m
/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Dist[(-d)^(m/2 - 1)/
(2*e^(2*p)*(q + 1)), Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e*x^2))*(2*(-d)^(-m/2 + 1)*e^(2*
p)*(q + 1)*(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2))], x], x],
x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]

Rule 1275

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] &&
NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]

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 ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {-1+x^2}{x^4 \left (a+b-b x^2\right )^3} \, dx,x,\text {sech}(c+d x)\right )}{d}\\ &=\frac {a b \text {sech}(c+d x)}{4 (a+b)^3 d \left (a+b-b \text {sech}^2(c+d x)\right )^2}+\frac {b \text {Subst}\left (\int \frac {-\frac {4}{b (a+b)}+\frac {4 a x^2}{b (a+b)^2}+\frac {3 a x^4}{(a+b)^3}}{x^4 \left (a+b-b x^2\right )^2} \, dx,x,\text {sech}(c+d x)\right )}{4 d}\\ &=\frac {a b \text {sech}(c+d x)}{4 (a+b)^3 d \left (a+b-b \text {sech}^2(c+d x)\right )^2}+\frac {(7 a-4 b) b \text {sech}(c+d x)}{8 (a+b)^4 d \left (a+b-b \text {sech}^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {-8 b (a+b)+8 (a-b) b x^2+\frac {(7 a-4 b) b^2 x^4}{a+b}}{x^4 \left (a+b-b x^2\right )} \, dx,x,\text {sech}(c+d x)\right )}{8 b (a+b)^3 d}\\ &=\frac {a b \text {sech}(c+d x)}{4 (a+b)^3 d \left (a+b-b \text {sech}^2(c+d x)\right )^2}+\frac {(7 a-4 b) b \text {sech}(c+d x)}{8 (a+b)^4 d \left (a+b-b \text {sech}^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \left (-\frac {8 b}{x^4}+\frac {8 (a-2 b) b}{(a+b) x^2}+\frac {5 (3 a-4 b) b^2}{(a+b) \left (a+b-b x^2\right )}\right ) \, dx,x,\text {sech}(c+d x)\right )}{8 b (a+b)^3 d}\\ &=-\frac {(a-2 b) \cosh (c+d x)}{(a+b)^4 d}+\frac {\cosh ^3(c+d x)}{3 (a+b)^3 d}+\frac {a b \text {sech}(c+d x)}{4 (a+b)^3 d \left (a+b-b \text {sech}^2(c+d x)\right )^2}+\frac {(7 a-4 b) b \text {sech}(c+d x)}{8 (a+b)^4 d \left (a+b-b \text {sech}^2(c+d x)\right )}+\frac {(5 (3 a-4 b) b) \text {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\text {sech}(c+d x)\right )}{8 (a+b)^4 d}\\ &=\frac {5 (3 a-4 b) \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{8 (a+b)^{9/2} d}-\frac {(a-2 b) \cosh (c+d x)}{(a+b)^4 d}+\frac {\cosh ^3(c+d x)}{3 (a+b)^3 d}+\frac {a b \text {sech}(c+d x)}{4 (a+b)^3 d \left (a+b-b \text {sech}^2(c+d x)\right )^2}+\frac {(7 a-4 b) b \text {sech}(c+d x)}{8 (a+b)^4 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 = 1.37, size = 227, normalized size = 1.37 \begin {gather*} \frac {\frac {15 i (3 a-4 b) \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)^{9/2}}-\frac {6 \cosh (c+d x) \left (3 a^3-24 a^2 b+30 a b^2-13 b^3+\left (6 a^3-27 a^2 b-11 a b^2+22 b^3\right ) \cosh (2 (c+d x))+3 (a-3 b) (a+b)^2 \cosh ^2(2 (c+d x))\right )}{(a+b)^4 (a-b+(a+b) \cosh (2 (c+d x)))^2}+\frac {2 \cosh (3 (c+d x))}{(a+b)^3}}{24 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((15*I)*(3*a - 4*b)*Sqrt[b]*(ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]] + ArcTan[((-I)*Sq
rt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]]))/(a + b)^(9/2) - (6*Cosh[c + d*x]*(3*a^3 - 24*a^2*b + 30*a*b^
2 - 13*b^3 + (6*a^3 - 27*a^2*b - 11*a*b^2 + 22*b^3)*Cosh[2*(c + d*x)] + 3*(a - 3*b)*(a + b)^2*Cosh[2*(c + d*x)
]^2))/((a + b)^4*(a - b + (a + b)*Cosh[2*(c + d*x)])^2) + (2*Cosh[3*(c + d*x)])/(a + b)^3)/(24*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(340\) vs. \(2(150)=300\).
time = 2.66, size = 341, normalized size = 2.05

method	result	size
derivativedivides	\(\frac {-\frac {2 b \left (\frac {-\frac {\left (9 a +20 b \right ) a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {\left (27 a^{3}+66 a^{2} b +56 a \,b^{2}-16 b^{3}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\left (-\frac {27}{8} a^{2}-\frac {11}{2} a b +2 b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {9 a^{2}}{8}+\frac {a b}{4}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}-\frac {5 \left (3 a -4 b \right ) \arctanh \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{16 \sqrt {a b +b^{2}}}\right )}{\left (a +b \right )^{4}}-\frac {1}{3 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a +5 b}{2 \left (a +b \right )^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {1}{3 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {a -5 b}{2 \left (a +b \right )^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\)	\(341\)
default	\(\frac {-\frac {2 b \left (\frac {-\frac {\left (9 a +20 b \right ) a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {\left (27 a^{3}+66 a^{2} b +56 a \,b^{2}-16 b^{3}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\left (-\frac {27}{8} a^{2}-\frac {11}{2} a b +2 b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {9 a^{2}}{8}+\frac {a b}{4}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}-\frac {5 \left (3 a -4 b \right ) \arctanh \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{16 \sqrt {a b +b^{2}}}\right )}{\left (a +b \right )^{4}}-\frac {1}{3 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a +5 b}{2 \left (a +b \right )^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {1}{3 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {a -5 b}{2 \left (a +b \right )^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\)	\(341\)
risch	\(\frac {{\mathrm e}^{3 d x +3 c}}{24 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}-\frac {3 \,{\mathrm e}^{d x +c} a}{8 \left (a +b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}+\frac {9 \,{\mathrm e}^{d x +c} b}{8 \left (a +b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}-\frac {3 \,{\mathrm e}^{-d x -c} a}{8 \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) d}+\frac {9 \,{\mathrm e}^{-d x -c} b}{8 \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) d}+\frac {{\mathrm e}^{-3 d x -3 c}}{24 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}+\frac {{\mathrm e}^{d x +c} b \left (9 a^{2} {\mathrm e}^{6 d x +6 c}+5 a b \,{\mathrm e}^{6 d x +6 c}-4 b^{2} {\mathrm e}^{6 d x +6 c}+27 a^{2} {\mathrm e}^{4 d x +4 c}-13 a b \,{\mathrm e}^{4 d x +4 c}+4 b^{2} {\mathrm e}^{4 d x +4 c}+27 a^{2} {\mathrm e}^{2 d x +2 c}-13 a b \,{\mathrm e}^{2 d x +2 c}+4 b^{2} {\mathrm e}^{2 d x +2 c}+9 a^{2}+5 a b -4 b^{2}\right )}{4 \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )^{2} d \left (a +b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {15 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {b \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) a}{16 \left (a +b \right )^{5} d}-\frac {5 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {b \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) b}{4 \left (a +b \right )^{5} d}-\frac {15 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {b \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) a}{16 \left (a +b \right )^{5} d}+\frac {5 \sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {b \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) b}{4 \left (a +b \right )^{5} d}\)	\(663\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-2*b/(a+b)^4*((-1/8*(9*a+20*b)*a*tanh(1/2*d*x+1/2*c)^6-1/8*(27*a^3+66*a^2*b+56*a*b^2-16*b^3)/a*tanh(1/2*d
*x+1/2*c)^4+(-27/8*a^2-11/2*a*b+2*b^2)*tanh(1/2*d*x+1/2*c)^2-9/8*a^2+1/4*a*b)/(a*tanh(1/2*d*x+1/2*c)^4+2*a*tan
h(1/2*d*x+1/2*c)^2+4*b*tanh(1/2*d*x+1/2*c)^2+a)^2-5/16*(3*a-4*b)/(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)))-1/3/(a+b)^3/(tanh(1/2*d*x+1/2*c)-1)^3-1/2/(a+b)^3/(tanh(1/2*d*x+1/2*c)-1)
^2-1/2/(a+b)^4*(-a+5*b)/(tanh(1/2*d*x+1/2*c)-1)+1/3/(a+b)^3/(tanh(1/2*d*x+1/2*c)+1)^3-1/2/(a+b)^3/(tanh(1/2*d*
x+1/2*c)+1)^2-1/2*(a-5*b)/(a+b)^4/(tanh(1/2*d*x+1/2*c)+1))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 7129 vs. \(2 (156) = 312\).
time = 0.51, size = 13095, normalized size = 78.89 \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)^3/(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[1/48*(2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^14 + 28*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)*s
inh(d*x + c)^13 + 2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sinh(d*x + c)^14 - 2*(5*a^3 - 13*a^2*b - 41*a*b^2 - 23*b^3
)*cosh(d*x + c)^12 - 2*(5*a^3 - 13*a^2*b - 41*a*b^2 - 23*b^3 - 91*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c
)^2)*sinh(d*x + c)^12 + 8*(91*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^3 - 3*(5*a^3 - 13*a^2*b - 41*a*b^2
- 23*b^3)*cosh(d*x + c))*sinh(d*x + c)^11 - 2*(39*a^3 - 173*a^2*b - 113*a*b^2 + 99*b^3)*cosh(d*x + c)^10 + 2*
(1001*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^4 - 39*a^3 + 173*a^2*b + 113*a*b^2 - 99*b^3 - 66*(5*a^3 -
13*a^2*b - 41*a*b^2 - 23*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 4*(1001*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh
(d*x + c)^5 - 110*(5*a^3 - 13*a^2*b - 41*a*b^2 - 23*b^3)*cosh(d*x + c)^3 - 5*(39*a^3 - 173*a^2*b - 113*a*b^2 +
99*b^3)*cosh(d*x + c))*sinh(d*x + c)^9 - 10*(17*a^3 - 95*a^2*b + 41*a*b^2 - 15*b^3)*cosh(d*x + c)^8 + 2*(3003
*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^6 - 495*(5*a^3 - 13*a^2*b - 41*a*b^2 - 23*b^3)*cosh(d*x + c)^4
- 85*a^3 + 475*a^2*b - 205*a*b^2 + 75*b^3 - 45*(39*a^3 - 173*a^2*b - 113*a*b^2 + 99*b^3)*cosh(d*x + c)^2)*sinh
(d*x + c)^8 + 16*(429*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^7 - 99*(5*a^3 - 13*a^2*b - 41*a*b^2 - 23*b
^3)*cosh(d*x + c)^5 - 15*(39*a^3 - 173*a^2*b - 113*a*b^2 + 99*b^3)*cosh(d*x + c)^3 - 5*(17*a^3 - 95*a^2*b + 41
*a*b^2 - 15*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 - 10*(17*a^3 - 95*a^2*b + 41*a*b^2 - 15*b^3)*cosh(d*x + c)^6 +
2*(3003*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^8 - 924*(5*a^3 - 13*a^2*b - 41*a*b^2 - 23*b^3)*cosh(d*x
+ c)^6 - 210*(39*a^3 - 173*a^2*b - 113*a*b^2 + 99*b^3)*cosh(d*x + c)^4 - 85*a^3 + 475*a^2*b - 205*a*b^2 + 75*
b^3 - 140*(17*a^3 - 95*a^2*b + 41*a*b^2 - 15*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(1001*(a^3 + 3*a^2*b +
3*a*b^2 + b^3)*cosh(d*x + c)^9 - 396*(5*a^3 - 13*a^2*b - 41*a*b^2 - 23*b^3)*cosh(d*x + c)^7 - 126*(39*a^3 - 17
3*a^2*b - 113*a*b^2 + 99*b^3)*cosh(d*x + c)^5 - 140*(17*a^3 - 95*a^2*b + 41*a*b^2 - 15*b^3)*cosh(d*x + c)^3 -
15*(17*a^3 - 95*a^2*b + 41*a*b^2 - 15*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(39*a^3 - 173*a^2*b - 113*a*b^2
+ 99*b^3)*cosh(d*x + c)^4 + 2*(1001*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^10 - 495*(5*a^3 - 13*a^2*b -
41*a*b^2 - 23*b^3)*cosh(d*x + c)^8 - 210*(39*a^3 - 173*a^2*b - 113*a*b^2 + 99*b^3)*cosh(d*x + c)^6 - 350*(17*
a^3 - 95*a^2*b + 41*a*b^2 - 15*b^3)*cosh(d*x + c)^4 - 39*a^3 + 173*a^2*b + 113*a*b^2 - 99*b^3 - 75*(17*a^3 - 9
5*a^2*b + 41*a*b^2 - 15*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(91*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cosh(d*x
+ c)^11 - 55*(5*a^3 - 13*a^2*b - 41*a*b^2 - 23*b^3)*cosh(d*x + c)^9 - 30*(39*a^3 - 173*a^2*b - 113*a*b^2 + 99
*b^3)*cosh(d*x + c)^7 - 70*(17*a^3 - 95*a^2*b + 41*a*b^2 - 15*b^3)*cosh(d*x + c)^5 - 25*(17*a^3 - 95*a^2*b + 4
1*a*b^2 - 15*b^3)*cosh(d*x + c)^3 - (39*a^3 - 173*a^2*b - 113*a*b^2 + 99*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 +
2*a^3 + 6*a^2*b + 6*a*b^2 + 2*b^3 - 2*(5*a^3 - 13*a^2*b - 41*a*b^2 - 23*b^3)*cosh(d*x + c)^2 + 2*(91*(a^3 + 3
*a^2*b + 3*a*b^2 + b^3)*cosh(d*x + c)^12 - 66*(5*a^3 - 13*a^2*b - 41*a*b^2 - 23*b^3)*cosh(d*x + c)^10 - 45*(39
*a^3 - 173*a^2*b - 113*a*b^2 + 99*b^3)*cosh(d*x + c)^8 - 140*(17*a^3 - 95*a^2*b + 41*a*b^2 - 15*b^3)*cosh(d*x
+ c)^6 - 75*(17*a^3 - 95*a^2*b + 41*a*b^2 - 15*b^3)*cosh(d*x + c)^4 - 5*a^3 + 13*a^2*b + 41*a*b^2 + 23*b^3 - 6
*(39*a^3 - 173*a^2*b - 113*a*b^2 + 99*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 15*((3*a^3 + 2*a^2*b - 5*a*b^2 -
4*b^3)*cosh(d*x + c)^11 + 11*(3*a^3 + 2*a^2*b - 5*a*b^2 - 4*b^3)*cosh(d*x + c)*sinh(d*x + c)^10 + (3*a^3 + 2*
a^2*b - 5*a*b^2 - 4*b^3)*sinh(d*x + c)^11 + 4*(3*a^3 - 4*a^2*b - 3*a*b^2 + 4*b^3)*cosh(d*x + c)^9 + (12*a^3 -
16*a^2*b - 12*a*b^2 + 16*b^3 + 55*(3*a^3 + 2*a^2*b - 5*a*b^2 - 4*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^9 + 3*(55
*(3*a^3 + 2*a^2*b - 5*a*b^2 - 4*b^3)*cosh(d*x + c)^3 + 12*(3*a^3 - 4*a^2*b - 3*a*b^2 + 4*b^3)*cosh(d*x + c))*s
inh(d*x + c)^8 + 2*(9*a^3 - 18*a^2*b + 17*a*b^2 - 12*b^3)*cosh(d*x + c)^7 + 2*(165*(3*a^3 + 2*a^2*b - 5*a*b^2
- 4*b^3)*cosh(d*x + c)^4 + 9*a^3 - 18*a^2*b + 17*a*b^2 - 12*b^3 + 72*(3*a^3 - 4*a^2*b - 3*a*b^2 + 4*b^3)*cosh(
d*x + c)^2)*sinh(d*x + c)^7 + 14*(33*(3*a^3 + 2*a^2*b - 5*a*b^2 - 4*b^3)*cosh(d*x + c)^5 + 24*(3*a^3 - 4*a^2*b
- 3*a*b^2 + 4*b^3)*cosh(d*x + c)^3 + (9*a^3 - 18*a^2*b + 17*a*b^2 - 12*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 +
4*(3*a^3 - 4*a^2*b - 3*a*b^2 + 4*b^3)*cosh(d*x + c)^5 + 2*(231*(3*a^3 + 2*a^2*b - 5*a*b^2 - 4*b^3)*cosh(d*x +
c)^6 + 252*(3*a^3 - 4*a^2*b - 3*a*b^2 + 4*b^3)*cosh(d*x + c)^4 + 6*a^3 - 8*a^2*b - 6*a*b^2 + 8*b^3 + 21*(9*a^3
- 18*a^2*b + 17*a*b^2 - 12*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 2*(165*(3*a^3 + 2*a^2*b - 5*a*b^2 - 4*b^3)
*cosh(d*x + c)^7 + 252*(3*a^3 - 4*a^2*b - 3*a*b^2 + 4*b^3)*cosh(d*x + c)^5 + 35*(9*a^3 - 18*a^2*b + 17*a*b^2 -
12*b^3)*cosh(d*x + c)^3 + 10*(3*a^3 - 4*a^2*b - 3*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + (3*a^3 + 2*
a^2*b - 5*a*b^2 - 4*b^3)*cosh(d*x + c)^3 + (165...

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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(sinh(d*x+c)**3/(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(sinh(d*x+c)^3/(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 {sinh}\left (c+d\,x\right )}^3}{{\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(sinh(c + d*x)^3/(a + b*tanh(c + d*x)^2)^3,x)

[Out]

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

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