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

Optimal. Leaf size=154 \[ \frac {5 \sqrt {b} (3 a+7 b) \text {ArcTan}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{8 a^{9/2} d}-\frac {(a+3 b) \cosh (c+d x)}{a^4 d}+\frac {\cosh ^3(c+d x)}{3 a^3 d}+\frac {b^2 (a+b) \cosh (c+d x)}{4 a^4 d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {b (9 a+13 b) \cosh (c+d x)}{8 a^4 d \left (b+a \cosh ^2(c+d x)\right )} \]

[Out]

-(a+3*b)*cosh(d*x+c)/a^4/d+1/3*cosh(d*x+c)^3/a^3/d+1/4*b^2*(a+b)*cosh(d*x+c)/a^4/d/(b+a*cosh(d*x+c)^2)^2-1/8*b
*(9*a+13*b)*cosh(d*x+c)/a^4/d/(b+a*cosh(d*x+c)^2)+5/8*(3*a+7*b)*arctan(cosh(d*x+c)*a^(1/2)/b^(1/2))*b^(1/2)/a^
(9/2)/d

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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 = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4218, 466, 1828, 1167, 211} \begin {gather*} \frac {5 \sqrt {b} (3 a+7 b) \text {ArcTan}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{8 a^{9/2} d}+\frac {b^2 (a+b) \cosh (c+d x)}{4 a^4 d \left (a \cosh ^2(c+d x)+b\right )^2}-\frac {b (9 a+13 b) \cosh (c+d x)}{8 a^4 d \left (a \cosh ^2(c+d x)+b\right )}-\frac {(a+3 b) \cosh (c+d x)}{a^4 d}+\frac {\cosh ^3(c+d x)}{3 a^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*Sqrt[b]*(3*a + 7*b)*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(8*a^(9/2)*d) - ((a + 3*b)*Cosh[c + d*x])/(a^4
*d) + Cosh[c + d*x]^3/(3*a^3*d) + (b^2*(a + b)*Cosh[c + d*x])/(4*a^4*d*(b + a*Cosh[c + d*x]^2)^2) - (b*(9*a +
13*b)*Cosh[c + d*x])/(8*a^4*d*(b + a*Cosh[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 466

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[(a + b*x^2)^(p + 1)*E
xpandToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)]
- (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[
m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1167

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, 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] && IGtQ[q, -2]

Rule 1828

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*
g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rule 4218

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = F
reeFactors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*(ff*x)^n)^p/(ff*x)^(n*p
)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p
]

Rubi steps

\begin {align*} \int \frac {\sinh ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx &=-\frac {\text {Subst}\left (\int \frac {x^6 \left (1-x^2\right )}{\left (b+a x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {b^2 (a+b) \cosh (c+d x)}{4 a^4 d \left (b+a \cosh ^2(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {-b^2 (a+b)+4 a b (a+b) x^2-4 a^2 (a+b) x^4+4 a^3 x^6}{\left (b+a x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{4 a^4 d}\\ &=\frac {b^2 (a+b) \cosh (c+d x)}{4 a^4 d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {b (9 a+13 b) \cosh (c+d x)}{8 a^4 d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {-b^2 (7 a+11 b)+8 a b (a+2 b) x^2-8 a^2 b x^4}{b+a x^2} \, dx,x,\cosh (c+d x)\right )}{8 a^4 b d}\\ &=\frac {b^2 (a+b) \cosh (c+d x)}{4 a^4 d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {b (9 a+13 b) \cosh (c+d x)}{8 a^4 d \left (b+a \cosh ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \left (8 b (a+3 b)-8 a b x^2-\frac {5 \left (3 a b^2+7 b^3\right )}{b+a x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{8 a^4 b d}\\ &=-\frac {(a+3 b) \cosh (c+d x)}{a^4 d}+\frac {\cosh ^3(c+d x)}{3 a^3 d}+\frac {b^2 (a+b) \cosh (c+d x)}{4 a^4 d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {b (9 a+13 b) \cosh (c+d x)}{8 a^4 d \left (b+a \cosh ^2(c+d x)\right )}+\frac {(5 b (3 a+7 b)) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\cosh (c+d x)\right )}{8 a^4 d}\\ &=\frac {5 \sqrt {b} (3 a+7 b) \tan ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{8 a^{9/2} d}-\frac {(a+3 b) \cosh (c+d x)}{a^4 d}+\frac {\cosh ^3(c+d x)}{3 a^3 d}+\frac {b^2 (a+b) \cosh (c+d x)}{4 a^4 d \left (b+a \cosh ^2(c+d x)\right )^2}-\frac {b (9 a+13 b) \cosh (c+d x)}{8 a^4 d \left (b+a \cosh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 7.06, size = 1217, normalized size = 7.90 \begin {gather*} \frac {(a+2 b+a \cosh (2 (c+d x)))^3 \text {sech}^6(c+d x) \left (\frac {24 (3 a-4 b) \left (\text {ArcTan}\left (\frac {\left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )+\text {ArcTan}\left (\frac {\left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )\right )}{a^{3/2} b^{5/2}}-\frac {54 \left (\text {ArcTan}\left (\frac {\sqrt {a}-i \sqrt {a+b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )+\text {ArcTan}\left (\frac {\sqrt {a}+i \sqrt {a+b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )\right )}{\sqrt {a} b^{5/2}}-\frac {36 \cosh (c+d x) (3 a+10 b+3 a \cosh (2 (c+d x)))}{b^2 (a+2 b+a \cosh (2 (c+d x)))^2}+\frac {48 \cosh (c+d x) \left (3 a^2+6 a b+8 b^2+a (3 a-4 b) \cosh (2 (c+d x))\right )}{a b^2 (a+2 b+a \cosh (2 (c+d x)))^2}+\frac {3 \left (3 a^4-40 a^3 b+720 a^2 b^2+6720 a b^3+8960 b^4\right ) \text {ArcTan}\left (\frac {\left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )+3 \left (3 a^4-40 a^3 b+720 a^2 b^2+6720 a b^3+8960 b^4\right ) \text {ArcTan}\left (\frac {\left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )+\frac {2 \sqrt {a} \sqrt {b} \cosh (c+d x) \left (9 a^5-90 a^4 b-10144 a^3 b^2-48672 a^2 b^3-85120 a b^4-53760 b^5+a \left (9 a^4-120 a^3 b-12432 a^2 b^2-47936 a b^3-44800 b^4\right ) \cosh (2 (c+d x))-128 a^2 b^2 (15 a+28 b) \cosh (4 (c+d x))+128 a^3 b^2 \cosh (6 (c+d x))\right )}{(a+2 b+a \cosh (2 (c+d x)))^2}}{a^{9/2} b^{5/2}}+\frac {9 \left (-\frac {3 \left (a^3-8 a^2 b+80 a b^2+320 b^3\right ) \text {ArcTan}\left (\frac {\left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )}{b^{5/2}}-\frac {3 \left (a^3-8 a^2 b+80 a b^2+320 b^3\right ) \text {ArcTan}\left (\frac {\left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )}{b^{5/2}}+512 \sqrt {a} \cosh (c) \cosh (d x)-\frac {8 \sqrt {a} \left (a^3+24 a^2 b+80 a b^2+64 b^3\right ) \cosh (c+d x)}{b (a+2 b+a \cosh (2 (c+d x)))^2}-\frac {2 \sqrt {a} \left (3 a^3-24 a^2 b-400 a b^2-576 b^3\right ) \cosh (c+d x)}{b^2 (a+2 b+a \cosh (2 (c+d x)))}+512 \sqrt {a} \sinh (c) \sinh (d x)\right )}{a^{7/2}}\right )}{49152 d \left (a+b \text {sech}^2(c+d x)\right )^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])^3*Sech[c + d*x]^6*((24*(3*a - 4*b)*(ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Co
sh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh
[(d*x)/2]))/Sqrt[b]] + ArcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + C
osh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]]))/(a^(3/2)*b^(5/2)) - (54
*(ArcTan[(Sqrt[a] - I*Sqrt[a + b]*Tanh[(c + d*x)/2])/Sqrt[b]] + ArcTan[(Sqrt[a] + I*Sqrt[a + b]*Tanh[(c + d*x)
/2])/Sqrt[b]]))/(Sqrt[a]*b^(5/2)) - (36*Cosh[c + d*x]*(3*a + 10*b + 3*a*Cosh[2*(c + d*x)]))/(b^2*(a + 2*b + a*
Cosh[2*(c + d*x)])^2) + (48*Cosh[c + d*x]*(3*a^2 + 6*a*b + 8*b^2 + a*(3*a - 4*b)*Cosh[2*(c + d*x)]))/(a*b^2*(a
+ 2*b + a*Cosh[2*(c + d*x)])^2) + (3*(3*a^4 - 40*a^3*b + 720*a^2*b^2 + 6720*a*b^3 + 8960*b^4)*ArcTan[((Sqrt[a
] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(
Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] + 3*(3*a^4 - 40*a^3*b + 720*a^2*b^2 + 6720*a*b^3 + 8960*b^4)*Ar
cTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[
a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] + (2*Sqrt[a]*Sqrt[b]*Cosh[c + d*x]*(9*a^5 - 90*a^4
*b - 10144*a^3*b^2 - 48672*a^2*b^3 - 85120*a*b^4 - 53760*b^5 + a*(9*a^4 - 120*a^3*b - 12432*a^2*b^2 - 47936*a*
b^3 - 44800*b^4)*Cosh[2*(c + d*x)] - 128*a^2*b^2*(15*a + 28*b)*Cosh[4*(c + d*x)] + 128*a^3*b^2*Cosh[6*(c + d*x
)]))/(a + 2*b + a*Cosh[2*(c + d*x)])^2)/(a^(9/2)*b^(5/2)) + (9*((-3*(a^3 - 8*a^2*b + 80*a*b^2 + 320*b^3)*ArcTa
n[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a +
b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]])/b^(5/2) - (3*(a^3 - 8*a^2*b + 80*a*b^2 + 320*b^3)*Ar
cTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[
a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]])/b^(5/2) + 512*Sqrt[a]*Cosh[c]*Cosh[d*x] - (8*Sqrt
[a]*(a^3 + 24*a^2*b + 80*a*b^2 + 64*b^3)*Cosh[c + d*x])/(b*(a + 2*b + a*Cosh[2*(c + d*x)])^2) - (2*Sqrt[a]*(3*
a^3 - 24*a^2*b - 400*a*b^2 - 576*b^3)*Cosh[c + d*x])/(b^2*(a + 2*b + a*Cosh[2*(c + d*x)])) + 512*Sqrt[a]*Sinh[
c]*Sinh[d*x]))/a^(7/2)))/(49152*d*(a + b*Sech[c + d*x]^2)^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(347\) vs. \(2(138)=276\).
time = 2.34, size = 348, normalized size = 2.26

method	result	size
derivativedivides	\(\frac {\frac {2 b \left (\frac {\left (-\frac {9}{8} a^{2}+\frac {1}{4} a b +\frac {11}{8} b^{2}\right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\left (27 a^{3}+15 a^{2} b +5 a \,b^{2}+33 b^{3}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )}+\left (-\frac {27}{8} a^{2}-\frac {5}{4} a b +\frac {33}{8} b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {9 a^{2}}{8}-\frac {5 a b}{2}-\frac {11 b^{2}}{8}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}+\frac {5 \left (3 a +7 b \right ) \arctan \left (\frac {2 \left (a +b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a -2 b}{4 \sqrt {a b}}\right )}{16 \sqrt {a b}}\right )}{a^{4}}-\frac {1}{3 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a -6 b}{2 a^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {1}{3 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{2 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {a +6 b}{2 a^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\)	\(348\)
default	\(\frac {\frac {2 b \left (\frac {\left (-\frac {9}{8} a^{2}+\frac {1}{4} a b +\frac {11}{8} b^{2}\right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\left (27 a^{3}+15 a^{2} b +5 a \,b^{2}+33 b^{3}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )}+\left (-\frac {27}{8} a^{2}-\frac {5}{4} a b +\frac {33}{8} b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {9 a^{2}}{8}-\frac {5 a b}{2}-\frac {11 b^{2}}{8}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}+\frac {5 \left (3 a +7 b \right ) \arctan \left (\frac {2 \left (a +b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a -2 b}{4 \sqrt {a b}}\right )}{16 \sqrt {a b}}\right )}{a^{4}}-\frac {1}{3 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a -6 b}{2 a^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {1}{3 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{2 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {a +6 b}{2 a^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\)	\(348\)
risch	\(\frac {{\mathrm e}^{3 d x +3 c}}{24 a^{3} d}-\frac {3 \,{\mathrm e}^{d x +c}}{8 a^{3} d}-\frac {3 \,{\mathrm e}^{d x +c} b}{2 a^{4} d}-\frac {3 \,{\mathrm e}^{-d x -c}}{8 a^{3} d}-\frac {3 \,{\mathrm e}^{-d x -c} b}{2 a^{4} d}+\frac {{\mathrm e}^{-3 d x -3 c}}{24 a^{3} d}-\frac {{\mathrm e}^{d x +c} b \left (9 a^{2} {\mathrm e}^{6 d x +6 c}+13 a b \,{\mathrm e}^{6 d x +6 c}+27 a^{2} {\mathrm e}^{4 d x +4 c}+67 a b \,{\mathrm e}^{4 d x +4 c}+44 b^{2} {\mathrm e}^{4 d x +4 c}+27 a^{2} {\mathrm e}^{2 d x +2 c}+67 a b \,{\mathrm e}^{2 d x +2 c}+44 b^{2} {\mathrm e}^{2 d x +2 c}+9 a^{2}+13 a b \right )}{4 a^{4} d \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2}}+\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right )}{16 a^{4} d}+\frac {35 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right ) b}{16 a^{5} d}-\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right )}{16 a^{4} d}-\frac {35 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right ) b}{16 a^{5} d}\)	\(447\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(2*b/a^4*(((-9/8*a^2+1/4*a*b+11/8*b^2)*tanh(1/2*d*x+1/2*c)^6-1/8*(27*a^3+15*a^2*b+5*a*b^2+33*b^3)/(a+b)*ta
nh(1/2*d*x+1/2*c)^4+(-27/8*a^2-5/4*a*b+33/8*b^2)*tanh(1/2*d*x+1/2*c)^2-9/8*a^2-5/2*a*b-11/8*b^2)/(a*tanh(1/2*d
*x+1/2*c)^4+b*tanh(1/2*d*x+1/2*c)^4+2*a*tanh(1/2*d*x+1/2*c)^2-2*b*tanh(1/2*d*x+1/2*c)^2+a+b)^2+5/16*(3*a+7*b)/
(a*b)^(1/2)*arctan(1/4*(2*(a+b)*tanh(1/2*d*x+1/2*c)^2+2*a-2*b)/(a*b)^(1/2)))-1/3/a^3/(tanh(1/2*d*x+1/2*c)-1)^3
-1/2/a^3/(tanh(1/2*d*x+1/2*c)-1)^2-1/2/a^4*(-a-6*b)/(tanh(1/2*d*x+1/2*c)-1)+1/3/a^3/(tanh(1/2*d*x+1/2*c)+1)^3-
1/2/a^3/(tanh(1/2*d*x+1/2*c)+1)^2-1/2*(a+6*b)/a^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*sech(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. 4793 vs. \(2 (138) = 276\).
time = 0.45, size = 8667, normalized size = 56.28 \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*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[1/48*(2*a^3*cosh(d*x + c)^14 + 28*a^3*cosh(d*x + c)*sinh(d*x + c)^13 + 2*a^3*sinh(d*x + c)^14 - 2*(5*a^3 + 28
*a^2*b)*cosh(d*x + c)^12 + 2*(91*a^3*cosh(d*x + c)^2 - 5*a^3 - 28*a^2*b)*sinh(d*x + c)^12 + 8*(91*a^3*cosh(d*x
+ c)^3 - 3*(5*a^3 + 28*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^11 - 2*(39*a^3 + 290*a^2*b + 350*a*b^2)*cosh(d*x +
c)^10 + 2*(1001*a^3*cosh(d*x + c)^4 - 39*a^3 - 290*a^2*b - 350*a*b^2 - 66*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^2)
*sinh(d*x + c)^10 + 4*(1001*a^3*cosh(d*x + c)^5 - 110*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^3 - 5*(39*a^3 + 290*a^2
*b + 350*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^9 - 10*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c)^8
+ 2*(3003*a^3*cosh(d*x + c)^6 - 495*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^4 - 85*a^3 - 730*a^2*b - 1410*a*b^2 - 84
0*b^3 - 45*(39*a^3 + 290*a^2*b + 350*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^8 + 16*(429*a^3*cosh(d*x + c)^7 - 9
9*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^5 - 15*(39*a^3 + 290*a^2*b + 350*a*b^2)*cosh(d*x + c)^3 - 5*(17*a^3 + 146*a
^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 - 10*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cos
h(d*x + c)^6 + 2*(3003*a^3*cosh(d*x + c)^8 - 924*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^6 - 210*(39*a^3 + 290*a^2*b
+ 350*a*b^2)*cosh(d*x + c)^4 - 85*a^3 - 730*a^2*b - 1410*a*b^2 - 840*b^3 - 140*(17*a^3 + 146*a^2*b + 282*a*b^2
+ 168*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(1001*a^3*cosh(d*x + c)^9 - 396*(5*a^3 + 28*a^2*b)*cosh(d*x +
c)^7 - 126*(39*a^3 + 290*a^2*b + 350*a*b^2)*cosh(d*x + c)^5 - 140*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*
cosh(d*x + c)^3 - 15*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(39*a^3 + 2
90*a^2*b + 350*a*b^2)*cosh(d*x + c)^4 + 2*(1001*a^3*cosh(d*x + c)^10 - 495*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^8
- 210*(39*a^3 + 290*a^2*b + 350*a*b^2)*cosh(d*x + c)^6 - 350*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d
*x + c)^4 - 39*a^3 - 290*a^2*b - 350*a*b^2 - 75*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c)^2)*si
nh(d*x + c)^4 + 8*(91*a^3*cosh(d*x + c)^11 - 55*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^9 - 30*(39*a^3 + 290*a^2*b +
350*a*b^2)*cosh(d*x + c)^7 - 70*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c)^5 - 25*(17*a^3 + 146*
a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c)^3 - (39*a^3 + 290*a^2*b + 350*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3
+ 2*a^3 - 2*(5*a^3 + 28*a^2*b)*cosh(d*x + c)^2 + 2*(91*a^3*cosh(d*x + c)^12 - 66*(5*a^3 + 28*a^2*b)*cosh(d*x
+ c)^10 - 45*(39*a^3 + 290*a^2*b + 350*a*b^2)*cosh(d*x + c)^8 - 140*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)
*cosh(d*x + c)^6 - 75*(17*a^3 + 146*a^2*b + 282*a*b^2 + 168*b^3)*cosh(d*x + c)^4 - 5*a^3 - 28*a^2*b - 6*(39*a^
3 + 290*a^2*b + 350*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 15*((3*a^3 + 7*a^2*b)*cosh(d*x + c)^11 + 11*(3*a
^3 + 7*a^2*b)*cosh(d*x + c)*sinh(d*x + c)^10 + (3*a^3 + 7*a^2*b)*sinh(d*x + c)^11 + 4*(3*a^3 + 13*a^2*b + 14*a
*b^2)*cosh(d*x + c)^9 + (12*a^3 + 52*a^2*b + 56*a*b^2 + 55*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^9
+ 3*(55*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^3 + 12*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^8 +
2*(9*a^3 + 45*a^2*b + 80*a*b^2 + 56*b^3)*cosh(d*x + c)^7 + 2*(165*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^4 + 9*a^3 +
45*a^2*b + 80*a*b^2 + 56*b^3 + 72*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 14*(33*(3*a
^3 + 7*a^2*b)*cosh(d*x + c)^5 + 24*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^3 + (9*a^3 + 45*a^2*b + 80*a*b^
2 + 56*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 + 4*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^5 + 2*(231*(3*a^3 +
7*a^2*b)*cosh(d*x + c)^6 + 252*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^4 + 6*a^3 + 26*a^2*b + 28*a*b^2 +
21*(9*a^3 + 45*a^2*b + 80*a*b^2 + 56*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 2*(165*(3*a^3 + 7*a^2*b)*cosh(d*x
+ c)^7 + 252*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^5 + 35*(9*a^3 + 45*a^2*b + 80*a*b^2 + 56*b^3)*cosh(d
*x + c)^3 + 10*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 + (3*a^3 + 7*a^2*b)*cosh(d*x + c)^
3 + (165*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^8 + 336*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^6 + 70*(9*a^3 + 4
5*a^2*b + 80*a*b^2 + 56*b^3)*cosh(d*x + c)^4 + 3*a^3 + 7*a^2*b + 40*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c
)^2)*sinh(d*x + c)^3 + (55*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^9 + 144*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c)
^7 + 42*(9*a^3 + 45*a^2*b + 80*a*b^2 + 56*b^3)*cosh(d*x + c)^5 + 40*(3*a^3 + 13*a^2*b + 14*a*b^2)*cosh(d*x + c
)^3 + 3*(3*a^3 + 7*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^2 + (11*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^10 + 36*(3*a^3
+ 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^8 + 14*(9*a^3 + 45*a^2*b + 80*a*b^2 + 56*b^3)*cosh(d*x + c)^6 + 20*(3*a^3
+ 13*a^2*b + 14*a*b^2)*cosh(d*x + c)^4 + 3*(3*a^3 + 7*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c))*sqrt(-b/a)*log((
a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 + 2*(a - 2*b)*cosh(d*x + c)^2 + 2*(3
*a*cosh(d*x + c)^2 + a - 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 + (a - 2*b)*cosh(d*x + c))*sinh(d*x + c)
+ 4*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh...

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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*sech(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*sech(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 )}^6\,{\mathrm {sinh}\left (c+d\,x\right )}^3}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\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/cosh(c + d*x)^2)^3,x)

[Out]

int((cosh(c + d*x)^6*sinh(c + d*x)^3)/(b + a*cosh(c + d*x)^2)^3, x)

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