3.1.34 \(\int \frac {\sinh ^3(c+d x)}{(a+b \text {sech}^2(c+d x))^2} \, dx\) [34]

Optimal. Leaf size=114 \[ \frac {\sqrt {b} (3 a+5 b) \text {ArcTan}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{2 a^{7/2} d}-\frac {(a+2 b) \cosh (c+d x)}{a^3 d}+\frac {\cosh ^3(c+d x)}{3 a^2 d}-\frac {b (a+b) \cosh (c+d x)}{2 a^3 d \left (b+a \cosh ^2(c+d x)\right )} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4218, 466, 1167, 211} \begin {gather*} \frac {\sqrt {b} (3 a+5 b) \text {ArcTan}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{2 a^{7/2} d}-\frac {b (a+b) \cosh (c+d x)}{2 a^3 d \left (a \cosh ^2(c+d x)+b\right )}-\frac {(a+2 b) \cosh (c+d x)}{a^3 d}+\frac {\cosh ^3(c+d x)}{3 a^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b]*(3*a + 5*b)*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(2*a^(7/2)*d) - ((a + 2*b)*Cosh[c + d*x])/(a^3*d
) + Cosh[c + d*x]^3/(3*a^2*d) - (b*(a + b)*Cosh[c + d*x])/(2*a^3*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 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 )^2} \, dx &=-\frac {\text {Subst}\left (\int \frac {x^4 \left (1-x^2\right )}{\left (b+a x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {b (a+b) \cosh (c+d x)}{2 a^3 d \left (b+a \cosh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {b (a+b)-2 a (a+b) x^2+2 a^2 x^4}{b+a x^2} \, dx,x,\cosh (c+d x)\right )}{2 a^3 d}\\ &=-\frac {b (a+b) \cosh (c+d x)}{2 a^3 d \left (b+a \cosh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \left (-2 (a+2 b)+2 a x^2+\frac {3 a b+5 b^2}{b+a x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{2 a^3 d}\\ &=-\frac {(a+2 b) \cosh (c+d x)}{a^3 d}+\frac {\cosh ^3(c+d x)}{3 a^2 d}-\frac {b (a+b) \cosh (c+d x)}{2 a^3 d \left (b+a \cosh ^2(c+d x)\right )}+\frac {(b (3 a+5 b)) \text {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\cosh (c+d x)\right )}{2 a^3 d}\\ &=\frac {\sqrt {b} (3 a+5 b) \tan ^{-1}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{2 a^{7/2} d}-\frac {(a+2 b) \cosh (c+d x)}{a^3 d}+\frac {\cosh ^3(c+d x)}{3 a^2 d}-\frac {b (a+b) \cosh (c+d x)}{2 a^3 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 = 3.36, size = 861, normalized size = 7.55 \begin {gather*} \frac {(a+2 b+a \cosh (2 (c+d x)))^2 \text {sech}^4(c+d x) \left (\frac {9 a^3 \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^{3/2}}+576 a \sqrt {b} \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 )+960 b^{3/2} \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 {9 a^3 \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^{3/2}}+576 a \sqrt {b} \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 )+960 b^{3/2} \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 {9 a^3 \text {ArcTan}\left (\frac {\sqrt {a}-i \sqrt {a+b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )}{b^{3/2}}-\frac {9 a^3 \text {ArcTan}\left (\frac {\sqrt {a}+i \sqrt {a+b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )}{b^{3/2}}-96 \sqrt {a} (3 a+8 b) \cosh (c) \cosh (d x)+32 a^{3/2} \cosh (3 c) \cosh (3 d x)-\frac {384 a^{3/2} b \cosh (c+d x)}{a+2 b+a \cosh (2 (c+d x))}-\frac {384 \sqrt {a} b^2 \cosh (c+d x)}{a+2 b+a \cosh (2 (c+d x))}-288 a^{3/2} \sinh (c) \sinh (d x)-768 \sqrt {a} b \sinh (c) \sinh (d x)+32 a^{3/2} \sinh (3 c) \sinh (3 d x)\right )}{1536 a^{7/2} d \left (a+b \text {sech}^2(c+d x)\right )^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x]^4*((9*a^3*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Si
nh[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^(3/2) + 576*a*Sqrt[b]*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]] + 960*b^(3/2
)*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*S
qrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] + (9*a^3*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]])/b^(3/2) + 576*a*Sqrt[b]*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]
] + 960*b^(3/2)*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]] - (9*a^3*ArcTan[(Sqrt[a] - I*Sqr
t[a + b]*Tanh[(c + d*x)/2])/Sqrt[b]])/b^(3/2) - (9*a^3*ArcTan[(Sqrt[a] + I*Sqrt[a + b]*Tanh[(c + d*x)/2])/Sqrt
[b]])/b^(3/2) - 96*Sqrt[a]*(3*a + 8*b)*Cosh[c]*Cosh[d*x] + 32*a^(3/2)*Cosh[3*c]*Cosh[3*d*x] - (384*a^(3/2)*b*C
osh[c + d*x])/(a + 2*b + a*Cosh[2*(c + d*x)]) - (384*Sqrt[a]*b^2*Cosh[c + d*x])/(a + 2*b + a*Cosh[2*(c + d*x)]
) - 288*a^(3/2)*Sinh[c]*Sinh[d*x] - 768*Sqrt[a]*b*Sinh[c]*Sinh[d*x] + 32*a^(3/2)*Sinh[3*c]*Sinh[3*d*x]))/(1536
*a^(7/2)*d*(a + b*Sech[c + d*x]^2)^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(263\) vs. \(2(100)=200\).
time = 2.25, size = 264, normalized size = 2.32

method	result	size
derivativedivides	\(\frac {-\frac {1}{3 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a -4 b}{2 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {4 b \left (\frac {\left (-\frac {a}{4}+\frac {b}{4}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {a}{4}-\frac {b}{4}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (3 a +5 b \right ) \arctan \left (\frac {2 \left (a +b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a -2 b}{4 \sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{3}}+\frac {1}{3 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{2 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {a +4 b}{2 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\)	\(264\)
default	\(\frac {-\frac {1}{3 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a -4 b}{2 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {4 b \left (\frac {\left (-\frac {a}{4}+\frac {b}{4}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {a}{4}-\frac {b}{4}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (3 a +5 b \right ) \arctan \left (\frac {2 \left (a +b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a -2 b}{4 \sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{3}}+\frac {1}{3 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{2 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {a +4 b}{2 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\)	\(264\)
risch	\(\frac {{\mathrm e}^{3 d x +3 c}}{24 a^{2} d}-\frac {3 \,{\mathrm e}^{d x +c}}{8 a^{2} d}-\frac {{\mathrm e}^{d x +c} b}{a^{3} d}-\frac {3 \,{\mathrm e}^{-d x -c}}{8 a^{2} d}-\frac {{\mathrm e}^{-d x -c} b}{a^{3} d}+\frac {{\mathrm e}^{-3 d x -3 c}}{24 a^{2} d}-\frac {b \left (a +b \right ) {\mathrm e}^{d x +c} \left (1+{\mathrm e}^{2 d x +2 c}\right )}{a^{3} d \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )}+\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right )}{4 a^{3} d}+\frac {5 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right ) b}{4 a^{4} d}-\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right )}{4 a^{3} d}-\frac {5 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right ) b}{4 a^{4} d}\)	\(342\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-1/3/a^2/(tanh(1/2*d*x+1/2*c)-1)^3-1/2/a^2/(tanh(1/2*d*x+1/2*c)-1)^2-1/2/a^3*(-a-4*b)/(tanh(1/2*d*x+1/2*c
)-1)+4/a^3*b*(((-1/4*a+1/4*b)*tanh(1/2*d*x+1/2*c)^2-1/4*a-1/4*b)/(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)+1/8*(3*a+5*b)/(a*b)^(1/2)*arctan(1/4*(2*(a+b)*tan
h(1/2*d*x+1/2*c)^2+2*a-2*b)/(a*b)^(1/2)))+1/3/a^2/(tanh(1/2*d*x+1/2*c)+1)^3-1/2/a^2/(tanh(1/2*d*x+1/2*c)+1)^2-
1/2*(a+4*b)/a^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(sinh(d*x+c)^3/(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

1/24*(a^2*e^(10*d*x + 10*c) + a^2 - (7*a^2*e^(8*c) + 20*a*b*e^(8*c))*e^(8*d*x) - 2*(13*a^2*e^(6*c) + 66*a*b*e^
(6*c) + 60*b^2*e^(6*c))*e^(6*d*x) - 2*(13*a^2*e^(4*c) + 66*a*b*e^(4*c) + 60*b^2*e^(4*c))*e^(4*d*x) - (7*a^2*e^
(2*c) + 20*a*b*e^(2*c))*e^(2*d*x))/(a^4*d*e^(7*d*x + 7*c) + a^4*d*e^(3*d*x + 3*c) + 2*(a^4*d*e^(5*c) + 2*a^3*b
*d*e^(5*c))*e^(5*d*x)) + 1/8*integrate(8*((3*a*b*e^(3*c) + 5*b^2*e^(3*c))*e^(3*d*x) - (3*a*b*e^c + 5*b^2*e^c)*
e^(d*x))/(a^4*e^(4*d*x + 4*c) + a^4 + 2*(a^4*e^(2*c) + 2*a^3*b*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. 2021 vs. \(2 (100) = 200\).
time = 0.39, size = 3804, normalized size = 33.37 \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)^2,x, algorithm="fricas")

[Out]

[1/24*(a^2*cosh(d*x + c)^10 + 10*a^2*cosh(d*x + c)*sinh(d*x + c)^9 + a^2*sinh(d*x + c)^10 - (7*a^2 + 20*a*b)*c
osh(d*x + c)^8 + (45*a^2*cosh(d*x + c)^2 - 7*a^2 - 20*a*b)*sinh(d*x + c)^8 + 8*(15*a^2*cosh(d*x + c)^3 - (7*a^
2 + 20*a*b)*cosh(d*x + c))*sinh(d*x + c)^7 - 2*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^6 + 2*(105*a^2*cosh(d*
x + c)^4 - 14*(7*a^2 + 20*a*b)*cosh(d*x + c)^2 - 13*a^2 - 66*a*b - 60*b^2)*sinh(d*x + c)^6 + 4*(63*a^2*cosh(d*
x + c)^5 - 14*(7*a^2 + 20*a*b)*cosh(d*x + c)^3 - 3*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 -
2*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^4 + 2*(105*a^2*cosh(d*x + c)^6 - 35*(7*a^2 + 20*a*b)*cosh(d*x + c)
^4 - 15*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^2 - 13*a^2 - 66*a*b - 60*b^2)*sinh(d*x + c)^4 + 8*(15*a^2*cos
h(d*x + c)^7 - 7*(7*a^2 + 20*a*b)*cosh(d*x + c)^5 - 5*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^3 - (13*a^2 + 6
6*a*b + 60*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - (7*a^2 + 20*a*b)*cosh(d*x + c)^2 + (45*a^2*cosh(d*x + c)^8 -
28*(7*a^2 + 20*a*b)*cosh(d*x + c)^6 - 30*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^4 - 12*(13*a^2 + 66*a*b + 60
*b^2)*cosh(d*x + c)^2 - 7*a^2 - 20*a*b)*sinh(d*x + c)^2 + 6*((3*a^2 + 5*a*b)*cosh(d*x + c)^7 + 7*(3*a^2 + 5*a*
b)*cosh(d*x + c)*sinh(d*x + c)^6 + (3*a^2 + 5*a*b)*sinh(d*x + c)^7 + 2*(3*a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)
^5 + (21*(3*a^2 + 5*a*b)*cosh(d*x + c)^2 + 6*a^2 + 22*a*b + 20*b^2)*sinh(d*x + c)^5 + 5*(7*(3*a^2 + 5*a*b)*cos
h(d*x + c)^3 + 2*(3*a^2 + 11*a*b + 10*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 + (3*a^2 + 5*a*b)*cosh(d*x + c)^3 +
(35*(3*a^2 + 5*a*b)*cosh(d*x + c)^4 + 20*(3*a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)^2 + 3*a^2 + 5*a*b)*sinh(d*x +
c)^3 + (21*(3*a^2 + 5*a*b)*cosh(d*x + c)^5 + 20*(3*a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)^3 + 3*(3*a^2 + 5*a*b)
*cosh(d*x + c))*sinh(d*x + c)^2 + (7*(3*a^2 + 5*a*b)*cosh(d*x + c)^6 + 10*(3*a^2 + 11*a*b + 10*b^2)*cosh(d*x +
c)^4 + 3*(3*a^2 + 5*a*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(d*x + c)^2 + a*sinh(d*x + c)^3 + a*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 + a)*sinh(d*x + c))*sqrt(-
b/a) + a)/(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))*sin
h(d*x + c) + a)) + a^2 + 2*(5*a^2*cosh(d*x + c)^9 - 4*(7*a^2 + 20*a*b)*cosh(d*x + c)^7 - 6*(13*a^2 + 66*a*b +
60*b^2)*cosh(d*x + c)^5 - 4*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^3 - (7*a^2 + 20*a*b)*cosh(d*x + c))*sinh(
d*x + c))/(a^4*d*cosh(d*x + c)^7 + 7*a^4*d*cosh(d*x + c)*sinh(d*x + c)^6 + a^4*d*sinh(d*x + c)^7 + a^4*d*cosh(
d*x + c)^3 + 2*(a^4 + 2*a^3*b)*d*cosh(d*x + c)^5 + (21*a^4*d*cosh(d*x + c)^2 + 2*(a^4 + 2*a^3*b)*d)*sinh(d*x +
c)^5 + 5*(7*a^4*d*cosh(d*x + c)^3 + 2*(a^4 + 2*a^3*b)*d*cosh(d*x + c))*sinh(d*x + c)^4 + (35*a^4*d*cosh(d*x +
c)^4 + a^4*d + 20*(a^4 + 2*a^3*b)*d*cosh(d*x + c)^2)*sinh(d*x + c)^3 + (21*a^4*d*cosh(d*x + c)^5 + 3*a^4*d*co
sh(d*x + c) + 20*(a^4 + 2*a^3*b)*d*cosh(d*x + c)^3)*sinh(d*x + c)^2 + (7*a^4*d*cosh(d*x + c)^6 + 3*a^4*d*cosh(
d*x + c)^2 + 10*(a^4 + 2*a^3*b)*d*cosh(d*x + c)^4)*sinh(d*x + c)), 1/24*(a^2*cosh(d*x + c)^10 + 10*a^2*cosh(d*
x + c)*sinh(d*x + c)^9 + a^2*sinh(d*x + c)^10 - (7*a^2 + 20*a*b)*cosh(d*x + c)^8 + (45*a^2*cosh(d*x + c)^2 - 7
*a^2 - 20*a*b)*sinh(d*x + c)^8 + 8*(15*a^2*cosh(d*x + c)^3 - (7*a^2 + 20*a*b)*cosh(d*x + c))*sinh(d*x + c)^7 -
2*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^6 + 2*(105*a^2*cosh(d*x + c)^4 - 14*(7*a^2 + 20*a*b)*cosh(d*x + c)
^2 - 13*a^2 - 66*a*b - 60*b^2)*sinh(d*x + c)^6 + 4*(63*a^2*cosh(d*x + c)^5 - 14*(7*a^2 + 20*a*b)*cosh(d*x + c)
^3 - 3*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^
4 + 2*(105*a^2*cosh(d*x + c)^6 - 35*(7*a^2 + 20*a*b)*cosh(d*x + c)^4 - 15*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x
+ c)^2 - 13*a^2 - 66*a*b - 60*b^2)*sinh(d*x + c)^4 + 8*(15*a^2*cosh(d*x + c)^7 - 7*(7*a^2 + 20*a*b)*cosh(d*x +
c)^5 - 5*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^3 - (13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c))*sinh(d*x + c)
^3 - (7*a^2 + 20*a*b)*cosh(d*x + c)^2 + (45*a^2*cosh(d*x + c)^8 - 28*(7*a^2 + 20*a*b)*cosh(d*x + c)^6 - 30*(13
*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^4 - 12*(13*a^2 + 66*a*b + 60*b^2)*cosh(d*x + c)^2 - 7*a^2 - 20*a*b)*sinh
(d*x + c)^2 - 12*((3*a^2 + 5*a*b)*cosh(d*x + c)^7 + 7*(3*a^2 + 5*a*b)*cosh(d*x + c)*sinh(d*x + c)^6 + (3*a^2 +
5*a*b)*sinh(d*x + c)^7 + 2*(3*a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)^5 + (21*(3*a^2 + 5*a*b)*cosh(d*x + c)^2 +
6*a^2 + 22*a*b + 20*b^2)*sinh(d*x + c)^5 + 5*(7*(3*a^2 + 5*a*b)*cosh(d*x + c)^3 + 2*(3*a^2 + 11*a*b + 10*b^2)*
cosh(d*x + c))*sinh(d*x + c)^4 + (3*a^2 + 5*a*b)*cosh(d*x + c)^3 + (35*(3*a^2 + 5*a*b)*cosh(d*x + c)^4 + 20*(3
*a^2 + 11*a*b + 10*b^2)*cosh(d*x + c)^2 + 3*a^2...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh ^{3}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{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)**3/(a+b*sech(d*x+c)**2)**2,x)

[Out]

Integral(sinh(c + d*x)**3/(a + b*sech(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)^3/(a+b*sech(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 )}^4\,{\mathrm {sinh}\left (c+d\,x\right )}^3}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^2} \,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)^2,x)

[Out]

int((cosh(c + d*x)^4*sinh(c + d*x)^3)/(b + a*cosh(c + d*x)^2)^2, x)

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