3.4.37 \(\int \frac {\text {sech}^3(c+d x)}{(a+b \sinh ^2(c+d x))^2} \, dx\) [337]

Optimal. Leaf size=157 \[ \frac {(a-5 b) \text {ArcTan}(\sinh (c+d x))}{2 (a-b)^3 d}+\frac {(5 a-b) b^{3/2} \text {ArcTan}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} (a-b)^3 d}+\frac {b (a+b) \sinh (c+d x)}{2 a (a-b)^2 d \left (a+b \sinh ^2(c+d x)\right )}+\frac {\text {sech}(c+d x) \tanh (c+d x)}{2 (a-b) d \left (a+b \sinh ^2(c+d x)\right )} \]

[Out]

1/2*(a-5*b)*arctan(sinh(d*x+c))/(a-b)^3/d+1/2*(5*a-b)*b^(3/2)*arctan(sinh(d*x+c)*b^(1/2)/a^(1/2))/a^(3/2)/(a-b
)^3/d+1/2*b*(a+b)*sinh(d*x+c)/a/(a-b)^2/d/(a+b*sinh(d*x+c)^2)+1/2*sech(d*x+c)*tanh(d*x+c)/(a-b)/d/(a+b*sinh(d*
x+c)^2)

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Rubi [A]
time = 0.13, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3269, 425, 541, 536, 209, 211} \begin {gather*} \frac {b^{3/2} (5 a-b) \text {ArcTan}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} d (a-b)^3}+\frac {(a-5 b) \text {ArcTan}(\sinh (c+d x))}{2 d (a-b)^3}+\frac {b (a+b) \sinh (c+d x)}{2 a d (a-b)^2 \left (a+b \sinh ^2(c+d x)\right )}+\frac {\tanh (c+d x) \text {sech}(c+d x)}{2 d (a-b) \left (a+b \sinh ^2(c+d x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a - 5*b)*ArcTan[Sinh[c + d*x]])/(2*(a - b)^3*d) + ((5*a - b)*b^(3/2)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]]
)/(2*a^(3/2)*(a - b)^3*d) + (b*(a + b)*Sinh[c + d*x])/(2*a*(a - b)^2*d*(a + b*Sinh[c + d*x]^2)) + (Sech[c + d*
x]*Tanh[c + d*x])/(2*(a - b)*d*(a + b*Sinh[c + d*x]^2))

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 3269

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

Rubi steps

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

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Mathematica [A]
time = 0.79, size = 230, normalized size = 1.46 \begin {gather*} \frac {2 \sqrt {a} (a-b) b^2 \sinh (c+d x)+(2 a-b) \left (b^{3/2} (-5 a+b) \text {ArcTan}\left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {b}}\right )+2 a^{3/2} (a-5 b) \text {ArcTan}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+a^{3/2} (a-b) \text {sech}(c+d x) \tanh (c+d x)\right )+b \cosh (2 (c+d x)) \left (b^{3/2} (-5 a+b) \text {ArcTan}\left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {b}}\right )+2 a^{3/2} (a-5 b) \text {ArcTan}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+a^{3/2} (a-b) \text {sech}(c+d x) \tanh (c+d x)\right )}{2 a^{3/2} (a-b)^3 d (2 a-b+b \cosh (2 (c+d x)))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(367\) vs. \(2(141)=282\).
time = 2.06, size = 368, normalized size = 2.34

method	result	size
derivativedivides	\(\frac {\frac {2 b^{2} \left (\frac {-\frac {\left (a -b \right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {\left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\left (5 a -b \right ) \left (\frac {\left (-a +\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (a +\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2}\right )}{\left (a -b \right )^{3}}+\frac {\frac {2 \left (\left (\frac {b}{2}-\frac {a}{2}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {b}{2}+\frac {a}{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\left (a -5 b \right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a -b \right )^{3}}}{d}\)	\(368\)
default	\(\frac {\frac {2 b^{2} \left (\frac {-\frac {\left (a -b \right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {\left (a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\left (5 a -b \right ) \left (\frac {\left (-a +\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (a +\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2}\right )}{\left (a -b \right )^{3}}+\frac {\frac {2 \left (\left (\frac {b}{2}-\frac {a}{2}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {b}{2}+\frac {a}{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\left (a -5 b \right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a -b \right )^{3}}}{d}\)	\(368\)
risch	\(\frac {{\mathrm e}^{d x +c} \left (a b \,{\mathrm e}^{6 d x +6 c}+b^{2} {\mathrm e}^{6 d x +6 c}+4 a^{2} {\mathrm e}^{4 d x +4 c}-3 a b \,{\mathrm e}^{4 d x +4 c}+b^{2} {\mathrm e}^{4 d x +4 c}-4 a^{2} {\mathrm e}^{2 d x +2 c}+3 a b \,{\mathrm e}^{2 d x +2 c}-b^{2} {\mathrm e}^{2 d x +2 c}-a b -b^{2}\right )}{d \left (a -b \right )^{2} \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2} a \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{2 \left (a -b \right )^{3} d}-\frac {5 i \ln \left ({\mathrm e}^{d x +c}+i\right ) b}{2 \left (a -b \right )^{3} d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{2 \left (a -b \right )^{3} d}+\frac {5 i \ln \left ({\mathrm e}^{d x +c}-i\right ) b}{2 \left (a -b \right )^{3} d}+\frac {5 \sqrt {-a b}\, b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{b}-1\right )}{4 a \left (a -b \right )^{3} d}-\frac {\sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{b}-1\right )}{4 a^{2} \left (a -b \right )^{3} d}-\frac {5 \sqrt {-a b}\, b \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{b}-1\right )}{4 a \left (a -b \right )^{3} d}+\frac {\sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{b}-1\right )}{4 a^{2} \left (a -b \right )^{3} d}\)	\(494\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(2*b^2/(a-b)^3*((-1/2*(a-b)/a*tanh(1/2*d*x+1/2*c)^3+1/2*(a-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)+1/2*(5*a-b)*(1/2*(-a+(-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
))-1/2*(a+(-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))))+2/(a-b)^3*(((1/2*b-1/2*a)*tanh(1/2*d*x+1/2*c)^3+(-1/2*b+1/2*a)*t
anh(1/2*d*x+1/2*c))/(tanh(1/2*d*x+1/2*c)^2+1)^2+1/2*(a-5*b)*arctan(tanh(1/2*d*x+1/2*c))))

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

[Out]

(a*e^c - 5*b*e^c)*arctan(e^(d*x + c))*e^(-c)/(a^3*d - 3*a^2*b*d + 3*a*b^2*d - b^3*d) + ((a*b*e^(7*c) + b^2*e^(
7*c))*e^(7*d*x) + (4*a^2*e^(5*c) - 3*a*b*e^(5*c) + b^2*e^(5*c))*e^(5*d*x) - (4*a^2*e^(3*c) - 3*a*b*e^(3*c) + b
^2*e^(3*c))*e^(3*d*x) - (a*b*e^c + b^2*e^c)*e^(d*x))/(a^3*b*d - 2*a^2*b^2*d + a*b^3*d + (a^3*b*d*e^(8*c) - 2*a
^2*b^2*d*e^(8*c) + a*b^3*d*e^(8*c))*e^(8*d*x) + 4*(a^4*d*e^(6*c) - 2*a^3*b*d*e^(6*c) + a^2*b^2*d*e^(6*c))*e^(6
*d*x) + 2*(4*a^4*d*e^(4*c) - 9*a^3*b*d*e^(4*c) + 6*a^2*b^2*d*e^(4*c) - a*b^3*d*e^(4*c))*e^(4*d*x) + 4*(a^4*d*e
^(2*c) - 2*a^3*b*d*e^(2*c) + a^2*b^2*d*e^(2*c))*e^(2*d*x)) + 8*integrate(1/8*((5*a*b^2*e^(3*c) - b^3*e^(3*c))*
e^(3*d*x) + (5*a*b^2*e^c - b^3*e^c)*e^(d*x))/(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4 + (a^4*b*e^(4*c) - 3*a^3*b
^2*e^(4*c) + 3*a^2*b^3*e^(4*c) - a*b^4*e^(4*c))*e^(4*d*x) + 2*(2*a^5*e^(2*c) - 7*a^4*b*e^(2*c) + 9*a^3*b^2*e^(
2*c) - 5*a^2*b^3*e^(2*c) + a*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. 3474 vs. \(2 (141) = 282\).
time = 0.70, size = 6548, normalized size = 41.71 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*(a^2*b - b^3)*cosh(d*x + c)^7 + 28*(a^2*b - b^3)*cosh(d*x + c)*sinh(d*x + c)^6 + 4*(a^2*b - b^3)*sinh(
d*x + c)^7 + 4*(4*a^3 - 7*a^2*b + 4*a*b^2 - b^3)*cosh(d*x + c)^5 + 4*(4*a^3 - 7*a^2*b + 4*a*b^2 - b^3 + 21*(a^
2*b - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 20*(7*(a^2*b - b^3)*cosh(d*x + c)^3 + (4*a^3 - 7*a^2*b + 4*a*b^2
- b^3)*cosh(d*x + c))*sinh(d*x + c)^4 - 4*(4*a^3 - 7*a^2*b + 4*a*b^2 - b^3)*cosh(d*x + c)^3 + 4*(35*(a^2*b -
b^3)*cosh(d*x + c)^4 - 4*a^3 + 7*a^2*b - 4*a*b^2 + b^3 + 10*(4*a^3 - 7*a^2*b + 4*a*b^2 - b^3)*cosh(d*x + c)^2)
*sinh(d*x + c)^3 + 4*(21*(a^2*b - b^3)*cosh(d*x + c)^5 + 10*(4*a^3 - 7*a^2*b + 4*a*b^2 - b^3)*cosh(d*x + c)^3
- 3*(4*a^3 - 7*a^2*b + 4*a*b^2 - b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + ((5*a*b^2 - b^3)*cosh(d*x + c)^8 + 8*(5
*a*b^2 - b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (5*a*b^2 - b^3)*sinh(d*x + c)^8 + 4*(5*a^2*b - a*b^2)*cosh(d*x +
c)^6 + 4*(5*a^2*b - a*b^2 + 7*(5*a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(5*a*b^2 - b^3)*cosh(d*
x + c)^3 + 3*(5*a^2*b - a*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(20*a^2*b - 9*a*b^2 + b^3)*cosh(d*x + c)^4 +
2*(35*(5*a*b^2 - b^3)*cosh(d*x + c)^4 + 20*a^2*b - 9*a*b^2 + b^3 + 30*(5*a^2*b - a*b^2)*cosh(d*x + c)^2)*sinh
(d*x + c)^4 + 8*(7*(5*a*b^2 - b^3)*cosh(d*x + c)^5 + 10*(5*a^2*b - a*b^2)*cosh(d*x + c)^3 + (20*a^2*b - 9*a*b^
2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 5*a*b^2 - b^3 + 4*(5*a^2*b - a*b^2)*cosh(d*x + c)^2 + 4*(7*(5*a*b^2
- b^3)*cosh(d*x + c)^6 + 15*(5*a^2*b - a*b^2)*cosh(d*x + c)^4 + 5*a^2*b - a*b^2 + 3*(20*a^2*b - 9*a*b^2 + b^3)
*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((5*a*b^2 - b^3)*cosh(d*x + c)^7 + 3*(5*a^2*b - a*b^2)*cosh(d*x + c)^5 +
(20*a^2*b - 9*a*b^2 + b^3)*cosh(d*x + c)^3 + (5*a^2*b - a*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-b/a)*log((
b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 - 2*(2*a + b)*cosh(d*x + c)^2 + 2*(3
*b*cosh(d*x + c)^2 - 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 - (2*a + 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) + b)/(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(
d*x + c)^4 + 2*(2*a - b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 + 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x +
c)^3 + (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + b)) + 4*((a^2*b - 5*a*b^2)*cosh(d*x + c)^8 + 8*(a^2*b - 5*a*b
^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2*b - 5*a*b^2)*sinh(d*x + c)^8 + 4*(a^3 - 5*a^2*b)*cosh(d*x + c)^6 + 4*
(a^3 - 5*a^2*b + 7*(a^2*b - 5*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a^2*b - 5*a*b^2)*cosh(d*x + c)^3
+ 3*(a^3 - 5*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(4*a^3 - 21*a^2*b + 5*a*b^2)*cosh(d*x + c)^4 + 2*(35*(
a^2*b - 5*a*b^2)*cosh(d*x + c)^4 + 4*a^3 - 21*a^2*b + 5*a*b^2 + 30*(a^3 - 5*a^2*b)*cosh(d*x + c)^2)*sinh(d*x +
c)^4 + 8*(7*(a^2*b - 5*a*b^2)*cosh(d*x + c)^5 + 10*(a^3 - 5*a^2*b)*cosh(d*x + c)^3 + (4*a^3 - 21*a^2*b + 5*a*
b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + a^2*b - 5*a*b^2 + 4*(a^3 - 5*a^2*b)*cosh(d*x + c)^2 + 4*(7*(a^2*b - 5*a*
b^2)*cosh(d*x + c)^6 + 15*(a^3 - 5*a^2*b)*cosh(d*x + c)^4 + a^3 - 5*a^2*b + 3*(4*a^3 - 21*a^2*b + 5*a*b^2)*cos
h(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^2*b - 5*a*b^2)*cosh(d*x + c)^7 + 3*(a^3 - 5*a^2*b)*cosh(d*x + c)^5 + (4*
a^3 - 21*a^2*b + 5*a*b^2)*cosh(d*x + c)^3 + (a^3 - 5*a^2*b)*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c)
+ sinh(d*x + c)) - 4*(a^2*b - b^3)*cosh(d*x + c) + 4*(7*(a^2*b - b^3)*cosh(d*x + c)^6 + 5*(4*a^3 - 7*a^2*b +
4*a*b^2 - b^3)*cosh(d*x + c)^4 - a^2*b + b^3 - 3*(4*a^3 - 7*a^2*b + 4*a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x +
c))/((a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cosh(d*x + c)^8 + 8*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*
cosh(d*x + c)*sinh(d*x + c)^7 + (a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*sinh(d*x + c)^8 + 4*(a^5 - 3*a^4*b +
3*a^3*b^2 - a^2*b^3)*d*cosh(d*x + c)^6 + 4*(7*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cosh(d*x + c)^2 + (a^
5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d)*sinh(d*x + c)^6 + 2*(4*a^5 - 13*a^4*b + 15*a^3*b^2 - 7*a^2*b^3 + a*b^4)*
d*cosh(d*x + c)^4 + 8*(7*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cosh(d*x + c)^3 + 3*(a^5 - 3*a^4*b + 3*a^3*
b^2 - a^2*b^3)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*d*cosh(d*x + c
)^4 + 30*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d*cosh(d*x + c)^2 + (4*a^5 - 13*a^4*b + 15*a^3*b^2 - 7*a^2*b^3
+ a*b^4)*d)*sinh(d*x + c)^4 + 4*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d*cosh(d*x + c)^2 + 8*(7*(a^4*b - 3*a^3*
b^2 + 3*a^2*b^3 - a*b^4)*d*cosh(d*x + c)^5 + 10*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d*cosh(d*x + c)^3 + (4*a
^5 - 13*a^4*b + 15*a^3*b^2 - 7*a^2*b^3 + a*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^4*b - 3*a^3*b^2 + 3
*a^2*b^3 - a*b^4)*d*cosh(d*x + c)^6 + 15*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d*cosh(d*x + c)^4 + 3*(4*a^5 -
13*a^4*b + 15*a^3*b^2 - 7*a^2*b^3 + a*b^4)*d*cosh(d*x + c)^2 + (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d)*sinh(d
*x + c)^2 + (a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)**3/(a+b*sinh(d*x+c)**2)**2,x)

[Out]

Integral(sech(c + d*x)**3/(a + b*sinh(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(sech(d*x+c)^3/(a+b*sinh(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 {1}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\left (b\,{\mathrm {sinh}\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(1/(cosh(c + d*x)^3*(a + b*sinh(c + d*x)^2)^2),x)

[Out]

int(1/(cosh(c + d*x)^3*(a + b*sinh(c + d*x)^2)^2), x)

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