3.4.45 \(\int \frac {\text {sech}(c+d x)}{(a+b \sinh ^2(c+d x))^3} \, dx\) [345]

Optimal. Leaf size=159 \[ \frac {\text {ArcTan}(\sinh (c+d x))}{(a-b)^3 d}-\frac {\sqrt {b} \left (15 a^2-10 a b+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} (a-b)^3 d}-\frac {b \sinh (c+d x)}{4 a (a-b) d \left (a+b \sinh ^2(c+d x)\right )^2}-\frac {(7 a-3 b) b \sinh (c+d x)}{8 a^2 (a-b)^2 d \left (a+b \sinh ^2(c+d x)\right )} \]

[Out]

arctan(sinh(d*x+c))/(a-b)^3/d-1/4*b*sinh(d*x+c)/a/(a-b)/d/(a+b*sinh(d*x+c)^2)^2-1/8*(7*a-3*b)*b*sinh(d*x+c)/a^
2/(a-b)^2/d/(a+b*sinh(d*x+c)^2)-1/8*(15*a^2-10*a*b+3*b^2)*arctan(sinh(d*x+c)*b^(1/2)/a^(1/2))*b^(1/2)/a^(5/2)/
(a-b)^3/d

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Rubi [A]
time = 0.12, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3269, 425, 541, 536, 209, 211} \begin {gather*} -\frac {b (7 a-3 b) \sinh (c+d x)}{8 a^2 d (a-b)^2 \left (a+b \sinh ^2(c+d x)\right )}-\frac {\sqrt {b} \left (15 a^2-10 a b+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} d (a-b)^3}+\frac {\text {ArcTan}(\sinh (c+d x))}{d (a-b)^3}-\frac {b \sinh (c+d x)}{4 a d (a-b) \left (a+b \sinh ^2(c+d x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[Sinh[c + d*x]]/((a - b)^3*d) - (Sqrt[b]*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a
]])/(8*a^(5/2)*(a - b)^3*d) - (b*Sinh[c + d*x])/(4*a*(a - b)*d*(a + b*Sinh[c + d*x]^2)^2) - ((7*a - 3*b)*b*Sin
h[c + d*x])/(8*a^2*(a - b)^2*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}(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=-\frac {b \sinh (c+d x)}{4 a (a-b) d \left (a+b \sinh ^2(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {4 a-3 b-3 b x^2}{\left (1+x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{4 a (a-b) d}\\ &=-\frac {b \sinh (c+d x)}{4 a (a-b) d \left (a+b \sinh ^2(c+d x)\right )^2}-\frac {(7 a-3 b) b \sinh (c+d x)}{8 a^2 (a-b)^2 d \left (a+b \sinh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {8 a^2-7 a b+3 b^2-(7 a-3 b) b x^2}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\sinh (c+d x)\right )}{8 a^2 (a-b)^2 d}\\ &=-\frac {b \sinh (c+d x)}{4 a (a-b) d \left (a+b \sinh ^2(c+d x)\right )^2}-\frac {(7 a-3 b) b \sinh (c+d x)}{8 a^2 (a-b)^2 d \left (a+b \sinh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{(a-b)^3 d}-\frac {\left (b \left (15 a^2-10 a b+3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sinh (c+d x)\right )}{8 a^2 (a-b)^3 d}\\ &=\frac {\tan ^{-1}(\sinh (c+d x))}{(a-b)^3 d}-\frac {\sqrt {b} \left (15 a^2-10 a b+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} (a-b)^3 d}-\frac {b \sinh (c+d x)}{4 a (a-b) d \left (a+b \sinh ^2(c+d x)\right )^2}-\frac {(7 a-3 b) b \sinh (c+d x)}{8 a^2 (a-b)^2 d \left (a+b \sinh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(321\) vs. \(2(159)=318\).
time = 0.55, size = 321, normalized size = 2.02 \begin {gather*} \frac {(-2 a+b)^2 \left (\sqrt {b} \left (15 a^2-10 a b+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {b}}\right )+16 a^{5/2} \text {ArcTan}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )\right )+\left (b^{5/2} \left (15 a^2-10 a b+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {b}}\right )+16 a^{5/2} b^2 \text {ArcTan}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )\right ) \cosh ^2(2 (c+d x))-2 \sqrt {a} b \left (18 a^3-35 a^2 b+20 a b^2-3 b^3\right ) \sinh (c+d x)-2 b \cosh (2 (c+d x)) \left (-\left ((2 a-b) \left (\sqrt {b} \left (15 a^2-10 a b+3 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {b}}\right )+16 a^{5/2} \text {ArcTan}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )+\sqrt {a} b \left (7 a^2-10 a b+3 b^2\right ) \sinh (c+d x)\right )}{8 a^{5/2} (a-b)^3 d (2 a-b+b \cosh (2 (c+d x)))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*a + b)^2*(Sqrt[b]*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]] + 16*a^(5/2)*ArcTan[T
anh[(c + d*x)/2]]) + (b^(5/2)*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]] + 16*a^(5/2)*b
^2*ArcTan[Tanh[(c + d*x)/2]])*Cosh[2*(c + d*x)]^2 - 2*Sqrt[a]*b*(18*a^3 - 35*a^2*b + 20*a*b^2 - 3*b^3)*Sinh[c
+ d*x] - 2*b*Cosh[2*(c + d*x)]*(-((2*a - b)*(Sqrt[b]*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/
Sqrt[b]] + 16*a^(5/2)*ArcTan[Tanh[(c + d*x)/2]])) + Sqrt[a]*b*(7*a^2 - 10*a*b + 3*b^2)*Sinh[c + d*x]))/(8*a^(5
/2)*(a - b)^3*d*(2*a - b + b*Cosh[2*(c + d*x)])^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(413\) vs. \(2(145)=290\).
time = 1.92, size = 414, normalized size = 2.60

method	result	size
derivativedivides	\(\frac {\frac {2 \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a -b \right )^{3}}-\frac {2 b \left (\frac {-\frac {\left (9 a^{2}-14 a b +5 b^{2}\right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {\left (27 a^{3}-70 a^{2} b +55 a \,b^{2}-12 b^{3}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}-\frac {\left (27 a^{3}-70 a^{2} b +55 a \,b^{2}-12 b^{3}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}+\frac {\left (9 a^{2}-14 a b +5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {\left (15 a^{2}-10 a b +3 b^{2}\right ) \left (\frac {\left (-a +\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (a +\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{8 a}\right )}{\left (a -b \right )^{3}}}{d}\)	\(414\)
default	\(\frac {\frac {2 \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a -b \right )^{3}}-\frac {2 b \left (\frac {-\frac {\left (9 a^{2}-14 a b +5 b^{2}\right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {\left (27 a^{3}-70 a^{2} b +55 a \,b^{2}-12 b^{3}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}-\frac {\left (27 a^{3}-70 a^{2} b +55 a \,b^{2}-12 b^{3}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}+\frac {\left (9 a^{2}-14 a b +5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {\left (15 a^{2}-10 a b +3 b^{2}\right ) \left (\frac {\left (-a +\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (a +\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{8 a}\right )}{\left (a -b \right )^{3}}}{d}\)	\(414\)
risch	\(-\frac {{\mathrm e}^{d x +c} b \left (7 a b \,{\mathrm e}^{6 d x +6 c}-3 b^{2} {\mathrm e}^{6 d x +6 c}+36 a^{2} {\mathrm e}^{4 d x +4 c}-41 a b \,{\mathrm e}^{4 d x +4 c}+9 b^{2} {\mathrm e}^{4 d x +4 c}-36 a^{2} {\mathrm e}^{2 d x +2 c}+41 a b \,{\mathrm e}^{2 d x +2 c}-9 b^{2} {\mathrm e}^{2 d x +2 c}-7 a b +3 b^{2}\right )}{4 \left (a -b \right )^{2} a^{2} d \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )^{2}}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right )}{\left (a -b \right )^{3} d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right )}{\left (a -b \right )^{3} d}+\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{b}-1\right )}{16 a \left (a -b \right )^{3} d}-\frac {5 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{b}-1\right ) b}{8 a^{2} \left (a -b \right )^{3} d}+\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{b}-1\right ) b^{2}}{16 a^{3} \left (a -b \right )^{3} d}-\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{b}-1\right )}{16 a \left (a -b \right )^{3} d}+\frac {5 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{b}-1\right ) b}{8 a^{2} \left (a -b \right )^{3} d}-\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{b}-1\right ) b^{2}}{16 a^{3} \left (a -b \right )^{3} d}\)	\(536\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(2/(a-b)^3*arctan(tanh(1/2*d*x+1/2*c))-2*b/(a-b)^3*((-1/8*(9*a^2-14*a*b+5*b^2)/a*tanh(1/2*d*x+1/2*c)^7+1/8
*(27*a^3-70*a^2*b+55*a*b^2-12*b^3)/a^2*tanh(1/2*d*x+1/2*c)^5-1/8*(27*a^3-70*a^2*b+55*a*b^2-12*b^3)/a^2*tanh(1/
2*d*x+1/2*c)^3+1/8*(9*a^2-14*a*b+5*b^2)/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+1/8/a*(15*a^2-10*a*b+3*b^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)*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)))))

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

[Out]

-1/4*((7*a*b^2*e^(7*c) - 3*b^3*e^(7*c))*e^(7*d*x) + (36*a^2*b*e^(5*c) - 41*a*b^2*e^(5*c) + 9*b^3*e^(5*c))*e^(5
*d*x) - (36*a^2*b*e^(3*c) - 41*a*b^2*e^(3*c) + 9*b^3*e^(3*c))*e^(3*d*x) - (7*a*b^2*e^c - 3*b^3*e^c)*e^(d*x))/(
a^4*b^2*d - 2*a^3*b^3*d + a^2*b^4*d + (a^4*b^2*d*e^(8*c) - 2*a^3*b^3*d*e^(8*c) + a^2*b^4*d*e^(8*c))*e^(8*d*x)
+ 4*(2*a^5*b*d*e^(6*c) - 5*a^4*b^2*d*e^(6*c) + 4*a^3*b^3*d*e^(6*c) - a^2*b^4*d*e^(6*c))*e^(6*d*x) + 2*(8*a^6*d
*e^(4*c) - 24*a^5*b*d*e^(4*c) + 27*a^4*b^2*d*e^(4*c) - 14*a^3*b^3*d*e^(4*c) + 3*a^2*b^4*d*e^(4*c))*e^(4*d*x) +
4*(2*a^5*b*d*e^(2*c) - 5*a^4*b^2*d*e^(2*c) + 4*a^3*b^3*d*e^(2*c) - a^2*b^4*d*e^(2*c))*e^(2*d*x)) + 2*arctan(e
^(d*x + c))/(a^3*d - 3*a^2*b*d + 3*a*b^2*d - b^3*d) - 2*integrate(1/8*((15*a^2*b*e^(3*c) - 10*a*b^2*e^(3*c) +
3*b^3*e^(3*c))*e^(3*d*x) + (15*a^2*b*e^c - 10*a*b^2*e^c + 3*b^3*e^c)*e^(d*x))/(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 -
a^2*b^4 + (a^5*b*e^(4*c) - 3*a^4*b^2*e^(4*c) + 3*a^3*b^3*e^(4*c) - a^2*b^4*e^(4*c))*e^(4*d*x) + 2*(2*a^6*e^(2
*c) - 7*a^5*b*e^(2*c) + 9*a^4*b^2*e^(2*c) - 5*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. 4384 vs. \(2 (145) = 290\).
time = 0.49, size = 8083, normalized size = 50.84 \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)/(a+b*sinh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[-1/16*(4*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^7 + 28*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)*sin
h(d*x + c)^6 + 4*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*sinh(d*x + c)^7 + 4*(36*a^3*b - 77*a^2*b^2 + 50*a*b^3 - 9*b^4)
*cosh(d*x + c)^5 + 4*(36*a^3*b - 77*a^2*b^2 + 50*a*b^3 - 9*b^4 + 21*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x +
c)^2)*sinh(d*x + c)^5 + 20*(7*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^3 + (36*a^3*b - 77*a^2*b^2 + 50*a*b
^3 - 9*b^4)*cosh(d*x + c))*sinh(d*x + c)^4 - 4*(36*a^3*b - 77*a^2*b^2 + 50*a*b^3 - 9*b^4)*cosh(d*x + c)^3 + 4*
(35*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^4 - 36*a^3*b + 77*a^2*b^2 - 50*a*b^3 + 9*b^4 + 10*(36*a^3*b -
77*a^2*b^2 + 50*a*b^3 - 9*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 4*(21*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d
*x + c)^5 + 10*(36*a^3*b - 77*a^2*b^2 + 50*a*b^3 - 9*b^4)*cosh(d*x + c)^3 - 3*(36*a^3*b - 77*a^2*b^2 + 50*a*b^
3 - 9*b^4)*cosh(d*x + c))*sinh(d*x + c)^2 + ((15*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^8 + 8*(15*a^2*b^2 -
10*a*b^3 + 3*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (15*a^2*b^2 - 10*a*b^3 + 3*b^4)*sinh(d*x + c)^8 + 4*(30*a^3
*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c)^6 + 4*(30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4 + 7*(15*a^2*
b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(15*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^
3 + 3*(30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(120*a^4 - 200*a^3*b + 149
*a^2*b^2 - 54*a*b^3 + 9*b^4)*cosh(d*x + c)^4 + 2*(35*(15*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^4 + 120*a^4
- 200*a^3*b + 149*a^2*b^2 - 54*a*b^3 + 9*b^4 + 30*(30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c)^2)
*sinh(d*x + c)^4 + 15*a^2*b^2 - 10*a*b^3 + 3*b^4 + 8*(7*(15*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^5 + 10*(
30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c)^3 + (120*a^4 - 200*a^3*b + 149*a^2*b^2 - 54*a*b^3 + 9*
b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c)^2 + 4*(7*(15*
a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^6 + 15*(30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c)^4 +
30*a^3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4 + 3*(120*a^4 - 200*a^3*b + 149*a^2*b^2 - 54*a*b^3 + 9*b^4)*cosh(d*x +
c)^2)*sinh(d*x + c)^2 + 8*((15*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^7 + 3*(30*a^3*b - 35*a^2*b^2 + 16*a*
b^3 - 3*b^4)*cosh(d*x + c)^5 + (120*a^4 - 200*a^3*b + 149*a^2*b^2 - 54*a*b^3 + 9*b^4)*cosh(d*x + c)^3 + (30*a^
3*b - 35*a^2*b^2 + 16*a*b^3 - 3*b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-b/a)*log((b*cosh(d*x + c)^4 + 4*b*cos
h(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)*co
sh(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)) - 32*(a^2*b^2*cosh(d*x + c)^8 + 8*a^2*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + a^2*b^2*si
nh(d*x + c)^8 + 4*(2*a^3*b - a^2*b^2)*cosh(d*x + c)^6 + 4*(7*a^2*b^2*cosh(d*x + c)^2 + 2*a^3*b - a^2*b^2)*sinh
(d*x + c)^6 + 8*(7*a^2*b^2*cosh(d*x + c)^3 + 3*(2*a^3*b - a^2*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(8*a^4 -
8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^4 + 2*(35*a^2*b^2*cosh(d*x + c)^4 + 8*a^4 - 8*a^3*b + 3*a^2*b^2 + 30*(2*a^
3*b - a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + a^2*b^2 + 8*(7*a^2*b^2*cosh(d*x + c)^5 + 10*(2*a^3*b - a^2*b
^2)*cosh(d*x + c)^3 + (8*a^4 - 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(2*a^3*b - a^2*b^2)*cos
h(d*x + c)^2 + 4*(7*a^2*b^2*cosh(d*x + c)^6 + 15*(2*a^3*b - a^2*b^2)*cosh(d*x + c)^4 + 2*a^3*b - a^2*b^2 + 3*(
8*a^4 - 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*(a^2*b^2*cosh(d*x + c)^7 + 3*(2*a^3*b - a^2*
b^2)*cosh(d*x + c)^5 + (8*a^4 - 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^3 + (2*a^3*b - a^2*b^2)*cosh(d*x + c))*sinh
(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) - 4*(7*a^2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c) + 4*(7*(7*a^
2*b^2 - 10*a*b^3 + 3*b^4)*cosh(d*x + c)^6 + 5*(36*a^3*b - 77*a^2*b^2 + 50*a*b^3 - 9*b^4)*cosh(d*x + c)^4 - 7*a
^2*b^2 + 10*a*b^3 - 3*b^4 - 3*(36*a^3*b - 77*a^2*b^2 + 50*a*b^3 - 9*b^4)*cosh(d*x + c)^2)*sinh(d*x + c))/((a^5
*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^8 + 8*(a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*d*co
sh(d*x + c)*sinh(d*x + c)^7 + (a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)*d*sinh(d*x + c)^8 + 4*(2*a^6*b - 7*a
^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^6 + 4*(7*(a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 - a^2*b^5)
*d*cosh(d*x + c)^2 + (2*a^6*b - 7*a^5*b^2 + 9*a^4*b^3 - 5*a^3*b^4 + a^2*b^5)*d)*sinh(d*x + c)^6 + 2*(8*a^7 - 3
2*a^6*b + 51*a^5*b^2 - 41*a^4*b^3 + 17*a^3*b^4 - 3*a^2*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a^5*b^2 - 3*a^4*b^3 + 3*
a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^3 + 3*(2*a^6...

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

[Out]

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

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