3.4.27 \(\int \frac {\text {sech}^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) [327]

Optimal. Leaf size=138 \[ \frac {\left (3 a^2-10 a b+15 b^2\right ) \text {ArcTan}(\sinh (c+d x))}{8 (a-b)^3 d}-\frac {b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a-b)^3 d}+\frac {(3 a-7 b) \text {sech}(c+d x) \tanh (c+d x)}{8 (a-b)^2 d}+\frac {\text {sech}^3(c+d x) \tanh (c+d x)}{4 (a-b) d} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 138, 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 {\left (3 a^2-10 a b+15 b^2\right ) \text {ArcTan}(\sinh (c+d x))}{8 d (a-b)^3}-\frac {b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} d (a-b)^3}+\frac {\tanh (c+d x) \text {sech}^3(c+d x)}{4 d (a-b)}+\frac {(3 a-7 b) \tanh (c+d x) \text {sech}(c+d x)}{8 d (a-b)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sech[c + d*x]^5/(a + b*Sinh[c + d*x]^2),x]

[Out]

((3*a^2 - 10*a*b + 15*b^2)*ArcTan[Sinh[c + d*x]])/(8*(a - b)^3*d) - (b^(5/2)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sq
rt[a]])/(Sqrt[a]*(a - b)^3*d) + ((3*a - 7*b)*Sech[c + d*x]*Tanh[c + d*x])/(8*(a - b)^2*d) + (Sech[c + d*x]^3*T
anh[c + d*x])/(4*(a - b)*d)

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

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Mathematica [A]
time = 0.56, size = 139, normalized size = 1.01 \begin {gather*} \frac {8 b^{5/2} \text {ArcTan}\left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {b}}\right )+2 \sqrt {a} \left (3 a^2-10 a b+15 b^2\right ) \text {ArcTan}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+\sqrt {a} \left (3 a^2-10 a b+7 b^2\right ) \text {sech}(c+d x) \tanh (c+d x)+2 \sqrt {a} (a-b)^2 \text {sech}^3(c+d x) \tanh (c+d x)}{8 \sqrt {a} (a-b)^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sech[c + d*x]^5/(a + b*Sinh[c + d*x]^2),x]

[Out]

(8*b^(5/2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]] + 2*Sqrt[a]*(3*a^2 - 10*a*b + 15*b^2)*ArcTan[Tanh[(c + d*x)
/2]] + Sqrt[a]*(3*a^2 - 10*a*b + 7*b^2)*Sech[c + d*x]*Tanh[c + d*x] + 2*Sqrt[a]*(a - b)^2*Sech[c + d*x]^3*Tanh
[c + d*x])/(8*Sqrt[a]*(a - b)^3*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(351\) vs. \(2(124)=248\).
time = 2.12, size = 352, normalized size = 2.55

method	result	size
derivativedivides	\(\frac {-\frac {2 b^{3} a \left (\frac {\left (-a +\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (a +\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{\left (a -b \right )^{3}}+\frac {\frac {2 \left (\left (-\frac {5}{8} a^{2}+\frac {7}{4} a b -\frac {9}{8} b^{2}\right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {3}{8} a^{2}-\frac {1}{4} a b -\frac {1}{8} b^{2}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {3}{8} a^{2}+\frac {1}{4} a b +\frac {1}{8} b^{2}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{8} a^{2}-\frac {7}{4} a b +\frac {9}{8} b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {\left (3 a^{2}-10 a b +15 b^{2}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{\left (a -b \right )^{3}}}{d}\)	\(352\)
default	\(\frac {-\frac {2 b^{3} a \left (\frac {\left (-a +\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (a +\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{\left (a -b \right )^{3}}+\frac {\frac {2 \left (\left (-\frac {5}{8} a^{2}+\frac {7}{4} a b -\frac {9}{8} b^{2}\right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {3}{8} a^{2}-\frac {1}{4} a b -\frac {1}{8} b^{2}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {3}{8} a^{2}+\frac {1}{4} a b +\frac {1}{8} b^{2}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{8} a^{2}-\frac {7}{4} a b +\frac {9}{8} b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {\left (3 a^{2}-10 a b +15 b^{2}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{\left (a -b \right )^{3}}}{d}\)	\(352\)
risch	\(\frac {{\mathrm e}^{d x +c} \left (3 a \,{\mathrm e}^{6 d x +6 c}-7 b \,{\mathrm e}^{6 d x +6 c}+11 a \,{\mathrm e}^{4 d x +4 c}-15 b \,{\mathrm e}^{4 d x +4 c}-11 a \,{\mathrm e}^{2 d x +2 c}+15 b \,{\mathrm e}^{2 d x +2 c}-3 a +7 b \right )}{4 d \left (a -b \right )^{2} \left (1+{\mathrm e}^{2 d x +2 c}\right )^{4}}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{8 \left (a -b \right )^{3} d}-\frac {5 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a b}{4 \left (a -b \right )^{3} d}+\frac {15 i \ln \left ({\mathrm e}^{d x +c}+i\right ) b^{2}}{8 \left (a -b \right )^{3} d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{8 \left (a -b \right )^{3} d}+\frac {5 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a b}{4 \left (a -b \right )^{3} d}-\frac {15 i \ln \left ({\mathrm e}^{d x +c}-i\right ) b^{2}}{8 \left (a -b \right )^{3} d}+\frac {\sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{b}-1\right )}{2 a \left (a -b \right )^{3} d}-\frac {\sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{b}-1\right )}{2 a \left (a -b \right )^{3} d}\)	\(372\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^5/(a+b*sinh(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

1/d*(-2*b^3/(a-b)^3*a*(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)*arc
tan(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*(((-5/8*a^2+7/4*a*b-9/8*b^2)*tanh(1/2*d*x+1/2*c)^7+(3/8*a^2-1/4*a*b-1/8*b^2)*tanh(1/2*d*x+1/2*c)^5+(-3/8
*a^2+1/4*a*b+1/8*b^2)*tanh(1/2*d*x+1/2*c)^3+(5/8*a^2-7/4*a*b+9/8*b^2)*tanh(1/2*d*x+1/2*c))/(tanh(1/2*d*x+1/2*c
)^2+1)^4+1/8*(3*a^2-10*a*b+15*b^2)*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)^5/(a+b*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

1/4*(3*a^2*e^c - 10*a*b*e^c + 15*b^2*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) +
1/4*((3*a*e^(7*c) - 7*b*e^(7*c))*e^(7*d*x) + (11*a*e^(5*c) - 15*b*e^(5*c))*e^(5*d*x) - (11*a*e^(3*c) - 15*b*e
^(3*c))*e^(3*d*x) - (3*a*e^c - 7*b*e^c)*e^(d*x))/(a^2*d - 2*a*b*d + b^2*d + (a^2*d*e^(8*c) - 2*a*b*d*e^(8*c) +
b^2*d*e^(8*c))*e^(8*d*x) + 4*(a^2*d*e^(6*c) - 2*a*b*d*e^(6*c) + b^2*d*e^(6*c))*e^(6*d*x) + 6*(a^2*d*e^(4*c) -
2*a*b*d*e^(4*c) + b^2*d*e^(4*c))*e^(4*d*x) + 4*(a^2*d*e^(2*c) - 2*a*b*d*e^(2*c) + b^2*d*e^(2*c))*e^(2*d*x)) -
32*integrate(1/16*(b^3*e^(3*d*x + 3*c) + b^3*e^(d*x + c))/(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4 + (a^3*b*e^(4*c)
- 3*a^2*b^2*e^(4*c) + 3*a*b^3*e^(4*c) - b^4*e^(4*c))*e^(4*d*x) + 2*(2*a^4*e^(2*c) - 7*a^3*b*e^(2*c) + 9*a^2*b
^2*e^(2*c) - 5*a*b^3*e^(2*c) + 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. 2827 vs. \(2 (124) = 248\).
time = 0.47, size = 5500, normalized size = 39.86 \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)^5/(a+b*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

[1/4*((3*a^2 - 10*a*b + 7*b^2)*cosh(d*x + c)^7 + 7*(3*a^2 - 10*a*b + 7*b^2)*cosh(d*x + c)*sinh(d*x + c)^6 + (3
*a^2 - 10*a*b + 7*b^2)*sinh(d*x + c)^7 + (11*a^2 - 26*a*b + 15*b^2)*cosh(d*x + c)^5 + (21*(3*a^2 - 10*a*b + 7*
b^2)*cosh(d*x + c)^2 + 11*a^2 - 26*a*b + 15*b^2)*sinh(d*x + c)^5 + 5*(7*(3*a^2 - 10*a*b + 7*b^2)*cosh(d*x + c)
^3 + (11*a^2 - 26*a*b + 15*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 - (11*a^2 - 26*a*b + 15*b^2)*cosh(d*x + c)^3 +
(35*(3*a^2 - 10*a*b + 7*b^2)*cosh(d*x + c)^4 + 10*(11*a^2 - 26*a*b + 15*b^2)*cosh(d*x + c)^2 - 11*a^2 + 26*a*b
- 15*b^2)*sinh(d*x + c)^3 + (21*(3*a^2 - 10*a*b + 7*b^2)*cosh(d*x + c)^5 + 10*(11*a^2 - 26*a*b + 15*b^2)*cosh
(d*x + c)^3 - 3*(11*a^2 - 26*a*b + 15*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 - 2*(b^2*cosh(d*x + c)^8 + 8*b^2*cos
h(d*x + c)*sinh(d*x + c)^7 + b^2*sinh(d*x + c)^8 + 4*b^2*cosh(d*x + c)^6 + 4*(7*b^2*cosh(d*x + c)^2 + b^2)*sin
h(d*x + c)^6 + 6*b^2*cosh(d*x + c)^4 + 8*(7*b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35
*b^2*cosh(d*x + c)^4 + 30*b^2*cosh(d*x + c)^2 + 3*b^2)*sinh(d*x + c)^4 + 4*b^2*cosh(d*x + c)^2 + 8*(7*b^2*cosh
(d*x + c)^5 + 10*b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*b^2*cosh(d*x + c)^6 + 15*b^
2*cosh(d*x + c)^4 + 9*b^2*cosh(d*x + c)^2 + b^2)*sinh(d*x + c)^2 + b^2 + 8*(b^2*cosh(d*x + c)^7 + 3*b^2*cosh(d
*x + c)^5 + 3*b^2*cosh(d*x + c)^3 + 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)) + ((3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^8 + 8*(3*a^2 - 10*a*b + 15*b^2)*cosh(d
*x + c)*sinh(d*x + c)^7 + (3*a^2 - 10*a*b + 15*b^2)*sinh(d*x + c)^8 + 4*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c
)^6 + 4*(7*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^2 + 3*a^2 - 10*a*b + 15*b^2)*sinh(d*x + c)^6 + 8*(7*(3*a^2
- 10*a*b + 15*b^2)*cosh(d*x + c)^3 + 3*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 6*(3*a^2 - 1
0*a*b + 15*b^2)*cosh(d*x + c)^4 + 2*(35*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^4 + 30*(3*a^2 - 10*a*b + 15*b^
2)*cosh(d*x + c)^2 + 9*a^2 - 30*a*b + 45*b^2)*sinh(d*x + c)^4 + 8*(7*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^5
+ 10*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^3 + 3*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 +
4*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^2 + 4*(7*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^6 + 15*(3*a^2 - 10
*a*b + 15*b^2)*cosh(d*x + c)^4 + 9*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^2 + 3*a^2 - 10*a*b + 15*b^2)*sinh(d
*x + c)^2 + 3*a^2 - 10*a*b + 15*b^2 + 8*((3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^7 + 3*(3*a^2 - 10*a*b + 15*b^
2)*cosh(d*x + c)^5 + 3*(3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c)^3 + (3*a^2 - 10*a*b + 15*b^2)*cosh(d*x + c))*si
nh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) - (3*a^2 - 10*a*b + 7*b^2)*cosh(d*x + c) + (7*(3*a^2 - 10*a
*b + 7*b^2)*cosh(d*x + c)^6 + 5*(11*a^2 - 26*a*b + 15*b^2)*cosh(d*x + c)^4 - 3*(11*a^2 - 26*a*b + 15*b^2)*cosh
(d*x + c)^2 - 3*a^2 + 10*a*b - 7*b^2)*sinh(d*x + c))/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^8 + 8*(a
^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sinh(d*x + c
)^8 + 4*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^6 + 4*(7*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x +
c)^2 + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d)*sinh(d*x + c)^6 + 6*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^
4 + 8*(7*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^3 + 3*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)
)*sinh(d*x + c)^5 + 2*(35*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^4 + 30*(a^3 - 3*a^2*b + 3*a*b^2 - b^
3)*d*cosh(d*x + c)^2 + 3*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d)*sinh(d*x + c)^4 + 4*(a^3 - 3*a^2*b + 3*a*b^2 - b^3
)*d*cosh(d*x + c)^2 + 8*(7*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^5 + 10*(a^3 - 3*a^2*b + 3*a*b^2 - b
^3)*d*cosh(d*x + c)^3 + 3*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^3 - 3*a^2
*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^6 + 15*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^4 + 9*(a^3 - 3*a^2*
b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^2 + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d)*sinh(d*x + c)^2 + (a^3 - 3*a^2*b + 3
*a*b^2 - b^3)*d + 8*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^7 + 3*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*c
osh(d*x + c)^5 + 3*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(d*x + c)^3 + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cosh(
d*x + c))*sinh(d*x + c)), 1/4*((3*a^2 - 10*a*b + 7*b^2)*cosh(d*x + c)^7 + 7*(3*a^2 - 10*a*b + 7*b^2)*cosh(d*x
+ c)*sinh(d*x + c)^6 + (3*a^2 - 10*a*b + 7*b^2)...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {sech}^{5}{\left (c + d x \right )}}{a + b \sinh ^{2}{\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)**5/(a+b*sinh(d*x+c)**2),x)

[Out]

Integral(sech(c + d*x)**5/(a + b*sinh(c + d*x)**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)^5/(a+b*sinh(d*x+c)^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 [B]
time = 12.05, size = 2500, normalized size = 18.12 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(c + d*x)^5*(a + b*sinh(c + d*x)^2)),x)

[Out]

(4*exp(c + d*x))/((a*d - b*d)*(4*exp(2*c + 2*d*x) + 6*exp(4*c + 4*d*x) + 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x)
+ 1)) - (6*exp(c + d*x))/((a*d - b*d)*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) + (at
an((exp(d*x)*exp(c)*(243*a^12*(a^6*d^2 + b^6*d^2 - 6*a*b^5*d^2 - 6*a^5*b*d^2 + 15*a^2*b^4*d^2 - 20*a^3*b^3*d^2
+ 15*a^4*b^2*d^2)^(1/2) + 3840*b^12*(a^6*d^2 + b^6*d^2 - 6*a*b^5*d^2 - 6*a^5*b*d^2 + 15*a^2*b^4*d^2 - 20*a^3*
b^3*d^2 + 15*a^4*b^2*d^2)^(1/2) - 110560*a*b^11*(a^6*d^2 + b^6*d^2 - 6*a*b^5*d^2 - 6*a^5*b*d^2 + 15*a^2*b^4*d^
2 - 20*a^3*b^3*d^2 + 15*a^4*b^2*d^2)^(1/2) - 4050*a^11*b*(a^6*d^2 + b^6*d^2 - 6*a*b^5*d^2 - 6*a^5*b*d^2 + 15*a
^2*b^4*d^2 - 20*a^3*b^3*d^2 + 15*a^4*b^2*d^2)^(1/2) + 976143*a^2*b^10*(a^6*d^2 + b^6*d^2 - 6*a*b^5*d^2 - 6*a^5
*b*d^2 + 15*a^2*b^4*d^2 - 20*a^3*b^3*d^2 + 15*a^4*b^2*d^2)^(1/2) - 2740050*a^3*b^9*(a^6*d^2 + b^6*d^2 - 6*a*b^
5*d^2 - 6*a^5*b*d^2 + 15*a^2*b^4*d^2 - 20*a^3*b^3*d^2 + 15*a^4*b^2*d^2)^(1/2) + 4252775*a^4*b^8*(a^6*d^2 + b^6
*d^2 - 6*a*b^5*d^2 - 6*a^5*b*d^2 + 15*a^2*b^4*d^2 - 20*a^3*b^3*d^2 + 15*a^4*b^2*d^2)^(1/2) - 4316760*a^5*b^7*(
a^6*d^2 + b^6*d^2 - 6*a*b^5*d^2 - 6*a^5*b*d^2 + 15*a^2*b^4*d^2 - 20*a^3*b^3*d^2 + 15*a^4*b^2*d^2)^(1/2) + 3087
390*a^6*b^6*(a^6*d^2 + b^6*d^2 - 6*a*b^5*d^2 - 6*a^5*b*d^2 + 15*a^2*b^4*d^2 - 20*a^3*b^3*d^2 + 15*a^4*b^2*d^2)
^(1/2) - 1608364*a^7*b^5*(a^6*d^2 + b^6*d^2 - 6*a*b^5*d^2 - 6*a^5*b*d^2 + 15*a^2*b^4*d^2 - 20*a^3*b^3*d^2 + 15
*a^4*b^2*d^2)^(1/2) + 615750*a^8*b^4*(a^6*d^2 + b^6*d^2 - 6*a*b^5*d^2 - 6*a^5*b*d^2 + 15*a^2*b^4*d^2 - 20*a^3*
b^3*d^2 + 15*a^4*b^2*d^2)^(1/2) - 171000*a^9*b^3*(a^6*d^2 + b^6*d^2 - 6*a*b^5*d^2 - 6*a^5*b*d^2 + 15*a^2*b^4*d
^2 - 20*a^3*b^3*d^2 + 15*a^4*b^2*d^2)^(1/2) + 33075*a^10*b^2*(a^6*d^2 + b^6*d^2 - 6*a*b^5*d^2 - 6*a^5*b*d^2 +
15*a^2*b^4*d^2 - 20*a^3*b^3*d^2 + 15*a^4*b^2*d^2)^(1/2)))/(81*a^13*d*(9*a^4 - 60*a^3*b - 300*a*b^3 + 225*b^4 +
190*a^2*b^2)^(1/2) - 256*b^13*d*(9*a^4 - 60*a^3*b - 300*a*b^3 + 225*b^4 + 190*a^2*b^2)^(1/2) - 82593*a^2*b^11
*d*(9*a^4 - 60*a^3*b - 300*a*b^3 + 225*b^4 + 190*a^2*b^2)^(1/2) + 343611*a^3*b^10*d*(9*a^4 - 60*a^3*b - 300*a*
b^3 + 225*b^4 + 190*a^2*b^2)^(1/2) - 788535*a^4*b^9*d*(9*a^4 - 60*a^3*b - 300*a*b^3 + 225*b^4 + 190*a^2*b^2)^(
1/2) + 1157013*a^5*b^8*d*(9*a^4 - 60*a^3*b - 300*a*b^3 + 225*b^4 + 190*a^2*b^2)^(1/2) - 1173354*a^6*b^7*d*(9*a
^4 - 60*a^3*b - 300*a*b^3 + 225*b^4 + 190*a^2*b^2)^(1/2) + 857934*a^7*b^6*d*(9*a^4 - 60*a^3*b - 300*a*b^3 + 22
5*b^4 + 190*a^2*b^2)^(1/2) - 461358*a^8*b^5*d*(9*a^4 - 60*a^3*b - 300*a*b^3 + 225*b^4 + 190*a^2*b^2)^(1/2) + 1
82890*a^9*b^4*d*(9*a^4 - 60*a^3*b - 300*a*b^3 + 225*b^4 + 190*a^2*b^2)^(1/2) - 52581*a^10*b^3*d*(9*a^4 - 60*a^
3*b - 300*a*b^3 + 225*b^4 + 190*a^2*b^2)^(1/2) + 10503*a^11*b^2*d*(9*a^4 - 60*a^3*b - 300*a*b^3 + 225*b^4 + 19
0*a^2*b^2)^(1/2) + 7968*a*b^12*d*(9*a^4 - 60*a^3*b - 300*a*b^3 + 225*b^4 + 190*a^2*b^2)^(1/2) - 1323*a^12*b*d*
(9*a^4 - 60*a^3*b - 300*a*b^3 + 225*b^4 + 190*a^2*b^2)^(1/2)))*(9*a^4 - 60*a^3*b - 300*a*b^3 + 225*b^4 + 190*a
^2*b^2)^(1/2))/(4*(a^6*d^2 + b^6*d^2 - 6*a*b^5*d^2 - 6*a^5*b*d^2 + 15*a^2*b^4*d^2 - 20*a^3*b^3*d^2 + 15*a^4*b^
2*d^2)^(1/2)) - ((2*atan((b^3*exp(d*x)*exp(c)*(a*d^2*(a - b)^6)^(1/2))/(2*a*d*(a - b)^3*(b^5)^(1/2))) - 2*atan
((exp(d*x)*exp(c)*((4*(4032*a^5*d*(b^5)^(5/2) - 74990*a^10*d*(b^5)^(3/2) + 18*a^15*d*(b^5)^(1/2) + 32*a*b^9*d*
(b^5)^(3/2) + 288*a^9*b*d*(b^5)^(3/2) - 282*a^14*b*d*(b^5)^(1/2) - 288*a^2*b^8*d*(b^5)^(3/2) + 1152*a^3*b^7*d*
(b^5)^(3/2) - 2688*a^4*b^6*d*(b^5)^(3/2) - 4032*a^6*b^4*d*(b^5)^(3/2) + 2688*a^7*b^3*d*(b^5)^(3/2) - 1152*a^8*
b^2*d*(b^5)^(3/2) - 450*a^2*b^13*d*(b^5)^(1/2) + 4650*a^3*b^12*d*(b^5)^(1/2) - 21980*a^4*b^11*d*(b^5)^(1/2) +
62940*a^5*b^10*d*(b^5)^(1/2) - 121878*a^6*b^9*d*(b^5)^(1/2) + 168702*a^7*b^8*d*(b^5)^(1/2) - 172008*a^8*b^7*d*
(b^5)^(1/2) + 131112*a^9*b^6*d*(b^5)^(1/2) + 31878*a^11*b^4*d*(b^5)^(1/2) - 9852*a^12*b^3*d*(b^5)^(1/2) + 2108
*a^13*b^2*d*(b^5)^(1/2)))/(a*b^4*(a - b)^7*(a*b - a^2)*(a*d^2*(a - b)^6)^(1/2)*(3*a*b^2 - 3*a^2*b + a^3 - b^3)
*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)*(225*a*b^4 - 60*a^4*b + 9*a^5 - 16*b^5 - 300*a^2*b^3 + 190*a^3*b^
2)*(a^7*d^2 + a*b^6*d^2 - 6*a^6*b*d^2 - 6*a^2*b^5*d^2 + 15*a^3*b^4*d^2 - 20*a^4*b^3*d^2 + 15*a^5*b^2*d^2)^(1/2
)*(a^6 - 6*a^5*b - 6*a*b^5 + b^6 + 15*a^2*b^4 - 20*a^3*b^3 + 15*a^4*b^2)) + (2*(16*b^14*(a^7*d^2 + a*b^6*d^2 -
6*a^6*b*d^2 - 6*a^2*b^5*d^2 + 15*a^3*b^4*d^2 - 20*a^4*b^3*d^2 + 15*a^5*b^2*d^2)^(1/2) - 321*a*b^13*(a^7*d^2 +
a*b^6*d^2 - 6*a^6*b*d^2 - 6*a^2*b^5*d^2 + 15*a^3*b^4*d^2 - 20*a^4*b^3*d^2 + 15*a^5*b^2*d^2)^(1/2) + 1890*a^2*
b^12*(a^7*d^2 + a*b^6*d^2 - 6*a^6*b*d^2 - 6*a^2*b^5*d^2 + 15*a^3*b^4*d^2 - 20*a^4*b^3*d^2 + 15*a^5*b^2*d^2)^(1
/2) - 5685*a^3*b^11*(a^7*d^2 + a*b^6*d^2 - 6*a^6*b*d^2 - 6*a^2*b^5*d^2 + 15*a^3*b^4*d^2 - 20*a^4*b^3*d^2 + 15*
a^5*b^2*d^2)^(1/2) + 10440*a^4*b^10*(a^7*d^2 + a*b^6*d^2 - 6*a^6*b*d^2 - 6*a^2*b^5*d^2 + 15*a^3*b^4*d^2 - 20*a
^4*b^3*d^2 + 15*a^5*b^2*d^2)^(1/2) - 12690*a^5*b^9*(a^7*d^2 + a*b^6*d^2 - 6*a^6*b*d^2 - 6*a^2*b^5*d^2 + 15*a^3
*b^4*d^2 - 20*a^4*b^3*d^2 + 15*a^5*b^2*d^2)^(1/...

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