3.2.1 \(\int \frac {\text {sech}^7(c+d x)}{(a+b \text {sech}^2(c+d x))^3} \, dx\) [101]

Optimal. Leaf size=153 \[ \frac {\text {ArcTan}(\sinh (c+d x))}{b^3 d}-\frac {\sqrt {a} \left (8 a^2+20 a b+15 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{8 b^3 (a+b)^{5/2} d}-\frac {a \sinh (c+d x)}{4 b (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )^2}-\frac {a (4 a+7 b) \sinh (c+d x)}{8 b^2 (a+b)^2 d \left (a+b+a \sinh ^2(c+d x)\right )} \]

[Out]

arctan(sinh(d*x+c))/b^3/d-1/4*a*sinh(d*x+c)/b/(a+b)/d/(a+b+a*sinh(d*x+c)^2)^2-1/8*a*(4*a+7*b)*sinh(d*x+c)/b^2/
(a+b)^2/d/(a+b+a*sinh(d*x+c)^2)-1/8*(8*a^2+20*a*b+15*b^2)*arctan(sinh(d*x+c)*a^(1/2)/(a+b)^(1/2))*a^(1/2)/b^3/
(a+b)^(5/2)/d

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Rubi [A]
time = 0.15, antiderivative size = 153, 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 = {4232, 425, 541, 536, 209, 211} \begin {gather*} -\frac {\sqrt {a} \left (8 a^2+20 a b+15 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{8 b^3 d (a+b)^{5/2}}-\frac {a (4 a+7 b) \sinh (c+d x)}{8 b^2 d (a+b)^2 \left (a \sinh ^2(c+d x)+a+b\right )}-\frac {a \sinh (c+d x)}{4 b d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )^2}+\frac {\text {ArcTan}(\sinh (c+d x))}{b^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sech[c + d*x]^7/(a + b*Sech[c + d*x]^2)^3,x]

[Out]

ArcTan[Sinh[c + d*x]]/(b^3*d) - (Sqrt[a]*(8*a^2 + 20*a*b + 15*b^2)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]]
)/(8*b^3*(a + b)^(5/2)*d) - (a*Sinh[c + d*x])/(4*b*(a + b)*d*(a + b + a*Sinh[c + d*x]^2)^2) - (a*(4*a + 7*b)*S
inh[c + d*x])/(8*b^2*(a + b)^2*d*(a + b + a*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 4232

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

Rubi steps

\begin {align*} \int \frac {\text {sech}^7(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (a+b+a x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=-\frac {a \sinh (c+d x)}{4 b (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {a+4 b-3 a x^2}{\left (1+x^2\right ) \left (a+b+a x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{4 b (a+b) d}\\ &=-\frac {a \sinh (c+d x)}{4 b (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )^2}-\frac {a (4 a+7 b) \sinh (c+d x)}{8 b^2 (a+b)^2 d \left (a+b+a \sinh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {4 a^2+9 a b+8 b^2-a (4 a+7 b) x^2}{\left (1+x^2\right ) \left (a+b+a x^2\right )} \, dx,x,\sinh (c+d x)\right )}{8 b^2 (a+b)^2 d}\\ &=-\frac {a \sinh (c+d x)}{4 b (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )^2}-\frac {a (4 a+7 b) \sinh (c+d x)}{8 b^2 (a+b)^2 d \left (a+b+a \sinh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{b^3 d}-\frac {\left (a \left (8 a^2+20 a b+15 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b+a x^2} \, dx,x,\sinh (c+d x)\right )}{8 b^3 (a+b)^2 d}\\ &=\frac {\tan ^{-1}(\sinh (c+d x))}{b^3 d}-\frac {\sqrt {a} \left (8 a^2+20 a b+15 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{8 b^3 (a+b)^{5/2} d}-\frac {a \sinh (c+d x)}{4 b (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )^2}-\frac {a (4 a+7 b) \sinh (c+d x)}{8 b^2 (a+b)^2 d \left (a+b+a \sinh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [A]
time = 3.15, size = 247, normalized size = 1.61 \begin {gather*} \frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^5(c+d x) \left (16 \text {ArcTan}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right ) (a+2 b+a \cosh (2 (c+d x)))^2 \text {sech}(c+d x)+\frac {\sqrt {a} \left (8 a^2+20 a b+15 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a+b} \text {csch}(c+d x) \sqrt {(\cosh (c)-\sinh (c))^2} (\cosh (c)+\sinh (c))}{\sqrt {a}}\right ) (a+2 b+a \cosh (2 (c+d x)))^2 \text {sech}(c+d x) (\cosh (c)-\sinh (c))}{(a+b)^{5/2} \sqrt {(\cosh (c)-\sinh (c))^2}}-\frac {8 a b^2 \tanh (c+d x)}{a+b}-\frac {2 a b (4 a+7 b) (a+2 b+a \cosh (2 (c+d x))) \tanh (c+d x)}{(a+b)^2}\right )}{64 b^3 d \left (a+b \text {sech}^2(c+d x)\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sech[c + d*x]^7/(a + b*Sech[c + d*x]^2)^3,x]

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^5*(16*ArcTan[Tanh[(c + d*x)/2]]*(a + 2*b + a*Cosh[2*(c + d*x)])
^2*Sech[c + d*x] + (Sqrt[a]*(8*a^2 + 20*a*b + 15*b^2)*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[c] - Sinh[c
])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x]*(Cosh[c] - Sinh[c]))/((a +
b)^(5/2)*Sqrt[(Cosh[c] - Sinh[c])^2]) - (8*a*b^2*Tanh[c + d*x])/(a + b) - (2*a*b*(4*a + 7*b)*(a + 2*b + a*Cos
h[2*(c + d*x)])*Tanh[c + d*x])/(a + b)^2))/(64*b^3*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. \(313\) vs. \(2(139)=278\).
time = 2.00, size = 314, normalized size = 2.05

method	result	size
derivativedivides	\(\frac {\frac {2 \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3}}-\frac {2 a \left (\frac {-\frac {b \left (9 b +4 a \right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )}-\frac {b \left (4 a^{2}-11 a b -27 b^{2}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )^{2}}+\frac {b \left (4 a^{2}-11 a b -27 b^{2}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )^{2}}+\frac {b \left (9 b +4 a \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a +8 b}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}+\frac {\left (8 a^{2}+20 a b +15 b^{2}\right ) \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}\right )}{8 a^{2}+16 a b +8 b^{2}}\right )}{b^{3}}}{d}\)	\(314\)
default	\(\frac {\frac {2 \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3}}-\frac {2 a \left (\frac {-\frac {b \left (9 b +4 a \right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )}-\frac {b \left (4 a^{2}-11 a b -27 b^{2}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )^{2}}+\frac {b \left (4 a^{2}-11 a b -27 b^{2}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )^{2}}+\frac {b \left (9 b +4 a \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a +8 b}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}+\frac {\left (8 a^{2}+20 a b +15 b^{2}\right ) \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}\right )}{8 a^{2}+16 a b +8 b^{2}}\right )}{b^{3}}}{d}\)	\(314\)
risch	\(-\frac {{\mathrm e}^{d x +c} a \left (4 a^{2} {\mathrm e}^{6 d x +6 c}+7 a b \,{\mathrm e}^{6 d x +6 c}+4 a^{2} {\mathrm e}^{4 d x +4 c}+31 a b \,{\mathrm e}^{4 d x +4 c}+36 b^{2} {\mathrm e}^{4 d x +4 c}-4 a^{2} {\mathrm e}^{2 d x +2 c}-31 a b \,{\mathrm e}^{2 d x +2 c}-36 b^{2} {\mathrm e}^{2 d x +2 c}-4 a^{2}-7 a b \right )}{4 b^{2} \left (a +b \right )^{2} d \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2}}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right )}{d \,b^{3}}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right )}{d \,b^{3}}+\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a}-1\right ) a^{2}}{2 \left (a +b \right )^{3} d \,b^{3}}+\frac {5 \sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a}-1\right ) a}{4 \left (a +b \right )^{3} d \,b^{2}}+\frac {15 \sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a}-1\right )}{16 \left (a +b \right )^{3} d b}-\frac {\sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a}-1\right ) a^{2}}{2 \left (a +b \right )^{3} d \,b^{3}}-\frac {5 \sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a}-1\right ) a}{4 \left (a +b \right )^{3} d \,b^{2}}-\frac {15 \sqrt {-a \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a}-1\right )}{16 \left (a +b \right )^{3} d b}\)	\(538\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^7/(a+b*sech(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(2/b^3*arctan(tanh(1/2*d*x+1/2*c))-2/b^3*a*((-1/8*b*(9*b+4*a)/(a+b)*tanh(1/2*d*x+1/2*c)^7-1/8*b*(4*a^2-11*
a*b-27*b^2)/(a+b)^2*tanh(1/2*d*x+1/2*c)^5+1/8*b*(4*a^2-11*a*b-27*b^2)/(a+b)^2*tanh(1/2*d*x+1/2*c)^3+1/8*b*(9*b
+4*a)/(a+b)*tanh(1/2*d*x+1/2*c))/(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+1/8*(8*a^2+20*a*b+15*b^2)/(a^2+2*a*b+b^2)*(1/2/(a+b)^(1/2)/a^(1/2)*arctan(1/2*(
2*(a+b)^(1/2)*tanh(1/2*d*x+1/2*c)+2*b^(1/2))/a^(1/2))+1/2/(a+b)^(1/2)/a^(1/2)*arctan(1/2*(2*(a+b)^(1/2)*tanh(1
/2*d*x+1/2*c)-2*b^(1/2))/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)^7/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

-1/4*((4*a^3*e^(7*c) + 7*a^2*b*e^(7*c))*e^(7*d*x) + (4*a^3*e^(5*c) + 31*a^2*b*e^(5*c) + 36*a*b^2*e^(5*c))*e^(5
*d*x) - (4*a^3*e^(3*c) + 31*a^2*b*e^(3*c) + 36*a*b^2*e^(3*c))*e^(3*d*x) - (4*a^3*e^c + 7*a^2*b*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*(a^4*b^2*d*e^(6*c) + 4*a^3*b^3*d*e^(6*c) + 5*a^2*b^4*d*e^(6*c) + 2*a*b^5*d*e^(6*c))*e^(6*d*x) + 2*(3*a^4*b
^2*d*e^(4*c) + 14*a^3*b^3*d*e^(4*c) + 27*a^2*b^4*d*e^(4*c) + 24*a*b^5*d*e^(4*c) + 8*b^6*d*e^(4*c))*e^(4*d*x) +
4*(a^4*b^2*d*e^(2*c) + 4*a^3*b^3*d*e^(2*c) + 5*a^2*b^4*d*e^(2*c) + 2*a*b^5*d*e^(2*c))*e^(2*d*x)) + 2*arctan(e
^(d*x + c))/(b^3*d) - 128*integrate(1/512*((8*a^3*e^(3*c) + 20*a^2*b*e^(3*c) + 15*a*b^2*e^(3*c))*e^(3*d*x) + (
8*a^3*e^c + 20*a^2*b*e^c + 15*a*b^2*e^c)*e^(d*x))/(a^3*b^3 + 2*a^2*b^4 + a*b^5 + (a^3*b^3*e^(4*c) + 2*a^2*b^4*
e^(4*c) + a*b^5*e^(4*c))*e^(4*d*x) + 2*(a^3*b^3*e^(2*c) + 4*a^2*b^4*e^(2*c) + 5*a*b^5*e^(2*c) + 2*b^6*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. 4335 vs. \(2 (139) = 278\).
time = 0.50, size = 7993, normalized size = 52.24 \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)^7/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[-1/16*(4*(4*a^3*b + 7*a^2*b^2)*cosh(d*x + c)^7 + 28*(4*a^3*b + 7*a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^6 + 4*(
4*a^3*b + 7*a^2*b^2)*sinh(d*x + c)^7 + 4*(4*a^3*b + 31*a^2*b^2 + 36*a*b^3)*cosh(d*x + c)^5 + 4*(4*a^3*b + 31*a
^2*b^2 + 36*a*b^3 + 21*(4*a^3*b + 7*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 20*(7*(4*a^3*b + 7*a^2*b^2)*co
sh(d*x + c)^3 + (4*a^3*b + 31*a^2*b^2 + 36*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 - 4*(4*a^3*b + 31*a^2*b^2 + 3
6*a*b^3)*cosh(d*x + c)^3 + 4*(35*(4*a^3*b + 7*a^2*b^2)*cosh(d*x + c)^4 - 4*a^3*b - 31*a^2*b^2 - 36*a*b^3 + 10*
(4*a^3*b + 31*a^2*b^2 + 36*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 4*(21*(4*a^3*b + 7*a^2*b^2)*cosh(d*x + c)
^5 + 10*(4*a^3*b + 31*a^2*b^2 + 36*a*b^3)*cosh(d*x + c)^3 - 3*(4*a^3*b + 31*a^2*b^2 + 36*a*b^3)*cosh(d*x + c))
*sinh(d*x + c)^2 - ((8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(d*x + c)^8 + 8*(8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(d
*x + c)*sinh(d*x + c)^7 + (8*a^4 + 20*a^3*b + 15*a^2*b^2)*sinh(d*x + c)^8 + 4*(8*a^4 + 36*a^3*b + 55*a^2*b^2 +
30*a*b^3)*cosh(d*x + c)^6 + 4*(8*a^4 + 36*a^3*b + 55*a^2*b^2 + 30*a*b^3 + 7*(8*a^4 + 20*a^3*b + 15*a^2*b^2)*c
osh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(d*x + c)^3 + 3*(8*a^4 + 36*a^3*b +
55*a^2*b^2 + 30*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(24*a^4 + 124*a^3*b + 269*a^2*b^2 + 280*a*b^3 + 120
*b^4)*cosh(d*x + c)^4 + 2*(35*(8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(d*x + c)^4 + 24*a^4 + 124*a^3*b + 269*a^2*b
^2 + 280*a*b^3 + 120*b^4 + 30*(8*a^4 + 36*a^3*b + 55*a^2*b^2 + 30*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*
a^4 + 20*a^3*b + 15*a^2*b^2 + 8*(7*(8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(d*x + c)^5 + 10*(8*a^4 + 36*a^3*b + 55
*a^2*b^2 + 30*a*b^3)*cosh(d*x + c)^3 + (24*a^4 + 124*a^3*b + 269*a^2*b^2 + 280*a*b^3 + 120*b^4)*cosh(d*x + c))
*sinh(d*x + c)^3 + 4*(8*a^4 + 36*a^3*b + 55*a^2*b^2 + 30*a*b^3)*cosh(d*x + c)^2 + 4*(7*(8*a^4 + 20*a^3*b + 15*
a^2*b^2)*cosh(d*x + c)^6 + 15*(8*a^4 + 36*a^3*b + 55*a^2*b^2 + 30*a*b^3)*cosh(d*x + c)^4 + 8*a^4 + 36*a^3*b +
55*a^2*b^2 + 30*a*b^3 + 3*(24*a^4 + 124*a^3*b + 269*a^2*b^2 + 280*a*b^3 + 120*b^4)*cosh(d*x + c)^2)*sinh(d*x +
c)^2 + 8*((8*a^4 + 20*a^3*b + 15*a^2*b^2)*cosh(d*x + c)^7 + 3*(8*a^4 + 36*a^3*b + 55*a^2*b^2 + 30*a*b^3)*cosh
(d*x + c)^5 + (24*a^4 + 124*a^3*b + 269*a^2*b^2 + 280*a*b^3 + 120*b^4)*cosh(d*x + c)^3 + (8*a^4 + 36*a^3*b + 5
5*a^2*b^2 + 30*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a/(a + b))*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*(3*a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 - 3*a - 2*b)*si
nh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - (3*a + 2*b)*cosh(d*x + c))*sinh(d*x + c) - 4*((a + b)*cosh(d*x + c)^3 +
3*(a + b)*cosh(d*x + c)*sinh(d*x + c)^2 + (a + b)*sinh(d*x + c)^3 - (a + b)*cosh(d*x + c) + (3*(a + b)*cosh(d
*x + c)^2 - a - b)*sinh(d*x + c))*sqrt(-a/(a + b)) + 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))*sinh(d*x + c) + a)) - 32*((a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^8
+ 8*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^4 + 2*a^3*b + a^2*b^2)*sinh(d*x + c)^8 + 4*(
a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^6 + 4*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3 + 7*(a^4 + 2*a^3
*b + a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^3 + 3*(a^4 + 4*a
^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b
^4)*cosh(d*x + c)^4 + 2*(35*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^4 + 3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b
^3 + 8*b^4 + 30*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + a^4 + 2*a^3*b + a^2*b
^2 + 8*(7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^5 + 10*(a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^3
+ (3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(a^4 + 4*a^3*b + 5*a^
2*b^2 + 2*a*b^3)*cosh(d*x + c)^2 + 4*(7*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^6 + 15*(a^4 + 4*a^3*b + 5*a^2*
b^2 + 2*a*b^3)*cosh(d*x + c)^4 + a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3 + 3*(3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a
*b^3 + 8*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^7 + 3*(a^4 + 4*a^3
*b + 5*a^2*b^2 + 2*a*b^3)*cosh(d*x + c)^5 + (3*a^4 + 14*a^3*b + 27*a^2*b^2 + 24*a*b^3 + 8*b^4)*cosh(d*x + c)^3
+ (a^4 + 4*a^3*b + 5*a^2*b^2 + 2*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) -
4*(4*a^3*b + 7*a^2*b^2)*cosh(d*x + c) + 4*(7*(4*a^3*b + 7*a^2*b^2)*cosh(d*x + c)^6 + 5*(4*a^3*b + 31*a^2*b^2
+ 36*a*b^3)*cosh(d*x + c)^4 - 4*a^3*b - 7*a^2*b^2 - 3*(4*a^3*b + 31*a^2*b^2 + 36*a*b^3)*cosh(d*x + c)^2)*sinh(
d*x + c))/((a^4*b^3 + 2*a^3*b^4 + a^2*b^5)*d*cosh(d*x + c)^8 + 8*(a^4*b^3 + 2*a^3*b^4 + a^2*b^5)*d*cosh(d*x +
c)*sinh(d*x + c)^7 + (a^4*b^3 + 2*a^3*b^4 + a^2*b^5)*d*sinh(d*x + c)^8 + 4*(a^4*b^3 + 4*a^3*b^4 + 5*a^2*b^5 +
2*a*b^6)*d*cosh(d*x + c)^6 + 4*(7*(a^4*b^3 + 2*...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {sech}^{7}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)**7/(a+b*sech(d*x+c)**2)**3,x)

[Out]

Integral(sech(c + d*x)**7/(a + b*sech(c + d*x)**2)**3, 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)^7/(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 {1}{{\mathrm {cosh}\left (c+d\,x\right )}^7\,{\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(c + d*x)^7*(a + b/cosh(c + d*x)^2)^3),x)

[Out]

int(1/(cosh(c + d*x)^7*(a + b/cosh(c + d*x)^2)^3), x)

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