3.92 \(\int \frac{\text{sech}^7(c+d x)}{(a+b \text{sech}^2(c+d x))^2} \, dx\)

Optimal. Leaf size=153 \[ \frac{a^{3/2} (4 a+5 b) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{2 b^3 d (a+b)^{3/2}}+\frac{a (2 a+b) \sinh (c+d x)}{2 b^2 d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )}-\frac{(4 a-b) \tan ^{-1}(\sinh (c+d x))}{2 b^3 d}+\frac{\tanh (c+d x) \text{sech}(c+d x)}{2 b d \left (a \sinh ^2(c+d x)+a+b\right )} \]

[Out]

-((4*a - b)*ArcTan[Sinh[c + d*x]])/(2*b^3*d) + (a^(3/2)*(4*a + 5*b)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]
])/(2*b^3*(a + b)^(3/2)*d) + (a*(2*a + b)*Sinh[c + d*x])/(2*b^2*(a + b)*d*(a + b + a*Sinh[c + d*x]^2)) + (Sech
[c + d*x]*Tanh[c + d*x])/(2*b*d*(a + b + a*Sinh[c + d*x]^2))

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Rubi [A]  time = 0.199911, antiderivative size = 153, normalized size of antiderivative = 1., 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 = {4147, 414, 527, 522, 203, 205} \[ \frac{a^{3/2} (4 a+5 b) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{2 b^3 d (a+b)^{3/2}}+\frac{a (2 a+b) \sinh (c+d x)}{2 b^2 d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )}-\frac{(4 a-b) \tan ^{-1}(\sinh (c+d x))}{2 b^3 d}+\frac{\tanh (c+d x) \text{sech}(c+d x)}{2 b d \left (a \sinh ^2(c+d x)+a+b\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((4*a - b)*ArcTan[Sinh[c + d*x]])/(2*b^3*d) + (a^(3/2)*(4*a + 5*b)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]
])/(2*b^3*(a + b)^(3/2)*d) + (a*(2*a + b)*Sinh[c + d*x])/(2*b^2*(a + b)*d*(a + b + a*Sinh[c + d*x]^2)) + (Sech
[c + d*x]*Tanh[c + d*x])/(2*b*d*(a + b + a*Sinh[c + d*x]^2))

Rule 4147

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]

Rule 414

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]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 527

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 522

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 203

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

Rule 205

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

Rubi steps

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

Mathematica [B]  time = 5.04762, size = 489, normalized size = 3.2 \[ \frac{\text{sech}(c) \text{sech}^3(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (a^{3/2} \sinh (2 c) \text{sech}(c+d x) \left (2 a^2 \cosh (2 (c+d x))+2 a^2+5 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text{csch}(c+d x)}{\sqrt{a}}\right )-\cosh (c) \text{sech}(c+d x) \left (2 \sqrt{a+b} \left (4 a^2+3 a b-b^2\right ) \sqrt{(\cosh (c)-\sinh (c))^2} \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right ) (a \cosh (2 (c+d x))+a+2 b)-a^{5/2} b \sinh (c) (5 \cosh (2 (c+d x))+13) \tan ^{-1}\left (\frac{\sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text{csch}(c+d x)}{\sqrt{a}}\right )\right )+2 a^2 b \sqrt{a+b} \cosh (c) \sqrt{(\cosh (c)-\sinh (c))^2} \tanh (c+d x)-a^{3/2} (4 a+5 b) \cosh ^2(c) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b) \tan ^{-1}\left (\frac{\sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text{csch}(c+d x)}{\sqrt{a}}\right )+b (a+b)^{3/2} \sqrt{(\cosh (c)-\sinh (c))^2} \sinh (d x) \text{sech}^3(c+d x) (a \cosh (2 (c+d x))+a+2 b)+b (a+b)^{3/2} \sinh (c) \sqrt{(\cosh (c)-\sinh (c))^2} \text{sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b)\right )}{8 b^3 d (a+b)^{3/2} \sqrt{(\cosh (c)-\sinh (c))^2} \left (a+b \text{sech}^2(c+d x)\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c]*Sech[c + d*x]^3*(-(a^(3/2)*(4*a + 5*b)*ArcTan[(Sqrt[a + b]*Csch[c + d
*x]*Sqrt[(Cosh[c] - Sinh[c])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*Cosh[c]^2*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c
+ d*x]) + b*(a + b)^(3/2)*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*Sqrt[(Cosh[c] - Sinh[c])^2]*Sinh[c]
- Cosh[c]*Sech[c + d*x]*(2*Sqrt[a + b]*(4*a^2 + 3*a*b - b^2)*ArcTan[Tanh[(c + d*x)/2]]*(a + 2*b + a*Cosh[2*(c
+ d*x)])*Sqrt[(Cosh[c] - Sinh[c])^2] - a^(5/2)*b*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[c] - Sinh[c])^2
]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*(13 + 5*Cosh[2*(c + d*x)])*Sinh[c]) + a^(3/2)*ArcTan[(Sqrt[a + b]*Csch[c + d*x
]*Sqrt[(Cosh[c] - Sinh[c])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*(2*a^2 + 5*b^2 + 2*a^2*Cosh[2*(c + d*x)])*Sech[c +
d*x]*Sinh[2*c] + b*(a + b)^(3/2)*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^3*Sqrt[(Cosh[c] - Sinh[c])^2]*
Sinh[d*x] + 2*a^2*b*Sqrt[a + b]*Cosh[c]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[c + d*x]))/(8*b^3*(a + b)^(3/2)*d*(a
+ b*Sech[c + d*x]^2)^2*Sqrt[(Cosh[c] - Sinh[c])^2])

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Maple [B]  time = 0.067, size = 448, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d*a^2/b^2/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^
2*b+a+b)/(a+b)*tanh(1/2*d*x+1/2*c)^3+1/d*a^2/b^2/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d
*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)/(a+b)*tanh(1/2*d*x+1/2*c)+2/d/b^3*a^(5/2)/(a+b)^(3/2)*arctan(1/2*
(2*tanh(1/2*d*x+1/2*c)*(a+b)^(1/2)+2*b^(1/2))/a^(1/2))-2/d/b^3*a^(5/2)/(a+b)^(3/2)*arctan(1/2*(-2*tanh(1/2*d*x
+1/2*c)*(a+b)^(1/2)+2*b^(1/2))/a^(1/2))+5/2/d*a^(3/2)/b^2/(a+b)^(3/2)*arctan(1/2*(2*tanh(1/2*d*x+1/2*c)*(a+b)^
(1/2)+2*b^(1/2))/a^(1/2))-5/2/d/b^2*a^(3/2)/(a+b)^(3/2)*arctan(1/2*(-2*tanh(1/2*d*x+1/2*c)*(a+b)^(1/2)+2*b^(1/
2))/a^(1/2))-1/d/b^2/(tanh(1/2*d*x+1/2*c)^2+1)^2*tanh(1/2*d*x+1/2*c)^3+1/d/b^2/(tanh(1/2*d*x+1/2*c)^2+1)^2*tan
h(1/2*d*x+1/2*c)+1/d/b^2*arctan(tanh(1/2*d*x+1/2*c))-4/d/b^3*arctan(tanh(1/2*d*x+1/2*c))*a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (2 \, a^{2} e^{\left (7 \, c\right )} + a b e^{\left (7 \, c\right )}\right )} e^{\left (7 \, d x\right )} +{\left (2 \, a^{2} e^{\left (5 \, c\right )} + 5 \, a b e^{\left (5 \, c\right )} + 4 \, b^{2} e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} -{\left (2 \, a^{2} e^{\left (3 \, c\right )} + 5 \, a b e^{\left (3 \, c\right )} + 4 \, b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} -{\left (2 \, a^{2} e^{c} + a b e^{c}\right )} e^{\left (d x\right )}}{a^{2} b^{2} d + a b^{3} d +{\left (a^{2} b^{2} d e^{\left (8 \, c\right )} + a b^{3} d e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} + 4 \,{\left (a^{2} b^{2} d e^{\left (6 \, c\right )} + 2 \, a b^{3} d e^{\left (6 \, c\right )} + b^{4} d e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 2 \,{\left (3 \, a^{2} b^{2} d e^{\left (4 \, c\right )} + 7 \, a b^{3} d e^{\left (4 \, c\right )} + 4 \, b^{4} d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 4 \,{\left (a^{2} b^{2} d e^{\left (2 \, c\right )} + 2 \, a b^{3} d e^{\left (2 \, c\right )} + b^{4} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}} - \frac{{\left (4 \, a e^{c} - b e^{c}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) e^{\left (-c\right )}}{b^{3} d} + 128 \, \int \frac{{\left (4 \, a^{3} e^{\left (3 \, c\right )} + 5 \, a^{2} b e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} +{\left (4 \, a^{3} e^{c} + 5 \, a^{2} b e^{c}\right )} e^{\left (d x\right )}}{128 \,{\left (a^{2} b^{3} + a b^{4} +{\left (a^{2} b^{3} e^{\left (4 \, c\right )} + a b^{4} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (a^{2} b^{3} e^{\left (2 \, c\right )} + 3 \, a b^{4} e^{\left (2 \, c\right )} + 2 \, b^{5} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^7/(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

((2*a^2*e^(7*c) + a*b*e^(7*c))*e^(7*d*x) + (2*a^2*e^(5*c) + 5*a*b*e^(5*c) + 4*b^2*e^(5*c))*e^(5*d*x) - (2*a^2*
e^(3*c) + 5*a*b*e^(3*c) + 4*b^2*e^(3*c))*e^(3*d*x) - (2*a^2*e^c + a*b*e^c)*e^(d*x))/(a^2*b^2*d + a*b^3*d + (a^
2*b^2*d*e^(8*c) + a*b^3*d*e^(8*c))*e^(8*d*x) + 4*(a^2*b^2*d*e^(6*c) + 2*a*b^3*d*e^(6*c) + b^4*d*e^(6*c))*e^(6*
d*x) + 2*(3*a^2*b^2*d*e^(4*c) + 7*a*b^3*d*e^(4*c) + 4*b^4*d*e^(4*c))*e^(4*d*x) + 4*(a^2*b^2*d*e^(2*c) + 2*a*b^
3*d*e^(2*c) + b^4*d*e^(2*c))*e^(2*d*x)) - (4*a*e^c - b*e^c)*arctan(e^(d*x + c))*e^(-c)/(b^3*d) + 128*integrate
(1/128*((4*a^3*e^(3*c) + 5*a^2*b*e^(3*c))*e^(3*d*x) + (4*a^3*e^c + 5*a^2*b*e^c)*e^(d*x))/(a^2*b^3 + a*b^4 + (a
^2*b^3*e^(4*c) + a*b^4*e^(4*c))*e^(4*d*x) + 2*(a^2*b^3*e^(2*c) + 3*a*b^4*e^(2*c) + 2*b^5*e^(2*c))*e^(2*d*x)),
x)

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Fricas [B]  time = 3.42556, size = 15336, normalized size = 100.24 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^7/(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[1/4*(4*(2*a^2*b + a*b^2)*cosh(d*x + c)^7 + 28*(2*a^2*b + a*b^2)*cosh(d*x + c)*sinh(d*x + c)^6 + 4*(2*a^2*b +
a*b^2)*sinh(d*x + c)^7 + 4*(2*a^2*b + 5*a*b^2 + 4*b^3)*cosh(d*x + c)^5 + 4*(2*a^2*b + 5*a*b^2 + 4*b^3 + 21*(2*
a^2*b + a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 20*(7*(2*a^2*b + a*b^2)*cosh(d*x + c)^3 + (2*a^2*b + 5*a*b^2
+ 4*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 - 4*(2*a^2*b + 5*a*b^2 + 4*b^3)*cosh(d*x + c)^3 + 4*(35*(2*a^2*b + a*
b^2)*cosh(d*x + c)^4 - 2*a^2*b - 5*a*b^2 - 4*b^3 + 10*(2*a^2*b + 5*a*b^2 + 4*b^3)*cosh(d*x + c)^2)*sinh(d*x +
c)^3 + 4*(21*(2*a^2*b + a*b^2)*cosh(d*x + c)^5 + 10*(2*a^2*b + 5*a*b^2 + 4*b^3)*cosh(d*x + c)^3 - 3*(2*a^2*b +
5*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + ((4*a^3 + 5*a^2*b)*cosh(d*x + c)^8 + 8*(4*a^3 + 5*a^2*b)*co
sh(d*x + c)*sinh(d*x + c)^7 + (4*a^3 + 5*a^2*b)*sinh(d*x + c)^8 + 4*(4*a^3 + 9*a^2*b + 5*a*b^2)*cosh(d*x + c)^
6 + 4*(4*a^3 + 9*a^2*b + 5*a*b^2 + 7*(4*a^3 + 5*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(4*a^3 + 5*a^2*
b)*cosh(d*x + c)^3 + 3*(4*a^3 + 9*a^2*b + 5*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(12*a^3 + 31*a^2*b + 20*
a*b^2)*cosh(d*x + c)^4 + 2*(35*(4*a^3 + 5*a^2*b)*cosh(d*x + c)^4 + 12*a^3 + 31*a^2*b + 20*a*b^2 + 30*(4*a^3 +
9*a^2*b + 5*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(4*a^3 + 5*a^2*b)*cosh(d*x + c)^5 + 10*(4*a^3 + 9*a
^2*b + 5*a*b^2)*cosh(d*x + c)^3 + (12*a^3 + 31*a^2*b + 20*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*a^3 + 5*a^
2*b + 4*(4*a^3 + 9*a^2*b + 5*a*b^2)*cosh(d*x + c)^2 + 4*(7*(4*a^3 + 5*a^2*b)*cosh(d*x + c)^6 + 15*(4*a^3 + 9*a
^2*b + 5*a*b^2)*cosh(d*x + c)^4 + 4*a^3 + 9*a^2*b + 5*a*b^2 + 3*(12*a^3 + 31*a^2*b + 20*a*b^2)*cosh(d*x + c)^2
)*sinh(d*x + c)^2 + 8*((4*a^3 + 5*a^2*b)*cosh(d*x + c)^7 + 3*(4*a^3 + 9*a^2*b + 5*a*b^2)*cosh(d*x + c)^5 + (12
*a^3 + 31*a^2*b + 20*a*b^2)*cosh(d*x + c)^3 + (4*a^3 + 9*a^2*b + 5*a*b^2)*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)*sinh(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)) - 4*((4*a^3 + 3*a^2*b - a*b^2)*cosh(d*x + c)^8 + 8*(4*a^3 + 3*a^2*b - a*b^2)*cosh(d*x + c)*sinh(d*x + c)^
7 + (4*a^3 + 3*a^2*b - a*b^2)*sinh(d*x + c)^8 + 4*(4*a^3 + 7*a^2*b + 2*a*b^2 - b^3)*cosh(d*x + c)^6 + 4*(4*a^3
+ 7*a^2*b + 2*a*b^2 - b^3 + 7*(4*a^3 + 3*a^2*b - a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(4*a^3 + 3*a^
2*b - a*b^2)*cosh(d*x + c)^3 + 3*(4*a^3 + 7*a^2*b + 2*a*b^2 - b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(12*a^3
+ 25*a^2*b + 9*a*b^2 - 4*b^3)*cosh(d*x + c)^4 + 2*(35*(4*a^3 + 3*a^2*b - a*b^2)*cosh(d*x + c)^4 + 12*a^3 + 25*
a^2*b + 9*a*b^2 - 4*b^3 + 30*(4*a^3 + 7*a^2*b + 2*a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(4*a^3
+ 3*a^2*b - a*b^2)*cosh(d*x + c)^5 + 10*(4*a^3 + 7*a^2*b + 2*a*b^2 - b^3)*cosh(d*x + c)^3 + (12*a^3 + 25*a^2*b
+ 9*a*b^2 - 4*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*a^3 + 3*a^2*b - a*b^2 + 4*(4*a^3 + 7*a^2*b + 2*a*b^2 -
b^3)*cosh(d*x + c)^2 + 4*(7*(4*a^3 + 3*a^2*b - a*b^2)*cosh(d*x + c)^6 + 15*(4*a^3 + 7*a^2*b + 2*a*b^2 - b^3)*c
osh(d*x + c)^4 + 4*a^3 + 7*a^2*b + 2*a*b^2 - b^3 + 3*(12*a^3 + 25*a^2*b + 9*a*b^2 - 4*b^3)*cosh(d*x + c)^2)*si
nh(d*x + c)^2 + 8*((4*a^3 + 3*a^2*b - a*b^2)*cosh(d*x + c)^7 + 3*(4*a^3 + 7*a^2*b + 2*a*b^2 - b^3)*cosh(d*x +
c)^5 + (12*a^3 + 25*a^2*b + 9*a*b^2 - 4*b^3)*cosh(d*x + c)^3 + (4*a^3 + 7*a^2*b + 2*a*b^2 - b^3)*cosh(d*x + c)
)*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) - 4*(2*a^2*b + a*b^2)*cosh(d*x + c) + 4*(7*(2*a^2*b + a
*b^2)*cosh(d*x + c)^6 + 5*(2*a^2*b + 5*a*b^2 + 4*b^3)*cosh(d*x + c)^4 - 2*a^2*b - a*b^2 - 3*(2*a^2*b + 5*a*b^2
+ 4*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/((a^2*b^3 + a*b^4)*d*cosh(d*x + c)^8 + 8*(a^2*b^3 + a*b^4)*d*cosh(d*
x + c)*sinh(d*x + c)^7 + (a^2*b^3 + a*b^4)*d*sinh(d*x + c)^8 + 4*(a^2*b^3 + 2*a*b^4 + b^5)*d*cosh(d*x + c)^6 +
4*(7*(a^2*b^3 + a*b^4)*d*cosh(d*x + c)^2 + (a^2*b^3 + 2*a*b^4 + b^5)*d)*sinh(d*x + c)^6 + 2*(3*a^2*b^3 + 7*a*
b^4 + 4*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a^2*b^3 + a*b^4)*d*cosh(d*x + c)^3 + 3*(a^2*b^3 + 2*a*b^4 + b^5)*d*cosh
(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^2*b^3 + a*b^4)*d*cosh(d*x + c)^4 + 30*(a^2*b^3 + 2*a*b^4 + b^5)*d*cosh(d
*x + c)^2 + (3*a^2*b^3 + 7*a*b^4 + 4*b^5)*d)*sinh(d*x + c)^4 + 4*(a^2*b^3 + 2*a*b^4 + b^5)*d*cosh(d*x + c)^2 +
8*(7*(a^2*b^3 + a*b^4)*d*cosh(d*x + c)^5 + 10*(a^2*b^3 + 2*a*b^4 + b^5)*d*cosh(d*x + c)^3 + (3*a^2*b^3 + 7*a*
b^4 + 4*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^2*b^3 + a*b^4)*d*cosh(d*x + c)^6 + 15*(a^2*b^3 + 2*a*b
^4 + b^5)*d*cosh(d*x + c)^4 + 3*(3*a^2*b^3 + 7*a*b^4 + 4*b^5)*d*cosh(d*x + c)^2 + (a^2*b^3 + 2*a*b^4 + b^5)*d)
*sinh(d*x + c)^2 + (a^2*b^3 + a*b^4)*d + 8*((a^2*b^3 + a*b^4)*d*cosh(d*x + c)^7 + 3*(a^2*b^3 + 2*a*b^4 + b^5)*
d*cosh(d*x + c)^5 + (3*a^2*b^3 + 7*a*b^4 + 4*b^5)*d*cosh(d*x + c)^3 + (a^2*b^3 + 2*a*b^4 + b^5)*d*cosh(d*x + c
))*sinh(d*x + c)), 1/2*(2*(2*a^2*b + a*b^2)*cosh(d*x + c)^7 + 14*(2*a^2*b + a*b^2)*cosh(d*x + c)*sinh(d*x + c)
^6 + 2*(2*a^2*b + a*b^2)*sinh(d*x + c)^7 + 2*(2*a^2*b + 5*a*b^2 + 4*b^3)*cosh(d*x + c)^5 + 2*(2*a^2*b + 5*a*b^
2 + 4*b^3 + 21*(2*a^2*b + a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 10*(7*(2*a^2*b + a*b^2)*cosh(d*x + c)^3 +
(2*a^2*b + 5*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 - 2*(2*a^2*b + 5*a*b^2 + 4*b^3)*cosh(d*x + c)^3 + 2
*(35*(2*a^2*b + a*b^2)*cosh(d*x + c)^4 - 2*a^2*b - 5*a*b^2 - 4*b^3 + 10*(2*a^2*b + 5*a*b^2 + 4*b^3)*cosh(d*x +
c)^2)*sinh(d*x + c)^3 + 2*(21*(2*a^2*b + a*b^2)*cosh(d*x + c)^5 + 10*(2*a^2*b + 5*a*b^2 + 4*b^3)*cosh(d*x + c
)^3 - 3*(2*a^2*b + 5*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + ((4*a^3 + 5*a^2*b)*cosh(d*x + c)^8 + 8*(4
*a^3 + 5*a^2*b)*cosh(d*x + c)*sinh(d*x + c)^7 + (4*a^3 + 5*a^2*b)*sinh(d*x + c)^8 + 4*(4*a^3 + 9*a^2*b + 5*a*b
^2)*cosh(d*x + c)^6 + 4*(4*a^3 + 9*a^2*b + 5*a*b^2 + 7*(4*a^3 + 5*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*
(7*(4*a^3 + 5*a^2*b)*cosh(d*x + c)^3 + 3*(4*a^3 + 9*a^2*b + 5*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(12*a^
3 + 31*a^2*b + 20*a*b^2)*cosh(d*x + c)^4 + 2*(35*(4*a^3 + 5*a^2*b)*cosh(d*x + c)^4 + 12*a^3 + 31*a^2*b + 20*a*
b^2 + 30*(4*a^3 + 9*a^2*b + 5*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(4*a^3 + 5*a^2*b)*cosh(d*x + c)^5
+ 10*(4*a^3 + 9*a^2*b + 5*a*b^2)*cosh(d*x + c)^3 + (12*a^3 + 31*a^2*b + 20*a*b^2)*cosh(d*x + c))*sinh(d*x + c
)^3 + 4*a^3 + 5*a^2*b + 4*(4*a^3 + 9*a^2*b + 5*a*b^2)*cosh(d*x + c)^2 + 4*(7*(4*a^3 + 5*a^2*b)*cosh(d*x + c)^6
+ 15*(4*a^3 + 9*a^2*b + 5*a*b^2)*cosh(d*x + c)^4 + 4*a^3 + 9*a^2*b + 5*a*b^2 + 3*(12*a^3 + 31*a^2*b + 20*a*b^
2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((4*a^3 + 5*a^2*b)*cosh(d*x + c)^7 + 3*(4*a^3 + 9*a^2*b + 5*a*b^2)*cos
h(d*x + c)^5 + (12*a^3 + 31*a^2*b + 20*a*b^2)*cosh(d*x + c)^3 + (4*a^3 + 9*a^2*b + 5*a*b^2)*cosh(d*x + c))*sin
h(d*x + c))*sqrt(a/(a + b))*arctan(1/2*sqrt(a/(a + b))*(cosh(d*x + c) + sinh(d*x + c))) + ((4*a^3 + 5*a^2*b)*c
osh(d*x + c)^8 + 8*(4*a^3 + 5*a^2*b)*cosh(d*x + c)*sinh(d*x + c)^7 + (4*a^3 + 5*a^2*b)*sinh(d*x + c)^8 + 4*(4*
a^3 + 9*a^2*b + 5*a*b^2)*cosh(d*x + c)^6 + 4*(4*a^3 + 9*a^2*b + 5*a*b^2 + 7*(4*a^3 + 5*a^2*b)*cosh(d*x + c)^2)
*sinh(d*x + c)^6 + 8*(7*(4*a^3 + 5*a^2*b)*cosh(d*x + c)^3 + 3*(4*a^3 + 9*a^2*b + 5*a*b^2)*cosh(d*x + c))*sinh(
d*x + c)^5 + 2*(12*a^3 + 31*a^2*b + 20*a*b^2)*cosh(d*x + c)^4 + 2*(35*(4*a^3 + 5*a^2*b)*cosh(d*x + c)^4 + 12*a
^3 + 31*a^2*b + 20*a*b^2 + 30*(4*a^3 + 9*a^2*b + 5*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(4*a^3 + 5*a
^2*b)*cosh(d*x + c)^5 + 10*(4*a^3 + 9*a^2*b + 5*a*b^2)*cosh(d*x + c)^3 + (12*a^3 + 31*a^2*b + 20*a*b^2)*cosh(d
*x + c))*sinh(d*x + c)^3 + 4*a^3 + 5*a^2*b + 4*(4*a^3 + 9*a^2*b + 5*a*b^2)*cosh(d*x + c)^2 + 4*(7*(4*a^3 + 5*a
^2*b)*cosh(d*x + c)^6 + 15*(4*a^3 + 9*a^2*b + 5*a*b^2)*cosh(d*x + c)^4 + 4*a^3 + 9*a^2*b + 5*a*b^2 + 3*(12*a^3
+ 31*a^2*b + 20*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((4*a^3 + 5*a^2*b)*cosh(d*x + c)^7 + 3*(4*a^3 + 9
*a^2*b + 5*a*b^2)*cosh(d*x + c)^5 + (12*a^3 + 31*a^2*b + 20*a*b^2)*cosh(d*x + c)^3 + (4*a^3 + 9*a^2*b + 5*a*b^
2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a/(a + b))*arctan(1/2*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d*x +
c)^2 + a*sinh(d*x + c)^3 + (3*a + 4*b)*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 + 3*a + 4*b)*sinh(d*x + c))*sqrt(a
/(a + b))/a) - 2*((4*a^3 + 3*a^2*b - a*b^2)*cosh(d*x + c)^8 + 8*(4*a^3 + 3*a^2*b - a*b^2)*cosh(d*x + c)*sinh(d
*x + c)^7 + (4*a^3 + 3*a^2*b - a*b^2)*sinh(d*x + c)^8 + 4*(4*a^3 + 7*a^2*b + 2*a*b^2 - b^3)*cosh(d*x + c)^6 +
4*(4*a^3 + 7*a^2*b + 2*a*b^2 - b^3 + 7*(4*a^3 + 3*a^2*b - a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(4*a^
3 + 3*a^2*b - a*b^2)*cosh(d*x + c)^3 + 3*(4*a^3 + 7*a^2*b + 2*a*b^2 - b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*
(12*a^3 + 25*a^2*b + 9*a*b^2 - 4*b^3)*cosh(d*x + c)^4 + 2*(35*(4*a^3 + 3*a^2*b - a*b^2)*cosh(d*x + c)^4 + 12*a
^3 + 25*a^2*b + 9*a*b^2 - 4*b^3 + 30*(4*a^3 + 7*a^2*b + 2*a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7
*(4*a^3 + 3*a^2*b - a*b^2)*cosh(d*x + c)^5 + 10*(4*a^3 + 7*a^2*b + 2*a*b^2 - b^3)*cosh(d*x + c)^3 + (12*a^3 +
25*a^2*b + 9*a*b^2 - 4*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*a^3 + 3*a^2*b - a*b^2 + 4*(4*a^3 + 7*a^2*b + 2*
a*b^2 - b^3)*cosh(d*x + c)^2 + 4*(7*(4*a^3 + 3*a^2*b - a*b^2)*cosh(d*x + c)^6 + 15*(4*a^3 + 7*a^2*b + 2*a*b^2
- b^3)*cosh(d*x + c)^4 + 4*a^3 + 7*a^2*b + 2*a*b^2 - b^3 + 3*(12*a^3 + 25*a^2*b + 9*a*b^2 - 4*b^3)*cosh(d*x +
c)^2)*sinh(d*x + c)^2 + 8*((4*a^3 + 3*a^2*b - a*b^2)*cosh(d*x + c)^7 + 3*(4*a^3 + 7*a^2*b + 2*a*b^2 - b^3)*cos
h(d*x + c)^5 + (12*a^3 + 25*a^2*b + 9*a*b^2 - 4*b^3)*cosh(d*x + c)^3 + (4*a^3 + 7*a^2*b + 2*a*b^2 - b^3)*cosh(
d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) - 2*(2*a^2*b + a*b^2)*cosh(d*x + c) + 2*(7*(2*a
^2*b + a*b^2)*cosh(d*x + c)^6 + 5*(2*a^2*b + 5*a*b^2 + 4*b^3)*cosh(d*x + c)^4 - 2*a^2*b - a*b^2 - 3*(2*a^2*b +
5*a*b^2 + 4*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/((a^2*b^3 + a*b^4)*d*cosh(d*x + c)^8 + 8*(a^2*b^3 + a*b^4)*d
*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2*b^3 + a*b^4)*d*sinh(d*x + c)^8 + 4*(a^2*b^3 + 2*a*b^4 + b^5)*d*cosh(d*x
+ c)^6 + 4*(7*(a^2*b^3 + a*b^4)*d*cosh(d*x + c)^2 + (a^2*b^3 + 2*a*b^4 + b^5)*d)*sinh(d*x + c)^6 + 2*(3*a^2*b^
3 + 7*a*b^4 + 4*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a^2*b^3 + a*b^4)*d*cosh(d*x + c)^3 + 3*(a^2*b^3 + 2*a*b^4 + b^5
)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^2*b^3 + a*b^4)*d*cosh(d*x + c)^4 + 30*(a^2*b^3 + 2*a*b^4 + b^5)*
d*cosh(d*x + c)^2 + (3*a^2*b^3 + 7*a*b^4 + 4*b^5)*d)*sinh(d*x + c)^4 + 4*(a^2*b^3 + 2*a*b^4 + b^5)*d*cosh(d*x
+ c)^2 + 8*(7*(a^2*b^3 + a*b^4)*d*cosh(d*x + c)^5 + 10*(a^2*b^3 + 2*a*b^4 + b^5)*d*cosh(d*x + c)^3 + (3*a^2*b^
3 + 7*a*b^4 + 4*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^2*b^3 + a*b^4)*d*cosh(d*x + c)^6 + 15*(a^2*b^3
+ 2*a*b^4 + b^5)*d*cosh(d*x + c)^4 + 3*(3*a^2*b^3 + 7*a*b^4 + 4*b^5)*d*cosh(d*x + c)^2 + (a^2*b^3 + 2*a*b^4 +
b^5)*d)*sinh(d*x + c)^2 + (a^2*b^3 + a*b^4)*d + 8*((a^2*b^3 + a*b^4)*d*cosh(d*x + c)^7 + 3*(a^2*b^3 + 2*a*b^4
+ b^5)*d*cosh(d*x + c)^5 + (3*a^2*b^3 + 7*a*b^4 + 4*b^5)*d*cosh(d*x + c)^3 + (a^2*b^3 + 2*a*b^4 + b^5)*d*cosh
(d*x + c))*sinh(d*x + c))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^7/(a+b*sech(d*x+c)^2)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError