3.90 \(\int \frac{\text{sech}^5(c+d x)}{(a+b \text{sech}^2(c+d x))^2} \, dx\)

Optimal. Leaf size=101 \[ -\frac{\sqrt{a} (2 a+3 b) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{2 b^2 d (a+b)^{3/2}}-\frac{a \sinh (c+d x)}{2 b d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )}+\frac{\tan ^{-1}(\sinh (c+d x))}{b^2 d} \]

[Out]

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

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Rubi [A]  time = 0.116827, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {4147, 414, 522, 203, 205} \[ -\frac{\sqrt{a} (2 a+3 b) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{2 b^2 d (a+b)^{3/2}}-\frac{a \sinh (c+d x)}{2 b d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )}+\frac{\tan ^{-1}(\sinh (c+d x))}{b^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [B]  time = 2.46963, size = 282, normalized size = 2.79 \[ \frac{\text{sech}^3(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (-2 a b \sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} \tanh (c+d x)+\sqrt{a} (2 a+3 b) \cosh (c) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b) \tan ^{-1}\left (\frac{\sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text{csch}(c+d x)}{\sqrt{a}}\right )-\text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (\sqrt{a} (2 a+3 b) \sinh (c) \tan ^{-1}\left (\frac{\sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text{csch}(c+d x)}{\sqrt{a}}\right )-4 (a+b)^{3/2} \sqrt{(\cosh (c)-\sinh (c))^2} \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{8 b^2 d (a+b)^{3/2} \sqrt{(\cosh (c)-\sinh (c))^2} \left (a+b \text{sech}^2(c+d x)\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^3*(Sqrt[a]*(2*a + 3*b)*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(
Cosh[c] - Sinh[c])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*Cosh[c]*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x] - (a
+ 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]*(-4*(a + b)^(3/2)*ArcTan[Tanh[(c + d*x)/2]]*Sqrt[(Cosh[c] - Sinh[c
])^2] + Sqrt[a]*(2*a + 3*b)*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[c] - Sinh[c])^2]*(Cosh[c] + Sinh[c]))
/Sqrt[a]]*Sinh[c]) - 2*a*b*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[c + d*x]))/(8*b^2*(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 = 0.064, size = 361, normalized size = 3.6 \begin{align*}{\frac{a}{bd \left ( a+b \right ) } \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}a+b \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a-2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a+b \right ) ^{-1}}-{\frac{a}{bd \left ( a+b \right ) }\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}a+b \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a-2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a+b \right ) ^{-1}}-{\frac{1}{d{b}^{2}}{a}^{{\frac{3}{2}}}\arctan \left ({\frac{1}{2} \left ( 2\,\tanh \left ( 1/2\,dx+c/2 \right ) \sqrt{a+b}+2\,\sqrt{b} \right ){\frac{1}{\sqrt{a}}}} \right ) \left ( a+b \right ) ^{-{\frac{3}{2}}}}-{\frac{1}{d{b}^{2}}{a}^{{\frac{3}{2}}}\arctan \left ({\frac{1}{2} \left ( 2\,\tanh \left ( 1/2\,dx+c/2 \right ) \sqrt{a+b}-2\,\sqrt{b} \right ){\frac{1}{\sqrt{a}}}} \right ) \left ( a+b \right ) ^{-{\frac{3}{2}}}}-{\frac{3}{2\,bd}\sqrt{a}\arctan \left ({\frac{1}{2} \left ( 2\,\tanh \left ( 1/2\,dx+c/2 \right ) \sqrt{a+b}+2\,\sqrt{b} \right ){\frac{1}{\sqrt{a}}}} \right ) \left ( a+b \right ) ^{-{\frac{3}{2}}}}-{\frac{3}{2\,bd}\sqrt{a}\arctan \left ({\frac{1}{2} \left ( 2\,\tanh \left ( 1/2\,dx+c/2 \right ) \sqrt{a+b}-2\,\sqrt{b} \right ){\frac{1}{\sqrt{a}}}} \right ) \left ( a+b \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{\arctan \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) }{d{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a e^{\left (3 \, d x + 3 \, c\right )} - a e^{\left (d x + c\right )}}{a^{2} b d + a b^{2} d +{\left (a^{2} b d e^{\left (4 \, c\right )} + a b^{2} d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (a^{2} b d e^{\left (2 \, c\right )} + 3 \, a b^{2} d e^{\left (2 \, c\right )} + 2 \, b^{3} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}} + \frac{2 \, \arctan \left (e^{\left (d x + c\right )}\right )}{b^{2} d} - 32 \, \int \frac{{\left (2 \, a^{2} e^{\left (3 \, c\right )} + 3 \, a b e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} +{\left (2 \, a^{2} e^{c} + 3 \, a b e^{c}\right )} e^{\left (d x\right )}}{32 \,{\left (a^{2} b^{2} + a b^{3} +{\left (a^{2} b^{2} e^{\left (4 \, c\right )} + a b^{3} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (a^{2} b^{2} e^{\left (2 \, c\right )} + 3 \, a b^{3} e^{\left (2 \, c\right )} + 2 \, b^{4} 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)^5/(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

-(a*e^(3*d*x + 3*c) - a*e^(d*x + c))/(a^2*b*d + a*b^2*d + (a^2*b*d*e^(4*c) + a*b^2*d*e^(4*c))*e^(4*d*x) + 2*(a
^2*b*d*e^(2*c) + 3*a*b^2*d*e^(2*c) + 2*b^3*d*e^(2*c))*e^(2*d*x)) + 2*arctan(e^(d*x + c))/(b^2*d) - 32*integrat
e(1/32*((2*a^2*e^(3*c) + 3*a*b*e^(3*c))*e^(3*d*x) + (2*a^2*e^c + 3*a*b*e^c)*e^(d*x))/(a^2*b^2 + a*b^3 + (a^2*b
^2*e^(4*c) + a*b^3*e^(4*c))*e^(4*d*x) + 2*(a^2*b^2*e^(2*c) + 3*a*b^3*e^(2*c) + 2*b^4*e^(2*c))*e^(2*d*x)), x)

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Fricas [B]  time = 2.67491, size = 5195, normalized size = 51.44 \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)^5/(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[-1/4*(4*a*b*cosh(d*x + c)^3 + 12*a*b*cosh(d*x + c)*sinh(d*x + c)^2 + 4*a*b*sinh(d*x + c)^3 - 4*a*b*cosh(d*x +
c) - ((2*a^2 + 3*a*b)*cosh(d*x + c)^4 + 4*(2*a^2 + 3*a*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a^2 + 3*a*b)*sin
h(d*x + c)^4 + 2*(2*a^2 + 7*a*b + 6*b^2)*cosh(d*x + c)^2 + 2*(3*(2*a^2 + 3*a*b)*cosh(d*x + c)^2 + 2*a^2 + 7*a*
b + 6*b^2)*sinh(d*x + c)^2 + 2*a^2 + 3*a*b + 4*((2*a^2 + 3*a*b)*cosh(d*x + c)^3 + (2*a^2 + 7*a*b + 6*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)*s
inh(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)) - 8*((a^2 + a*b)*cosh(d*x + c)^4 + 4*(a^2 + a*b)*cosh(d*x + c)*sinh(d*x +
c)^3 + (a^2 + a*b)*sinh(d*x + c)^4 + 2*(a^2 + 3*a*b + 2*b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 + a*b)*cosh(d*x + c)
^2 + a^2 + 3*a*b + 2*b^2)*sinh(d*x + c)^2 + a^2 + a*b + 4*((a^2 + a*b)*cosh(d*x + c)^3 + (a^2 + 3*a*b + 2*b^2)
*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) + 4*(3*a*b*cosh(d*x + c)^2 - a*b)*sinh(d*
x + c))/((a^2*b^2 + a*b^3)*d*cosh(d*x + c)^4 + 4*(a^2*b^2 + a*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2*b^2
+ a*b^3)*d*sinh(d*x + c)^4 + 2*(a^2*b^2 + 3*a*b^3 + 2*b^4)*d*cosh(d*x + c)^2 + 2*(3*(a^2*b^2 + a*b^3)*d*cosh(d
*x + c)^2 + (a^2*b^2 + 3*a*b^3 + 2*b^4)*d)*sinh(d*x + c)^2 + (a^2*b^2 + a*b^3)*d + 4*((a^2*b^2 + a*b^3)*d*cosh
(d*x + c)^3 + (a^2*b^2 + 3*a*b^3 + 2*b^4)*d*cosh(d*x + c))*sinh(d*x + c)), -1/2*(2*a*b*cosh(d*x + c)^3 + 6*a*b
*cosh(d*x + c)*sinh(d*x + c)^2 + 2*a*b*sinh(d*x + c)^3 - 2*a*b*cosh(d*x + c) + ((2*a^2 + 3*a*b)*cosh(d*x + c)^
4 + 4*(2*a^2 + 3*a*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a^2 + 3*a*b)*sinh(d*x + c)^4 + 2*(2*a^2 + 7*a*b + 6*b
^2)*cosh(d*x + c)^2 + 2*(3*(2*a^2 + 3*a*b)*cosh(d*x + c)^2 + 2*a^2 + 7*a*b + 6*b^2)*sinh(d*x + c)^2 + 2*a^2 +
3*a*b + 4*((2*a^2 + 3*a*b)*cosh(d*x + c)^3 + (2*a^2 + 7*a*b + 6*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a/(a +
b))*arctan(1/2*sqrt(a/(a + b))*(cosh(d*x + c) + sinh(d*x + c))) + ((2*a^2 + 3*a*b)*cosh(d*x + c)^4 + 4*(2*a^2
+ 3*a*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a^2 + 3*a*b)*sinh(d*x + c)^4 + 2*(2*a^2 + 7*a*b + 6*b^2)*cosh(d*x
+ c)^2 + 2*(3*(2*a^2 + 3*a*b)*cosh(d*x + c)^2 + 2*a^2 + 7*a*b + 6*b^2)*sinh(d*x + c)^2 + 2*a^2 + 3*a*b + 4*((
2*a^2 + 3*a*b)*cosh(d*x + c)^3 + (2*a^2 + 7*a*b + 6*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) - 4*((a^2 + a*b)*cosh(d*x + c)^4 + 4*(a^2 +
a*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + a*b)*sinh(d*x + c)^4 + 2*(a^2 + 3*a*b + 2*b^2)*cosh(d*x + c)^2 +
2*(3*(a^2 + a*b)*cosh(d*x + c)^2 + a^2 + 3*a*b + 2*b^2)*sinh(d*x + c)^2 + a^2 + a*b + 4*((a^2 + a*b)*cosh(d*x
+ c)^3 + (a^2 + 3*a*b + 2*b^2)*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) + 2*(3*a*b*
cosh(d*x + c)^2 - a*b)*sinh(d*x + c))/((a^2*b^2 + a*b^3)*d*cosh(d*x + c)^4 + 4*(a^2*b^2 + a*b^3)*d*cosh(d*x +
c)*sinh(d*x + c)^3 + (a^2*b^2 + a*b^3)*d*sinh(d*x + c)^4 + 2*(a^2*b^2 + 3*a*b^3 + 2*b^4)*d*cosh(d*x + c)^2 + 2
*(3*(a^2*b^2 + a*b^3)*d*cosh(d*x + c)^2 + (a^2*b^2 + 3*a*b^3 + 2*b^4)*d)*sinh(d*x + c)^2 + (a^2*b^2 + a*b^3)*d
+ 4*((a^2*b^2 + a*b^3)*d*cosh(d*x + c)^3 + (a^2*b^2 + 3*a*b^3 + 2*b^4)*d*cosh(d*x + c))*sinh(d*x + c))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)^5/(a+b*sech(d*x+c)^2)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError