3.1.81 \(\int \frac {\text {sech}^5(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx\) [81]

Optimal. Leaf size=86 \[ -\frac {(2 a-b) \text {ArcTan}(\sinh (c+d x))}{2 b^2 d}+\frac {a^{3/2} \text {ArcTan}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{b^2 \sqrt {a+b} d}+\frac {\text {sech}(c+d x) \tanh (c+d x)}{2 b d} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 86, normalized size of antiderivative = 1.00, 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 = {4232, 425, 536, 209, 211} \begin {gather*} \frac {a^{3/2} \text {ArcTan}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{b^2 d \sqrt {a+b}}-\frac {(2 a-b) \text {ArcTan}(\sinh (c+d x))}{2 b^2 d}+\frac {\tanh (c+d x) \text {sech}(c+d x)}{2 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(213\) vs. \(2(86)=172\).
time = 1.27, size = 213, normalized size = 2.48 \begin {gather*} \frac {\cosh (c) (a+2 b+a \cosh (2 (c+d x))) \text {sech}^2(c+d x) \left (b \sqrt {a+b} \text {sech}^2(c) \text {sech}^2(c+d x) \sqrt {(\cosh (c)-\sinh (c))^2} \sinh (d x)+2 a^{3/2} \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 ) (-1+\tanh (c))-\sqrt {a+b} \text {sech}(c) \sqrt {(\cosh (c)-\sinh (c))^2} \left (2 (2 a-b) \text {ArcTan}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )-b \text {sech}(c+d x) \tanh (c)\right )\right )}{4 b^2 \sqrt {a+b} d \left (a+b \text {sech}^2(c+d x)\right ) \sqrt {(\cosh (c)-\sinh (c))^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Cosh[c]*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*(b*Sqrt[a + b]*Sech[c]^2*Sech[c + d*x]^2*Sqrt[(Cosh[c
] - Sinh[c])^2]*Sinh[d*x] + 2*a^(3/2)*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[c] - Sinh[c])^2]*(Cosh[c] +
Sinh[c]))/Sqrt[a]]*(-1 + Tanh[c]) - Sqrt[a + b]*Sech[c]*Sqrt[(Cosh[c] - Sinh[c])^2]*(2*(2*a - b)*ArcTan[Tanh[
(c + d*x)/2]] - b*Sech[c + d*x]*Tanh[c])))/(4*b^2*Sqrt[a + b]*d*(a + b*Sech[c + d*x]^2)*Sqrt[(Cosh[c] - Sinh[c
])^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(158\) vs. \(2(74)=148\).
time = 1.95, size = 159, normalized size = 1.85

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (\frac {\frac {b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}}{\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (-b +2 a \right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{2}}+\frac {2 a^{2} \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}\right )}{b^{2}}}{d}\)	\(159\)
default	\(\frac {-\frac {2 \left (\frac {\frac {b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}}{\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (-b +2 a \right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{2}}+\frac {2 a^{2} \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}\right )}{b^{2}}}{d}\)	\(159\)
risch	\(\frac {{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d b \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right )}{2 d b}+\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{d \,b^{2}}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right )}{2 d b}-\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{d \,b^{2}}+\frac {\sqrt {-a \left (a +b \right )}\, a \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a}-1\right )}{2 \left (a +b \right ) d \,b^{2}}-\frac {\sqrt {-a \left (a +b \right )}\, a \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a \left (a +b \right )}\, {\mathrm e}^{d x +c}}{a}-1\right )}{2 \left (a +b \right ) d \,b^{2}}\)	\(223\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-2/b^2*((1/2*b*tanh(1/2*d*x+1/2*c)^3-1/2*b*tanh(1/2*d*x+1/2*c))/(tanh(1/2*d*x+1/2*c)^2+1)^2+1/2*(-b+2*a)*
arctan(tanh(1/2*d*x+1/2*c)))+2*a^2/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)^5/(a+b*sech(d*x+c)^2),x, algorithm="maxima")

[Out]

(e^(3*d*x + 3*c) - e^(d*x + c))/(b*d*e^(4*d*x + 4*c) + 2*b*d*e^(2*d*x + 2*c) + b*d) - (2*a*e^c - b*e^c)*arctan
(e^(d*x + c))*e^(-c)/(b^2*d) + 32*integrate(1/16*(a^2*e^(3*d*x + 3*c) + a^2*e^(d*x + c))/(a*b^2*e^(4*d*x + 4*c
) + a*b^2 + 2*(a*b^2*e^(2*c) + 2*b^3*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. 706 vs. \(2 (74) = 148\).
time = 0.40, size = 1518, normalized size = 17.65 \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*sech(d*x+c)^2),x, algorithm="fricas")

[Out]

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

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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 \operatorname {sech}^{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*sech(d*x+c)**2),x)

[Out]

Integral(sech(c + d*x)**5/(a + b*sech(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*sech(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 = 3.50, size = 946, normalized size = 11.00 \begin {gather*} \frac {\sqrt {a^3}\,\left (2\,\mathrm {atan}\left (\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {64\,\left (6\,b^3\,d\,{\left (a^3\right )}^{3/2}+2\,b^6\,d\,\sqrt {a^3}+6\,a\,b^2\,d\,{\left (a^3\right )}^{3/2}-4\,a\,b^5\,d\,\sqrt {a^3}-6\,a^2\,b^4\,d\,\sqrt {a^3}\right )}{a^4\,b^4\,\left (a+b\right )\,\left (b^2+a\,b\right )\,\sqrt {b^5\,d^2+a\,b^4\,d^2}\,\sqrt {b^4\,d^2\,\left (a+b\right )}\,\left (3\,a^3-3\,a\,b^2+b^3\right )}-\frac {32\,\left (3\,a^5\,\sqrt {b^5\,d^2+a\,b^4\,d^2}+a^2\,b^3\,\sqrt {b^5\,d^2+a\,b^4\,d^2}-3\,a^3\,b^2\,\sqrt {b^5\,d^2+a\,b^4\,d^2}\right )}{a^2\,b^6\,d\,{\left (a+b\right )}^2\,\left (b^2+a\,b\right )\,\sqrt {b^5\,d^2+a\,b^4\,d^2}\,\sqrt {a^3}\,\left (3\,a^3-3\,a\,b^2+b^3\right )}\right )+\frac {32\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\left (3\,a^5\,\sqrt {b^5\,d^2+a\,b^4\,d^2}+a^2\,b^3\,\sqrt {b^5\,d^2+a\,b^4\,d^2}-3\,a^3\,b^2\,\sqrt {b^5\,d^2+a\,b^4\,d^2}\right )}{a^2\,b^6\,d\,{\left (a+b\right )}^2\,\left (b^2+a\,b\right )\,\sqrt {b^5\,d^2+a\,b^4\,d^2}\,\sqrt {a^3}\,\left (3\,a^3-3\,a\,b^2+b^3\right )}\right )\,\left (\frac {a^2\,b^7\,\sqrt {b^5\,d^2+a\,b^4\,d^2}}{64}+\frac {a^3\,b^6\,\sqrt {b^5\,d^2+a\,b^4\,d^2}}{32}+\frac {a^4\,b^5\,\sqrt {b^5\,d^2+a\,b^4\,d^2}}{64}\right )\right )+2\,\mathrm {atan}\left (\frac {a^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {b^4\,d^2\,\left (a+b\right )}}{2\,b^2\,d\,\left (a+b\right )\,\sqrt {a^3}}\right )\right )}{2\,\sqrt {b^5\,d^2+a\,b^4\,d^2}}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (18\,a^7\,\sqrt {b^4\,d^2}-b^7\,\sqrt {b^4\,d^2}-21\,a^2\,b^5\,\sqrt {b^4\,d^2}+12\,a^3\,b^4\,\sqrt {b^4\,d^2}+30\,a^4\,b^3\,\sqrt {b^4\,d^2}-36\,a^5\,b^2\,\sqrt {b^4\,d^2}+8\,a\,b^6\,\sqrt {b^4\,d^2}-9\,a^6\,b\,\sqrt {b^4\,d^2}\right )}{b^8\,d\,\sqrt {4\,a^2-4\,a\,b+b^2}+9\,a^2\,b^6\,d\,\sqrt {4\,a^2-4\,a\,b+b^2}+6\,a^3\,b^5\,d\,\sqrt {4\,a^2-4\,a\,b+b^2}-18\,a^4\,b^4\,d\,\sqrt {4\,a^2-4\,a\,b+b^2}+9\,a^6\,b^2\,d\,\sqrt {4\,a^2-4\,a\,b+b^2}-6\,a\,b^7\,d\,\sqrt {4\,a^2-4\,a\,b+b^2}}\right )\,\sqrt {4\,a^2-4\,a\,b+b^2}}{\sqrt {b^4\,d^2}}+\frac {{\mathrm {e}}^{c+d\,x}}{b\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}}{b\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a^3)^(1/2)*(2*atan((exp(d*x)*exp(c)*((64*(6*b^3*d*(a^3)^(3/2) + 2*b^6*d*(a^3)^(1/2) + 6*a*b^2*d*(a^3)^(3/2)
- 4*a*b^5*d*(a^3)^(1/2) - 6*a^2*b^4*d*(a^3)^(1/2)))/(a^4*b^4*(a + b)*(a*b + b^2)*(b^5*d^2 + a*b^4*d^2)^(1/2)*(
b^4*d^2*(a + b))^(1/2)*(3*a^3 - 3*a*b^2 + b^3)) - (32*(3*a^5*(b^5*d^2 + a*b^4*d^2)^(1/2) + a^2*b^3*(b^5*d^2 +
a*b^4*d^2)^(1/2) - 3*a^3*b^2*(b^5*d^2 + a*b^4*d^2)^(1/2)))/(a^2*b^6*d*(a + b)^2*(a*b + b^2)*(b^5*d^2 + a*b^4*d
^2)^(1/2)*(a^3)^(1/2)*(3*a^3 - 3*a*b^2 + b^3))) + (32*exp(3*c)*exp(3*d*x)*(3*a^5*(b^5*d^2 + a*b^4*d^2)^(1/2) +
a^2*b^3*(b^5*d^2 + a*b^4*d^2)^(1/2) - 3*a^3*b^2*(b^5*d^2 + a*b^4*d^2)^(1/2)))/(a^2*b^6*d*(a + b)^2*(a*b + b^2
)*(b^5*d^2 + a*b^4*d^2)^(1/2)*(a^3)^(1/2)*(3*a^3 - 3*a*b^2 + b^3)))*((a^2*b^7*(b^5*d^2 + a*b^4*d^2)^(1/2))/64
+ (a^3*b^6*(b^5*d^2 + a*b^4*d^2)^(1/2))/32 + (a^4*b^5*(b^5*d^2 + a*b^4*d^2)^(1/2))/64)) + 2*atan((a^2*exp(d*x)
*exp(c)*(b^4*d^2*(a + b))^(1/2))/(2*b^2*d*(a + b)*(a^3)^(1/2)))))/(2*(b^5*d^2 + a*b^4*d^2)^(1/2)) - (atan((exp
(d*x)*exp(c)*(18*a^7*(b^4*d^2)^(1/2) - b^7*(b^4*d^2)^(1/2) - 21*a^2*b^5*(b^4*d^2)^(1/2) + 12*a^3*b^4*(b^4*d^2)
^(1/2) + 30*a^4*b^3*(b^4*d^2)^(1/2) - 36*a^5*b^2*(b^4*d^2)^(1/2) + 8*a*b^6*(b^4*d^2)^(1/2) - 9*a^6*b*(b^4*d^2)
^(1/2)))/(b^8*d*(4*a^2 - 4*a*b + b^2)^(1/2) + 9*a^2*b^6*d*(4*a^2 - 4*a*b + b^2)^(1/2) + 6*a^3*b^5*d*(4*a^2 - 4
*a*b + b^2)^(1/2) - 18*a^4*b^4*d*(4*a^2 - 4*a*b + b^2)^(1/2) + 9*a^6*b^2*d*(4*a^2 - 4*a*b + b^2)^(1/2) - 6*a*b
^7*d*(4*a^2 - 4*a*b + b^2)^(1/2)))*(4*a^2 - 4*a*b + b^2)^(1/2))/(b^4*d^2)^(1/2) + exp(c + d*x)/(b*d*(exp(2*c +
2*d*x) + 1)) - (2*exp(c + d*x))/(b*d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1))

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