3.1.73 \(\int \frac {\cosh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx\) [73]

Optimal. Leaf size=117 \[ \frac {\left (3 a^2-4 a b+8 b^2\right ) x}{8 a^3}-\frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a^3 \sqrt {a+b} d}+\frac {(3 a-4 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d} \]

[Out]

1/8*(3*a^2-4*a*b+8*b^2)*x/a^3+1/8*(3*a-4*b)*cosh(d*x+c)*sinh(d*x+c)/a^2/d+1/4*cosh(d*x+c)^3*sinh(d*x+c)/a/d-b^
(5/2)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))/a^3/d/(a+b)^(1/2)

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Rubi [A]
time = 0.15, antiderivative size = 117, 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 = {4231, 425, 541, 536, 212, 214} \begin {gather*} -\frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a^3 d \sqrt {a+b}}+\frac {(3 a-4 b) \sinh (c+d x) \cosh (c+d x)}{8 a^2 d}+\frac {x \left (3 a^2-4 a b+8 b^2\right )}{8 a^3}+\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]^4/(a + b*Sech[c + d*x]^2),x]

[Out]

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

Rule 212

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

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[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 4231

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

Rubi steps

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

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Mathematica [A]
time = 0.36, size = 95, normalized size = 0.81 \begin {gather*} \frac {4 \left (3 a^2-4 a b+8 b^2\right ) (c+d x)-\frac {32 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{\sqrt {a+b}}+8 a (a-b) \sinh (2 (c+d x))+a^2 \sinh (4 (c+d x))}{32 a^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]^4/(a + b*Sech[c + d*x]^2),x]

[Out]

(4*(3*a^2 - 4*a*b + 8*b^2)*(c + d*x) - (32*b^(5/2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/Sqrt[a + b] +
8*a*(a - b)*Sinh[2*(c + d*x)] + a^2*Sinh[4*(c + d*x)])/(32*a^3*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(346\) vs. \(2(103)=206\).
time = 3.35, size = 347, normalized size = 2.97

method	result	size
risch	\(\frac {3 x}{8 a}-\frac {b x}{2 a^{2}}+\frac {x \,b^{2}}{a^{3}}+\frac {{\mathrm e}^{4 d x +4 c}}{64 a d}+\frac {{\mathrm e}^{2 d x +2 c}}{8 a d}-\frac {{\mathrm e}^{2 d x +2 c} b}{8 a^{2} d}-\frac {{\mathrm e}^{-2 d x -2 c}}{8 a d}+\frac {{\mathrm e}^{-2 d x -2 c} b}{8 a^{2} d}-\frac {{\mathrm e}^{-4 d x -4 c}}{64 a d}+\frac {\sqrt {b \left (a +b \right )}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {b \left (a +b \right )}+a +2 b}{a}\right )}{2 \left (a +b \right ) d \,a^{3}}-\frac {\sqrt {b \left (a +b \right )}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {b \left (a +b \right )}-a -2 b}{a}\right )}{2 \left (a +b \right ) d \,a^{3}}\)	\(234\)
derivativedivides	\(\frac {\frac {2 b^{3} \left (-\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{a^{3}}+\frac {1}{4 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-7 a +4 b}{8 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-5 a +4 b}{8 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-3 a^{2}+4 a b -8 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 a^{3}}-\frac {1}{4 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{2 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-5 a +4 b}{8 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {7 a -4 b}{8 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (3 a^{2}-4 a b +8 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 a^{3}}}{d}\)	\(347\)
default	\(\frac {\frac {2 b^{3} \left (-\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{a^{3}}+\frac {1}{4 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-7 a +4 b}{8 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-5 a +4 b}{8 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-3 a^{2}+4 a b -8 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 a^{3}}-\frac {1}{4 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{2 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-5 a +4 b}{8 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {7 a -4 b}{8 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (3 a^{2}-4 a b +8 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 a^{3}}}{d}\)	\(347\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^4/(a+b*sech(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

1/d*(2*b^3/a^3*(-1/4/b^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a
+b)^(1/2))+1/4/b^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2-2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1
/2)))+1/4/a/(tanh(1/2*d*x+1/2*c)-1)^4+1/2/a/(tanh(1/2*d*x+1/2*c)-1)^3-1/8*(-7*a+4*b)/a^2/(tanh(1/2*d*x+1/2*c)-
1)^2-1/8*(-5*a+4*b)/a^2/(tanh(1/2*d*x+1/2*c)-1)+1/8/a^3*(-3*a^2+4*a*b-8*b^2)*ln(tanh(1/2*d*x+1/2*c)-1)-1/4/a/(
tanh(1/2*d*x+1/2*c)+1)^4+1/2/a/(tanh(1/2*d*x+1/2*c)+1)^3-1/8*(-5*a+4*b)/a^2/(tanh(1/2*d*x+1/2*c)+1)-1/8*(7*a-4
*b)/a^2/(tanh(1/2*d*x+1/2*c)+1)^2+1/8*(3*a^2-4*a*b+8*b^2)/a^3*ln(tanh(1/2*d*x+1/2*c)+1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (103) = 206\).
time = 0.54, size = 526, normalized size = 4.50 \begin {gather*} \frac {3 \, b \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{16 \, \sqrt {{\left (a + b\right )} b} a d} + \frac {3 \, {\left (d x + c\right )}}{8 \, a d} - \frac {{\left (8 \, b e^{\left (-2 \, d x - 2 \, c\right )} - a\right )} e^{\left (4 \, d x + 4 \, c\right )}}{64 \, a^{2} d} + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{8 \, a d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, a d} - \frac {b \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (a + 2 \, b\right )} e^{\left (2 \, d x + 2 \, c\right )} + a\right )}{4 \, a^{2} d} + \frac {b \log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{4 \, a^{2} d} + \frac {{\left (a b + 2 \, b^{2}\right )} \log \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{8 \, \sqrt {{\left (a + b\right )} b} a^{2} d} - \frac {{\left (a b + 2 \, b^{2}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{8 \, \sqrt {{\left (a + b\right )} b} a^{2} d} + \frac {{\left (a b + 2 \, b^{2}\right )} {\left (d x + c\right )}}{2 \, a^{3} d} + \frac {8 \, b e^{\left (-2 \, d x - 2 \, c\right )} - a e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, a^{2} d} + \frac {{\left (a^{2} b + 8 \, a b^{2} + 8 \, b^{3}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{16 \, \sqrt {{\left (a + b\right )} b} a^{3} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^4/(a+b*sech(d*x+c)^2),x, algorithm="maxima")

[Out]

3/16*b*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b
)))/(sqrt((a + b)*b)*a*d) + 3/8*(d*x + c)/(a*d) - 1/64*(8*b*e^(-2*d*x - 2*c) - a)*e^(4*d*x + 4*c)/(a^2*d) + 1/
8*e^(2*d*x + 2*c)/(a*d) - 1/8*e^(-2*d*x - 2*c)/(a*d) - 1/4*b*log(a*e^(4*d*x + 4*c) + 2*(a + 2*b)*e^(2*d*x + 2*
c) + a)/(a^2*d) + 1/4*b*log(2*(a + 2*b)*e^(-2*d*x - 2*c) + a*e^(-4*d*x - 4*c) + a)/(a^2*d) + 1/8*(a*b + 2*b^2)
*log((a*e^(2*d*x + 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(2*d*x + 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/(sqr
t((a + b)*b)*a^2*d) - 1/8*(a*b + 2*b^2)*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x -
2*c) + a + 2*b + 2*sqrt((a + b)*b)))/(sqrt((a + b)*b)*a^2*d) + 1/2*(a*b + 2*b^2)*(d*x + c)/(a^3*d) + 1/64*(8*b
*e^(-2*d*x - 2*c) - a*e^(-4*d*x - 4*c))/(a^2*d) + 1/16*(a^2*b + 8*a*b^2 + 8*b^3)*log((a*e^(-2*d*x - 2*c) + a +
2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/(sqrt((a + b)*b)*a^3*d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 718 vs. \(2 (103) = 206\).
time = 0.41, size = 1713, normalized size = 14.64 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^4/(a+b*sech(d*x+c)^2),x, algorithm="fricas")

[Out]

[1/64*(a^2*cosh(d*x + c)^8 + 8*a^2*cosh(d*x + c)*sinh(d*x + c)^7 + a^2*sinh(d*x + c)^8 + 8*(3*a^2 - 4*a*b + 8*
b^2)*d*x*cosh(d*x + c)^4 + 8*(a^2 - a*b)*cosh(d*x + c)^6 + 4*(7*a^2*cosh(d*x + c)^2 + 2*a^2 - 2*a*b)*sinh(d*x
+ c)^6 + 8*(7*a^2*cosh(d*x + c)^3 + 6*(a^2 - a*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*a^2*cosh(d*x + c)^4 +
4*(3*a^2 - 4*a*b + 8*b^2)*d*x + 60*(a^2 - a*b)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*a^2*cosh(d*x + c)^5 +
4*(3*a^2 - 4*a*b + 8*b^2)*d*x*cosh(d*x + c) + 20*(a^2 - a*b)*cosh(d*x + c)^3)*sinh(d*x + c)^3 - 8*(a^2 - a*b)*
cosh(d*x + c)^2 + 4*(7*a^2*cosh(d*x + c)^6 + 12*(3*a^2 - 4*a*b + 8*b^2)*d*x*cosh(d*x + c)^2 + 30*(a^2 - a*b)*c
osh(d*x + c)^4 - 2*a^2 + 2*a*b)*sinh(d*x + c)^2 + 32*(b^2*cosh(d*x + c)^4 + 4*b^2*cosh(d*x + c)^3*sinh(d*x + c
) + 6*b^2*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*b^2*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*sinh(d*x + c)^4)*sqrt(b/
(a + b))*log((a^2*cosh(d*x + c)^4 + 4*a^2*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*sinh(d*x + c)^4 + 2*(a^2 + 2*a*b
)*cosh(d*x + c)^2 + 2*(3*a^2*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 8*a*b + 8*b^2 + 4*(a^2*cos
h(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c) + 4*((a^2 + a*b)*cosh(d*x + c)^2 + 2*(a^2 + a*b)*cos
h(d*x + c)*sinh(d*x + c) + (a^2 + a*b)*sinh(d*x + c)^2 + a^2 + 3*a*b + 2*b^2)*sqrt(b/(a + b)))/(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)) - a^2 +
8*(a^2*cosh(d*x + c)^7 + 4*(3*a^2 - 4*a*b + 8*b^2)*d*x*cosh(d*x + c)^3 + 6*(a^2 - a*b)*cosh(d*x + c)^5 - 2*(a^
2 - a*b)*cosh(d*x + c))*sinh(d*x + c))/(a^3*d*cosh(d*x + c)^4 + 4*a^3*d*cosh(d*x + c)^3*sinh(d*x + c) + 6*a^3*
d*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*a^3*d*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*d*sinh(d*x + c)^4), 1/64*(a^2*
cosh(d*x + c)^8 + 8*a^2*cosh(d*x + c)*sinh(d*x + c)^7 + a^2*sinh(d*x + c)^8 + 8*(3*a^2 - 4*a*b + 8*b^2)*d*x*co
sh(d*x + c)^4 + 8*(a^2 - a*b)*cosh(d*x + c)^6 + 4*(7*a^2*cosh(d*x + c)^2 + 2*a^2 - 2*a*b)*sinh(d*x + c)^6 + 8*
(7*a^2*cosh(d*x + c)^3 + 6*(a^2 - a*b)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*a^2*cosh(d*x + c)^4 + 4*(3*a^2 -
4*a*b + 8*b^2)*d*x + 60*(a^2 - a*b)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*a^2*cosh(d*x + c)^5 + 4*(3*a^2 -
4*a*b + 8*b^2)*d*x*cosh(d*x + c) + 20*(a^2 - a*b)*cosh(d*x + c)^3)*sinh(d*x + c)^3 - 8*(a^2 - a*b)*cosh(d*x +
c)^2 + 4*(7*a^2*cosh(d*x + c)^6 + 12*(3*a^2 - 4*a*b + 8*b^2)*d*x*cosh(d*x + c)^2 + 30*(a^2 - a*b)*cosh(d*x + c
)^4 - 2*a^2 + 2*a*b)*sinh(d*x + c)^2 - 64*(b^2*cosh(d*x + c)^4 + 4*b^2*cosh(d*x + c)^3*sinh(d*x + c) + 6*b^2*c
osh(d*x + c)^2*sinh(d*x + c)^2 + 4*b^2*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*sinh(d*x + c)^4)*sqrt(-b/(a + b))*a
rctan(1/2*(a*cosh(d*x + c)^2 + 2*a*cosh(d*x + c)*sinh(d*x + c) + a*sinh(d*x + c)^2 + a + 2*b)*sqrt(-b/(a + b))
/b) - a^2 + 8*(a^2*cosh(d*x + c)^7 + 4*(3*a^2 - 4*a*b + 8*b^2)*d*x*cosh(d*x + c)^3 + 6*(a^2 - a*b)*cosh(d*x +
c)^5 - 2*(a^2 - a*b)*cosh(d*x + c))*sinh(d*x + c))/(a^3*d*cosh(d*x + c)^4 + 4*a^3*d*cosh(d*x + c)^3*sinh(d*x +
c) + 6*a^3*d*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*a^3*d*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*d*sinh(d*x + c)^4)
]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cosh ^{4}{\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(cosh(d*x+c)**4/(a+b*sech(d*x+c)**2),x)

[Out]

Integral(cosh(c + d*x)**4/(a + b*sech(c + d*x)**2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (103) = 206\).
time = 1.41, size = 208, normalized size = 1.78 \begin {gather*} -\frac {\frac {64 \, b^{3} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}} a^{3}} - \frac {8 \, {\left (3 \, a^{2} - 4 \, a b + 8 \, b^{2}\right )} {\left (d x + c\right )}}{a^{3}} - \frac {a e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a e^{\left (2 \, d x + 2 \, c\right )} - 8 \, b e^{\left (2 \, d x + 2 \, c\right )}}{a^{2}} + \frac {{\left (18 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 24 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 48 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 8 \, a b e^{\left (2 \, d x + 2 \, c\right )} + a^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{a^{3}}}{64 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^4/(a+b*sech(d*x+c)^2),x, algorithm="giac")

[Out]

-1/64*(64*b^3*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/(sqrt(-a*b - b^2)*a^3) - 8*(3*a^2 - 4
*a*b + 8*b^2)*(d*x + c)/a^3 - (a*e^(4*d*x + 4*c) + 8*a*e^(2*d*x + 2*c) - 8*b*e^(2*d*x + 2*c))/a^2 + (18*a^2*e^
(4*d*x + 4*c) - 24*a*b*e^(4*d*x + 4*c) + 48*b^2*e^(4*d*x + 4*c) + 8*a^2*e^(2*d*x + 2*c) - 8*a*b*e^(2*d*x + 2*c
) + a^2)*e^(-4*d*x - 4*c)/a^3)/d

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Mupad [B]
time = 2.00, size = 260, normalized size = 2.22 \begin {gather*} \frac {x\,\left (3\,a^2-4\,a\,b+8\,b^2\right )}{8\,a^3}-\frac {{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,a\,d}+\frac {{\mathrm {e}}^{4\,c+4\,d\,x}}{64\,a\,d}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (a-b\right )}{8\,a^2\,d}+\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a-b\right )}{8\,a^2\,d}+\frac {b^{5/2}\,\ln \left (\frac {4\,b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}}{a^4}-\frac {2\,b^{5/2}\,\left (a\,d+a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}+2\,b\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^4\,d\,\sqrt {a+b}}\right )}{2\,a^3\,d\,\sqrt {a+b}}-\frac {b^{5/2}\,\ln \left (\frac {4\,b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}}{a^4}+\frac {2\,b^{5/2}\,\left (a\,d+a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}+2\,b\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^4\,d\,\sqrt {a+b}}\right )}{2\,a^3\,d\,\sqrt {a+b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)^4/(a + b/cosh(c + d*x)^2),x)

[Out]

(x*(3*a^2 - 4*a*b + 8*b^2))/(8*a^3) - exp(- 4*c - 4*d*x)/(64*a*d) + exp(4*c + 4*d*x)/(64*a*d) - (exp(- 2*c - 2
*d*x)*(a - b))/(8*a^2*d) + (exp(2*c + 2*d*x)*(a - b))/(8*a^2*d) + (b^(5/2)*log((4*b^3*exp(2*c + 2*d*x))/a^4 -
(2*b^(5/2)*(a*d + a*d*exp(2*c + 2*d*x) + 2*b*d*exp(2*c + 2*d*x)))/(a^4*d*(a + b)^(1/2))))/(2*a^3*d*(a + b)^(1/
2)) - (b^(5/2)*log((4*b^3*exp(2*c + 2*d*x))/a^4 + (2*b^(5/2)*(a*d + a*d*exp(2*c + 2*d*x) + 2*b*d*exp(2*c + 2*d
*x)))/(a^4*d*(a + b)^(1/2))))/(2*a^3*d*(a + b)^(1/2))

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