3.2.5 \(\int \frac {\cosh ^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx\) [105]

Optimal. Leaf size=120 \[ \frac {\left (3 a^2+10 a b+15 b^2\right ) x}{8 (a+b)^3}+\frac {b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^3 d}+\frac {(3 a+7 b) \cosh (c+d x) \sinh (c+d x)}{8 (a+b)^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d} \]

[Out]

1/8*(3*a^2+10*a*b+15*b^2)*x/(a+b)^3+1/8*(3*a+7*b)*cosh(d*x+c)*sinh(d*x+c)/(a+b)^2/d+1/4*cosh(d*x+c)^3*sinh(d*x
+c)/(a+b)/d+b^(5/2)*arctan(b^(1/2)*tanh(d*x+c)/a^(1/2))/(a+b)^3/d/a^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 120, 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 = {3756, 425, 541, 536, 212, 211} \begin {gather*} \frac {x \left (3 a^2+10 a b+15 b^2\right )}{8 (a+b)^3}+\frac {b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} d (a+b)^3}+\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d (a+b)}+\frac {(3 a+7 b) \sinh (c+d x) \cosh (c+d x)}{8 d (a+b)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((3*a^2 + 10*a*b + 15*b^2)*x)/(8*(a + b)^3) + (b^(5/2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a +
b)^3*d) + ((3*a + 7*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*(a + b)^2*d) + (Cosh[c + d*x]^3*Sinh[c + d*x])/(4*(a +
b)*d)

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 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 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 3756

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/(c^(m - 1)*f), Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)
^n)^p, x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2] && (IntegersQ[n, p
] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rubi steps

\begin {align*} \int \frac {\cosh ^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^3 \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d}+\frac {\text {Subst}\left (\int \frac {3 a+4 b+3 b x^2}{\left (1-x^2\right )^2 \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{4 (a+b) d}\\ &=\frac {(3 a+7 b) \cosh (c+d x) \sinh (c+d x)}{8 (a+b)^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d}+\frac {\text {Subst}\left (\int \frac {3 a^2+7 a b+8 b^2+b (3 a+7 b) x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 (a+b)^2 d}\\ &=\frac {(3 a+7 b) \cosh (c+d x) \sinh (c+d x)}{8 (a+b)^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d}+\frac {b^3 \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{(a+b)^3 d}+\frac {\left (3 a^2+10 a b+15 b^2\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 (a+b)^3 d}\\ &=\frac {\left (3 a^2+10 a b+15 b^2\right ) x}{8 (a+b)^3}+\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^3 d}+\frac {(3 a+7 b) \cosh (c+d x) \sinh (c+d x)}{8 (a+b)^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d}\\ \end {align*}

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Mathematica [A]
time = 0.22, size = 115, normalized size = 0.96 \begin {gather*} \frac {\left (3 a^2+10 a b+15 b^2\right ) (c+d x)}{8 (a+b)^3 d}+\frac {b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^3 d}+\frac {(a+2 b) \sinh (2 (c+d x))}{4 (a+b)^2 d}+\frac {\sinh (4 (c+d x))}{32 (a+b) d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((3*a^2 + 10*a*b + 15*b^2)*(c + d*x))/(8*(a + b)^3*d) + (b^(5/2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(Sqr
t[a]*(a + b)^3*d) + ((a + 2*b)*Sinh[2*(c + d*x)])/(4*(a + b)^2*d) + Sinh[4*(c + d*x)]/(32*(a + b)*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(437\) vs. \(2(106)=212\).
time = 2.74, size = 438, normalized size = 3.65

method	result	size
risch	\(\frac {3 a^{2} x}{8 \left (a +b \right )^{3}}+\frac {5 a x b}{4 \left (a +b \right )^{3}}+\frac {15 x \,b^{2}}{8 \left (a +b \right )^{3}}+\frac {{\mathrm e}^{4 d x +4 c}}{64 \left (a +b \right ) d}+\frac {{\mathrm e}^{2 d x +2 c} b}{4 \left (a +b \right )^{2} d}+\frac {{\mathrm e}^{2 d x +2 c} a}{8 \left (a +b \right )^{2} d}-\frac {{\mathrm e}^{-2 d x -2 c} b}{4 \left (a^{2}+2 a b +b^{2}\right ) d}-\frac {{\mathrm e}^{-2 d x -2 c} a}{8 \left (a^{2}+2 a b +b^{2}\right ) d}-\frac {{\mathrm e}^{-4 d x -4 c}}{64 \left (a +b \right ) d}+\frac {\sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {-a b}-b}{a +b}\right )}{2 a \left (a +b \right )^{3} d}-\frac {\sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{2 a \left (a +b \right )^{3} d}\)	\(273\)
derivativedivides	\(\frac {-\frac {2 b^{3} a \left (-\frac {\left (-a +\sqrt {b \left (a +b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\left (a +\sqrt {b \left (a +b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{\left (a +b \right )^{3}}+\frac {1}{2 \left (2 b +2 a \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {2}{\left (4 a +4 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-7 a -11 b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-5 a -9 b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-3 a^{2}-10 a b -15 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 \left (a +b \right )^{3}}-\frac {1}{2 \left (2 b +2 a \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {2}{\left (4 a +4 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-5 a -9 b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {7 a +11 b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (3 a^{2}+10 a b +15 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 \left (a +b \right )^{3}}}{d}\)	\(438\)
default	\(\frac {-\frac {2 b^{3} a \left (-\frac {\left (-a +\sqrt {b \left (a +b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\left (a +\sqrt {b \left (a +b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{\left (a +b \right )^{3}}+\frac {1}{2 \left (2 b +2 a \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {2}{\left (4 a +4 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-7 a -11 b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-5 a -9 b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-3 a^{2}-10 a b -15 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 \left (a +b \right )^{3}}-\frac {1}{2 \left (2 b +2 a \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {2}{\left (4 a +4 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-5 a -9 b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {7 a +11 b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (3 a^{2}+10 a b +15 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 \left (a +b \right )^{3}}}{d}\)	\(438\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-2*b^3/(a+b)^3*a*(-1/2*(-a+(b*(a+b))^(1/2)-b)/a/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arcta
nh(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))+1/2*(a+(b*(a+b))^(1/2)+b)/a/(b*(a+b))^(1/2)/((2*
(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)))+1/2/(2*b+2*
a)/(tanh(1/2*d*x+1/2*c)-1)^4+2/(4*a+4*b)/(tanh(1/2*d*x+1/2*c)-1)^3-1/8*(-7*a-11*b)/(a+b)^2/(tanh(1/2*d*x+1/2*c
)-1)^2-1/8*(-5*a-9*b)/(a+b)^2/(tanh(1/2*d*x+1/2*c)-1)+1/8/(a+b)^3*(-3*a^2-10*a*b-15*b^2)*ln(tanh(1/2*d*x+1/2*c
)-1)-1/2/(2*b+2*a)/(tanh(1/2*d*x+1/2*c)+1)^4+2/(4*a+4*b)/(tanh(1/2*d*x+1/2*c)+1)^3-1/8*(-5*a-9*b)/(a+b)^2/(tan
h(1/2*d*x+1/2*c)+1)-1/8*(7*a+11*b)/(a+b)^2/(tanh(1/2*d*x+1/2*c)+1)^2+1/8*(3*a^2+10*a*b+15*b^2)/(a+b)^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. 514 vs. \(2 (106) = 212\).
time = 0.56, size = 514, normalized size = 4.28 \begin {gather*} -\frac {{\left (a b - b^{2}\right )} {\left (d x + c\right )}}{2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d} + \frac {{\left (8 \, b e^{\left (-2 \, d x - 2 \, c\right )} + a + b\right )} e^{\left (4 \, d x + 4 \, c\right )}}{64 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d} + \frac {b \log \left ({\left (a + b\right )} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (a - b\right )} e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d} - \frac {b \log \left (2 \, {\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )} + a + b\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d} - \frac {{\left (a b - b^{2}\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a b} d} - \frac {{\left (a^{2} b - 6 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{8 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {a b} d} + \frac {{\left (a b - b^{2}\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a b} d} - \frac {3 \, b \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{8 \, \sqrt {a b} {\left (a + b\right )} d} - \frac {8 \, b e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d} + \frac {3 \, {\left (d x + c\right )}}{8 \, {\left (a + b\right )} d} + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{8 \, {\left (a + b\right )} d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, {\left (a + b\right )} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a*b - b^2)*(d*x + c)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) + 1/64*(8*b*e^(-2*d*x - 2*c) + a + b)*e^(4*d*x
+ 4*c)/((a^2 + 2*a*b + b^2)*d) + 1/4*b*log((a + b)*e^(4*d*x + 4*c) + 2*(a - b)*e^(2*d*x + 2*c) + a + b)/((a^2
+ 2*a*b + b^2)*d) - 1/4*b*log(2*(a - b)*e^(-2*d*x - 2*c) + (a + b)*e^(-4*d*x - 4*c) + a + b)/((a^2 + 2*a*b + b
^2)*d) - 1/4*(a*b - b^2)*arctan(1/2*((a + b)*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/((a^2 + 2*a*b + b^2)*sqrt(a*b
)*d) - 1/8*(a^2*b - 6*a*b^2 + b^3)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt(a*b))/((a^3 + 3*a^2*b +
3*a*b^2 + b^3)*sqrt(a*b)*d) + 1/4*(a*b - b^2)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt(a*b))/((a^2 +
2*a*b + b^2)*sqrt(a*b)*d) - 3/8*b*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt(a*b))/(sqrt(a*b)*(a + b)
*d) - 1/64*(8*b*e^(-2*d*x - 2*c) + (a + b)*e^(-4*d*x - 4*c))/((a^2 + 2*a*b + b^2)*d) + 3/8*(d*x + c)/((a + b)*
d) + 1/8*e^(2*d*x + 2*c)/((a + b)*d) - 1/8*e^(-2*d*x - 2*c)/((a + b)*d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 929 vs. \(2 (106) = 212\).
time = 0.49, size = 2180, normalized size = 18.17 \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*tanh(d*x+c)^2),x, algorithm="fricas")

[Out]

[1/64*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^8 + 8*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2 + 2*a*
b + b^2)*sinh(d*x + c)^8 + 8*(3*a^2 + 10*a*b + 15*b^2)*d*x*cosh(d*x + c)^4 + 8*(a^2 + 3*a*b + 2*b^2)*cosh(d*x
+ c)^6 + 4*(7*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + 2*a^2 + 6*a*b + 4*b^2)*sinh(d*x + c)^6 + 8*(7*(a^2 + 2*a*b
+ b^2)*cosh(d*x + c)^3 + 6*(a^2 + 3*a*b + 2*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^2 + 2*a*b + b^2)*c
osh(d*x + c)^4 + 4*(3*a^2 + 10*a*b + 15*b^2)*d*x + 60*(a^2 + 3*a*b + 2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 +
8*(7*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^5 + 4*(3*a^2 + 10*a*b + 15*b^2)*d*x*cosh(d*x + c) + 20*(a^2 + 3*a*b +
2*b^2)*cosh(d*x + c)^3)*sinh(d*x + c)^3 - 8*(a^2 + 3*a*b + 2*b^2)*cosh(d*x + c)^2 + 4*(7*(a^2 + 2*a*b + b^2)*c
osh(d*x + c)^6 + 12*(3*a^2 + 10*a*b + 15*b^2)*d*x*cosh(d*x + c)^2 + 30*(a^2 + 3*a*b + 2*b^2)*cosh(d*x + c)^4 -
2*a^2 - 6*a*b - 4*b^2)*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)*log(
((a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 4*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + 2*a*b + b^
2)*sinh(d*x + c)^4 + 2*(a^2 - b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 - b^2)*sin
h(d*x + c)^2 + a^2 - 6*a*b + b^2 + 4*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + (a^2 - b^2)*cosh(d*x + c))*sinh(d*
x + c) + 4*((a^2 + a*b)*cosh(d*x + c)^2 + 2*(a^2 + a*b)*cosh(d*x + c)*sinh(d*x + c) + (a^2 + a*b)*sinh(d*x + c
)^2 + a^2 - a*b)*sqrt(-b/a))/((a + b)*cosh(d*x + c)^4 + 4*(a + b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + b)*sinh
(d*x + c)^4 + 2*(a - b)*cosh(d*x + c)^2 + 2*(3*(a + b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c)^2 + 4*((a + b)*c
osh(d*x + c)^3 + (a - b)*cosh(d*x + c))*sinh(d*x + c) + a + b)) - a^2 - 2*a*b - b^2 + 8*((a^2 + 2*a*b + b^2)*c
osh(d*x + c)^7 + 4*(3*a^2 + 10*a*b + 15*b^2)*d*x*cosh(d*x + c)^3 + 6*(a^2 + 3*a*b + 2*b^2)*cosh(d*x + c)^5 - 2
*(a^2 + 3*a*b + 2*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c)^4 + 4*(a
^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c)^3*sinh(d*x + c) + 6*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x +
c)^2*sinh(d*x + c)^2 + 4*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^3 + 3*a^2*b + 3
*a*b^2 + b^3)*d*sinh(d*x + c)^4), 1/64*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^8 + 8*(a^2 + 2*a*b + b^2)*cosh(d*x +
c)*sinh(d*x + c)^7 + (a^2 + 2*a*b + b^2)*sinh(d*x + c)^8 + 8*(3*a^2 + 10*a*b + 15*b^2)*d*x*cosh(d*x + c)^4 +
8*(a^2 + 3*a*b + 2*b^2)*cosh(d*x + c)^6 + 4*(7*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + 2*a^2 + 6*a*b + 4*b^2)*si
nh(d*x + c)^6 + 8*(7*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + 6*(a^2 + 3*a*b + 2*b^2)*cosh(d*x + c))*sinh(d*x + c
)^5 + 2*(35*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 4*(3*a^2 + 10*a*b + 15*b^2)*d*x + 60*(a^2 + 3*a*b + 2*b^2)*c
osh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^5 + 4*(3*a^2 + 10*a*b + 15*b^2)*d*x*c
osh(d*x + c) + 20*(a^2 + 3*a*b + 2*b^2)*cosh(d*x + c)^3)*sinh(d*x + c)^3 - 8*(a^2 + 3*a*b + 2*b^2)*cosh(d*x +
c)^2 + 4*(7*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 + 12*(3*a^2 + 10*a*b + 15*b^2)*d*x*cosh(d*x + c)^2 + 30*(a^2 +
3*a*b + 2*b^2)*cosh(d*x + c)^4 - 2*a^2 - 6*a*b - 4*b^2)*sinh(d*x + c)^2 + 64*(b^2*cosh(d*x + c)^4 + 4*b^2*cos
h(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)*arctan(1/2*((a + b)*cosh(d*x + c)^2 + 2*(a + b)*cosh(d*x + c)*sinh(d*x + c) + (a +
b)*sinh(d*x + c)^2 + a - b)*sqrt(b/a)/b) - a^2 - 2*a*b - b^2 + 8*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^7 + 4*(3*
a^2 + 10*a*b + 15*b^2)*d*x*cosh(d*x + c)^3 + 6*(a^2 + 3*a*b + 2*b^2)*cosh(d*x + c)^5 - 2*(a^2 + 3*a*b + 2*b^2)
*cosh(d*x + c))*sinh(d*x + c))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c)^4 + 4*(a^3 + 3*a^2*b + 3*a*b^2
+ b^3)*d*cosh(d*x + c)^3*sinh(d*x + c) + 6*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c)^2*sinh(d*x + c)^2
+ 4*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^3 + 3*a^2*b + 3*a*b^2 + b^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 \tanh ^{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*tanh(d*x+c)**2),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (106) = 212\).
time = 2.01, size = 301, normalized size = 2.51 \begin {gather*} \frac {\frac {64 \, b^{3} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {a b}} + \frac {8 \, {\left (3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} {\left (d x + c\right )}}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {{\left (18 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 60 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 90 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 24 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 16 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{2} + 2 \, a b + b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a e^{\left (2 \, d x + 2 \, c\right )} + 16 \, b e^{\left (2 \, d x + 2 \, c\right )}}{a^{2} + 2 \, a b + b^{2}}}{64 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(64*b^3*arctan(1/2*(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/((a^3 + 3*a^2*b + 3*a*b^2 +
b^3)*sqrt(a*b)) + 8*(3*a^2 + 10*a*b + 15*b^2)*(d*x + c)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) - (18*a^2*e^(4*d*x +
4*c) + 60*a*b*e^(4*d*x + 4*c) + 90*b^2*e^(4*d*x + 4*c) + 8*a^2*e^(2*d*x + 2*c) + 24*a*b*e^(2*d*x + 2*c) + 16*b
^2*e^(2*d*x + 2*c) + a^2 + 2*a*b + b^2)*e^(-4*d*x - 4*c)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + (a*e^(4*d*x + 4*c)
+ b*e^(4*d*x + 4*c) + 8*a*e^(2*d*x + 2*c) + 16*b*e^(2*d*x + 2*c))/(a^2 + 2*a*b + b^2))/d

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Mupad [B]
time = 1.93, size = 967, normalized size = 8.06 \begin {gather*} \frac {x\,\left (3\,a^2+10\,a\,b+15\,b^2\right )}{8\,{\left (a+b\right )}^3}-\frac {{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,d\,\left (a+b\right )}+\frac {{\mathrm {e}}^{4\,c+4\,d\,x}}{64\,d\,\left (a+b\right )}+\frac {\mathrm {atan}\left (\left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (\frac {4\,b^3}{d\,{\left (a+b\right )}^5\,\sqrt {b^5}\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}+\frac {\left (a-b\right )\,\left (a^4\,d\,\sqrt {b^5}-b^4\,d\,\sqrt {b^5}-2\,a\,b^3\,d\,\sqrt {b^5}+2\,a^3\,b\,d\,\sqrt {b^5}\right )}{b^3\,{\left (a+b\right )}^2\,\sqrt {a\,d^2\,{\left (a+b\right )}^6}\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )\,\sqrt {a^7\,d^2+6\,a^6\,b\,d^2+15\,a^5\,b^2\,d^2+20\,a^4\,b^3\,d^2+15\,a^3\,b^4\,d^2+6\,a^2\,b^5\,d^2+a\,b^6\,d^2}}\right )+\frac {\left (a-b\right )\,\left (a^4\,d\,\sqrt {b^5}+b^4\,d\,\sqrt {b^5}+4\,a\,b^3\,d\,\sqrt {b^5}+4\,a^3\,b\,d\,\sqrt {b^5}+6\,a^2\,b^2\,d\,\sqrt {b^5}\right )}{b^3\,{\left (a+b\right )}^2\,\sqrt {a\,d^2\,{\left (a+b\right )}^6}\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )\,\sqrt {a^7\,d^2+6\,a^6\,b\,d^2+15\,a^5\,b^2\,d^2+20\,a^4\,b^3\,d^2+15\,a^3\,b^4\,d^2+6\,a^2\,b^5\,d^2+a\,b^6\,d^2}}\right )\,\left (\frac {a^4\,\sqrt {a^7\,d^2+6\,a^6\,b\,d^2+15\,a^5\,b^2\,d^2+20\,a^4\,b^3\,d^2+15\,a^3\,b^4\,d^2+6\,a^2\,b^5\,d^2+a\,b^6\,d^2}}{2}+\frac {b^4\,\sqrt {a^7\,d^2+6\,a^6\,b\,d^2+15\,a^5\,b^2\,d^2+20\,a^4\,b^3\,d^2+15\,a^3\,b^4\,d^2+6\,a^2\,b^5\,d^2+a\,b^6\,d^2}}{2}+2\,a\,b^3\,\sqrt {a^7\,d^2+6\,a^6\,b\,d^2+15\,a^5\,b^2\,d^2+20\,a^4\,b^3\,d^2+15\,a^3\,b^4\,d^2+6\,a^2\,b^5\,d^2+a\,b^6\,d^2}+2\,a^3\,b\,\sqrt {a^7\,d^2+6\,a^6\,b\,d^2+15\,a^5\,b^2\,d^2+20\,a^4\,b^3\,d^2+15\,a^3\,b^4\,d^2+6\,a^2\,b^5\,d^2+a\,b^6\,d^2}+3\,a^2\,b^2\,\sqrt {a^7\,d^2+6\,a^6\,b\,d^2+15\,a^5\,b^2\,d^2+20\,a^4\,b^3\,d^2+15\,a^3\,b^4\,d^2+6\,a^2\,b^5\,d^2+a\,b^6\,d^2}\right )\right )\,\sqrt {b^5}}{\sqrt {a^7\,d^2+6\,a^6\,b\,d^2+15\,a^5\,b^2\,d^2+20\,a^4\,b^3\,d^2+15\,a^3\,b^4\,d^2+6\,a^2\,b^5\,d^2+a\,b^6\,d^2}}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (a+2\,b\right )}{8\,d\,{\left (a+b\right )}^2}+\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+2\,b\right )}{8\,d\,{\left (a+b\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*(10*a*b + 3*a^2 + 15*b^2))/(8*(a + b)^3) - exp(- 4*c - 4*d*x)/(64*d*(a + b)) + exp(4*c + 4*d*x)/(64*d*(a +
b)) + (atan((exp(2*c)*exp(2*d*x)*((4*b^3)/(d*(a + b)^5*(b^5)^(1/2)*(3*a*b^2 + 3*a^2*b + a^3 + b^3)) + ((a - b)
*(a^4*d*(b^5)^(1/2) - b^4*d*(b^5)^(1/2) - 2*a*b^3*d*(b^5)^(1/2) + 2*a^3*b*d*(b^5)^(1/2)))/(b^3*(a + b)^2*(a*d^
2*(a + b)^6)^(1/2)*(3*a*b^2 + 3*a^2*b + a^3 + b^3)*(a^7*d^2 + a*b^6*d^2 + 6*a^6*b*d^2 + 6*a^2*b^5*d^2 + 15*a^3
*b^4*d^2 + 20*a^4*b^3*d^2 + 15*a^5*b^2*d^2)^(1/2))) + ((a - b)*(a^4*d*(b^5)^(1/2) + b^4*d*(b^5)^(1/2) + 4*a*b^
3*d*(b^5)^(1/2) + 4*a^3*b*d*(b^5)^(1/2) + 6*a^2*b^2*d*(b^5)^(1/2)))/(b^3*(a + b)^2*(a*d^2*(a + b)^6)^(1/2)*(3*
a*b^2 + 3*a^2*b + a^3 + b^3)*(a^7*d^2 + a*b^6*d^2 + 6*a^6*b*d^2 + 6*a^2*b^5*d^2 + 15*a^3*b^4*d^2 + 20*a^4*b^3*
d^2 + 15*a^5*b^2*d^2)^(1/2)))*((a^4*(a^7*d^2 + a*b^6*d^2 + 6*a^6*b*d^2 + 6*a^2*b^5*d^2 + 15*a^3*b^4*d^2 + 20*a
^4*b^3*d^2 + 15*a^5*b^2*d^2)^(1/2))/2 + (b^4*(a^7*d^2 + a*b^6*d^2 + 6*a^6*b*d^2 + 6*a^2*b^5*d^2 + 15*a^3*b^4*d
^2 + 20*a^4*b^3*d^2 + 15*a^5*b^2*d^2)^(1/2))/2 + 2*a*b^3*(a^7*d^2 + a*b^6*d^2 + 6*a^6*b*d^2 + 6*a^2*b^5*d^2 +
15*a^3*b^4*d^2 + 20*a^4*b^3*d^2 + 15*a^5*b^2*d^2)^(1/2) + 2*a^3*b*(a^7*d^2 + a*b^6*d^2 + 6*a^6*b*d^2 + 6*a^2*b
^5*d^2 + 15*a^3*b^4*d^2 + 20*a^4*b^3*d^2 + 15*a^5*b^2*d^2)^(1/2) + 3*a^2*b^2*(a^7*d^2 + a*b^6*d^2 + 6*a^6*b*d^
2 + 6*a^2*b^5*d^2 + 15*a^3*b^4*d^2 + 20*a^4*b^3*d^2 + 15*a^5*b^2*d^2)^(1/2)))*(b^5)^(1/2))/(a^7*d^2 + a*b^6*d^
2 + 6*a^6*b*d^2 + 6*a^2*b^5*d^2 + 15*a^3*b^4*d^2 + 20*a^4*b^3*d^2 + 15*a^5*b^2*d^2)^(1/2) - (exp(- 2*c - 2*d*x
)*(a + 2*b))/(8*d*(a + b)^2) + (exp(2*c + 2*d*x)*(a + 2*b))/(8*d*(a + b)^2)

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