\(\int \frac {\cosh ^3(c+d x)}{(a+b \tanh ^2(c+d x))^2} \, dx\) [115]

Optimal result

Integrand size = 23, antiderivative size = 128 \[ \int \frac {\cosh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {b^2 (6 a+b) \arctan \left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^{7/2} d}+\frac {(a+3 b) \sinh (c+d x)}{(a+b)^3 d}+\frac {\sinh ^3(c+d x)}{3 (a+b)^2 d}+\frac {b^3 \sinh (c+d x)}{2 a (a+b)^3 d \left (a+(a+b) \sinh ^2(c+d x)\right )} \]

[Out]

1/2*b^2*(6*a+b)*arctan(sinh(d*x+c)*(a+b)^(1/2)/a^(1/2))/a^(3/2)/(a+b)^(7/2)/d+(a+3*b)*sinh(d*x+c)/(a+b)^3/d+1/
3*sinh(d*x+c)^3/(a+b)^2/d+1/2*b^3*sinh(d*x+c)/a/(a+b)^3/d/(a+(a+b)*sinh(d*x+c)^2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3757, 398, 393, 211} \[ \int \frac {\cosh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {b^2 (6 a+b) \arctan \left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} d (a+b)^{7/2}}+\frac {b^3 \sinh (c+d x)}{2 a d (a+b)^3 \left ((a+b) \sinh ^2(c+d x)+a\right )}+\frac {\sinh ^3(c+d x)}{3 d (a+b)^2}+\frac {(a+3 b) \sinh (c+d x)}{d (a+b)^3} \]

[In]

Int[Cosh[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^2,x]

[Out]

(b^2*(6*a + b)*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^(7/2)*d) + ((a + 3*b)*Sinh[c +
d*x])/((a + b)^3*d) + Sinh[c + d*x]^3/(3*(a + b)^2*d) + (b^3*Sinh[c + d*x])/(2*a*(a + b)^3*d*(a + (a + b)*Sinh
[c + d*x]^2))

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 393

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 398

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 3757

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> With[{ff = F
reeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[ExpandToSum[b*(ff*x)^n + 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*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{\left (a+(a+b) x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a+3 b}{(a+b)^3}+\frac {x^2}{(a+b)^2}+\frac {b^2 (3 a+b)+3 b^2 (a+b) x^2}{(a+b)^3 \left (a+(a+b) x^2\right )^2}\right ) \, dx,x,\sinh (c+d x)\right )}{d} \\ & = \frac {(a+3 b) \sinh (c+d x)}{(a+b)^3 d}+\frac {\sinh ^3(c+d x)}{3 (a+b)^2 d}+\frac {\text {Subst}\left (\int \frac {b^2 (3 a+b)+3 b^2 (a+b) x^2}{\left (a+(a+b) x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{(a+b)^3 d} \\ & = \frac {(a+3 b) \sinh (c+d x)}{(a+b)^3 d}+\frac {\sinh ^3(c+d x)}{3 (a+b)^2 d}+\frac {b^3 \sinh (c+d x)}{2 a (a+b)^3 d \left (a+(a+b) \sinh ^2(c+d x)\right )}+\frac {\left (b^2 (6 a+b)\right ) \text {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\sinh (c+d x)\right )}{2 a (a+b)^3 d} \\ & = \frac {b^2 (6 a+b) \arctan \left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^{7/2} d}+\frac {(a+3 b) \sinh (c+d x)}{(a+b)^3 d}+\frac {\sinh ^3(c+d x)}{3 (a+b)^2 d}+\frac {b^3 \sinh (c+d x)}{2 a (a+b)^3 d \left (a+(a+b) \sinh ^2(c+d x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.30 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.87 \[ \int \frac {\cosh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\frac {-\frac {6 b^2 (6 a+b) \arctan \left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {a+b}}\right )}{a^{3/2} (a+b)^{7/2}}+\frac {3 \left (3 a+11 b+\frac {4 b^3}{a (a-b+(a+b) \cosh (2 (c+d x)))}\right ) \sinh (c+d x)}{(a+b)^3}+\frac {\sinh (3 (c+d x))}{(a+b)^2}}{12 d} \]

[In]

Integrate[Cosh[c + d*x]^3/(a + b*Tanh[c + d*x]^2)^2,x]

[Out]

((-6*b^2*(6*a + b)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[a + b]])/(a^(3/2)*(a + b)^(7/2)) + (3*(3*a + 11*b + (4*
b^3)/(a*(a - b + (a + b)*Cosh[2*(c + d*x)])))*Sinh[c + d*x])/(a + b)^3 + Sinh[3*(c + d*x)]/(a + b)^2)/(12*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(377\) vs. \(2(114)=228\).

Time = 26.64 (sec) , antiderivative size = 378, normalized size of antiderivative = 2.95

method	result	size
derivativedivides	\(\frac {-\frac {1}{3 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {a +3 b}{\left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{3 \left (a +b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {1}{2 \left (a +b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {a +3 b}{\left (a +b \right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {2 b^{2} \left (\frac {-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a}+\frac {\left (6 a +b \right ) \left (\frac {\left (\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2}\right )}{\left (a +b \right )^{3}}}{d}\)	\(378\)
default	\(\frac {-\frac {1}{3 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {a +3 b}{\left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{3 \left (a +b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {1}{2 \left (a +b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {a +3 b}{\left (a +b \right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {2 b^{2} \left (\frac {-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a}+\frac {\left (6 a +b \right ) \left (\frac {\left (\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2}\right )}{\left (a +b \right )^{3}}}{d}\)	\(378\)
risch	\(\frac {{\mathrm e}^{3 d x +3 c}}{24 d \left (a^{2}+2 a b +b^{2}\right )}+\frac {3 \,{\mathrm e}^{d x +c} a}{8 \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right ) d}+\frac {11 \,{\mathrm e}^{d x +c} b}{8 \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right ) d}-\frac {3 \,{\mathrm e}^{-d x -c} a}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}-\frac {11 \,{\mathrm e}^{-d x -c} b}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}-\frac {{\mathrm e}^{-3 d x -3 c}}{24 d \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left ({\mathrm e}^{2 d x +2 c}-1\right ) b^{3} {\mathrm e}^{d x +c}}{\left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right ) d \left (a +b \right ) a \left (a^{2}+2 a b +b^{2}\right )}-\frac {3 b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3} d}-\frac {b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3} d a}+\frac {3 b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3} d}+\frac {b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3} d a}\)	\(522\)

[In]

int(cosh(d*x+c)^3/(a+b*tanh(d*x+c)^2)^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/3/(a+b)^2/(tanh(1/2*d*x+1/2*c)-1)^3-1/2/(a+b)^2/(tanh(1/2*d*x+1/2*c)-1)^2-(a+3*b)/(a+b)^3/(tanh(1/2*d*
x+1/2*c)-1)-1/3/(a+b)^2/(1+tanh(1/2*d*x+1/2*c))^3+1/2/(a+b)^2/(1+tanh(1/2*d*x+1/2*c))^2-(a+3*b)/(a+b)^3/(1+tan
h(1/2*d*x+1/2*c))+2/(a+b)^3*b^2*((-1/2*b/a*tanh(1/2*d*x+1/2*c)^3+1/2*b/a*tanh(1/2*d*x+1/2*c))/(tanh(1/2*d*x+1/
2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)+1/2*(6*a+b)*(1/2*(((a+b)*b)^(1/2)+b)/a/((a+b)*
b)^(1/2)/((2*((a+b)*b)^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*((a+b)*b)^(1/2)+a+2*b)*a)^(1/2))
-1/2*(((a+b)*b)^(1/2)-b)/a/((a+b)*b)^(1/2)/((2*((a+b)*b)^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/(
(2*((a+b)*b)^(1/2)-a-2*b)*a)^(1/2)))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3642 vs. \(2 (114) = 228\).

Time = 0.34 (sec) , antiderivative size = 6934, normalized size of antiderivative = 54.17 \[ \int \frac {\cosh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate(cosh(d*x+c)^3/(a+b*tanh(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {\cosh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int \frac {\cosh ^{3}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \]

[In]

integrate(cosh(d*x+c)**3/(a+b*tanh(d*x+c)**2)**2,x)

[Out]

Integral(cosh(c + d*x)**3/(a + b*tanh(c + d*x)**2)**2, x)

Maxima [F]

\[ \int \frac {\cosh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int { \frac {\cosh \left (d x + c\right )^{3}}{{\left (b \tanh \left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]

[In]

integrate(cosh(d*x+c)^3/(a+b*tanh(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

-1/24*(a^3 + 2*a^2*b + a*b^2 - (a^3*e^(10*c) + 2*a^2*b*e^(10*c) + a*b^2*e^(10*c))*e^(10*d*x) - (11*a^3*e^(8*c)
+ 42*a^2*b*e^(8*c) + 31*a*b^2*e^(8*c))*e^(8*d*x) - 2*(5*a^3*e^(6*c) + 4*a^2*b*e^(6*c) - 49*a*b^2*e^(6*c) + 12
*b^3*e^(6*c))*e^(6*d*x) + 2*(5*a^3*e^(4*c) + 4*a^2*b*e^(4*c) - 49*a*b^2*e^(4*c) + 12*b^3*e^(4*c))*e^(4*d*x) +
(11*a^3*e^(2*c) + 42*a^2*b*e^(2*c) + 31*a*b^2*e^(2*c))*e^(2*d*x))/((a^5*d*e^(7*c) + 4*a^4*b*d*e^(7*c) + 6*a^3*
b^2*d*e^(7*c) + 4*a^2*b^3*d*e^(7*c) + a*b^4*d*e^(7*c))*e^(7*d*x) + 2*(a^5*d*e^(5*c) + 2*a^4*b*d*e^(5*c) - 2*a^
2*b^3*d*e^(5*c) - a*b^4*d*e^(5*c))*e^(5*d*x) + (a^5*d*e^(3*c) + 4*a^4*b*d*e^(3*c) + 6*a^3*b^2*d*e^(3*c) + 4*a^
2*b^3*d*e^(3*c) + a*b^4*d*e^(3*c))*e^(3*d*x)) + 1/8*integrate(8*((6*a*b^2*e^(3*c) + b^3*e^(3*c))*e^(3*d*x) + (
6*a*b^2*e^c + b^3*e^c)*e^(d*x))/(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4 + (a^5*e^(4*c) + 4*a^4*b*e^(4*c
) + 6*a^3*b^2*e^(4*c) + 4*a^2*b^3*e^(4*c) + a*b^4*e^(4*c))*e^(4*d*x) + 2*(a^5*e^(2*c) + 2*a^4*b*e^(2*c) - 2*a^
2*b^3*e^(2*c) - a*b^4*e^(2*c))*e^(2*d*x)), x)

Giac [F]

\[ \int \frac {\cosh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int { \frac {\cosh \left (d x + c\right )^{3}}{{\left (b \tanh \left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]

[In]

integrate(cosh(d*x+c)^3/(a+b*tanh(d*x+c)^2)^2,x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\cosh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \]

[In]

int(cosh(c + d*x)^3/(a + b*tanh(c + d*x)^2)^2,x)

[Out]

int(cosh(c + d*x)^3/(a + b*tanh(c + d*x)^2)^2, x)