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

Optimal. Leaf size=80 \[ \frac {b^2 \tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} d (a+b)^{5/2}}+\frac {\sinh ^3(c+d x)}{3 d (a+b)}+\frac {(a+2 b) \sinh (c+d x)}{d (a+b)^2} \]

[Out]

(a+2*b)*sinh(d*x+c)/(a+b)^2/d+1/3*sinh(d*x+c)^3/(a+b)/d+b^2*arctan(sinh(d*x+c)*(a+b)^(1/2)/a^(1/2))/(a+b)^(5/2
)/d/a^(1/2)

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Rubi [A]  time = 0.11, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3676, 390, 205} \[ \frac {b^2 \tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} d (a+b)^{5/2}}+\frac {\sinh ^3(c+d x)}{3 d (a+b)}+\frac {(a+2 b) \sinh (c+d x)}{d (a+b)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*ArcTan[(Sqrt[a + b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a + b)^(5/2)*d) + ((a + 2*b)*Sinh[c + d*x])/((a +
b)^2*d) + Sinh[c + d*x]^3/(3*(a + b)*d)

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]

Rule 390

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 3676

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

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Mathematica [A]  time = 0.43, size = 79, normalized size = 0.99 \[ \frac {-\frac {12 b^2 \tan ^{-1}\left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {a+b}}\right )}{\sqrt {a} (a+b)^{5/2}}+\frac {3 (3 a+7 b) \sinh (c+d x)}{(a+b)^2}+\frac {\sinh (3 (c+d x))}{a+b}}{12 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-12*b^2*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(5/2)) + (3*(3*a + 7*b)*Sinh[c + d*x])
/(a + b)^2 + Sinh[3*(c + d*x)]/(a + b))/(12*d)

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fricas [B]  time = 0.49, size = 1850, normalized size = 23.12 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/24*((a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^6 + 6*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)*sinh(d*x + c)^5 + (a^
3 + 2*a^2*b + a*b^2)*sinh(d*x + c)^6 + 3*(3*a^3 + 10*a^2*b + 7*a*b^2)*cosh(d*x + c)^4 + 3*(3*a^3 + 10*a^2*b +
7*a*b^2 + 5*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x +
c)^3 + 3*(3*a^3 + 10*a^2*b + 7*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - a^3 - 2*a^2*b - a*b^2 - 3*(3*a^3 + 10*
a^2*b + 7*a*b^2)*cosh(d*x + c)^2 + 3*(5*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^4 - 3*a^3 - 10*a^2*b - 7*a*b^2 +
6*(3*a^3 + 10*a^2*b + 7*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 12*(b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x +
c)^2*sinh(d*x + c) + 3*b^2*cosh(d*x + c)*sinh(d*x + c)^2 + b^2*sinh(d*x + c)^3)*sqrt(-a^2 - a*b)*log(((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*(3*a + b)*cosh(d*x + c
)^2 + 2*(3*(a + b)*cosh(d*x + c)^2 - 3*a - b)*sinh(d*x + c)^2 + 4*((a + b)*cosh(d*x + c)^3 - (3*a + b)*cosh(d*
x + c))*sinh(d*x + c) - 4*(cosh(d*x + c)^3 + 3*cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x +
c)^2 - 1)*sinh(d*x + c) - cosh(d*x + c))*sqrt(-a^2 - a*b) + a + b)/((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)*cosh(d*x + c)^3 + (a - b)*cosh(d*x + c))*sinh(d*x + c) + a + b)) + 6*((a
^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^5 + 2*(3*a^3 + 10*a^2*b + 7*a*b^2)*cosh(d*x + c)^3 - (3*a^3 + 10*a^2*b + 7
*a*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^3 + 3*(a^4 + 3*a^3*
b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)
*sinh(d*x + c)^2 + (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*sinh(d*x + c)^3), 1/24*((a^3 + 2*a^2*b + a*b^2)*cosh(
d*x + c)^6 + 6*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)*sinh(d*x + c)^5 + (a^3 + 2*a^2*b + a*b^2)*sinh(d*x + c)^6
+ 3*(3*a^3 + 10*a^2*b + 7*a*b^2)*cosh(d*x + c)^4 + 3*(3*a^3 + 10*a^2*b + 7*a*b^2 + 5*(a^3 + 2*a^2*b + a*b^2)*
cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^3 + 3*(3*a^3 + 10*a^2*b + 7*a*b^
2)*cosh(d*x + c))*sinh(d*x + c)^3 - a^3 - 2*a^2*b - a*b^2 - 3*(3*a^3 + 10*a^2*b + 7*a*b^2)*cosh(d*x + c)^2 + 3
*(5*(a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^4 - 3*a^3 - 10*a^2*b - 7*a*b^2 + 6*(3*a^3 + 10*a^2*b + 7*a*b^2)*cosh
(d*x + c)^2)*sinh(d*x + c)^2 + 24*(b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x + c)^2*sinh(d*x + c) + 3*b^2*cosh(d*x
+ c)*sinh(d*x + c)^2 + b^2*sinh(d*x + c)^3)*sqrt(a^2 + a*b)*arctan(1/2*((a + b)*cosh(d*x + c)^3 + 3*(a + b)*co
sh(d*x + c)*sinh(d*x + c)^2 + (a + b)*sinh(d*x + c)^3 + (3*a - b)*cosh(d*x + c) + (3*(a + b)*cosh(d*x + c)^2 +
3*a - b)*sinh(d*x + c))/sqrt(a^2 + a*b)) + 24*(b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x + c)^2*sinh(d*x + c) + 3*
b^2*cosh(d*x + c)*sinh(d*x + c)^2 + b^2*sinh(d*x + c)^3)*sqrt(a^2 + a*b)*arctan(1/2*sqrt(a^2 + a*b)*(cosh(d*x
+ c) + sinh(d*x + c))/a) + 6*((a^3 + 2*a^2*b + a*b^2)*cosh(d*x + c)^5 + 2*(3*a^3 + 10*a^2*b + 7*a*b^2)*cosh(d*
x + c)^3 - (3*a^3 + 10*a^2*b + 7*a*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*c
osh(d*x + c)^3 + 3*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^4 + 3*a^3*b + 3*
a^2*b^2 + a*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^2 + (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*sinh(d*x + c)^3)]

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giac [B]  time = 0.49, size = 829, normalized size = 10.36 \[ \frac {\frac {24 \, {\left (a b^{2} + \sqrt {-a b} b^{2}\right )} \sqrt {a^{2} - b^{2} + 2 \, \sqrt {-a b} {\left (a + b\right )}} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} \arctan \left (\frac {e^{\left (d x\right )}}{\sqrt {\frac {a^{3} e^{\left (2 \, c\right )} + a^{2} b e^{\left (2 \, c\right )} - a b^{2} e^{\left (2 \, c\right )} - b^{3} e^{\left (2 \, c\right )} + \sqrt {{\left (a^{3} e^{\left (2 \, c\right )} + a^{2} b e^{\left (2 \, c\right )} - a b^{2} e^{\left (2 \, c\right )} - b^{3} e^{\left (2 \, c\right )}\right )}^{2} - {\left (a^{3} e^{\left (4 \, c\right )} + 3 \, a^{2} b e^{\left (4 \, c\right )} + 3 \, a b^{2} e^{\left (4 \, c\right )} + b^{3} e^{\left (4 \, c\right )}\right )} {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}}}{a^{3} e^{\left (4 \, c\right )} + 3 \, a^{2} b e^{\left (4 \, c\right )} + 3 \, a b^{2} e^{\left (4 \, c\right )} + b^{3} e^{\left (4 \, c\right )}}}}\right ) e^{\left (-2 \, c\right )}}{a^{6} + 3 \, a^{5} b + 2 \, a^{4} b^{2} - 2 \, a^{3} b^{3} - 3 \, a^{2} b^{4} - a b^{5} + 2 \, {\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} \sqrt {-a b}} + \frac {24 \, {\left (a^{2} b^{2} - 3 \, a b^{3} - {\left (3 \, a b^{2} - b^{3}\right )} \sqrt {-a b}\right )} {\left | a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )} \right |} \arctan \left (\frac {e^{\left (d x\right )}}{\sqrt {\frac {a^{3} e^{\left (2 \, c\right )} + a^{2} b e^{\left (2 \, c\right )} - a b^{2} e^{\left (2 \, c\right )} - b^{3} e^{\left (2 \, c\right )} - \sqrt {{\left (a^{3} e^{\left (2 \, c\right )} + a^{2} b e^{\left (2 \, c\right )} - a b^{2} e^{\left (2 \, c\right )} - b^{3} e^{\left (2 \, c\right )}\right )}^{2} - {\left (a^{3} e^{\left (4 \, c\right )} + 3 \, a^{2} b e^{\left (4 \, c\right )} + 3 \, a b^{2} e^{\left (4 \, c\right )} + b^{3} e^{\left (4 \, c\right )}\right )} {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}}}{a^{3} e^{\left (4 \, c\right )} + 3 \, a^{2} b e^{\left (4 \, c\right )} + 3 \, a b^{2} e^{\left (4 \, c\right )} + b^{3} e^{\left (4 \, c\right )}}}}\right ) e^{\left (-2 \, c\right )}}{{\left (a^{5} + 2 \, a^{4} b - 2 \, a^{2} b^{3} - a b^{4} - 2 \, {\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \sqrt {-a b}\right )} \sqrt {a^{2} - b^{2} - 2 \, \sqrt {-a b} {\left (a + b\right )}}} - \frac {{\left (9 \, a e^{\left (2 \, d x + 2 \, c\right )} + 21 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )} e^{\left (-3 \, d x\right )}}{a^{2} e^{\left (3 \, c\right )} + 2 \, a b e^{\left (3 \, c\right )} + b^{2} e^{\left (3 \, c\right )}} + \frac {a^{2} e^{\left (3 \, d x + 24 \, c\right )} + 2 \, a b e^{\left (3 \, d x + 24 \, c\right )} + b^{2} e^{\left (3 \, d x + 24 \, c\right )} + 9 \, a^{2} e^{\left (d x + 22 \, c\right )} + 30 \, a b e^{\left (d x + 22 \, c\right )} + 21 \, b^{2} e^{\left (d x + 22 \, c\right )}}{a^{3} e^{\left (21 \, c\right )} + 3 \, a^{2} b e^{\left (21 \, c\right )} + 3 \, a b^{2} e^{\left (21 \, c\right )} + b^{3} e^{\left (21 \, c\right )}}}{24 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(24*(a*b^2 + sqrt(-a*b)*b^2)*sqrt(a^2 - b^2 + 2*sqrt(-a*b)*(a + b))*abs(a*e^(2*c) + b*e^(2*c))*arctan(e^(
d*x)/sqrt((a^3*e^(2*c) + a^2*b*e^(2*c) - a*b^2*e^(2*c) - b^3*e^(2*c) + sqrt((a^3*e^(2*c) + a^2*b*e^(2*c) - a*b
^2*e^(2*c) - b^3*e^(2*c))^2 - (a^3*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) + b^3*e^(4*c))*(a^3 + 3*a^2*b +
3*a*b^2 + b^3)))/(a^3*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) + b^3*e^(4*c))))*e^(-2*c)/(a^6 + 3*a^5*b +
2*a^4*b^2 - 2*a^3*b^3 - 3*a^2*b^4 - a*b^5 + 2*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*sqrt(-a*b)) + 24
*(a^2*b^2 - 3*a*b^3 - (3*a*b^2 - b^3)*sqrt(-a*b))*abs(a*e^(2*c) + b*e^(2*c))*arctan(e^(d*x)/sqrt((a^3*e^(2*c)
+ a^2*b*e^(2*c) - a*b^2*e^(2*c) - b^3*e^(2*c) - sqrt((a^3*e^(2*c) + a^2*b*e^(2*c) - a*b^2*e^(2*c) - b^3*e^(2*c
))^2 - (a^3*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) + b^3*e^(4*c))*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)))/(a^3*
e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) + b^3*e^(4*c))))*e^(-2*c)/((a^5 + 2*a^4*b - 2*a^2*b^3 - a*b^4 - 2*
(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*sqrt(-a*b))*sqrt(a^2 - b^2 - 2*sqrt(-a*b)*(a + b))) - (9*a*e^(2*d*x + 2*c)
+ 21*b*e^(2*d*x + 2*c) + a + b)*e^(-3*d*x)/(a^2*e^(3*c) + 2*a*b*e^(3*c) + b^2*e^(3*c)) + (a^2*e^(3*d*x + 24*c
) + 2*a*b*e^(3*d*x + 24*c) + b^2*e^(3*d*x + 24*c) + 9*a^2*e^(d*x + 22*c) + 30*a*b*e^(d*x + 22*c) + 21*b^2*e^(d
*x + 22*c))/(a^3*e^(21*c) + 3*a^2*b*e^(21*c) + 3*a*b^2*e^(21*c) + b^3*e^(21*c)))/d

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maple [B]  time = 0.44, size = 468, normalized size = 5.85 \[ -\frac {2}{3 d \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (2 b +2 a \right )}-\frac {1}{d \left (2 b +2 a \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {a}{d \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2 b}{d \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2}{3 d \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3} \left (2 b +2 a \right )}+\frac {1}{d \left (2 b +2 a \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {a}{d \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2 b}{d \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {b^{2} \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{d \left (a +b \right )^{2} \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {b^{3} \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{d \left (a +b \right )^{2} \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {b^{2} \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{d \left (a +b \right )^{2} \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}+\frac {b^{3} \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{d \left (a +b \right )^{2} \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left ({\left (a e^{\left (6 \, c\right )} + b e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 3 \, {\left (3 \, a e^{\left (4 \, c\right )} + 7 \, b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} - 3 \, {\left (3 \, a e^{\left (2 \, c\right )} + 7 \, b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - a - b\right )} e^{\left (-3 \, d x\right )}}{24 \, {\left (a^{2} d e^{\left (3 \, c\right )} + 2 \, a b d e^{\left (3 \, c\right )} + b^{2} d e^{\left (3 \, c\right )}\right )}} + \frac {1}{8} \, \int \frac {16 \, {\left (b^{2} e^{\left (3 \, d x + 3 \, c\right )} + b^{2} e^{\left (d x + c\right )}\right )}}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3} + {\left (a^{3} e^{\left (4 \, c\right )} + 3 \, a^{2} b e^{\left (4 \, c\right )} + 3 \, a b^{2} e^{\left (4 \, c\right )} + b^{3} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \, {\left (a^{3} e^{\left (2 \, c\right )} + a^{2} b e^{\left (2 \, c\right )} - a b^{2} e^{\left (2 \, c\right )} - b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*((a*e^(6*c) + b*e^(6*c))*e^(6*d*x) + 3*(3*a*e^(4*c) + 7*b*e^(4*c))*e^(4*d*x) - 3*(3*a*e^(2*c) + 7*b*e^(2*
c))*e^(2*d*x) - a - b)*e^(-3*d*x)/(a^2*d*e^(3*c) + 2*a*b*d*e^(3*c) + b^2*d*e^(3*c)) + 1/8*integrate(16*(b^2*e^
(3*d*x + 3*c) + b^2*e^(d*x + c))/(a^3 + 3*a^2*b + 3*a*b^2 + b^3 + (a^3*e^(4*c) + 3*a^2*b*e^(4*c) + 3*a*b^2*e^(
4*c) + b^3*e^(4*c))*e^(4*d*x) + 2*(a^3*e^(2*c) + a^2*b*e^(2*c) - a*b^2*e^(2*c) - b^3*e^(2*c))*e^(2*d*x)), x)

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mupad [B]  time = 2.88, size = 2194, normalized size = 27.42 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(3*c + 3*d*x)/(24*d*(a + b)) - exp(- 3*c - 3*d*x)/(24*d*(a + b)) + ((b^4)^(1/2)*(2*atan((exp(d*x)*exp(c)*((
4*(10*a^2*d*(b^4)^(5/2) + 12*a^6*d*(b^4)^(3/2) + 2*a*b^9*d*(b^4)^(1/2) + 10*a^3*b^3*d*(b^4)^(3/2) + 2*a^2*b^8*
d*(b^4)^(1/2) + 20*a^3*b^7*d*(b^4)^(1/2) + 40*a^4*b^6*d*(b^4)^(1/2) + 30*a^5*b^5*d*(b^4)^(1/2) + 2*a^7*b^3*d*(
b^4)^(1/2)))/(a*b^5*(a + b)^5*(a*d^2*(a + b)^5)^(1/2)*(4*a*b^3 + 4*a^3*b + a^4 + b^4 + 6*a^2*b^2)*(5*a*b^4 + 5
*a^4*b + a^5 + b^5 + 10*a^2*b^3 + 10*a^3*b^2)*(a^6*d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*
d^2 + 10*a^4*b^2*d^2)^(1/2)) - (2*(b^9*(a^6*d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 1
0*a^4*b^2*d^2)^(1/2) + 4*a*b^8*(a^6*d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^
2*d^2)^(1/2) + 6*a^2*b^7*(a^6*d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)
^(1/2) + 4*a^3*b^6*(a^6*d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2)
+ a^4*b^5*(a^6*d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2)))/(a^2*
b^3*d*(a + b)^7*(b^4)^(1/2)*(4*a*b^3 + 4*a^3*b + a^4 + b^4 + 6*a^2*b^2)*(5*a*b^4 + 5*a^4*b + a^5 + b^5 + 10*a^
2*b^3 + 10*a^3*b^2)*(a^6*d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2
))) + (2*exp(3*c)*exp(3*d*x)*(b^9*(a^6*d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4
*b^2*d^2)^(1/2) + 4*a*b^8*(a^6*d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2
)^(1/2) + 6*a^2*b^7*(a^6*d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2
) + 4*a^3*b^6*(a^6*d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2) + a^
4*b^5*(a^6*d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2)))/(a^2*b^3*d
*(a + b)^7*(b^4)^(1/2)*(4*a*b^3 + 4*a^3*b + a^4 + b^4 + 6*a^2*b^2)*(5*a*b^4 + 5*a^4*b + a^5 + b^5 + 10*a^2*b^3
+ 10*a^3*b^2)*(a^6*d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2)))*(
(a^11*(a^6*d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2))/4 + (a*b^10
*(a^6*d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2))/4 + (5*a^10*b*(a
^6*d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2))/2 + (5*a^2*b^9*(a^6
*d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2))/2 + (45*a^3*b^8*(a^6*
d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2))/4 + 30*a^4*b^7*(a^6*d^
2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2) + (105*a^5*b^6*(a^6*d^2 +
a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2))/2 + 63*a^6*b^5*(a^6*d^2 + a
*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2) + (105*a^7*b^4*(a^6*d^2 + a*b^
5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2))/2 + 30*a^8*b^3*(a^6*d^2 + a*b^5*
d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2) + (45*a^9*b^2*(a^6*d^2 + a*b^5*d^2
+ 5*a^5*b*d^2 + 5*a^2*b^4*d^2 + 10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2))/4)) + 2*atan((b^2*exp(d*x)*exp(c)*(a*d
^2*(a + b)^5)^(1/2))/(2*a*d*(a + b)^2*(b^4)^(1/2)))))/(2*(a^6*d^2 + a*b^5*d^2 + 5*a^5*b*d^2 + 5*a^2*b^4*d^2 +
10*a^3*b^3*d^2 + 10*a^4*b^2*d^2)^(1/2)) - (exp(- c - d*x)*(3*a + 7*b))/(8*d*(a + b)^2) + (exp(c + d*x)*(3*a +
7*b))/(8*d*(a + b)^2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{3}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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