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

Optimal. Leaf size=124 \[ \frac {\cosh ^3(c+d x)}{3 d (a+b)^2}-\frac {(a-b) \cosh (c+d x)}{d (a+b)^3}+\frac {a b \text {sech}(c+d x)}{2 d (a+b)^3 \left (a-b \text {sech}^2(c+d x)+b\right )}+\frac {\sqrt {b} (3 a-2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{2 d (a+b)^{7/2}} \]

[Out]

-(a-b)*cosh(d*x+c)/(a+b)^3/d+1/3*cosh(d*x+c)^3/(a+b)^2/d+1/2*a*b*sech(d*x+c)/(a+b)^3/d/(a+b-b*sech(d*x+c)^2)+1
/2*(3*a-2*b)*arctanh(sech(d*x+c)*b^(1/2)/(a+b)^(1/2))*b^(1/2)/(a+b)^(7/2)/d

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Rubi [A]  time = 0.22, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3664, 456, 1261, 208} \[ \frac {\cosh ^3(c+d x)}{3 d (a+b)^2}-\frac {(a-b) \cosh (c+d x)}{d (a+b)^3}+\frac {a b \text {sech}(c+d x)}{2 d (a+b)^3 \left (a-b \text {sech}^2(c+d x)+b\right )}+\frac {\sqrt {b} (3 a-2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{2 d (a+b)^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*a - 2*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(2*(a + b)^(7/2)*d) - ((a - b)*Cosh[c + d*x
])/((a + b)^3*d) + Cosh[c + d*x]^3/(3*(a + b)^2*d) + (a*b*Sech[c + d*x])/(2*(a + b)^3*d*(a + b - b*Sech[c + d*
x]^2))

Rule 208

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

Rule 456

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[x^m*(a + b*x^2)^(p +
1)*ExpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)]
- ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &
& ILtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1261

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] &&
NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 3664

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sec[e + f*x], x]}, Dist[1/(f*ff^m), Subst[Int[((-1 + ff^2*x^2)^((m - 1)/2)*(a - b + b*ff^2*x^2)^p)/x^(
m + 1), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

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

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Mathematica [C]  time = 1.37, size = 160, normalized size = 1.29 \[ \frac {\frac {3 \cosh (c+d x) \left (a \left (\frac {4 b}{(a+b) \cosh (2 (c+d x))+a-b}-3\right )+5 b\right )}{(a+b)^3}+\frac {\cosh (3 (c+d x))}{(a+b)^2}+\frac {6 i \sqrt {b} (3 a-2 b) \left (\tan ^{-1}\left (\frac {-\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )-i \sqrt {a+b}}{\sqrt {b}}\right )+\tan ^{-1}\left (\frac {\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )-i \sqrt {a+b}}{\sqrt {b}}\right )\right )}{(a+b)^{7/2}}}{12 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((6*I)*(3*a - 2*b)*Sqrt[b]*(ArcTan[((-I)*Sqrt[a + b] - Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]] + ArcTan[((-I)*Sqr
t[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b]]))/(a + b)^(7/2) + (3*Cosh[c + d*x]*(5*b + a*(-3 + (4*b)/(a - b
+ (a + b)*Cosh[2*(c + d*x)]))))/(a + b)^3 + Cosh[3*(c + d*x)]/(a + b)^2)/(12*d)

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fricas [B]  time = 0.75, size = 5025, normalized size = 40.52 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/24*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^10 + 10*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^9 + (a^2 + 2*
a*b + b^2)*sinh(d*x + c)^10 - (7*a^2 - 6*a*b - 13*b^2)*cosh(d*x + c)^8 + (45*(a^2 + 2*a*b + b^2)*cosh(d*x + c)
^2 - 7*a^2 + 6*a*b + 13*b^2)*sinh(d*x + c)^8 + 8*(15*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 - (7*a^2 - 6*a*b - 13
*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 - 2*(13*a^2 - 40*a*b + 7*b^2)*cosh(d*x + c)^6 + 2*(105*(a^2 + 2*a*b + b^2
)*cosh(d*x + c)^4 - 14*(7*a^2 - 6*a*b - 13*b^2)*cosh(d*x + c)^2 - 13*a^2 + 40*a*b - 7*b^2)*sinh(d*x + c)^6 + 4
*(63*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^5 - 14*(7*a^2 - 6*a*b - 13*b^2)*cosh(d*x + c)^3 - 3*(13*a^2 - 40*a*b +
7*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(13*a^2 - 40*a*b + 7*b^2)*cosh(d*x + c)^4 + 2*(105*(a^2 + 2*a*b + b^
2)*cosh(d*x + c)^6 - 35*(7*a^2 - 6*a*b - 13*b^2)*cosh(d*x + c)^4 - 15*(13*a^2 - 40*a*b + 7*b^2)*cosh(d*x + c)^
2 - 13*a^2 + 40*a*b - 7*b^2)*sinh(d*x + c)^4 + 8*(15*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^7 - 7*(7*a^2 - 6*a*b -
13*b^2)*cosh(d*x + c)^5 - 5*(13*a^2 - 40*a*b + 7*b^2)*cosh(d*x + c)^3 - (13*a^2 - 40*a*b + 7*b^2)*cosh(d*x + c
))*sinh(d*x + c)^3 - (7*a^2 - 6*a*b - 13*b^2)*cosh(d*x + c)^2 + (45*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^8 - 28*(
7*a^2 - 6*a*b - 13*b^2)*cosh(d*x + c)^6 - 30*(13*a^2 - 40*a*b + 7*b^2)*cosh(d*x + c)^4 - 12*(13*a^2 - 40*a*b +
7*b^2)*cosh(d*x + c)^2 - 7*a^2 + 6*a*b + 13*b^2)*sinh(d*x + c)^2 - 6*((3*a^2 + a*b - 2*b^2)*cosh(d*x + c)^7 +
7*(3*a^2 + a*b - 2*b^2)*cosh(d*x + c)*sinh(d*x + c)^6 + (3*a^2 + a*b - 2*b^2)*sinh(d*x + c)^7 + 2*(3*a^2 - 5*
a*b + 2*b^2)*cosh(d*x + c)^5 + (21*(3*a^2 + a*b - 2*b^2)*cosh(d*x + c)^2 + 6*a^2 - 10*a*b + 4*b^2)*sinh(d*x +
c)^5 + 5*(7*(3*a^2 + a*b - 2*b^2)*cosh(d*x + c)^3 + 2*(3*a^2 - 5*a*b + 2*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 +
(3*a^2 + a*b - 2*b^2)*cosh(d*x + c)^3 + (35*(3*a^2 + a*b - 2*b^2)*cosh(d*x + c)^4 + 20*(3*a^2 - 5*a*b + 2*b^2
)*cosh(d*x + c)^2 + 3*a^2 + a*b - 2*b^2)*sinh(d*x + c)^3 + (21*(3*a^2 + a*b - 2*b^2)*cosh(d*x + c)^5 + 20*(3*a
^2 - 5*a*b + 2*b^2)*cosh(d*x + c)^3 + 3*(3*a^2 + a*b - 2*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 + (7*(3*a^2 + a*b
- 2*b^2)*cosh(d*x + c)^6 + 10*(3*a^2 - 5*a*b + 2*b^2)*cosh(d*x + c)^4 + 3*(3*a^2 + a*b - 2*b^2)*cosh(d*x + c)
^2)*sinh(d*x + c))*sqrt(b/(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*(a + 3*b)*cosh(d*x + c)^2 + 2*(3*(a + b)*cosh(d*x + c)^2 + a + 3*b)*sinh(d*x + c)^2
+ 4*((a + b)*cosh(d*x + c)^3 + (a + 3*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(b/(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)) + a^2 + 2*a*b + b^2 + 2*(5
*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^9 - 4*(7*a^2 - 6*a*b - 13*b^2)*cosh(d*x + c)^7 - 6*(13*a^2 - 40*a*b + 7*b^2
)*cosh(d*x + c)^5 - 4*(13*a^2 - 40*a*b + 7*b^2)*cosh(d*x + c)^3 - (7*a^2 - 6*a*b - 13*b^2)*cosh(d*x + c))*sinh
(d*x + c))/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)^7 + 7*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a
*b^3 + b^4)*d*cosh(d*x + c)*sinh(d*x + c)^6 + (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*sinh(d*x + c)^7 +
2*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*d*cosh(d*x + c)^5 + (21*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d
*x + c)^2 + 2*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*d)*sinh(d*x + c)^5 + (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)
*d*cosh(d*x + c)^3 + 5*(7*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)^3 + 2*(a^4 + 2*a^3*b - 2
*a*b^3 - b^4)*d*cosh(d*x + c))*sinh(d*x + c)^4 + (35*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x +
c)^4 + 20*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*d*cosh(d*x + c)^2 + (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d)*s
inh(d*x + c)^3 + (21*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)^5 + 20*(a^4 + 2*a^3*b - 2*a*b
^3 - b^4)*d*cosh(d*x + c)^3 + 3*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c))*sinh(d*x + c)^2 +
(7*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)^6 + 10*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*d*cosh(
d*x + c)^4 + 3*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)^2)*sinh(d*x + c)), 1/24*((a^2 + 2*a
*b + b^2)*cosh(d*x + c)^10 + 10*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^9 + (a^2 + 2*a*b + b^2)*sinh(d
*x + c)^10 - (7*a^2 - 6*a*b - 13*b^2)*cosh(d*x + c)^8 + (45*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 - 7*a^2 + 6*a*
b + 13*b^2)*sinh(d*x + c)^8 + 8*(15*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 - (7*a^2 - 6*a*b - 13*b^2)*cosh(d*x +
c))*sinh(d*x + c)^7 - 2*(13*a^2 - 40*a*b + 7*b^2)*cosh(d*x + c)^6 + 2*(105*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^4
- 14*(7*a^2 - 6*a*b - 13*b^2)*cosh(d*x + c)^2 - 13*a^2 + 40*a*b - 7*b^2)*sinh(d*x + c)^6 + 4*(63*(a^2 + 2*a*b
+ b^2)*cosh(d*x + c)^5 - 14*(7*a^2 - 6*a*b - 13*b^2)*cosh(d*x + c)^3 - 3*(13*a^2 - 40*a*b + 7*b^2)*cosh(d*x +
c))*sinh(d*x + c)^5 - 2*(13*a^2 - 40*a*b + 7*b^2)*cosh(d*x + c)^4 + 2*(105*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^
6 - 35*(7*a^2 - 6*a*b - 13*b^2)*cosh(d*x + c)^4 - 15*(13*a^2 - 40*a*b + 7*b^2)*cosh(d*x + c)^2 - 13*a^2 + 40*a
*b - 7*b^2)*sinh(d*x + c)^4 + 8*(15*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^7 - 7*(7*a^2 - 6*a*b - 13*b^2)*cosh(d*x
+ c)^5 - 5*(13*a^2 - 40*a*b + 7*b^2)*cosh(d*x + c)^3 - (13*a^2 - 40*a*b + 7*b^2)*cosh(d*x + c))*sinh(d*x + c)^
3 - (7*a^2 - 6*a*b - 13*b^2)*cosh(d*x + c)^2 + (45*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^8 - 28*(7*a^2 - 6*a*b - 1
3*b^2)*cosh(d*x + c)^6 - 30*(13*a^2 - 40*a*b + 7*b^2)*cosh(d*x + c)^4 - 12*(13*a^2 - 40*a*b + 7*b^2)*cosh(d*x
+ c)^2 - 7*a^2 + 6*a*b + 13*b^2)*sinh(d*x + c)^2 + 12*((3*a^2 + a*b - 2*b^2)*cosh(d*x + c)^7 + 7*(3*a^2 + a*b
- 2*b^2)*cosh(d*x + c)*sinh(d*x + c)^6 + (3*a^2 + a*b - 2*b^2)*sinh(d*x + c)^7 + 2*(3*a^2 - 5*a*b + 2*b^2)*cos
h(d*x + c)^5 + (21*(3*a^2 + a*b - 2*b^2)*cosh(d*x + c)^2 + 6*a^2 - 10*a*b + 4*b^2)*sinh(d*x + c)^5 + 5*(7*(3*a
^2 + a*b - 2*b^2)*cosh(d*x + c)^3 + 2*(3*a^2 - 5*a*b + 2*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 + (3*a^2 + a*b -
2*b^2)*cosh(d*x + c)^3 + (35*(3*a^2 + a*b - 2*b^2)*cosh(d*x + c)^4 + 20*(3*a^2 - 5*a*b + 2*b^2)*cosh(d*x + c)^
2 + 3*a^2 + a*b - 2*b^2)*sinh(d*x + c)^3 + (21*(3*a^2 + a*b - 2*b^2)*cosh(d*x + c)^5 + 20*(3*a^2 - 5*a*b + 2*b
^2)*cosh(d*x + c)^3 + 3*(3*a^2 + a*b - 2*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 + (7*(3*a^2 + a*b - 2*b^2)*cosh(d
*x + c)^6 + 10*(3*a^2 - 5*a*b + 2*b^2)*cosh(d*x + c)^4 + 3*(3*a^2 + a*b - 2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c
))*sqrt(-b/(a + b))*arctan(1/2*((a + b)*cosh(d*x + c)^3 + 3*(a + b)*cosh(d*x + c)*sinh(d*x + c)^2 + (a + b)*si
nh(d*x + c)^3 + (a - 3*b)*cosh(d*x + c) + (3*(a + b)*cosh(d*x + c)^2 + a - 3*b)*sinh(d*x + c))*sqrt(-b/(a + b)
)/b) - 12*((3*a^2 + a*b - 2*b^2)*cosh(d*x + c)^7 + 7*(3*a^2 + a*b - 2*b^2)*cosh(d*x + c)*sinh(d*x + c)^6 + (3*
a^2 + a*b - 2*b^2)*sinh(d*x + c)^7 + 2*(3*a^2 - 5*a*b + 2*b^2)*cosh(d*x + c)^5 + (21*(3*a^2 + a*b - 2*b^2)*cos
h(d*x + c)^2 + 6*a^2 - 10*a*b + 4*b^2)*sinh(d*x + c)^5 + 5*(7*(3*a^2 + a*b - 2*b^2)*cosh(d*x + c)^3 + 2*(3*a^2
- 5*a*b + 2*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 + (3*a^2 + a*b - 2*b^2)*cosh(d*x + c)^3 + (35*(3*a^2 + a*b -
2*b^2)*cosh(d*x + c)^4 + 20*(3*a^2 - 5*a*b + 2*b^2)*cosh(d*x + c)^2 + 3*a^2 + a*b - 2*b^2)*sinh(d*x + c)^3 + (
21*(3*a^2 + a*b - 2*b^2)*cosh(d*x + c)^5 + 20*(3*a^2 - 5*a*b + 2*b^2)*cosh(d*x + c)^3 + 3*(3*a^2 + a*b - 2*b^2
)*cosh(d*x + c))*sinh(d*x + c)^2 + (7*(3*a^2 + a*b - 2*b^2)*cosh(d*x + c)^6 + 10*(3*a^2 - 5*a*b + 2*b^2)*cosh(
d*x + c)^4 + 3*(3*a^2 + a*b - 2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c))*sqrt(-b/(a + b))*arctan(1/2*((a + b)*cosh
(d*x + c) + (a + b)*sinh(d*x + c))*sqrt(-b/(a + b))/b) + a^2 + 2*a*b + b^2 + 2*(5*(a^2 + 2*a*b + b^2)*cosh(d*x
+ c)^9 - 4*(7*a^2 - 6*a*b - 13*b^2)*cosh(d*x + c)^7 - 6*(13*a^2 - 40*a*b + 7*b^2)*cosh(d*x + c)^5 - 4*(13*a^2
- 40*a*b + 7*b^2)*cosh(d*x + c)^3 - (7*a^2 - 6*a*b - 13*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a^4 + 4*a^3*b +
6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)^7 + 7*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)*s
inh(d*x + c)^6 + (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*sinh(d*x + c)^7 + 2*(a^4 + 2*a^3*b - 2*a*b^3 -
b^4)*d*cosh(d*x + c)^5 + (21*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)^2 + 2*(a^4 + 2*a^3*b
- 2*a*b^3 - b^4)*d)*sinh(d*x + c)^5 + (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)^3 + 5*(7*(a^
4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)^3 + 2*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*d*cosh(d*x + c)
)*sinh(d*x + c)^4 + (35*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)^4 + 20*(a^4 + 2*a^3*b - 2*
a*b^3 - b^4)*d*cosh(d*x + c)^2 + (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d)*sinh(d*x + c)^3 + (21*(a^4 + 4
*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)^5 + 20*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*d*cosh(d*x + c)^3 +
3*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c))*sinh(d*x + c)^2 + (7*(a^4 + 4*a^3*b + 6*a^2*b^
2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)^6 + 10*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*d*cosh(d*x + c)^4 + 3*(a^4 + 4*a^3*b
+ 6*a^2*b^2 + 4*a*b^3 + b^4)*d*cosh(d*x + c)^2)*sinh(d*x + c))]

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giac [B]  time = 0.94, size = 1929, normalized size = 15.56 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*((9*a*e^(2*d*x + 2*c) - 15*b*e^(2*d*x + 2*c) - a - b)*e^(-3*d*x)/(a^3*e^(3*c) + 3*a^2*b*e^(3*c) + 3*a*b^
2*e^(3*c) + b^3*e^(3*c)) - 12*(2*(6*a^2*b^2 - 4*a*b^3 + (3*a^2*b - 5*a*b^2 + 2*b^3)*sqrt(-a*b))*(a^3*e^(2*c) +
3*a^2*b*e^(2*c) + 3*a*b^2*e^(2*c) + b^3*e^(2*c))^2*abs(a*e^(2*c) + b*e^(2*c)) + (3*a^6*b + 7*a^5*b^2 - 10*a^3
*b^4 - 5*a^2*b^5 + 3*a*b^6 + 2*b^7 - 2*(3*a^5*b + 10*a^4*b^2 + 10*a^3*b^3 - 5*a*b^5 - 2*b^6)*sqrt(-a*b))*abs(-
a^3*e^(2*c) - 3*a^2*b*e^(2*c) - 3*a*b^2*e^(2*c) - b^3*e^(2*c))*abs(a*e^(2*c) + b*e^(2*c))*e^(2*c) + (6*a^9*b +
26*a^8*b^2 + 34*a^7*b^3 - 6*a^6*b^4 - 50*a^5*b^5 - 34*a^4*b^6 + 6*a^3*b^7 + 14*a^2*b^8 + 4*a*b^9 + (3*a^9 + 1
0*a^8*b + 4*a^7*b^2 - 20*a^6*b^3 - 22*a^5*b^4 + 8*a^4*b^5 + 20*a^3*b^6 + 4*a^2*b^7 - 5*a*b^8 - 2*b^9)*sqrt(-a*
b))*abs(a*e^(2*c) + b*e^(2*c))*e^(4*c))*arctan(e^(d*x)/sqrt((a^4*e^(2*c) + 2*a^3*b*e^(2*c) - 2*a*b^3*e^(2*c) -
b^4*e^(2*c) + sqrt((a^4*e^(2*c) + 2*a^3*b*e^(2*c) - 2*a*b^3*e^(2*c) - b^4*e^(2*c))^2 - (a^4*e^(4*c) + 4*a^3*b
*e^(4*c) + 6*a^2*b^2*e^(4*c) + 4*a*b^3*e^(4*c) + b^4*e^(4*c))*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)))/(a
^4*e^(4*c) + 4*a^3*b*e^(4*c) + 6*a^2*b^2*e^(4*c) + 4*a*b^3*e^(4*c) + b^4*e^(4*c))))*e^(-4*c)/((a^9 + 9*a^8*b +
36*a^7*b^2 + 84*a^6*b^3 + 126*a^5*b^4 + 126*a^4*b^5 + 84*a^3*b^6 + 36*a^2*b^7 + 9*a*b^8 + b^9)*sqrt(a^2 - b^2
- 2*sqrt(-a*b)*(a + b))*abs(-a^3*e^(2*c) - 3*a^2*b*e^(2*c) - 3*a*b^2*e^(2*c) - b^3*e^(2*c))) - 12*(2*(6*a^2*b
^2 - 4*a*b^3 - (3*a^2*b - 5*a*b^2 + 2*b^3)*sqrt(-a*b))*(a^3*e^(2*c) + 3*a^2*b*e^(2*c) + 3*a*b^2*e^(2*c) + b^3*
e^(2*c))^2*abs(a*e^(2*c) + b*e^(2*c)) + (3*a^6*b + 7*a^5*b^2 - 10*a^3*b^4 - 5*a^2*b^5 + 3*a*b^6 + 2*b^7 + 2*(3
*a^5*b + 10*a^4*b^2 + 10*a^3*b^3 - 5*a*b^5 - 2*b^6)*sqrt(-a*b))*abs(-a^3*e^(2*c) - 3*a^2*b*e^(2*c) - 3*a*b^2*e
^(2*c) - b^3*e^(2*c))*abs(a*e^(2*c) + b*e^(2*c))*e^(2*c) + (6*a^9*b + 26*a^8*b^2 + 34*a^7*b^3 - 6*a^6*b^4 - 50
*a^5*b^5 - 34*a^4*b^6 + 6*a^3*b^7 + 14*a^2*b^8 + 4*a*b^9 - (3*a^9 + 10*a^8*b + 4*a^7*b^2 - 20*a^6*b^3 - 22*a^5
*b^4 + 8*a^4*b^5 + 20*a^3*b^6 + 4*a^2*b^7 - 5*a*b^8 - 2*b^9)*sqrt(-a*b))*abs(a*e^(2*c) + b*e^(2*c))*e^(4*c))*a
rctan(e^(d*x)/sqrt((a^4*e^(2*c) + 2*a^3*b*e^(2*c) - 2*a*b^3*e^(2*c) - b^4*e^(2*c) - sqrt((a^4*e^(2*c) + 2*a^3*
b*e^(2*c) - 2*a*b^3*e^(2*c) - b^4*e^(2*c))^2 - (a^4*e^(4*c) + 4*a^3*b*e^(4*c) + 6*a^2*b^2*e^(4*c) + 4*a*b^3*e^
(4*c) + b^4*e^(4*c))*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)))/(a^4*e^(4*c) + 4*a^3*b*e^(4*c) + 6*a^2*b^2*
e^(4*c) + 4*a*b^3*e^(4*c) + b^4*e^(4*c))))*e^(-4*c)/((a^9 + 9*a^8*b + 36*a^7*b^2 + 84*a^6*b^3 + 126*a^5*b^4 +
126*a^4*b^5 + 84*a^3*b^6 + 36*a^2*b^7 + 9*a*b^8 + b^9)*sqrt(a^2 - b^2 + 2*sqrt(-a*b)*(a + b))*abs(-a^3*e^(2*c)
- 3*a^2*b*e^(2*c) - 3*a*b^2*e^(2*c) - b^3*e^(2*c))) - (a^4*e^(3*d*x + 36*c) + 4*a^3*b*e^(3*d*x + 36*c) + 6*a^
2*b^2*e^(3*d*x + 36*c) + 4*a*b^3*e^(3*d*x + 36*c) + b^4*e^(3*d*x + 36*c) - 9*a^4*e^(d*x + 34*c) - 12*a^3*b*e^(
d*x + 34*c) + 18*a^2*b^2*e^(d*x + 34*c) + 36*a*b^3*e^(d*x + 34*c) + 15*b^4*e^(d*x + 34*c))/(a^6*e^(33*c) + 6*a
^5*b*e^(33*c) + 15*a^4*b^2*e^(33*c) + 20*a^3*b^3*e^(33*c) + 15*a^2*b^4*e^(33*c) + 6*a*b^5*e^(33*c) + b^6*e^(33
*c)) - 24*(a*b*e^(3*d*x + 3*c) + a*b*e^(d*x + 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) + 2*a*e^(2*d*x + 2*c) - 2*b*e^(2*d*x + 2*c) + a + b)))/d

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maple [B]  time = 0.32, size = 267, normalized size = 2.15 \[ \frac {-\frac {1}{3 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a +3 b}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {1}{3 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {a -3 b}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {4 b \left (\frac {\left (-\frac {a}{4}-\frac {b}{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {a}{4}}{\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a}-\frac {\left (3 a -2 b \right ) \arctanh \left (\frac {2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{8 \sqrt {a b +b^{2}}}\right )}{\left (a +b \right )^{3}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{2} + 2 \, a b + b^{2} + {\left (a^{2} e^{\left (10 \, c\right )} + 2 \, a b e^{\left (10 \, c\right )} + b^{2} e^{\left (10 \, c\right )}\right )} e^{\left (10 \, d x\right )} - {\left (7 \, a^{2} e^{\left (8 \, c\right )} - 6 \, a b e^{\left (8 \, c\right )} - 13 \, b^{2} e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} - 2 \, {\left (13 \, a^{2} e^{\left (6 \, c\right )} - 40 \, a b e^{\left (6 \, c\right )} + 7 \, b^{2} e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} - 2 \, {\left (13 \, a^{2} e^{\left (4 \, c\right )} - 40 \, a b e^{\left (4 \, c\right )} + 7 \, b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} - {\left (7 \, a^{2} e^{\left (2 \, c\right )} - 6 \, a b e^{\left (2 \, c\right )} - 13 \, b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}{24 \, {\left ({\left (a^{4} d e^{\left (7 \, c\right )} + 4 \, a^{3} b d e^{\left (7 \, c\right )} + 6 \, a^{2} b^{2} d e^{\left (7 \, c\right )} + 4 \, a b^{3} d e^{\left (7 \, c\right )} + b^{4} d e^{\left (7 \, c\right )}\right )} e^{\left (7 \, d x\right )} + 2 \, {\left (a^{4} d e^{\left (5 \, c\right )} + 2 \, a^{3} b d e^{\left (5 \, c\right )} - 2 \, a b^{3} d e^{\left (5 \, c\right )} - b^{4} d e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} + {\left (a^{4} d e^{\left (3 \, c\right )} + 4 \, a^{3} b d e^{\left (3 \, c\right )} + 6 \, a^{2} b^{2} d e^{\left (3 \, c\right )} + 4 \, a b^{3} d e^{\left (3 \, c\right )} + b^{4} d e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )}\right )}} - \frac {1}{8} \, \int \frac {8 \, {\left ({\left (3 \, a b e^{\left (3 \, c\right )} - 2 \, b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} - {\left (3 \, a b e^{c} - 2 \, b^{2} e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} + {\left (a^{4} e^{\left (4 \, c\right )} + 4 \, a^{3} b e^{\left (4 \, c\right )} + 6 \, a^{2} b^{2} e^{\left (4 \, c\right )} + 4 \, a b^{3} e^{\left (4 \, c\right )} + b^{4} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \, {\left (a^{4} e^{\left (2 \, c\right )} + 2 \, a^{3} b e^{\left (2 \, c\right )} - 2 \, a b^{3} e^{\left (2 \, c\right )} - b^{4} 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(sinh(d*x+c)^3/(a+b*tanh(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

1/24*(a^2 + 2*a*b + b^2 + (a^2*e^(10*c) + 2*a*b*e^(10*c) + b^2*e^(10*c))*e^(10*d*x) - (7*a^2*e^(8*c) - 6*a*b*e
^(8*c) - 13*b^2*e^(8*c))*e^(8*d*x) - 2*(13*a^2*e^(6*c) - 40*a*b*e^(6*c) + 7*b^2*e^(6*c))*e^(6*d*x) - 2*(13*a^2
*e^(4*c) - 40*a*b*e^(4*c) + 7*b^2*e^(4*c))*e^(4*d*x) - (7*a^2*e^(2*c) - 6*a*b*e^(2*c) - 13*b^2*e^(2*c))*e^(2*d
*x))/((a^4*d*e^(7*c) + 4*a^3*b*d*e^(7*c) + 6*a^2*b^2*d*e^(7*c) + 4*a*b^3*d*e^(7*c) + b^4*d*e^(7*c))*e^(7*d*x)
+ 2*(a^4*d*e^(5*c) + 2*a^3*b*d*e^(5*c) - 2*a*b^3*d*e^(5*c) - b^4*d*e^(5*c))*e^(5*d*x) + (a^4*d*e^(3*c) + 4*a^3
*b*d*e^(3*c) + 6*a^2*b^2*d*e^(3*c) + 4*a*b^3*d*e^(3*c) + b^4*d*e^(3*c))*e^(3*d*x)) - 1/8*integrate(8*((3*a*b*e
^(3*c) - 2*b^2*e^(3*c))*e^(3*d*x) - (3*a*b*e^c - 2*b^2*e^c)*e^(d*x))/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^
4 + (a^4*e^(4*c) + 4*a^3*b*e^(4*c) + 6*a^2*b^2*e^(4*c) + 4*a*b^3*e^(4*c) + b^4*e^(4*c))*e^(4*d*x) + 2*(a^4*e^(
2*c) + 2*a^3*b*e^(2*c) - 2*a*b^3*e^(2*c) - b^4*e^(2*c))*e^(2*d*x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^3}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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