\(\int \frac {\sinh ^4(c+d x)}{(a+b \tanh ^2(c+d x))^3} \, dx\) [41]

Optimal result

Integrand size = 23, antiderivative size = 240 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {3 \left (a^2-10 a b+5 b^2\right ) x}{8 (a+b)^5}+\frac {3 \sqrt {b} \left (5 a^2-10 a b+b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 \sqrt {a} (a+b)^5 d}-\frac {(5 a-3 b) \cosh (c+d x) \sinh (c+d x)}{8 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {(7 a-5 b) b \tanh (c+d x)}{8 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {3 (a-b) b \tanh (c+d x)}{2 (a+b)^4 d \left (a+b \tanh ^2(c+d x)\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3744, 481, 541, 536, 212, 211} \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {3 \sqrt {b} \left (5 a^2-10 a b+b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 \sqrt {a} d (a+b)^5}+\frac {3 x \left (a^2-10 a b+5 b^2\right )}{8 (a+b)^5}+\frac {3 b (a-b) \tanh (c+d x)}{2 d (a+b)^4 \left (a+b \tanh ^2(c+d x)\right )}+\frac {b (7 a-5 b) \tanh (c+d x)}{8 d (a+b)^3 \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(5 a-3 b) \sinh (c+d x) \cosh (c+d x)}{8 d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )^2} \]

[In]

Int[Sinh[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

(3*(a^2 - 10*a*b + 5*b^2)*x)/(8*(a + b)^5) + (3*Sqrt[b]*(5*a^2 - 10*a*b + b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/
Sqrt[a]])/(8*Sqrt[a]*(a + b)^5*d) - ((5*a - 3*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*(a + b)^2*d*(a + b*Tanh[c + d
*x]^2)^2) + (Cosh[c + d*x]^3*Sinh[c + d*x])/(4*(a + b)*d*(a + b*Tanh[c + d*x]^2)^2) + ((7*a - 5*b)*b*Tanh[c +
d*x])/(8*(a + b)^3*d*(a + b*Tanh[c + d*x]^2)^2) + (3*(a - b)*b*Tanh[c + d*x])/(2*(a + b)^4*d*(a + b*Tanh[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 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 481

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-a)*e^(
2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Dist[e^
(2*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1)
+ (a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, 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 3744

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^3 \left (a+b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {a+(4 a-3 b) x^2}{\left (1-x^2\right )^2 \left (a+b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{4 (a+b) d} \\ & = -\frac {(5 a-3 b) \cosh (c+d x) \sinh (c+d x)}{8 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-a (3 a-5 b)+5 (5 a-3 b) b x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{8 (a+b)^2 d} \\ & = -\frac {(5 a-3 b) \cosh (c+d x) \sinh (c+d x)}{8 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {(7 a-5 b) b \tanh (c+d x)}{8 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {12 a^2 (a-3 b)-12 a (7 a-5 b) b x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{32 a (a+b)^3 d} \\ & = -\frac {(5 a-3 b) \cosh (c+d x) \sinh (c+d x)}{8 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {(7 a-5 b) b \tanh (c+d x)}{8 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {3 (a-b) b \tanh (c+d x)}{2 (a+b)^4 d \left (a+b \tanh ^2(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {-24 a^2 \left (a^2-6 a b+b^2\right )+96 a^2 (a-b) b x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{64 a^2 (a+b)^4 d} \\ & = -\frac {(5 a-3 b) \cosh (c+d x) \sinh (c+d x)}{8 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {(7 a-5 b) b \tanh (c+d x)}{8 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {3 (a-b) b \tanh (c+d x)}{2 (a+b)^4 d \left (a+b \tanh ^2(c+d x)\right )}+\frac {\left (3 b \left (5 a^2-10 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{8 (a+b)^5 d}+\frac {\left (3 \left (a^2-10 a b+5 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 (a+b)^5 d} \\ & = \frac {3 \left (a^2-10 a b+5 b^2\right ) x}{8 (a+b)^5}+\frac {3 \sqrt {b} \left (5 a^2-10 a b+b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 \sqrt {a} (a+b)^5 d}-\frac {(5 a-3 b) \cosh (c+d x) \sinh (c+d x)}{8 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {(7 a-5 b) b \tanh (c+d x)}{8 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {3 (a-b) b \tanh (c+d x)}{2 (a+b)^4 d \left (a+b \tanh ^2(c+d x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.64 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.77 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {12 \left (a^2-10 a b+5 b^2\right ) (c+d x)+\frac {12 \sqrt {b} \left (5 a^2-10 a b+b^2\right ) \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a}}-8 (a-2 b) (a+b) \sinh (2 (c+d x))+\frac {16 a b^2 (a+b) \sinh (2 (c+d x))}{(a-b+(a+b) \cosh (2 (c+d x)))^2}+\frac {4 (9 a-5 b) b (a+b) \sinh (2 (c+d x))}{a-b+(a+b) \cosh (2 (c+d x))}+(a+b)^2 \sinh (4 (c+d x))}{32 (a+b)^5 d} \]

[In]

Integrate[Sinh[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

(12*(a^2 - 10*a*b + 5*b^2)*(c + d*x) + (12*Sqrt[b]*(5*a^2 - 10*a*b + b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[
a]])/Sqrt[a] - 8*(a - 2*b)*(a + b)*Sinh[2*(c + d*x)] + (16*a*b^2*(a + b)*Sinh[2*(c + d*x)])/(a - b + (a + b)*C
osh[2*(c + d*x)])^2 + (4*(9*a - 5*b)*b*(a + b)*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)]) + (a + b
)^2*Sinh[4*(c + d*x)])/(32*(a + b)^5*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(609\) vs. \(2(220)=440\).

Time = 193.92 (sec) , antiderivative size = 610, normalized size of antiderivative = 2.54

method	result	size
derivativedivides	\(\frac {-\frac {1}{4 \left (a +b \right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {1}{2 \left (a +b \right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {3 a -9 b}{8 \left (a +b \right )^{4} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {11 b -a}{8 \left (a +b \right )^{4} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (3 a^{2}-30 a b +15 b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )^{5}}+\frac {1}{4 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {3 a -9 b}{8 \left (a +b \right )^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {-11 b +a}{8 \left (a +b \right )^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-3 a^{2}+30 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 )^{5}}-\frac {2 b \left (\frac {-\frac {3 a \left (3 a^{2}+2 a b -b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8}+\left (-\frac {27}{8} a^{3}-\frac {23}{4} a^{2} b +\frac {1}{8} a \,b^{2}+\frac {5}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {27}{8} a^{3}-\frac {23}{4} a^{2} b +\frac {1}{8} a \,b^{2}+\frac {5}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-\frac {9}{8} a^{3}-\frac {3}{4} a^{2} b +\frac {3}{8} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\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 \right )^{2}}+\frac {\left (15 a^{2}-30 a b +3 b^{2}\right ) a \left (\frac {\left (a +\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 (-a +\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 )}{8}\right )}{\left (a +b \right )^{5}}}{d}\)	\(610\)
default	\(\frac {-\frac {1}{4 \left (a +b \right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {1}{2 \left (a +b \right )^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {3 a -9 b}{8 \left (a +b \right )^{4} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {11 b -a}{8 \left (a +b \right )^{4} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (3 a^{2}-30 a b +15 b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )^{5}}+\frac {1}{4 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {3 a -9 b}{8 \left (a +b \right )^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {-11 b +a}{8 \left (a +b \right )^{4} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\left (-3 a^{2}+30 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 )^{5}}-\frac {2 b \left (\frac {-\frac {3 a \left (3 a^{2}+2 a b -b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8}+\left (-\frac {27}{8} a^{3}-\frac {23}{4} a^{2} b +\frac {1}{8} a \,b^{2}+\frac {5}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {27}{8} a^{3}-\frac {23}{4} a^{2} b +\frac {1}{8} a \,b^{2}+\frac {5}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-\frac {9}{8} a^{3}-\frac {3}{4} a^{2} b +\frac {3}{8} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\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 \right )^{2}}+\frac {\left (15 a^{2}-30 a b +3 b^{2}\right ) a \left (\frac {\left (a +\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 (-a +\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 )}{8}\right )}{\left (a +b \right )^{5}}}{d}\)	\(610\)
risch	\(\frac {3 x \,a^{2}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left (a +b \right )^{2}}-\frac {15 x a b}{4 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left (a +b \right )^{2}}+\frac {15 x \,b^{2}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left (a +b \right )^{2}}+\frac {{\mathrm e}^{4 d x +4 c}}{64 \left (a +b \right )^{3} d}-\frac {{\mathrm e}^{2 d x +2 c} a}{8 \left (a^{2}+2 a b +b^{2}\right ) \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 ) \left (a +b \right )^{2} d}+\frac {{\mathrm e}^{-2 d x -2 c} a}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left (a +b \right ) d}-\frac {{\mathrm e}^{-2 d x -2 c} b}{4 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \left (a +b \right ) d}-\frac {{\mathrm e}^{-4 d x -4 c}}{64 \left (a^{2}+2 a b +b^{2}\right ) \left (a +b \right ) d}-\frac {b \left (9 a^{3} {\mathrm e}^{6 d x +6 c}-9 a^{2} b \,{\mathrm e}^{6 d x +6 c}-13 a \,b^{2} {\mathrm e}^{6 d x +6 c}+5 \,{\mathrm e}^{6 d x +6 c} b^{3}+27 a^{3} {\mathrm e}^{4 d x +4 c}-33 a^{2} b \,{\mathrm e}^{4 d x +4 c}+37 a \,b^{2} {\mathrm e}^{4 d x +4 c}-15 \,{\mathrm e}^{4 d x +4 c} b^{3}+27 a^{3} {\mathrm e}^{2 d x +2 c}-11 a^{2} b \,{\mathrm e}^{2 d x +2 c}-23 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}+15 \,{\mathrm e}^{2 d x +2 c} b^{3}+9 a^{3}+13 a^{2} b -a \,b^{2}-5 b^{3}\right )}{4 \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 )^{2} d \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) \left (a +b \right )}+\frac {15 a \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right )}{16 \left (a +b \right )^{5} d}-\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right ) b}{8 \left (a +b \right )^{5} d}+\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right ) b^{2}}{16 a \left (a +b \right )^{5} d}-\frac {15 a \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{16 \left (a +b \right )^{5} d}+\frac {15 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right ) b}{8 \left (a +b \right )^{5} d}-\frac {3 \sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right ) b^{2}}{16 a \left (a +b \right )^{5} d}\)	\(879\)

[In]

int(sinh(d*x+c)^4/(a+b*tanh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/4/(a+b)^3/(1+tanh(1/2*d*x+1/2*c))^4+1/2/(a+b)^3/(1+tanh(1/2*d*x+1/2*c))^3-1/8*(3*a-9*b)/(a+b)^4/(1+tan
h(1/2*d*x+1/2*c))-1/8*(11*b-a)/(a+b)^4/(1+tanh(1/2*d*x+1/2*c))^2+1/8/(a+b)^5*(3*a^2-30*a*b+15*b^2)*ln(1+tanh(1
/2*d*x+1/2*c))+1/4/(a+b)^3/(tanh(1/2*d*x+1/2*c)-1)^4+1/2/(a+b)^3/(tanh(1/2*d*x+1/2*c)-1)^3-1/8*(3*a-9*b)/(a+b)
^4/(tanh(1/2*d*x+1/2*c)-1)-1/8*(-11*b+a)/(a+b)^4/(tanh(1/2*d*x+1/2*c)-1)^2+1/8/(a+b)^5*(-3*a^2+30*a*b-15*b^2)*
ln(tanh(1/2*d*x+1/2*c)-1)-2*b/(a+b)^5*((-3/8*a*(3*a^2+2*a*b-b^2)*tanh(1/2*d*x+1/2*c)^7+(-27/8*a^3-23/4*a^2*b+1
/8*a*b^2+5/2*b^3)*tanh(1/2*d*x+1/2*c)^5+(-27/8*a^3-23/4*a^2*b+1/8*a*b^2+5/2*b^3)*tanh(1/2*d*x+1/2*c)^3+(-9/8*a
^3-3/4*a^2*b+3/8*a*b^2)*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)^2+1/8*(15*a^2-30*a*b+3*b^2)*a*(1/2*(a+((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+((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. 9248 vs. \(2 (220) = 440\).

Time = 0.53 (sec) , antiderivative size = 18818, normalized size of antiderivative = 78.41 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(sinh(d*x+c)^4/(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Timed out} \]

[In]

integrate(sinh(d*x+c)**4/(a+b*tanh(d*x+c)**2)**3,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3392 vs. \(2 (220) = 440\).

Time = 0.73 (sec) , antiderivative size = 3392, normalized size of antiderivative = 14.13 \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(sinh(d*x+c)^4/(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

-3/8*(a*b - 3*b^2)*log((a + b)*e^(4*d*x + 4*c) + 2*(a - b)*e^(2*d*x + 2*c) + a + b)/((a^5 + 5*a^4*b + 10*a^3*b
^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d) - 3/4*b*log((a + b)*e^(4*d*x + 4*c) + 2*(a - b)*e^(2*d*x + 2*c) + a + b)/(
(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d) + 3/8*(a*b - 3*b^2)*log(2*(a - b)*e^(-2*d*x - 2*c) + (a + b)*e^
(-4*d*x - 4*c) + a + b)/((a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d) + 3/4*b*log(2*(a - b)*e^
(-2*d*x - 2*c) + (a + b)*e^(-4*d*x - 4*c) + a + b)/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d) + 3/128*(5*
a^4*b - 80*a^3*b^2 + 50*a^2*b^3 + 8*a*b^4 + b^5)*arctan(1/2*((a + b)*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/((a^7
+ 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*sqrt(a*b)*d) + 3/32*(5*a^3*b - 15*a^2*b^2 - 5*a*b^
3 - b^4)*arctan(1/2*((a + b)*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2
*b^4)*sqrt(a*b)*d) - 3/128*(5*a^4*b - 80*a^3*b^2 + 50*a^2*b^3 + 8*a*b^4 + b^5)*arctan(1/2*((a + b)*e^(-2*d*x -
2*c) + a - b)/sqrt(a*b))/((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*sqrt(a*b)*d) - 3/32
*(5*a^3*b - 15*a^2*b^2 - 5*a*b^3 - b^4)*arctan(1/2*((a + b)*e^(-2*d*x - 2*c) + a - b)/sqrt(a*b))/((a^6 + 4*a^5
*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*sqrt(a*b)*d) - 3/64*(15*a^2*b + 10*a*b^2 + 3*b^3)*arctan(1/2*((a + b)*e^
(-2*d*x - 2*c) + a - b)/sqrt(a*b))/((a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)*sqrt(a*b)*d) - 1/64*(9*a^5*b - 65*a^
4*b^2 - 134*a^3*b^3 - 34*a^2*b^4 + 29*a*b^5 + 3*b^6 + (9*a^5*b - 183*a^4*b^2 + 98*a^3*b^3 + 266*a^2*b^4 - 27*a
*b^5 - 3*b^6)*e^(6*d*x + 6*c) + (27*a^5*b - 459*a^4*b^2 + 710*a^3*b^3 - 542*a^2*b^4 + 63*a*b^5 + 9*b^6)*e^(4*d
*x + 4*c) + (27*a^5*b - 341*a^4*b^2 + 86*a^3*b^3 + 398*a^2*b^4 - 65*a*b^5 - 9*b^6)*e^(2*d*x + 2*c))/((a^9 + 7*
a^8*b + 21*a^7*b^2 + 35*a^6*b^3 + 35*a^5*b^4 + 21*a^4*b^5 + 7*a^3*b^6 + a^2*b^7 + (a^9 + 7*a^8*b + 21*a^7*b^2
+ 35*a^6*b^3 + 35*a^5*b^4 + 21*a^4*b^5 + 7*a^3*b^6 + a^2*b^7)*e^(8*d*x + 8*c) + 4*(a^9 + 5*a^8*b + 9*a^7*b^2 +
5*a^6*b^3 - 5*a^5*b^4 - 9*a^4*b^5 - 5*a^3*b^6 - a^2*b^7)*e^(6*d*x + 6*c) + 2*(3*a^9 + 13*a^8*b + 23*a^7*b^2 +
25*a^6*b^3 + 25*a^5*b^4 + 23*a^4*b^5 + 13*a^3*b^6 + 3*a^2*b^7)*e^(4*d*x + 4*c) + 4*(a^9 + 5*a^8*b + 9*a^7*b^2
+ 5*a^6*b^3 - 5*a^5*b^4 - 9*a^4*b^5 - 5*a^3*b^6 - a^2*b^7)*e^(2*d*x + 2*c))*d) + 1/64*(9*a^5*b - 65*a^4*b^2 -
134*a^3*b^3 - 34*a^2*b^4 + 29*a*b^5 + 3*b^6 + (27*a^5*b - 341*a^4*b^2 + 86*a^3*b^3 + 398*a^2*b^4 - 65*a*b^5 -
9*b^6)*e^(-2*d*x - 2*c) + (27*a^5*b - 459*a^4*b^2 + 710*a^3*b^3 - 542*a^2*b^4 + 63*a*b^5 + 9*b^6)*e^(-4*d*x -
4*c) + (9*a^5*b - 183*a^4*b^2 + 98*a^3*b^3 + 266*a^2*b^4 - 27*a*b^5 - 3*b^6)*e^(-6*d*x - 6*c))/((a^9 + 7*a^8*
b + 21*a^7*b^2 + 35*a^6*b^3 + 35*a^5*b^4 + 21*a^4*b^5 + 7*a^3*b^6 + a^2*b^7 + 4*(a^9 + 5*a^8*b + 9*a^7*b^2 + 5
*a^6*b^3 - 5*a^5*b^4 - 9*a^4*b^5 - 5*a^3*b^6 - a^2*b^7)*e^(-2*d*x - 2*c) + 2*(3*a^9 + 13*a^8*b + 23*a^7*b^2 +
25*a^6*b^3 + 25*a^5*b^4 + 23*a^4*b^5 + 13*a^3*b^6 + 3*a^2*b^7)*e^(-4*d*x - 4*c) + 4*(a^9 + 5*a^8*b + 9*a^7*b^2
+ 5*a^6*b^3 - 5*a^5*b^4 - 9*a^4*b^5 - 5*a^3*b^6 - a^2*b^7)*e^(-6*d*x - 6*c) + (a^9 + 7*a^8*b + 21*a^7*b^2 + 3
5*a^6*b^3 + 35*a^5*b^4 + 21*a^4*b^5 + 7*a^3*b^6 + a^2*b^7)*e^(-8*d*x - 8*c))*d) - 1/16*(9*a^4*b + 4*a^3*b^2 -
22*a^2*b^3 - 20*a*b^4 - 3*b^5 + 3*(3*a^4*b - 22*a^3*b^2 - 20*a^2*b^3 + 6*a*b^4 + b^5)*e^(6*d*x + 6*c) + (27*a^
4*b - 156*a^3*b^2 + 110*a^2*b^3 - 36*a*b^4 - 9*b^5)*e^(4*d*x + 4*c) + (27*a^4*b - 86*a^3*b^2 - 84*a^2*b^3 + 38
*a*b^4 + 9*b^5)*e^(2*d*x + 2*c))/((a^8 + 6*a^7*b + 15*a^6*b^2 + 20*a^5*b^3 + 15*a^4*b^4 + 6*a^3*b^5 + a^2*b^6
+ (a^8 + 6*a^7*b + 15*a^6*b^2 + 20*a^5*b^3 + 15*a^4*b^4 + 6*a^3*b^5 + a^2*b^6)*e^(8*d*x + 8*c) + 4*(a^8 + 4*a^
7*b + 5*a^6*b^2 - 5*a^4*b^4 - 4*a^3*b^5 - a^2*b^6)*e^(6*d*x + 6*c) + 2*(3*a^8 + 10*a^7*b + 13*a^6*b^2 + 12*a^5
*b^3 + 13*a^4*b^4 + 10*a^3*b^5 + 3*a^2*b^6)*e^(4*d*x + 4*c) + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 - 5*a^4*b^4 - 4*a^3
*b^5 - a^2*b^6)*e^(2*d*x + 2*c))*d) + 1/16*(9*a^4*b + 4*a^3*b^2 - 22*a^2*b^3 - 20*a*b^4 - 3*b^5 + (27*a^4*b -
86*a^3*b^2 - 84*a^2*b^3 + 38*a*b^4 + 9*b^5)*e^(-2*d*x - 2*c) + (27*a^4*b - 156*a^3*b^2 + 110*a^2*b^3 - 36*a*b^
4 - 9*b^5)*e^(-4*d*x - 4*c) + 3*(3*a^4*b - 22*a^3*b^2 - 20*a^2*b^3 + 6*a*b^4 + b^5)*e^(-6*d*x - 6*c))/((a^8 +
6*a^7*b + 15*a^6*b^2 + 20*a^5*b^3 + 15*a^4*b^4 + 6*a^3*b^5 + a^2*b^6 + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 - 5*a^4*b^
4 - 4*a^3*b^5 - a^2*b^6)*e^(-2*d*x - 2*c) + 2*(3*a^8 + 10*a^7*b + 13*a^6*b^2 + 12*a^5*b^3 + 13*a^4*b^4 + 10*a^
3*b^5 + 3*a^2*b^6)*e^(-4*d*x - 4*c) + 4*(a^8 + 4*a^7*b + 5*a^6*b^2 - 5*a^4*b^4 - 4*a^3*b^5 - a^2*b^6)*e^(-6*d*
x - 6*c) + (a^8 + 6*a^7*b + 15*a^6*b^2 + 20*a^5*b^3 + 15*a^4*b^4 + 6*a^3*b^5 + a^2*b^6)*e^(-8*d*x - 8*c))*d) +
3/32*(9*a^3*b + 21*a^2*b^2 + 15*a*b^3 + 3*b^4 + (27*a^3*b + 13*a^2*b^2 - 23*a*b^3 - 9*b^4)*e^(-2*d*x - 2*c) +
3*(9*a^3*b - 3*a^2*b^2 + 7*a*b^3 + 3*b^4)*e^(-4*d*x - 4*c) + (9*a^3*b - a^2*b^2 - 13*a*b^3 - 3*b^4)*e^(-6*d*x
- 6*c))/((a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5 + 4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^
4*b^3 - 3*a^3*b^4 - a^2*b^5)*e^(-2*d*x - 2*c) + 2*(3*a^7 + 7*a^6*b + 6*a^5*b^2 + 6*a^4*b^3 + 7*a^3*b^4 + 3*a^2
*b^5)*e^(-4*d*x - 4*c) + 4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4*b^3 - 3*a^3*b^4 - a^2*b^5)*e^(-6*d*x - 6*c) + (a
^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*e^(-8*d*x - 8*c))*d) + 3/8*(d*x + c)/((a^3 + 3*a
^2*b + 3*a*b^2 + b^3)*d) + 1/64*((a + b)*e^(4*d*x + 4*c) + 24*b*e^(2*d*x + 2*c))/((a^4 + 4*a^3*b + 6*a^2*b^2 +
4*a*b^3 + b^4)*d) - 1/64*(24*b*e^(-2*d*x - 2*c) + (a + b)*e^(-4*d*x - 4*c))/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a
*b^3 + b^4)*d) - 1/8*e^(2*d*x + 2*c)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d) + 1/8*e^(-2*d*x - 2*c)/((a^3 + 3*a^2*
b + 3*a*b^2 + b^3)*d)

Giac [F]

\[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int { \frac {\sinh \left (d x + c\right )^{4}}{{\left (b \tanh \left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]

[In]

integrate(sinh(d*x+c)^4/(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^4}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \]

[In]

int(sinh(c + d*x)^4/(a + b*tanh(c + d*x)^2)^3,x)

[Out]

int(sinh(c + d*x)^4/(a + b*tanh(c + d*x)^2)^3, x)