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

Optimal. Leaf size=240 \[ \frac {3 \sqrt {b} \left (5 a^2-10 a b+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 \sqrt {a} d (a+b)^5}+\frac {3 x \left (a^2-10 a b+5 b^2\right )}{8 (a+b)^5}+\frac {3 b (a-b) \tanh (c+d x)}{2 d (a+b)^4 \left (a+b \tanh ^2(c+d x)\right )}+\frac {b (7 a-5 b) \tanh (c+d x)}{8 d (a+b)^3 \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(5 a-3 b) \sinh (c+d x) \cosh (c+d x)}{8 d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )^2} \]

[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)

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Rubi [A]  time = 0.35, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3663, 470, 527, 522, 206, 205} \[ \frac {3 \sqrt {b} \left (5 a^2-10 a b+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 \sqrt {a} d (a+b)^5}+\frac {3 x \left (a^2-10 a b+5 b^2\right )}{8 (a+b)^5}+\frac {3 b (a-b) \tanh (c+d x)}{2 d (a+b)^4 \left (a+b \tanh ^2(c+d x)\right )}+\frac {b (7 a-5 b) \tanh (c+d x)}{8 d (a+b)^3 \left (a+b \tanh ^2(c+d x)\right )^2}+\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d (a+b) \left (a+b \tanh ^2(c+d x)\right )^2}-\frac {(5 a-3 b) \sinh (c+d x) \cosh (c+d x)}{8 d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )^2} \]

Antiderivative was successfully verified.

[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 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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 470

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 522

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 527

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 3663

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*} \int \frac {\sinh ^4(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx &=\frac {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \tan ^{-1}\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*}

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Mathematica [A]  time = 0.86, size = 184, normalized size = 0.77 \[ \frac {12 \left (a^2-10 a b+5 b^2\right ) (c+d x)+\frac {12 \sqrt {b} \left (5 a^2-10 a b+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {16 a b^2 (a+b) \sinh (2 (c+d x))}{((a+b) \cosh (2 (c+d x))+a-b)^2}-8 (a-2 b) (a+b) \sinh (2 (c+d x))+(a+b)^2 \sinh (4 (c+d x))+\frac {4 b (9 a-5 b) (a+b) \sinh (2 (c+d x))}{(a+b) \cosh (2 (c+d x))+a-b}}{32 d (a+b)^5} \]

Antiderivative was successfully verified.

[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)

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

[Out]

Timed out

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giac [B]  time = 3.00, size = 923, normalized size = 3.85 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(24*(a^2 - 10*a*b + 5*b^2)*d*x/(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5) + 24*(5*a^2*b*e^
(2*c) - 10*a*b^2*e^(2*c) + b^3*e^(2*c))*arctan(1/2*(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))*
e^(-2*c)/((a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*sqrt(a*b)) + (a^3*e^(4*d*x + 36*c) + 3*a^2
*b*e^(4*d*x + 36*c) + 3*a*b^2*e^(4*d*x + 36*c) + b^3*e^(4*d*x + 36*c) - 8*a^3*e^(2*d*x + 34*c) + 24*a*b^2*e^(2
*d*x + 34*c) + 16*b^3*e^(2*d*x + 34*c))/(a^6*e^(32*c) + 6*a^5*b*e^(32*c) + 15*a^4*b^2*e^(32*c) + 20*a^3*b^3*e^
(32*c) + 15*a^2*b^4*e^(32*c) + 6*a*b^5*e^(32*c) + b^6*e^(32*c)) - (6*a^4*e^(12*d*x + 12*c) - 48*a^3*b*e^(12*d*
x + 12*c) - 84*a^2*b^2*e^(12*d*x + 12*c) + 30*b^4*e^(12*d*x + 12*c) + 16*a^4*e^(10*d*x + 10*c) - 104*a^3*b*e^(
10*d*x + 10*c) - 24*a^2*b^2*e^(10*d*x + 10*c) + 72*a*b^3*e^(10*d*x + 10*c) - 24*b^4*e^(10*d*x + 10*c) + 5*a^4*
e^(8*d*x + 8*c) + 84*a^3*b*e^(8*d*x + 8*c) + 30*a^2*b^2*e^(8*d*x + 8*c) + 84*a*b^3*e^(8*d*x + 8*c) - 123*b^4*e
^(8*d*x + 8*c) - 20*a^4*e^(6*d*x + 6*c) + 280*a^3*b*e^(6*d*x + 6*c) - 64*a^2*b^2*e^(6*d*x + 6*c) - 152*a*b^3*e
^(6*d*x + 6*c) + 212*b^4*e^(6*d*x + 6*c) - 20*a^4*e^(4*d*x + 4*c) + 136*a^3*b*e^(4*d*x + 4*c) + 224*a^2*b^2*e^
(4*d*x + 4*c) - 40*a*b^3*e^(4*d*x + 4*c) - 108*b^4*e^(4*d*x + 4*c) - 4*a^4*e^(2*d*x + 2*c) + 24*a^2*b^2*e^(2*d
*x + 2*c) + 32*a*b^3*e^(2*d*x + 2*c) + 12*b^4*e^(2*d*x + 2*c) + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)/((a
^5*e^(4*c) + 5*a^4*b*e^(4*c) + 10*a^3*b^2*e^(4*c) + 10*a^2*b^3*e^(4*c) + 5*a*b^4*e^(4*c) + b^5*e^(4*c))*(a*e^(
2*d*x) + b*e^(2*d*x) + a*e^(6*d*x + 4*c) + b*e^(6*d*x + 4*c) + 2*a*e^(4*d*x + 2*c) - 2*b*e^(4*d*x + 2*c))^2))/
d

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maple [B]  time = 0.39, size = 2366, normalized size = 9.86 \[ \text {Expression too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/d/(a+b)^3/(tanh(1/2*d*x+1/2*c)+1)^4+1/4/d/(a+b)^3/(tanh(1/2*d*x+1/2*c)-1)^4+1/2/d/(a+b)^3/(tanh(1/2*d*x+1
/2*c)+1)^3-15/8/d*b/(a+b)^5/(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))*a^3-15/8/d*b/(a+b)^5/(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))*a^3+15/8/d*b^2/(a+b)^5/(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))*a^2+27/8/d*b
^3/(a+b)^5*a/(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))+15/8/d*b^2/(a+b)^5/(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))*a^2+27/8/d*b^3/(a+b)^5*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/d/(a+b)^3/(tanh(1/2*d*x+
1/2*c)-1)^3-1/8/d/(a+b)^4/(tanh(1/2*d*x+1/2*c)-1)^2*a+11/8/d/(a+b)^4/(tanh(1/2*d*x+1/2*c)-1)^2*b-3/8/d/(a+b)^4
/(tanh(1/2*d*x+1/2*c)-1)*a+9/8/d/(a+b)^4/(tanh(1/2*d*x+1/2*c)-1)*b-3/8/d/(a+b)^5*ln(tanh(1/2*d*x+1/2*c)-1)*a^2
-15/8/d/(a+b)^5*ln(tanh(1/2*d*x+1/2*c)-1)*b^2+1/8/d/(a+b)^4/(tanh(1/2*d*x+1/2*c)+1)^2*a-11/8/d/(a+b)^4/(tanh(1
/2*d*x+1/2*c)+1)^2*b-3/8/d/(a+b)^4/(tanh(1/2*d*x+1/2*c)+1)*a+9/8/d/(a+b)^4/(tanh(1/2*d*x+1/2*c)+1)*b+3/8/d/(a+
b)^5*ln(tanh(1/2*d*x+1/2*c)+1)*a^2+15/8/d/(a+b)^5*ln(tanh(1/2*d*x+1/2*c)+1)*b^2-5/d*b^4/(a+b)^5/(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*tanh(1/2*d*x+1/2*c)^5-5/d*b^4/(a+b)^5/(tan
h(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*tanh(1/2*d*x+1/2*c)^3+3/8/d*b^3/
(a+b)^5/((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))
-3/8/d*b^3/(a+b)^5/((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))+15/4/d/(a+b)^5*ln(tanh(1/2*d*x+1/2*c)-1)*a*b-15/4/d/(a+b)^5*ln(tanh(1/2*d*x+1/2*c)+1)*a*b-15/8/d*b/
(a+b)^5*a^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))+9/4/d*b/(a+b)^5/(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*tanh(1/2
*d*x+1/2*c)^7*a^3+9/4/d*b/(a+b)^5/(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*tanh(1/2*d*x+1/2*c)*a^3+27/4/d*b/(a+b)^5/(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*tanh(1/2*d*x+1/2*c)^5*a^3+27/4/d*b/(a+b)^5/(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*tanh(1/2*d*x+1/2*c)^3*a^3-15/4/d*b^2/(a+b)^5*a/((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))+3/2/d*b^2/(a+b)^5/(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*tanh(1/2*d*x+1/2*c)^7*a^2-3/4/d*b^3/(a+b)^5/(t
anh(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*tanh(1/2*d*x+1/2*c)^7*a+23/2/d
*b^2/(a+b)^5/(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*tanh(1/2*d*x+1/
2*c)^5*a^2-1/4/d*b^3/(a+b)^5/(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
*tanh(1/2*d*x+1/2*c)^5*a+23/2/d*b^2/(a+b)^5/(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*tanh(1/2*d*x+1/2*c)^3*a^2-1/4/d*b^3/(a+b)^5/(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*tanh(1/2*d*x+1/2*c)^3*a+3/2/d*b^2/(a+b)^5/(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*tanh(1/2*d*x+1/2*c)*a^2-3/4/d*b^3/(a+b)^5/(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*tanh(1/2*d*x+1/2*c)*a+15/8/d*b/(a+b)^5*a^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))-3/8/d*b^4/(a+b
)^5/(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))-3/8/d*b^4/(a+b)^5/(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))+15/4/d*b^2/(a+b)^5*a/((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 [B]  time = 1.12, size = 3392, normalized size = 14.13 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^4}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

Timed out

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