3.256 \(\int \frac {\sinh ^3(c+d x)}{(a-b \sinh ^4(c+d x))^3} \, dx\)

Optimal. Leaf size=288 \[ -\frac {\left (5 \sqrt {a}-2 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{3/2} b^{3/4} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}+\frac {\left (5 \sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{3/2} b^{3/4} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}-\frac {\cosh (c+d x) \left (-\left ((5 a+b) \cosh ^2(c+d x)\right )+11 a+b\right )}{32 a d (a-b)^2 \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 d (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2} \]

[Out]

-1/8*cosh(d*x+c)*(2-cosh(d*x+c)^2)/(a-b)/d/(a-b+2*b*cosh(d*x+c)^2-b*cosh(d*x+c)^4)^2-1/32*cosh(d*x+c)*(11*a+b-
(5*a+b)*cosh(d*x+c)^2)/a/(a-b)^2/d/(a-b+2*b*cosh(d*x+c)^2-b*cosh(d*x+c)^4)-1/64*arctan(b^(1/4)*cosh(d*x+c)/(a^
(1/2)-b^(1/2))^(1/2))*(5*a^(1/2)-2*b^(1/2))/a^(3/2)/b^(3/4)/d/(a^(1/2)-b^(1/2))^(5/2)+1/64*arctanh(b^(1/4)*cos
h(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))*(5*a^(1/2)+2*b^(1/2))/a^(3/2)/b^(3/4)/d/(a^(1/2)+b^(1/2))^(5/2)

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Rubi [A]  time = 0.52, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3215, 1178, 1166, 205, 208} \[ -\frac {\left (5 \sqrt {a}-2 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{3/2} b^{3/4} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}+\frac {\left (5 \sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{3/2} b^{3/4} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}-\frac {\cosh (c+d x) \left (-(5 a+b) \cosh ^2(c+d x)+11 a+b\right )}{32 a d (a-b)^2 \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 d (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((5*Sqrt[a] - 2*Sqrt[b])*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*a^(3/2)*(Sqrt[a] - Sqrt
[b])^(5/2)*b^(3/4)*d) + ((5*Sqrt[a] + 2*Sqrt[b])*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64
*a^(3/2)*(Sqrt[a] + Sqrt[b])^(5/2)*b^(3/4)*d) - (Cosh[c + d*x]*(2 - Cosh[c + d*x]^2))/(8*(a - b)*d*(a - b + 2*
b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4)^2) - (Cosh[c + d*x]*(11*a + b - (5*a + b)*Cosh[c + d*x]^2))/(32*a*(a -
b)^2*d*(a - b + 2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4))

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 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 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1178

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(a*b*e - d*(b^2 - 2*
a*c) - c*(b*d - 2*a*e)*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 3215

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[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 \sinh ^4(c+d x)\right )^3} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1-x^2}{\left (a-b+2 b x^2-b x^4\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {-12 a b+10 a b x^2}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cosh (c+d x)\right )}{16 a (a-b) b d}\\ &=-\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac {\cosh (c+d x) \left (11 a+b-(5 a+b) \cosh ^2(c+d x)\right )}{32 a (a-b)^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {4 a (13 a-b) b^2-4 a b^2 (5 a+b) x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{128 a^2 (a-b)^2 b^2 d}\\ &=-\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac {\cosh (c+d x) \left (11 a+b-(5 a+b) \cosh ^2(c+d x)\right )}{32 a (a-b)^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}+\frac {\left (5 \sqrt {a}-2 \sqrt {b}\right ) \operatorname {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{64 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^2 d}+\frac {\left (5 \sqrt {a}+2 \sqrt {b}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{64 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^2 d}\\ &=-\frac {\left (5 \sqrt {a}-2 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{3/4} d}+\frac {\left (5 \sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{3/4} d}-\frac {\cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}-\frac {\cosh (c+d x) \left (11 a+b-(5 a+b) \cosh ^2(c+d x)\right )}{32 a (a-b)^2 d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}\\ \end {align*}

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Mathematica [C]  time = 1.32, size = 802, normalized size = 2.78 \[ \frac {\frac {32 \cosh (c+d x) (-17 a-b+(5 a+b) \cosh (2 (c+d x)))}{a (8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x)))}+\frac {\text {RootSum}\left [b \text {$\#$1}^8-4 b \text {$\#$1}^6-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^2+b\& ,\frac {-5 a c \text {$\#$1}^6-b c \text {$\#$1}^6-5 a d x \text {$\#$1}^6-b d x \text {$\#$1}^6-10 a \log \left (\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6-2 b \log \left (\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6+47 a c \text {$\#$1}^4-5 b c \text {$\#$1}^4+47 a d x \text {$\#$1}^4-5 b d x \text {$\#$1}^4+94 a \log \left (\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-10 b \log \left (\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-47 a c \text {$\#$1}^2+5 b c \text {$\#$1}^2-47 a d x \text {$\#$1}^2+5 b d x \text {$\#$1}^2-94 a \log \left (\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+10 b \log \left (\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+5 a c+b c+5 a d x+b d x+10 a \log \left (\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+2 b \log \left (\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )}{b \text {$\#$1}^7-3 b \text {$\#$1}^5-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-b \text {$\#$1}}\& \right ]}{a}+\frac {512 (a-b) (\cosh (3 (c+d x))-5 \cosh (c+d x))}{(-8 a+3 b-4 b \cosh (2 (c+d x))+b \cosh (4 (c+d x)))^2}}{256 (a-b)^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((32*Cosh[c + d*x]*(-17*a - b + (5*a + b)*Cosh[2*(c + d*x)]))/(a*(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4
*(c + d*x)])) + (512*(a - b)*(-5*Cosh[c + d*x] + Cosh[3*(c + d*x)]))/(-8*a + 3*b - 4*b*Cosh[2*(c + d*x)] + b*C
osh[4*(c + d*x)])^2 + RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (5*a*c + b*c + 5*a*d
*x + b*d*x + 10*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] +
2*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 47*a*c*#1^2 +
5*b*c*#1^2 - 47*a*d*x*#1^2 + 5*b*d*x*#1^2 - 94*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2
]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 10*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - S
inh[(c + d*x)/2]*#1]*#1^2 + 47*a*c*#1^4 - 5*b*c*#1^4 + 47*a*d*x*#1^4 - 5*b*d*x*#1^4 + 94*a*Log[-Cosh[(c + d*x)
/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 10*b*Log[-Cosh[(c + d*x)/2] - Si
nh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 5*a*c*#1^6 - b*c*#1^6 - 5*a*d*x*#1^6 - b
*d*x*#1^6 - 10*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^
6 - 2*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^6)/(-(b*#
1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ]/a)/(256*(a - b)^2*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)^3/(a-b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 1.05, size = 1576, normalized size = 5.47 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64*(((5*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^2 + 41*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a*b + 8*sqrt(a*b)*sq
rt(-b^2 - sqrt(a*b)*b)*b^2)*(a^3 - 2*a^2*b + a*b^2)^2*abs(b) + (13*sqrt(-b^2 - sqrt(a*b)*b)*a^5*b + 77*sqrt(-b
^2 - sqrt(a*b)*b)*a^4*b^2 - 201*sqrt(-b^2 - sqrt(a*b)*b)*a^3*b^3 + 119*sqrt(-b^2 - sqrt(a*b)*b)*a^2*b^4 - 8*sq
rt(-b^2 - sqrt(a*b)*b)*a*b^5)*abs(a^3 - 2*a^2*b + a*b^2)*abs(b) + 2*(4*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^7*
b + 15*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^6*b^2 - 108*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^5*b^3 + 202*sqrt(
a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^4*b^4 - 168*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^3*b^5 + 63*sqrt(a*b)*sqrt(-b^
2 - sqrt(a*b)*b)*a^2*b^6 - 8*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a*b^7)*abs(b))*arctan(1/2*(e^(d*x + c) + e^(-d
*x - c))/sqrt(-(a^3*b - 2*a^2*b^2 + a*b^3 + sqrt((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*(a^3*b - 2*a^2*b^2 + a*b^
3) + (a^3*b - 2*a^2*b^2 + a*b^3)^2))/(a^3*b - 2*a^2*b^2 + a*b^3)))/((a^8*b^3 + 3*a^7*b^4 - 30*a^6*b^5 + 70*a^5
*b^6 - 75*a^4*b^7 + 39*a^3*b^8 - 8*a^2*b^9)*abs(a^3 - 2*a^2*b + a*b^2)) + ((5*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*
b)*a^2 + 41*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a*b + 8*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*b^2)*(a^3 - 2*a^2*b
+ a*b^2)^2*abs(b) + (13*sqrt(-b^2 + sqrt(a*b)*b)*a^5*b + 77*sqrt(-b^2 + sqrt(a*b)*b)*a^4*b^2 - 201*sqrt(-b^2 +
sqrt(a*b)*b)*a^3*b^3 + 119*sqrt(-b^2 + sqrt(a*b)*b)*a^2*b^4 - 8*sqrt(-b^2 + sqrt(a*b)*b)*a*b^5)*abs(a^3 - 2*a
^2*b + a*b^2)*abs(b) + 2*(4*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^7*b + 15*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a
^6*b^2 - 108*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^5*b^3 + 202*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^4*b^4 - 168
*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^3*b^5 + 63*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^2*b^6 - 8*sqrt(a*b)*sqrt
(-b^2 + sqrt(a*b)*b)*a*b^7)*abs(b))*arctan(1/2*(e^(d*x + c) + e^(-d*x - c))/sqrt(-(a^3*b - 2*a^2*b^2 + a*b^3 -
sqrt((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*(a^3*b - 2*a^2*b^2 + a*b^3) + (a^3*b - 2*a^2*b^2 + a*b^3)^2))/(a^3*b
- 2*a^2*b^2 + a*b^3)))/((a^8*b^3 + 3*a^7*b^4 - 30*a^6*b^5 + 70*a^5*b^6 - 75*a^4*b^7 + 39*a^3*b^8 - 8*a^2*b^9)
*abs(a^3 - 2*a^2*b + a*b^2)) + 4*(5*a*b*(e^(d*x + c) + e^(-d*x - c))^7 + b^2*(e^(d*x + c) + e^(-d*x - c))^7 -
84*a*b*(e^(d*x + c) + e^(-d*x - c))^5 - 12*b^2*(e^(d*x + c) + e^(-d*x - c))^5 - 144*a^2*(e^(d*x + c) + e^(-d*x
- c))^3 + 480*a*b*(e^(d*x + c) + e^(-d*x - c))^3 + 48*b^2*(e^(d*x + c) + e^(-d*x - c))^3 + 1216*a^2*(e^(d*x +
c) + e^(-d*x - c)) - 1152*a*b*(e^(d*x + c) + e^(-d*x - c)) - 64*b^2*(e^(d*x + c) + e^(-d*x - c)))/((b*(e^(d*x
+ c) + e^(-d*x - c))^4 - 8*b*(e^(d*x + c) + e^(-d*x - c))^2 - 16*a + 16*b)^2*(a^3 - 2*a^2*b + a*b^2)))/d

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maple [B]  time = 0.13, size = 2916, normalized size = 10.12 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4
*tanh(1/2*d*x+1/2*c)^2*a+a)^2/(a^2-2*a*b+b^2)/a*tanh(1/2*d*x+1/2*c)^12*b^2+32/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tan
h(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/a^2/(
a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^8*b^3-4/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x
+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4*
b^2+1/8/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c
)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^12-104/d/(tanh(1/2*d*x+1/2*c)^8*a-4*t
anh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/a*b
^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^8-5/8/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d
*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*a/(a^2-2*a*b+b^2)+15/2/d/(tanh(1/2*d*x
+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*
c)^2*a+a)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^10*a-175/8/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*
a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/(a^2-2*a*b+b^2)*tanh(1/2
*d*x+1/2*c)^8*a-219/8/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh
(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^10+275/4/d/(tanh(1/2*d*
x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2
*c)^2*a+a)^2*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^8-533/8/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*
a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*b/(a^2-2*a*b+b^2)*tanh(1
/2*d*x+1/2*c)^6+55/2/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(
1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^6*a-141/8/d/(tanh(1/2*d*x+
1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c
)^2*a+a)^2*a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4+11/2/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6
*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*a/(a^2-2*a*b+b^2)*tanh(1/2*
d*x+1/2*c)^2-1/2/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*
d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^14-5/64/d/b/(a^2-2*a*b+b^2)/
(-a*b+(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)-2*a)/(-a*b+(a*b)^(1/2)*a)^(1/2)
)*(a*b)^(1/2)-5/64/d/b/(a^2-2*a*b+b^2)/(-a*b-(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(-2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*
b)^(1/2)+2*a)/(-a*b-(a*b)^(1/2)*a)^(1/2))*(a*b)^(1/2)+79/4/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*
a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*b/(a^2-2*a*b+b^2)*tanh(1
/2*d*x+1/2*c)^4+1/4/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1
/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2/(a^2-2*a*b+b^2)*b-1/8/d/(a^2-2*a*b+b^2)/(-a*b+(a*b)^(1/2)*a)^(1
/2)*arctan(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)-2*a)/(-a*b+(a*b)^(1/2)*a)^(1/2))+1/8/d/(a^2-2*a*b+b^2)
/(-a*b-(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(-2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)+2*a)/(-a*b-(a*b)^(1/2)*a)^(1/
2))+1/8/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c
)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^14+29/4/d/(tanh(1/2*d*x+1/2*c)^8*a-4*
tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*b/
(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^12-17/8/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*
x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^2
-1/32/d*b/(a^2-2*a*b+b^2)/a/(-a*b-(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(-2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)+2*
a)/(-a*b-(a*b)^(1/2)*a)^(1/2))+18/d/(tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4
*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(1/2*d*x+1/2*c)^2*a+a)^2*b^2/(a^2-2*a*b+b^2)/a*tanh(1/2*d*x+1/2*c)^10-1/64
/d/(a^2-2*a*b+b^2)/a/(-a*b+(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)-2*a)/(-a*b
+(a*b)^(1/2)*a)^(1/2))*(a*b)^(1/2)-1/64/d/(a^2-2*a*b+b^2)/a/(-a*b-(a*b)^(1/2)*a)^(1/2)*arctan(1/4*(-2*tanh(1/2
*d*x+1/2*c)^2*a+4*(a*b)^(1/2)+2*a)/(-a*b-(a*b)^(1/2)*a)^(1/2))*(a*b)^(1/2)+1/32/d*b/(a^2-2*a*b+b^2)/a/(-a*b+(a
*b)^(1/2)*a)^(1/2)*arctan(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a+4*(a*b)^(1/2)-2*a)/(-a*b+(a*b)^(1/2)*a)^(1/2))+14/d/(
tanh(1/2*d*x+1/2*c)^8*a-4*tanh(1/2*d*x+1/2*c)^6*a+6*tanh(1/2*d*x+1/2*c)^4*a-16*b*tanh(1/2*d*x+1/2*c)^4-4*tanh(
1/2*d*x+1/2*c)^2*a+a)^2/a*b^2/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^6

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*((5*a*b*e^(15*c) + b^2*e^(15*c))*e^(15*d*x) - (49*a*b*e^(13*c) + 5*b^2*e^(13*c))*e^(13*d*x) - 3*(48*a^2*
e^(11*c) - 55*a*b*e^(11*c) - 3*b^2*e^(11*c))*e^(11*d*x) + (784*a^2*e^(9*c) - 377*a*b*e^(9*c) - 5*b^2*e^(9*c))*
e^(9*d*x) + (784*a^2*e^(7*c) - 377*a*b*e^(7*c) - 5*b^2*e^(7*c))*e^(7*d*x) - 3*(48*a^2*e^(5*c) - 55*a*b*e^(5*c)
- 3*b^2*e^(5*c))*e^(5*d*x) - (49*a*b*e^(3*c) + 5*b^2*e^(3*c))*e^(3*d*x) + (5*a*b*e^c + b^2*e^c)*e^(d*x))/(a^3
*b^2*d - 2*a^2*b^3*d + a*b^4*d + (a^3*b^2*d*e^(16*c) - 2*a^2*b^3*d*e^(16*c) + a*b^4*d*e^(16*c))*e^(16*d*x) - 8
*(a^3*b^2*d*e^(14*c) - 2*a^2*b^3*d*e^(14*c) + a*b^4*d*e^(14*c))*e^(14*d*x) - 4*(8*a^4*b*d*e^(12*c) - 23*a^3*b^
2*d*e^(12*c) + 22*a^2*b^3*d*e^(12*c) - 7*a*b^4*d*e^(12*c))*e^(12*d*x) + 8*(16*a^4*b*d*e^(10*c) - 39*a^3*b^2*d*
e^(10*c) + 30*a^2*b^3*d*e^(10*c) - 7*a*b^4*d*e^(10*c))*e^(10*d*x) + 2*(128*a^5*d*e^(8*c) - 352*a^4*b*d*e^(8*c)
+ 355*a^3*b^2*d*e^(8*c) - 166*a^2*b^3*d*e^(8*c) + 35*a*b^4*d*e^(8*c))*e^(8*d*x) + 8*(16*a^4*b*d*e^(6*c) - 39*
a^3*b^2*d*e^(6*c) + 30*a^2*b^3*d*e^(6*c) - 7*a*b^4*d*e^(6*c))*e^(6*d*x) - 4*(8*a^4*b*d*e^(4*c) - 23*a^3*b^2*d*
e^(4*c) + 22*a^2*b^3*d*e^(4*c) - 7*a*b^4*d*e^(4*c))*e^(4*d*x) - 8*(a^3*b^2*d*e^(2*c) - 2*a^2*b^3*d*e^(2*c) + a
*b^4*d*e^(2*c))*e^(2*d*x)) - 1/8*integrate(1/2*((5*a*e^(7*c) + b*e^(7*c))*e^(7*d*x) - (47*a*e^(5*c) - 5*b*e^(5
*c))*e^(5*d*x) + (47*a*e^(3*c) - 5*b*e^(3*c))*e^(3*d*x) - (5*a*e^c + b*e^c)*e^(d*x))/(a^3*b - 2*a^2*b^2 + a*b^
3 + (a^3*b*e^(8*c) - 2*a^2*b^2*e^(8*c) + a*b^3*e^(8*c))*e^(8*d*x) - 4*(a^3*b*e^(6*c) - 2*a^2*b^2*e^(6*c) + a*b
^3*e^(6*c))*e^(6*d*x) - 2*(8*a^4*e^(4*c) - 19*a^3*b*e^(4*c) + 14*a^2*b^2*e^(4*c) - 3*a*b^3*e^(4*c))*e^(4*d*x)
- 4*(a^3*b*e^(2*c) - 2*a^2*b^2*e^(2*c) + a*b^3*e^(2*c))*e^(2*d*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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