3.3.56 \(\int \frac {\sinh ^3(c+d x)}{(a-b \sinh ^4(c+d x))^3} \, dx\) [256]

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

[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 = 0.38, 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 = {3294, 1192, 1180, 211, 214} \begin {gather*} -\frac {\left (5 \sqrt {a}-2 \sqrt {b}\right ) \text {ArcTan}\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/64*((5*Sqrt[a] - 2*Sqrt[b])*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(a^(3/2)*(Sqrt[a] - Sq
rt[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 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 214

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

Rule 1180

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 1192

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 3294

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 {\text {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 {\text {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 {\text {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 ) \text {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 ) \text {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] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.89, size = 802, normalized size = 2.78 \begin {gather*} \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 {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 \cosh (4 (c+d x)))^2}+\frac {\text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {5 a c+b c+5 a d x+b d x+10 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+2 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )-47 a c \text {$\#$1}^2+5 b c \text {$\#$1}^2-47 a d x \text {$\#$1}^2+5 b d x \text {$\#$1}^2-94 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+10 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+47 a c \text {$\#$1}^4-5 b c \text {$\#$1}^4+47 a d x \text {$\#$1}^4-5 b d x \text {$\#$1}^4+94 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-10 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-5 a c \text {$\#$1}^6-b c \text {$\#$1}^6-5 a d x \text {$\#$1}^6-b d x \text {$\#$1}^6-10 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6-2 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{a}}{256 (a-b)^2 d} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(592\) vs. \(2(236)=472\).
time = 10.73, size = 593, normalized size = 2.06 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,method=_RETURNVERBOSE)

[Out]

1/d*(8*(-1/64*(4*a-b)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^14+1/64*(a^2+58*a*b-32*b^2)/(a^2-2*a*b+b^2)/a*tanh(1
/2*d*x+1/2*c)^12+3/64/a*(20*a^2-73*a*b+48*b^2)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^10-1/64/a^2*(175*a^3-550*a^
2*b+832*a*b^2-256*b^3)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^8+1/64/a*(220*a^2-533*a*b+112*b^2)/(a^2-2*a*b+b^2)*
tanh(1/2*d*x+1/2*c)^6-1/64*(141*a^2-158*a*b+32*b^2)/a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4+1/64*(44*a-17*b)/(
a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^2-1/64*(5*a-2*b)/(a^2-2*a*b+b^2))/(a*tanh(1/2*d*x+1/2*c)^8-4*a*tanh(1/2*d*x
+1/2*c)^6+6*a*tanh(1/2*d*x+1/2*c)^4-16*b*tanh(1/2*d*x+1/2*c)^4-4*a*tanh(1/2*d*x+1/2*c)^2+a)^2+1/8/(a^2-2*a*b+b
^2)*(-1/8*(5*(a*b)^(1/2)*a+(a*b)^(1/2)*b-8*a*b+2*b^2)/a/b/(-(a*b)^(1/2)*a-a*b)^(1/2)*arctan(1/4*(-2*a*tanh(1/2
*d*x+1/2*c)^2+4*(a*b)^(1/2)+2*a)/(-(a*b)^(1/2)*a-a*b)^(1/2))+1/8*(-5*(a*b)^(1/2)*a-(a*b)^(1/2)*b-8*a*b+2*b^2)/
a/b/((a*b)^(1/2)*a-a*b)^(1/2)*arctan(1/4*(2*a*tanh(1/2*d*x+1/2*c)^2+4*(a*b)^(1/2)-2*a)/((a*b)^(1/2)*a-a*b)^(1/
2))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 20961 vs. \(2 (233) = 466\).
time = 0.74, size = 20961, normalized size = 72.78 \begin {gather*} \text {too large to display} \end {gather*}

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]

-1/128*(8*(5*a*b + b^2)*cosh(d*x + c)^15 + 120*(5*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^14 + 8*(5*a*b + b^2)*
sinh(d*x + c)^15 - 8*(49*a*b + 5*b^2)*cosh(d*x + c)^13 + 8*(105*(5*a*b + b^2)*cosh(d*x + c)^2 - 49*a*b - 5*b^2
)*sinh(d*x + c)^13 + 104*(35*(5*a*b + b^2)*cosh(d*x + c)^3 - (49*a*b + 5*b^2)*cosh(d*x + c))*sinh(d*x + c)^12
- 24*(48*a^2 - 55*a*b - 3*b^2)*cosh(d*x + c)^11 + 24*(455*(5*a*b + b^2)*cosh(d*x + c)^4 - 26*(49*a*b + 5*b^2)*
cosh(d*x + c)^2 - 48*a^2 + 55*a*b + 3*b^2)*sinh(d*x + c)^11 + 88*(273*(5*a*b + b^2)*cosh(d*x + c)^5 - 26*(49*a
*b + 5*b^2)*cosh(d*x + c)^3 - 3*(48*a^2 - 55*a*b - 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^10 + 8*(784*a^2 - 377*a
*b - 5*b^2)*cosh(d*x + c)^9 + 8*(5005*(5*a*b + b^2)*cosh(d*x + c)^6 - 715*(49*a*b + 5*b^2)*cosh(d*x + c)^4 - 1
65*(48*a^2 - 55*a*b - 3*b^2)*cosh(d*x + c)^2 + 784*a^2 - 377*a*b - 5*b^2)*sinh(d*x + c)^9 + 72*(715*(5*a*b + b
^2)*cosh(d*x + c)^7 - 143*(49*a*b + 5*b^2)*cosh(d*x + c)^5 - 55*(48*a^2 - 55*a*b - 3*b^2)*cosh(d*x + c)^3 + (7
84*a^2 - 3 ...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1580 vs. \(2 (233) = 466\).
time = 0.75, size = 1580, normalized size = 5.49 \begin {gather*} \frac {\frac {{\left ({\left (20 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{2} + 29 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b + 5 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} b^{2}\right )} {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )}^{2} {\left | b \right |} - {\left (52 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{5} b - 43 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{4} b^{2} - 75 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{3} b^{3} + 71 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b^{4} - 5 \, \sqrt {-b^{2} - \sqrt {a b} b} a b^{5}\right )} {\left | a^{3} - 2 \, a^{2} b + a b^{2} \right |} {\left | b \right |} + 2 \, {\left (16 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{7} b - 48 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{6} b^{2} + 27 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{5} b^{3} + 52 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{4} b^{4} - 78 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{3} b^{5} + 36 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b^{6} - 5 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b^{7}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a^{3} b - 2 \, a^{2} b^{2} + a b^{3} + \sqrt {{\left (a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right )} {\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} + {\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )}^{2}}}{a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}}}\right )}{{\left (4 \, a^{8} b^{3} - 15 \, a^{7} b^{4} + 15 \, a^{6} b^{5} + 10 \, a^{5} b^{6} - 30 \, a^{4} b^{7} + 21 \, a^{3} b^{8} - 5 \, a^{2} b^{9}\right )} {\left | a^{3} - 2 \, a^{2} b + a b^{2} \right |}} - \frac {{\left ({\left (20 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{2} + 29 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b + 5 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} b^{2}\right )} {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )}^{2} {\left | b \right |} + {\left (52 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{5} b - 43 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{4} b^{2} - 75 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{3} b^{3} + 71 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b^{4} - 5 \, \sqrt {-b^{2} + \sqrt {a b} b} a b^{5}\right )} {\left | a^{3} - 2 \, a^{2} b + a b^{2} \right |} {\left | b \right |} + 2 \, {\left (16 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{7} b - 48 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{6} b^{2} + 27 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{5} b^{3} + 52 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{4} b^{4} - 78 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{3} b^{5} + 36 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b^{6} - 5 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b^{7}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a^{3} b - 2 \, a^{2} b^{2} + a b^{3} - \sqrt {{\left (a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right )} {\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )} + {\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\right )}^{2}}}{a^{3} b - 2 \, a^{2} b^{2} + a b^{3}}}}\right )}{{\left (4 \, a^{8} b^{3} - 15 \, a^{7} b^{4} + 15 \, a^{6} b^{5} + 10 \, a^{5} b^{6} - 30 \, a^{4} b^{7} + 21 \, a^{3} b^{8} - 5 \, a^{2} b^{9}\right )} {\left | a^{3} - 2 \, a^{2} b + a b^{2} \right |}} - \frac {4 \, {\left (5 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} + b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} - 84 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 12 \, b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 144 \, a^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 480 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 48 \, b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 1216 \, a^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 1152 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 64 \, b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}\right )}}{{\left (b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} - 8 \, b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 16 \, a + 16 \, b\right )}^{2} {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )}}}{64 \, d} \end {gather*}

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*(((20*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^2 + 29*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a*b + 5*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) - (52*sqrt(-b^2 - sqrt(a*b)*b)*a^5*b - 43*sqrt(-b
^2 - sqrt(a*b)*b)*a^4*b^2 - 75*sqrt(-b^2 - sqrt(a*b)*b)*a^3*b^3 + 71*sqrt(-b^2 - sqrt(a*b)*b)*a^2*b^4 - 5*sqrt
(-b^2 - sqrt(a*b)*b)*a*b^5)*abs(a^3 - 2*a^2*b + a*b^2)*abs(b) + 2*(16*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^7*b
- 48*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^6*b^2 + 27*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^5*b^3 + 52*sqrt(a*b
)*sqrt(-b^2 - sqrt(a*b)*b)*a^4*b^4 - 78*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^3*b^5 + 36*sqrt(a*b)*sqrt(-b^2 -
sqrt(a*b)*b)*a^2*b^6 - 5*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)))/((4*a^8*b^3 - 15*a^7*b^4 + 15*a^6*b^5 + 10*a^5*
b^6 - 30*a^4*b^7 + 21*a^3*b^8 - 5*a^2*b^9)*abs(a^3 - 2*a^2*b + a*b^2)) - ((20*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*
b)*a^2 + 29*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a*b + 5*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*b^2)*(a^3 - 2*a^2*b
+ a*b^2)^2*abs(b) + (52*sqrt(-b^2 + sqrt(a*b)*b)*a^5*b - 43*sqrt(-b^2 + sqrt(a*b)*b)*a^4*b^2 - 75*sqrt(-b^2 +
sqrt(a*b)*b)*a^3*b^3 + 71*sqrt(-b^2 + sqrt(a*b)*b)*a^2*b^4 - 5*sqrt(-b^2 + sqrt(a*b)*b)*a*b^5)*abs(a^3 - 2*a^2
*b + a*b^2)*abs(b) + 2*(16*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^7*b - 48*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^
6*b^2 + 27*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^5*b^3 + 52*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^4*b^4 - 78*sqr
t(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^3*b^5 + 36*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^2*b^6 - 5*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 - sqr
t((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)))/((4*a^8*b^3 - 15*a^7*b^4 + 15*a^6*b^5 + 10*a^5*b^6 - 30*a^4*b^7 + 21*a^3*b^8 - 5*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 - 8
4*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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

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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