3.3.54 \(\int \frac {\sinh ^7(c+d x)}{(a-b \sinh ^4(c+d x))^3} \, dx\) [254]

Optimal. Leaf size=290 \[ \frac {3 \left (\sqrt {a}-2 \sqrt {b}\right ) \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 \sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{7/4} d}-\frac {3 \left (\sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{7/4} d}-\frac {a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )^2}+\frac {\cosh (c+d x) \left (5 a-17 b-3 (a-3 b) \cosh ^2(c+d x)\right )}{32 (a-b)^2 b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )} \]

[Out]

-1/8*a*cosh(d*x+c)*(2-cosh(d*x+c)^2)/(a-b)/b/d/(a-b+2*b*cosh(d*x+c)^2-b*cosh(d*x+c)^4)^2+1/32*cosh(d*x+c)*(5*a
-17*b-3*(a-3*b)*cosh(d*x+c)^2)/(a-b)^2/b/d/(a-b+2*b*cosh(d*x+c)^2-b*cosh(d*x+c)^4)+3/64*arctan(b^(1/4)*cosh(d*
x+c)/(a^(1/2)-b^(1/2))^(1/2))*(a^(1/2)-2*b^(1/2))/b^(7/4)/d/a^(1/2)/(a^(1/2)-b^(1/2))^(5/2)-3/64*arctanh(b^(1/
4)*cosh(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))*(a^(1/2)+2*b^(1/2))/b^(7/4)/d/a^(1/2)/(a^(1/2)+b^(1/2))^(5/2)

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Rubi [A]
time = 0.36, antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3294, 1219, 1192, 1180, 211, 214} \begin {gather*} \frac {3 \left (\sqrt {a}-2 \sqrt {b}\right ) \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 \sqrt {a} b^{7/4} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}-\frac {3 \left (\sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 \sqrt {a} b^{7/4} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}+\frac {\cosh (c+d x) \left (-3 (a-3 b) \cosh ^2(c+d x)+5 a-17 b\right )}{32 b d (a-b)^2 \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )}-\frac {a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 b 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]^7/(a - b*Sinh[c + d*x]^4)^3,x]

[Out]

(3*(Sqrt[a] - 2*Sqrt[b])*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*Sqrt[a]*(Sqrt[a] - Sqrt[
b])^(5/2)*b^(7/4)*d) - (3*(Sqrt[a] + 2*Sqrt[b])*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(64*
Sqrt[a]*(Sqrt[a] + Sqrt[b])^(5/2)*b^(7/4)*d) - (a*Cosh[c + d*x]*(2 - Cosh[c + d*x]^2))/(8*(a - b)*b*d*(a - b +
2*b*Cosh[c + d*x]^2 - b*Cosh[c + d*x]^4)^2) + (Cosh[c + d*x]*(5*a - 17*b - 3*(a - 3*b)*Cosh[c + d*x]^2))/(32*
(a - b)^2*b*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 1219

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{f = Coeff[Polynom
ialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x
^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a*b*g - f*(b^2 - 2*a*c) - c*(b*f - 2*a*g)*x^2)/(2
*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToS
um[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c
*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)*x^2, x], 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] && IGtQ[q, 1] && LtQ[p, -1]

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 ^7(c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^3} \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{\left (a-b+2 b x^2-b x^4\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) 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 {4 a (a-4 b)-2 a (3 a-8 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 {a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) 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 (5 a-17 b-3 (a-3 b) \cosh ^2(c+d x)\right )}{32 (a-b)^2 b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {-12 a^2 (a-5 b) b+12 a^2 (a-3 b) 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 {a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) 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 (5 a-17 b-3 (a-3 b) \cosh ^2(c+d x)\right )}{32 (a-b)^2 b d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}-\frac {\left (3 \left (\sqrt {a}+2 \sqrt {b}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{64 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )^2 b d}-\frac {\left (3 \left (a^{3/2}-3 \sqrt {a} b-2 b^{3/2}\right )\right ) \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{64 \sqrt {a} (a-b)^2 b d}\\ &=\frac {3 \left (\sqrt {a}-2 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{64 \sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{7/4} d}-\frac {3 \left (\sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{64 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{7/4} d}-\frac {a \cosh (c+d x) \left (2-\cosh ^2(c+d x)\right )}{8 (a-b) 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 (5 a-17 b-3 (a-3 b) \cosh ^2(c+d x)\right )}{32 (a-b)^2 b 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 = 0.96, size = 802, normalized size = 2.77 \begin {gather*} \frac {-\frac {32 \cosh (c+d x) (-7 a+25 b+3 (a-3 b) \cosh (2 (c+d x)))}{8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))}+\frac {512 a (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}-3 \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {a c-3 b c+a d x-3 b d x+2 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )-6 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )-3 a c \text {$\#$1}^2+17 b c \text {$\#$1}^2-3 a d x \text {$\#$1}^2+17 b d x \text {$\#$1}^2-6 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+34 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+3 a c \text {$\#$1}^4-17 b c \text {$\#$1}^4+3 a d x \text {$\#$1}^4-17 b d x \text {$\#$1}^4+6 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-34 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-a c \text {$\#$1}^6+3 b c \text {$\#$1}^6-a d x \text {$\#$1}^6+3 b d x \text {$\#$1}^6-2 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6+6 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{256 (a-b)^2 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^7/(a - b*Sinh[c + d*x]^4)^3,x]

[Out]

((-32*Cosh[c + d*x]*(-7*a + 25*b + 3*(a - 3*b)*Cosh[2*(c + d*x)]))/(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh
[4*(c + d*x)]) + (512*a*(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 - 3*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (a*c - 3*b*c +
a*d*x - 3*b*d*x + 2*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1
] - 6*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 3*a*c*#1^2
+ 17*b*c*#1^2 - 3*a*d*x*#1^2 + 17*b*d*x*#1^2 - 6*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 + 34*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1
- Sinh[(c + d*x)/2]*#1]*#1^2 + 3*a*c*#1^4 - 17*b*c*#1^4 + 3*a*d*x*#1^4 - 17*b*d*x*#1^4 + 6*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 - 34*b*Log[-Cosh[(c + d*x)/2] -
Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - a*c*#1^6 + 3*b*c*#1^6 - a*d*x*#1^6 + 3
*b*d*x*#1^6 - 2*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 + 6*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) & ])/(256*(a - b)^2*b*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(584\) vs. \(2(238)=476\).
time = 11.00, size = 585, normalized size = 2.02 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(128*(-3/1024*a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^14-3/1024/b*(a-10*b)*a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/
2*c)^12+1/1024/b*(16*a^2-111*a*b+80*b^2)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^10-1/1024*(35*a^3-26*a^2*b-64*a*b
^2+256*b^3)/a/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^8+1/1024*(40*a^2+95*a*b-336*b^2)/b/(a^2-2*a*b+b^2)*tanh(1/
2*d*x+1/2*c)^6-1/1024*(25*a^2+54*a*b-64*b^2)/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4+1/1024*(8*a+19*b)*a/b/(a^
2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^2-1/1024*(2*b+a)*a/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+3/8/b/(a^2-2*a
*b+b^2)*a*(1/8*((a*b)^(1/2)*a-3*(a*b)^(1/2)*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*(-(a*b)^(1/2)*a+3*(a*b)^(1/2)*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)^7/(a-b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

1/16*(3*(a*b*e^(15*c) - 3*b^2*e^(15*c))*e^(15*d*x) - (23*a*b*e^(13*c) - 77*b^2*e^(13*c))*e^(13*d*x) + (16*a^2*
e^(11*c) + 131*a*b*e^(11*c) - 177*b^2*e^(11*c))*e^(11*d*x) - (144*a^2*e^(9*c) + 367*a*b*e^(9*c) - 109*b^2*e^(9
*c))*e^(9*d*x) - (144*a^2*e^(7*c) + 367*a*b*e^(7*c) - 109*b^2*e^(7*c))*e^(7*d*x) + (16*a^2*e^(5*c) + 131*a*b*e
^(5*c) - 177*b^2*e^(5*c))*e^(5*d*x) - (23*a*b*e^(3*c) - 77*b^2*e^(3*c))*e^(3*d*x) + 3*(a*b*e^c - 3*b^2*e^c)*e^
(d*x))/(a^2*b^3*d - 2*a*b^4*d + b^5*d + (a^2*b^3*d*e^(16*c) - 2*a*b^4*d*e^(16*c) + b^5*d*e^(16*c))*e^(16*d*x)
- 8*(a^2*b^3*d*e^(14*c) - 2*a*b^4*d*e^(14*c) + b^5*d*e^(14*c))*e^(14*d*x) - 4*(8*a^3*b^2*d*e^(12*c) - 23*a^2*b
^3*d*e^(12*c) + 22*a*b^4*d*e^(12*c) - 7*b^5*d*e^(12*c))*e^(12*d*x) + 8*(16*a^3*b^2*d*e^(10*c) - 39*a^2*b^3*d*e
^(10*c) + 30*a*b^4*d*e^(10*c) - 7*b^5*d*e^(10*c))*e^(10*d*x) + 2*(128*a^4*b*d*e^(8*c) - 352*a^3*b^2*d*e^(8*c)
+ 355*a^2*b^3*d*e^(8*c) - 166*a*b^4*d*e^(8*c) + 35*b^5*d*e^(8*c))*e^(8*d*x) + 8*(16*a^3*b^2*d*e^(6*c) - 39*a^2
*b^3*d*e^(6*c) + 30*a*b^4*d*e^(6*c) - 7*b^5*d*e^(6*c))*e^(6*d*x) - 4*(8*a^3*b^2*d*e^(4*c) - 23*a^2*b^3*d*e^(4*
c) + 22*a*b^4*d*e^(4*c) - 7*b^5*d*e^(4*c))*e^(4*d*x) - 8*(a^2*b^3*d*e^(2*c) - 2*a*b^4*d*e^(2*c) + b^5*d*e^(2*c
))*e^(2*d*x)) + 1/128*integrate(24*((a*e^(7*c) - 3*b*e^(7*c))*e^(7*d*x) - (3*a*e^(5*c) - 17*b*e^(5*c))*e^(5*d*
x) + (3*a*e^(3*c) - 17*b*e^(3*c))*e^(3*d*x) - (a*e^c - 3*b*e^c)*e^(d*x))/(a^2*b^2 - 2*a*b^3 + b^4 + (a^2*b^2*e
^(8*c) - 2*a*b^3*e^(8*c) + b^4*e^(8*c))*e^(8*d*x) - 4*(a^2*b^2*e^(6*c) - 2*a*b^3*e^(6*c) + b^4*e^(6*c))*e^(6*d
*x) - 2*(8*a^3*b*e^(4*c) - 19*a^2*b^2*e^(4*c) + 14*a*b^3*e^(4*c) - 3*b^4*e^(4*c))*e^(4*d*x) - 4*(a^2*b^2*e^(2*
c) - 2*a*b^3*e^(2*c) + b^4*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. 20362 vs. \(2 (234) = 468\).
time = 0.74, size = 20362, normalized size = 70.21 \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)^7/(a-b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

1/128*(24*(a*b - 3*b^2)*cosh(d*x + c)^15 + 360*(a*b - 3*b^2)*cosh(d*x + c)*sinh(d*x + c)^14 + 24*(a*b - 3*b^2)
*sinh(d*x + c)^15 - 8*(23*a*b - 77*b^2)*cosh(d*x + c)^13 + 8*(315*(a*b - 3*b^2)*cosh(d*x + c)^2 - 23*a*b + 77*
b^2)*sinh(d*x + c)^13 + 104*(105*(a*b - 3*b^2)*cosh(d*x + c)^3 - (23*a*b - 77*b^2)*cosh(d*x + c))*sinh(d*x + c
)^12 + 8*(16*a^2 + 131*a*b - 177*b^2)*cosh(d*x + c)^11 + 8*(4095*(a*b - 3*b^2)*cosh(d*x + c)^4 - 78*(23*a*b -
77*b^2)*cosh(d*x + c)^2 + 16*a^2 + 131*a*b - 177*b^2)*sinh(d*x + c)^11 + 88*(819*(a*b - 3*b^2)*cosh(d*x + c)^5
- 26*(23*a*b - 77*b^2)*cosh(d*x + c)^3 + (16*a^2 + 131*a*b - 177*b^2)*cosh(d*x + c))*sinh(d*x + c)^10 - 8*(14
4*a^2 + 367*a*b - 109*b^2)*cosh(d*x + c)^9 + 8*(15015*(a*b - 3*b^2)*cosh(d*x + c)^6 - 715*(23*a*b - 77*b^2)*co
sh(d*x + c)^4 + 55*(16*a^2 + 131*a*b - 177*b^2)*cosh(d*x + c)^2 - 144*a^2 - 367*a*b + 109*b^2)*sinh(d*x + c)^9
+ 24*(6435*(a*b - 3*b^2)*cosh(d*x + c)^7 - 429*(23*a*b - 77*b^2)*cosh(d*x + c)^5 + 55*(16*a^2 + 131*a*b - 177
*b^2)*cosh ...

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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)**7/(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. 1515 vs. \(2 (234) = 468\).
time = 1.03, size = 1515, normalized size = 5.22 \begin {gather*} -\frac {\frac {3 \, {\left ({\left (4 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{3} - 7 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b - 15 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b^{2}\right )} {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )}^{2} {\left | b \right |} - {\left (4 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{5} b^{2} - 23 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{4} b^{3} + 9 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{3} b^{4} + 35 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b^{5} - 25 \, \sqrt {-b^{2} - \sqrt {a b} b} a b^{6}\right )} {\left | a^{2} b - 2 \, a b^{2} + b^{3} \right |} {\left | b \right |} - 2 \, {\left (4 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{5} b^{4} - 11 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{4} b^{5} + 4 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{3} b^{6} + 14 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b^{7} - 16 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b^{8} + 5 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} b^{9}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a^{2} b^{2} - 2 \, a b^{3} + b^{4} + \sqrt {{\left (a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right )} {\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} + {\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )}^{2}}}{a^{2} b^{2} - 2 \, a b^{3} + b^{4}}}}\right )}{{\left (4 \, a^{7} b^{5} - 15 \, a^{6} b^{6} + 15 \, a^{5} b^{7} + 10 \, a^{4} b^{8} - 30 \, a^{3} b^{9} + 21 \, a^{2} b^{10} - 5 \, a b^{11}\right )} {\left | a^{2} b - 2 \, a b^{2} + b^{3} \right |}} - \frac {3 \, {\left ({\left (4 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{3} - 7 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b - 15 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b^{2}\right )} {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )}^{2} {\left | b \right |} + {\left (4 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{5} b^{2} - 23 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{4} b^{3} + 9 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{3} b^{4} + 35 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b^{5} - 25 \, \sqrt {-b^{2} + \sqrt {a b} b} a b^{6}\right )} {\left | a^{2} b - 2 \, a b^{2} + b^{3} \right |} {\left | b \right |} - 2 \, {\left (4 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{5} b^{4} - 11 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{4} b^{5} + 4 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{3} b^{6} + 14 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b^{7} - 16 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b^{8} + 5 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} b^{9}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a^{2} b^{2} - 2 \, a b^{3} + b^{4} - \sqrt {{\left (a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right )} {\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} + {\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )}^{2}}}{a^{2} b^{2} - 2 \, a b^{3} + b^{4}}}}\right )}{{\left (4 \, a^{7} b^{5} - 15 \, a^{6} b^{6} + 15 \, a^{5} b^{7} + 10 \, a^{4} b^{8} - 30 \, a^{3} b^{9} + 21 \, a^{2} b^{10} - 5 \, a b^{11}\right )} {\left | a^{2} b - 2 \, a b^{2} + b^{3} \right |}} - \frac {4 \, {\left (3 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} - 9 \, b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} - 44 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 140 \, b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 16 \, a^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 288 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 688 \, b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 192 \, a^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 896 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 1088 \, 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^{2} b - 2 \, a b^{2} + b^{3}\right )}}}{64 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^7/(a-b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

-1/64*(3*((4*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^3 - 7*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^2*b - 15*sqrt(a*b
)*sqrt(-b^2 - sqrt(a*b)*b)*a*b^2)*(a^2*b - 2*a*b^2 + b^3)^2*abs(b) - (4*sqrt(-b^2 - sqrt(a*b)*b)*a^5*b^2 - 23*
sqrt(-b^2 - sqrt(a*b)*b)*a^4*b^3 + 9*sqrt(-b^2 - sqrt(a*b)*b)*a^3*b^4 + 35*sqrt(-b^2 - sqrt(a*b)*b)*a^2*b^5 -
25*sqrt(-b^2 - sqrt(a*b)*b)*a*b^6)*abs(a^2*b - 2*a*b^2 + b^3)*abs(b) - 2*(4*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)
*a^5*b^4 - 11*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^4*b^5 + 4*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^3*b^6 + 14*s
qrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^2*b^7 - 16*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a*b^8 + 5*sqrt(a*b)*sqrt(-b^
2 - sqrt(a*b)*b)*b^9)*abs(b))*arctan(1/2*(e^(d*x + c) + e^(-d*x - c))/sqrt(-(a^2*b^2 - 2*a*b^3 + b^4 + sqrt((a
^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*(a^2*b^2 - 2*a*b^3 + b^4) + (a^2*b^2 - 2*a*b^3 + b^4)^2))/(a^2*b^2 - 2*a*b^3
+ b^4)))/((4*a^7*b^5 - 15*a^6*b^6 + 15*a^5*b^7 + 10*a^4*b^8 - 30*a^3*b^9 + 21*a^2*b^10 - 5*a*b^11)*abs(a^2*b
- 2*a*b^2 + b^3)) - 3*((4*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^3 - 7*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^2*b
- 15*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a*b^2)*(a^2*b - 2*a*b^2 + b^3)^2*abs(b) + (4*sqrt(-b^2 + sqrt(a*b)*b)*
a^5*b^2 - 23*sqrt(-b^2 + sqrt(a*b)*b)*a^4*b^3 + 9*sqrt(-b^2 + sqrt(a*b)*b)*a^3*b^4 + 35*sqrt(-b^2 + sqrt(a*b)*
b)*a^2*b^5 - 25*sqrt(-b^2 + sqrt(a*b)*b)*a*b^6)*abs(a^2*b - 2*a*b^2 + b^3)*abs(b) - 2*(4*sqrt(a*b)*sqrt(-b^2 +
sqrt(a*b)*b)*a^5*b^4 - 11*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^4*b^5 + 4*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a
^3*b^6 + 14*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^2*b^7 - 16*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a*b^8 + 5*sqrt(
a*b)*sqrt(-b^2 + sqrt(a*b)*b)*b^9)*abs(b))*arctan(1/2*(e^(d*x + c) + e^(-d*x - c))/sqrt(-(a^2*b^2 - 2*a*b^3 +
b^4 - sqrt((a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*(a^2*b^2 - 2*a*b^3 + b^4) + (a^2*b^2 - 2*a*b^3 + b^4)^2))/(a^2*
b^2 - 2*a*b^3 + b^4)))/((4*a^7*b^5 - 15*a^6*b^6 + 15*a^5*b^7 + 10*a^4*b^8 - 30*a^3*b^9 + 21*a^2*b^10 - 5*a*b^1
1)*abs(a^2*b - 2*a*b^2 + b^3)) - 4*(3*a*b*(e^(d*x + c) + e^(-d*x - c))^7 - 9*b^2*(e^(d*x + c) + e^(-d*x - c))^
7 - 44*a*b*(e^(d*x + c) + e^(-d*x - c))^5 + 140*b^2*(e^(d*x + c) + e^(-d*x - c))^5 + 16*a^2*(e^(d*x + c) + e^(
-d*x - c))^3 + 288*a*b*(e^(d*x + c) + e^(-d*x - c))^3 - 688*b^2*(e^(d*x + c) + e^(-d*x - c))^3 - 192*a^2*(e^(d
*x + c) + e^(-d*x - c)) - 896*a*b*(e^(d*x + c) + e^(-d*x - c)) + 1088*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^2*b - 2*a*b^2 + b^3)))/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 )}^7}{{\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)^7/(a - b*sinh(c + d*x)^4)^3,x)

[Out]

int(sinh(c + d*x)^7/(a - b*sinh(c + d*x)^4)^3, x)

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