3.1.52 \(\int \frac {\sinh ^3(c+d x)}{(a+b \sinh ^2(c+d x))^3} \, dx\) [52]

Optimal. Leaf size=135 \[ \frac {(a-4 b) \text {ArcTan}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{8 (a-b)^{5/2} b^{3/2} d}-\frac {a \cosh (c+d x)}{4 (a-b) b d \left (a-b+b \cosh ^2(c+d x)\right )^2}+\frac {(a-4 b) \cosh (c+d x)}{8 (a-b)^2 b d \left (a-b+b \cosh ^2(c+d x)\right )} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3265, 393, 205, 211} \begin {gather*} \frac {(a-4 b) \text {ArcTan}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{8 b^{3/2} d (a-b)^{5/2}}+\frac {(a-4 b) \cosh (c+d x)}{8 b d (a-b)^2 \left (a+b \cosh ^2(c+d x)-b\right )}-\frac {a \cosh (c+d x)}{4 b d (a-b) \left (a+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]^2)^3,x]

[Out]

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

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

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 393

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 3265

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(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 - b*ff^2*x^2)^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 ^2(c+d x)\right )^3} \, dx &=-\frac {\text {Subst}\left (\int \frac {1-x^2}{\left (a-b+b x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a \cosh (c+d x)}{4 (a-b) b d \left (a-b+b \cosh ^2(c+d x)\right )^2}+\frac {(a-4 b) \text {Subst}\left (\int \frac {1}{\left (a-b+b x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{4 (a-b) b d}\\ &=-\frac {a \cosh (c+d x)}{4 (a-b) b d \left (a-b+b \cosh ^2(c+d x)\right )^2}+\frac {(a-4 b) \cosh (c+d x)}{8 (a-b)^2 b d \left (a-b+b \cosh ^2(c+d x)\right )}+\frac {(a-4 b) \text {Subst}\left (\int \frac {1}{a-b+b x^2} \, dx,x,\cosh (c+d x)\right )}{8 (a-b)^2 b d}\\ &=\frac {(a-4 b) \tan ^{-1}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{8 (a-b)^{5/2} b^{3/2} d}-\frac {a \cosh (c+d x)}{4 (a-b) b d \left (a-b+b \cosh ^2(c+d x)\right )^2}+\frac {(a-4 b) \cosh (c+d x)}{8 (a-b)^2 b d \left (a-b+b \cosh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.90, size = 170, normalized size = 1.26 \begin {gather*} \frac {\frac {(a-4 b) \left (\text {ArcTan}\left (\frac {\sqrt {b}-i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )+\text {ArcTan}\left (\frac {\sqrt {b}+i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )\right )}{(a-b)^{5/2}}+\frac {2 \sqrt {b} \cosh (c+d x) \left (-2 a^2-5 a b+4 b^2+(a-4 b) b \cosh (2 (c+d x))\right )}{(a-b)^2 (2 a-b+b \cosh (2 (c+d x)))^2}}{8 b^{3/2} d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(280\) vs. \(2(121)=242\).
time = 1.41, size = 281, normalized size = 2.08

method	result	size
derivativedivides	\(\frac {\frac {\frac {a \left (a -4 b \right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (3 a^{3}-2 a^{2} b -8 a \,b^{2}+16 b^{3}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (3 a^{2}+4 a b -16 b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (2 b +a \right ) a}{4 b \left (a^{2}-2 a b +b^{2}\right )}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {\left (a -4 b \right ) \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{8 b \left (a^{2}-2 a b +b^{2}\right ) \sqrt {a b -b^{2}}}}{d}\)	\(281\)
default	\(\frac {\frac {\frac {a \left (a -4 b \right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (3 a^{3}-2 a^{2} b -8 a \,b^{2}+16 b^{3}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a b \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (3 a^{2}+4 a b -16 b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (2 b +a \right ) a}{4 b \left (a^{2}-2 a b +b^{2}\right )}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {\left (a -4 b \right ) \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{8 b \left (a^{2}-2 a b +b^{2}\right ) \sqrt {a b -b^{2}}}}{d}\)	\(281\)
risch	\(-\frac {{\mathrm e}^{d x +c} \left (-a b \,{\mathrm e}^{6 d x +6 c}+4 b^{2} {\mathrm e}^{6 d x +6 c}+4 a^{2} {\mathrm e}^{4 d x +4 c}+9 a b \,{\mathrm e}^{4 d x +4 c}-4 b^{2} {\mathrm e}^{4 d x +4 c}+4 a^{2} {\mathrm e}^{2 d x +2 c}+9 a b \,{\mathrm e}^{2 d x +2 c}-4 b^{2} {\mathrm e}^{2 d x +2 c}-a b +4 b^{2}\right )}{4 b \left (a -b \right )^{2} d \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )^{2}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a -b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a b +b^{2}}}+1\right ) a}{16 \sqrt {-a b +b^{2}}\, \left (a -b \right )^{2} d b}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a -b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a b +b^{2}}}+1\right )}{4 \sqrt {-a b +b^{2}}\, \left (a -b \right )^{2} d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a b +b^{2}}}+1\right ) a}{16 \sqrt {-a b +b^{2}}\, \left (a -b \right )^{2} d b}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a b +b^{2}}}+1\right )}{4 \sqrt {-a b +b^{2}}\, \left (a -b \right )^{2} d}\)	\(417\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(8*(1/32*a*(a-4*b)/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^6-1/32*(3*a^3-2*a^2*b-8*a*b^2+16*b^3)/a/b/(a^2-2*
a*b+b^2)*tanh(1/2*d*x+1/2*c)^4+1/32*(3*a^2+4*a*b-16*b^2)/b/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^2-1/32*(2*b+a)*
a/b/(a^2-2*a*b+b^2))/(a*tanh(1/2*d*x+1/2*c)^4-2*a*tanh(1/2*d*x+1/2*c)^2+4*b*tanh(1/2*d*x+1/2*c)^2+a)^2+1/8*(a-
4*b)/b/(a^2-2*a*b+b^2)/(a*b-b^2)^(1/2)*arctan(1/4*(2*a*tanh(1/2*d*x+1/2*c)^2-2*a+4*b)/(a*b-b^2)^(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)^2)^3,x, algorithm="maxima")

[Out]

1/4*((a*b*e^(7*c) - 4*b^2*e^(7*c))*e^(7*d*x) - (4*a^2*e^(5*c) + 9*a*b*e^(5*c) - 4*b^2*e^(5*c))*e^(5*d*x) - (4*
a^2*e^(3*c) + 9*a*b*e^(3*c) - 4*b^2*e^(3*c))*e^(3*d*x) + (a*b*e^c - 4*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^(8*c) - 2*a*b^4*d*e^(8*c) + b^5*d*e^(8*c))*e^(8*d*x) + 4*(2*a^3*b^2*d*e^(6*c) - 5*a^2*
b^3*d*e^(6*c) + 4*a*b^4*d*e^(6*c) - b^5*d*e^(6*c))*e^(6*d*x) + 2*(8*a^4*b*d*e^(4*c) - 24*a^3*b^2*d*e^(4*c) + 2
7*a^2*b^3*d*e^(4*c) - 14*a*b^4*d*e^(4*c) + 3*b^5*d*e^(4*c))*e^(4*d*x) + 4*(2*a^3*b^2*d*e^(2*c) - 5*a^2*b^3*d*e
^(2*c) + 4*a*b^4*d*e^(2*c) - b^5*d*e^(2*c))*e^(2*d*x)) + 1/8*integrate(2*((a*e^(3*c) - 4*b*e^(3*c))*e^(3*d*x)
- (a*e^c - 4*b*e^c)*e^(d*x))/(a^2*b^2 - 2*a*b^3 + b^4 + (a^2*b^2*e^(4*c) - 2*a*b^3*e^(4*c) + b^4*e^(4*c))*e^(4
*d*x) + 2*(2*a^3*b*e^(2*c) - 5*a^2*b^2*e^(2*c) + 4*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. 3293 vs. \(2 (121) = 242\).
time = 0.46, size = 6087, normalized size = 45.09 \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)^2)^3,x, algorithm="fricas")

[Out]

[1/16*(4*(a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(d*x + c)^7 + 28*(a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(d*x + c)*sinh(d*x +
c)^6 + 4*(a^2*b^2 - 5*a*b^3 + 4*b^4)*sinh(d*x + c)^7 - 4*(4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b^4)*cosh(d*x +
c)^5 - 4*(4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b^4 - 21*(a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(d*x + c)^2)*sinh(d*x +
c)^5 + 20*(7*(a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(d*x + c)^3 - (4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b^4)*cosh(d*x +
c))*sinh(d*x + c)^4 - 4*(4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b^4)*cosh(d*x + c)^3 + 4*(35*(a^2*b^2 - 5*a*b^3 +
4*b^4)*cosh(d*x + c)^4 - 4*a^3*b - 5*a^2*b^2 + 13*a*b^3 - 4*b^4 - 10*(4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b^4)
*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 4*(21*(a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(d*x + c)^5 - 10*(4*a^3*b + 5*a^2*b^
2 - 13*a*b^3 + 4*b^4)*cosh(d*x + c)^3 - 3*(4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b^4)*cosh(d*x + c))*sinh(d*x + c
)^2 + ((a*b^2 - 4*b^3)*cosh(d*x + c)^8 + 8*(a*b^2 - 4*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (a*b^2 - 4*b^3)*sin
h(d*x + c)^8 + 4*(2*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^6 + 4*(2*a^2*b - 9*a*b^2 + 4*b^3 + 7*(a*b^2 - 4*b^3
)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a*b^2 - 4*b^3)*cosh(d*x + c)^3 + 3*(2*a^2*b - 9*a*b^2 + 4*b^3)*cosh
(d*x + c))*sinh(d*x + c)^5 + 2*(8*a^3 - 40*a^2*b + 35*a*b^2 - 12*b^3)*cosh(d*x + c)^4 + 2*(35*(a*b^2 - 4*b^3)*
cosh(d*x + c)^4 + 8*a^3 - 40*a^2*b + 35*a*b^2 - 12*b^3 + 30*(2*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^2)*sinh(
d*x + c)^4 + 8*(7*(a*b^2 - 4*b^3)*cosh(d*x + c)^5 + 10*(2*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^3 + (8*a^3 -
40*a^2*b + 35*a*b^2 - 12*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + a*b^2 - 4*b^3 + 4*(2*a^2*b - 9*a*b^2 + 4*b^3)*c
osh(d*x + c)^2 + 4*(7*(a*b^2 - 4*b^3)*cosh(d*x + c)^6 + 15*(2*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^4 + 2*a^2
*b - 9*a*b^2 + 4*b^3 + 3*(8*a^3 - 40*a^2*b + 35*a*b^2 - 12*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a*b^2 -
4*b^3)*cosh(d*x + c)^7 + 3*(2*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c)^5 + (8*a^3 - 40*a^2*b + 35*a*b^2 - 12*b^
3)*cosh(d*x + c)^3 + (2*a^2*b - 9*a*b^2 + 4*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a*b + b^2)*log((b*cosh(d*
x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 - 2*(2*a - 3*b)*cosh(d*x + c)^2 + 2*(3*b*cosh
(d*x + c)^2 - 2*a + 3*b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 - (2*a - 3*b)*cosh(d*x + c))*sinh(d*x + c) + 4
*(cosh(d*x + c)^3 + 3*cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x + c)^2 + 1)*sinh(d*x + c)
+ cosh(d*x + c))*sqrt(-a*b + b^2) + b)/(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c
)^4 + 2*(2*a - b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 + 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 +
(2*a - b)*cosh(d*x + c))*sinh(d*x + c) + b)) + 4*(a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(d*x + c) + 4*(7*(a^2*b^2 -
5*a*b^3 + 4*b^4)*cosh(d*x + c)^6 - 5*(4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b^4)*cosh(d*x + c)^4 + a^2*b^2 - 5*a*
b^3 + 4*b^4 - 3*(4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4*b^4)*cosh(d*x + c)^2)*sinh(d*x + c))/((a^3*b^4 - 3*a^2*b^5
+ 3*a*b^6 - b^7)*d*cosh(d*x + c)^8 + 8*(a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d*cosh(d*x + c)*sinh(d*x + c)^7
+ (a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d*sinh(d*x + c)^8 + 4*(2*a^4*b^3 - 7*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b
^7)*d*cosh(d*x + c)^6 + 4*(7*(a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d*cosh(d*x + c)^2 + (2*a^4*b^3 - 7*a^3*b^4
+ 9*a^2*b^5 - 5*a*b^6 + b^7)*d)*sinh(d*x + c)^6 + 2*(8*a^5*b^2 - 32*a^4*b^3 + 51*a^3*b^4 - 41*a^2*b^5 + 17*a*b
^6 - 3*b^7)*d*cosh(d*x + c)^4 + 8*(7*(a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d*cosh(d*x + c)^3 + 3*(2*a^4*b^3 -
7*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^3*b^4 - 3*a^2*b^5 + 3*a*b^6
- b^7)*d*cosh(d*x + c)^4 + 30*(2*a^4*b^3 - 7*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7)*d*cosh(d*x + c)^2 + (8*a^5*
b^2 - 32*a^4*b^3 + 51*a^3*b^4 - 41*a^2*b^5 + 17*a*b^6 - 3*b^7)*d)*sinh(d*x + c)^4 + 4*(2*a^4*b^3 - 7*a^3*b^4 +
9*a^2*b^5 - 5*a*b^6 + b^7)*d*cosh(d*x + c)^2 + 8*(7*(a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d*cosh(d*x + c)^5 +
10*(2*a^4*b^3 - 7*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7)*d*cosh(d*x + c)^3 + (8*a^5*b^2 - 32*a^4*b^3 + 51*a^3*b
^4 - 41*a^2*b^5 + 17*a*b^6 - 3*b^7)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b
^7)*d*cosh(d*x + c)^6 + 15*(2*a^4*b^3 - 7*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7)*d*cosh(d*x + c)^4 + 3*(8*a^5*b^
2 - 32*a^4*b^3 + 51*a^3*b^4 - 41*a^2*b^5 + 17*a*b^6 - 3*b^7)*d*cosh(d*x + c)^2 + (2*a^4*b^3 - 7*a^3*b^4 + 9*a^
2*b^5 - 5*a*b^6 + b^7)*d)*sinh(d*x + c)^2 + (a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d + 8*((a^3*b^4 - 3*a^2*b^5
+ 3*a*b^6 - b^7)*d*cosh(d*x + c)^7 + 3*(2*a^4*b^3 - 7*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7)*d*cosh(d*x + c)^5 +
(8*a^5*b^2 - 32*a^4*b^3 + 51*a^3*b^4 - 41*a^2*b^5 + 17*a*b^6 - 3*b^7)*d*cosh(d*x + c)^3 + (2*a^4*b^3 - 7*a^3*
b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7)*d*cosh(d*x + c))*sinh(d*x + c)), 1/8*(2*(a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(d*x
+ c)^7 + 14*(a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(d*x + c)*sinh(d*x + c)^6 + 2*(a^2*b^2 - 5*a*b^3 + 4*b^4)*sinh(d*x
+ c)^7 - 2*(4*a^3*b + 5*a^2*b^2 - 13*a*b^3 + 4...

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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)**2)**3,x)

[Out]

Timed out

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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)^2)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^3}{{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\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)^2)^3,x)

[Out]

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

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