3.1.54 \(\int \frac {\sinh (c+d x)}{(a+b \sinh ^2(c+d x))^3} \, dx\) [54]

Optimal. Leaf size=118 \[ \frac {3 \text {ArcTan}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{8 (a-b)^{5/2} \sqrt {b} d}+\frac {\cosh (c+d x)}{4 (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )^2}+\frac {3 \cosh (c+d x)}{8 (a-b)^2 d \left (a-b+b \cosh ^2(c+d x)\right )} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3265, 205, 211} \begin {gather*} \frac {3 \text {ArcTan}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{8 \sqrt {b} d (a-b)^{5/2}}+\frac {3 \cosh (c+d x)}{8 d (a-b)^2 \left (a+b \cosh ^2(c+d x)-b\right )}+\frac {\cosh (c+d x)}{4 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]/(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

(3*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(8*(a - b)^(5/2)*Sqrt[b]*d) + Cosh[c + d*x]/(4*(a - b)*d*(a -
b + b*Cosh[c + d*x]^2)^2) + (3*Cosh[c + d*x])/(8*(a - b)^2*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 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 (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (a-b+b x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\cosh (c+d x)}{4 (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )^2}+\frac {3 \text {Subst}\left (\int \frac {1}{\left (a-b+b x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{4 (a-b) d}\\ &=\frac {\cosh (c+d x)}{4 (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )^2}+\frac {3 \cosh (c+d x)}{8 (a-b)^2 d \left (a-b+b \cosh ^2(c+d x)\right )}+\frac {3 \text {Subst}\left (\int \frac {1}{a-b+b x^2} \, dx,x,\cosh (c+d x)\right )}{8 (a-b)^2 d}\\ &=\frac {3 \tan ^{-1}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{8 (a-b)^{5/2} \sqrt {b} d}+\frac {\cosh (c+d x)}{4 (a-b) d \left (a-b+b \cosh ^2(c+d x)\right )^2}+\frac {3 \cosh (c+d x)}{8 (a-b)^2 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 = 0.53, size = 149, normalized size = 1.26 \begin {gather*} \frac {\frac {3 \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} \sqrt {b}}+\frac {2 \cosh (c+d x) (10 a-7 b+3 b \cosh (2 (c+d x)))}{(a-b)^2 (2 a-b+b \cosh (2 (c+d x)))^2}}{8 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((3*(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)*Sqrt[b]) + (2*Cosh[c + d*x]*(10*a - 7*b + 3*b*Cosh[2*(c + d*x)]))/((a - b)^2
*(2*a - b + b*Cosh[2*(c + d*x)])^2))/(8*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(276\) vs. \(2(104)=208\).
time = 0.95, size = 277, normalized size = 2.35

method	result	size
risch	\(\frac {{\mathrm e}^{d x +c} \left (3 b \,{\mathrm e}^{6 d x +6 c}+20 a \,{\mathrm e}^{4 d x +4 c}-11 b \,{\mathrm e}^{4 d x +4 c}+20 a \,{\mathrm e}^{2 d x +2 c}-11 b \,{\mathrm e}^{2 d x +2 c}+3 b \right )}{4 \left (a -b \right )^{2} d \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )^{2}}-\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a -b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a b +b^{2}}}+1\right )}{16 \sqrt {-a b +b^{2}}\, \left (a -b \right )^{2} d}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a b +b^{2}}}+1\right )}{16 \sqrt {-a b +b^{2}}\, \left (a -b \right )^{2} d}\)	\(237\)
derivativedivides	\(\frac {\frac {-\frac {\left (5 a^{2}-16 a b +8 b^{2}\right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (15 a^{3}-46 a^{2} b +56 a \,b^{2}-16 b^{3}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (15 a^{2}-32 a b +8 b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a \left (a^{2}-2 a b +b^{2}\right )}+\frac {2 \left (5 a -2 b \right )}{8 a^{2}-16 a b +8 b^{2}}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {3 \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{8 \left (a^{2}-2 a b +b^{2}\right ) \sqrt {a b -b^{2}}}}{d}\)	\(277\)
default	\(\frac {\frac {-\frac {\left (5 a^{2}-16 a b +8 b^{2}\right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (15 a^{3}-46 a^{2} b +56 a \,b^{2}-16 b^{3}\right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (15 a^{2}-32 a b +8 b^{2}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a \left (a^{2}-2 a b +b^{2}\right )}+\frac {2 \left (5 a -2 b \right )}{8 a^{2}-16 a b +8 b^{2}}}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {3 \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{8 \left (a^{2}-2 a b +b^{2}\right ) \sqrt {a b -b^{2}}}}{d}\)	\(277\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(2*(-1/8*(5*a^2-16*a*b+8*b^2)/a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^6+1/8/a^2*(15*a^3-46*a^2*b+56*a*b^2-16
*b^3)/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^4-1/8*(15*a^2-32*a*b+8*b^2)/a/(a^2-2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^2+
1/8*(5*a-2*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+3/8/(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)/(a+b*sinh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

1/4*((20*a*e^(5*c) - 11*b*e^(5*c))*e^(5*d*x) + (20*a*e^(3*c) - 11*b*e^(3*c))*e^(3*d*x) + 3*b*e^(7*d*x + 7*c) +
3*b*e^(d*x + c))/(a^2*b^2*d - 2*a*b^3*d + b^4*d + (a^2*b^2*d*e^(8*c) - 2*a*b^3*d*e^(8*c) + b^4*d*e^(8*c))*e^(
8*d*x) + 4*(2*a^3*b*d*e^(6*c) - 5*a^2*b^2*d*e^(6*c) + 4*a*b^3*d*e^(6*c) - b^4*d*e^(6*c))*e^(6*d*x) + 2*(8*a^4*
d*e^(4*c) - 24*a^3*b*d*e^(4*c) + 27*a^2*b^2*d*e^(4*c) - 14*a*b^3*d*e^(4*c) + 3*b^4*d*e^(4*c))*e^(4*d*x) + 4*(2
*a^3*b*d*e^(2*c) - 5*a^2*b^2*d*e^(2*c) + 4*a*b^3*d*e^(2*c) - b^4*d*e^(2*c))*e^(2*d*x)) + 1/2*integrate(3/2*(e^
(3*d*x + 3*c) - e^(d*x + c))/(a^2*b - 2*a*b^2 + b^3 + (a^2*b*e^(4*c) - 2*a*b^2*e^(4*c) + b^3*e^(4*c))*e^(4*d*x
) + 2*(2*a^3*e^(2*c) - 5*a^2*b*e^(2*c) + 4*a*b^2*e^(2*c) - 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. 2726 vs. \(2 (104) = 208\).
time = 0.47, size = 5152, normalized size = 43.66 \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)/(a+b*sinh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[1/16*(12*(a*b^2 - b^3)*cosh(d*x + c)^7 + 84*(a*b^2 - b^3)*cosh(d*x + c)*sinh(d*x + c)^6 + 12*(a*b^2 - b^3)*si
nh(d*x + c)^7 + 4*(20*a^2*b - 31*a*b^2 + 11*b^3)*cosh(d*x + c)^5 + 4*(20*a^2*b - 31*a*b^2 + 11*b^3 + 63*(a*b^2
- b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 20*(21*(a*b^2 - b^3)*cosh(d*x + c)^3 + (20*a^2*b - 31*a*b^2 + 11*b^
3)*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(20*a^2*b - 31*a*b^2 + 11*b^3)*cosh(d*x + c)^3 + 4*(105*(a*b^2 - b^3)*co
sh(d*x + c)^4 + 20*a^2*b - 31*a*b^2 + 11*b^3 + 10*(20*a^2*b - 31*a*b^2 + 11*b^3)*cosh(d*x + c)^2)*sinh(d*x + c
)^3 + 4*(63*(a*b^2 - b^3)*cosh(d*x + c)^5 + 10*(20*a^2*b - 31*a*b^2 + 11*b^3)*cosh(d*x + c)^3 + 3*(20*a^2*b -
31*a*b^2 + 11*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 - 3*(b^2*cosh(d*x + c)^8 + 8*b^2*cosh(d*x + c)*sinh(d*x + c)
^7 + b^2*sinh(d*x + c)^8 + 4*(2*a*b - b^2)*cosh(d*x + c)^6 + 4*(7*b^2*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x
+ c)^6 + 8*(7*b^2*cosh(d*x + c)^3 + 3*(2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(8*a^2 - 8*a*b + 3*b^2)
*cosh(d*x + c)^4 + 2*(35*b^2*cosh(d*x + c)^4 + 30*(2*a*b - b^2)*cosh(d*x + c)^2 + 8*a^2 - 8*a*b + 3*b^2)*sinh(
d*x + c)^4 + 8*(7*b^2*cosh(d*x + c)^5 + 10*(2*a*b - b^2)*cosh(d*x + c)^3 + (8*a^2 - 8*a*b + 3*b^2)*cosh(d*x +
c))*sinh(d*x + c)^3 + 4*(2*a*b - b^2)*cosh(d*x + c)^2 + 4*(7*b^2*cosh(d*x + c)^6 + 15*(2*a*b - b^2)*cosh(d*x +
c)^4 + 3*(8*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)^2 + b^2 + 8*(b^2*cosh(d*x + c)^
7 + 3*(2*a*b - b^2)*cosh(d*x + c)^5 + (8*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c)^3 + (2*a*b - b^2)*cosh(d*x + c))*s
inh(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*c
osh(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)) + 12*(a*b^2 - b^3)*
cosh(d*x + c) + 4*(21*(a*b^2 - b^3)*cosh(d*x + c)^6 + 5*(20*a^2*b - 31*a*b^2 + 11*b^3)*cosh(d*x + c)^4 + 3*a*b
^2 - 3*b^3 + 3*(20*a^2*b - 31*a*b^2 + 11*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5
- b^6)*d*cosh(d*x + c)^8 + 8*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^3*b^3
- 3*a^2*b^4 + 3*a*b^5 - b^6)*d*sinh(d*x + c)^8 + 4*(2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4 - 5*a*b^5 + b^6)*d*cosh(
d*x + c)^6 + 4*(7*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cosh(d*x + c)^2 + (2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4
- 5*a*b^5 + b^6)*d)*sinh(d*x + c)^6 + 2*(8*a^5*b - 32*a^4*b^2 + 51*a^3*b^3 - 41*a^2*b^4 + 17*a*b^5 - 3*b^6)*d
*cosh(d*x + c)^4 + 8*(7*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cosh(d*x + c)^3 + 3*(2*a^4*b^2 - 7*a^3*b^3 + 9
*a^2*b^4 - 5*a*b^5 + b^6)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cos
h(d*x + c)^4 + 30*(2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4 - 5*a*b^5 + b^6)*d*cosh(d*x + c)^2 + (8*a^5*b - 32*a^4*b^
2 + 51*a^3*b^3 - 41*a^2*b^4 + 17*a*b^5 - 3*b^6)*d)*sinh(d*x + c)^4 + 4*(2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4 - 5*
a*b^5 + b^6)*d*cosh(d*x + c)^2 + 8*(7*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cosh(d*x + c)^5 + 10*(2*a^4*b^2
- 7*a^3*b^3 + 9*a^2*b^4 - 5*a*b^5 + b^6)*d*cosh(d*x + c)^3 + (8*a^5*b - 32*a^4*b^2 + 51*a^3*b^3 - 41*a^2*b^4 +
17*a*b^5 - 3*b^6)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*cosh(d*x +
c)^6 + 15*(2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4 - 5*a*b^5 + b^6)*d*cosh(d*x + c)^4 + 3*(8*a^5*b - 32*a^4*b^2 + 51
*a^3*b^3 - 41*a^2*b^4 + 17*a*b^5 - 3*b^6)*d*cosh(d*x + c)^2 + (2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4 - 5*a*b^5 + b
^6)*d)*sinh(d*x + c)^2 + (a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d + 8*((a^3*b^3 - 3*a^2*b^4 + 3*a*b^5 - b^6)*d*
cosh(d*x + c)^7 + 3*(2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4 - 5*a*b^5 + b^6)*d*cosh(d*x + c)^5 + (8*a^5*b - 32*a^4*
b^2 + 51*a^3*b^3 - 41*a^2*b^4 + 17*a*b^5 - 3*b^6)*d*cosh(d*x + c)^3 + (2*a^4*b^2 - 7*a^3*b^3 + 9*a^2*b^4 - 5*a
*b^5 + b^6)*d*cosh(d*x + c))*sinh(d*x + c)), 1/8*(6*(a*b^2 - b^3)*cosh(d*x + c)^7 + 42*(a*b^2 - b^3)*cosh(d*x
+ c)*sinh(d*x + c)^6 + 6*(a*b^2 - b^3)*sinh(d*x + c)^7 + 2*(20*a^2*b - 31*a*b^2 + 11*b^3)*cosh(d*x + c)^5 + 2*
(20*a^2*b - 31*a*b^2 + 11*b^3 + 63*(a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 10*(21*(a*b^2 - b^3)*cosh(
d*x + c)^3 + (20*a^2*b - 31*a*b^2 + 11*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 2*(20*a^2*b - 31*a*b^2 + 11*b^3)*
cosh(d*x + c)^3 + 2*(105*(a*b^2 - b^3)*cosh(d*x + c)^4 + 20*a^2*b - 31*a*b^2 + 11*b^3 + 10*(20*a^2*b - 31*a*b^
2 + 11*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(63*(a*b^2 - b^3)*cosh(d*x + c)^5 + 10*(20*a^2*b - 31*a*b^2 +
11*b^3)*cosh(d*x + c)^3 + 3*(20*a^2*b - 31*a*b^2 + 11*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + 3*(b^2*cosh(d*x +
c)^8 + 8*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + b...

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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)/(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)/(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 )}{{\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)/(a + b*sinh(c + d*x)^2)^3,x)

[Out]

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

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