3.3.45 \(\int \frac {\sinh (c+d x)}{(a-b \sinh ^4(c+d x))^2} \, dx\) [245]

Optimal. Leaf size=221 \[ \frac {\left (3 \sqrt {a}-2 \sqrt {b}\right ) \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} \sqrt [4]{b} d}+\frac {\left (3 \sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} \sqrt [4]{b} d}+\frac {\cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{4 a (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )} \]

[Out]

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

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Rubi [A]
time = 0.22, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {3294, 1106, 1180, 211, 214} \begin {gather*} \frac {\left (3 \sqrt {a}-2 \sqrt {b}\right ) \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{3/2} \sqrt [4]{b} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}+\frac {\left (3 \sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{3/2} \sqrt [4]{b} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}+\frac {\cosh (c+d x) \left (a-b \cosh ^2(c+d x)+b\right )}{4 a d (a-b) \left (a-b \cosh ^4(c+d x)+2 b \cosh ^2(c+d x)-b\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]/(a - b*Sinh[c + d*x]^4)^2,x]

[Out]

((3*Sqrt[a] - 2*Sqrt[b])*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*a^(3/2)*(Sqrt[a] - Sqrt[b
])^(3/2)*b^(1/4)*d) + ((3*Sqrt[a] + 2*Sqrt[b])*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(8*a^
(3/2)*(Sqrt[a] + Sqrt[b])^(3/2)*b^(1/4)*d) + (Cosh[c + d*x]*(a + b - b*Cosh[c + d*x]^2))/(4*a*(a - 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 1106

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

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 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 (c+d x)}{\left (a-b \sinh ^4(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{4 a (a-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 {2 (a-b) b+4 b^2-2 \left (4 (a-b) b+4 b^2\right )+2 b^2 x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{8 a (a-b) b d}\\ &=\frac {\cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{4 a (a-b) d \left (a-b+2 b \cosh ^2(c+d x)-b \cosh ^4(c+d x)\right )}-\frac {\left (\left (3 \sqrt {a}-2 \sqrt {b}\right ) \sqrt {b}\right ) \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{8 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right ) d}-\frac {\left (b^2-\frac {-4 b^3-2 b \left (2 (a-b) b+4 b^2-2 \left (4 (a-b) b+4 b^2\right )\right )}{4 \sqrt {a} \sqrt {b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{8 a (a-b) b d}\\ &=\frac {\left (3 \sqrt {a}-2 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{8 a^{3/2} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} \sqrt [4]{b} d}+\frac {\left (3 \sqrt {a}+2 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{8 a^{3/2} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} \sqrt [4]{b} d}+\frac {\cosh (c+d x) \left (a+b-b \cosh ^2(c+d x)\right )}{4 a (a-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 = 0.27, size = 597, normalized size = 2.70 \begin {gather*} \frac {\frac {32 \cosh (c+d x) (2 a+b-b \cosh (2 (c+d x)))}{8 a-3 b+4 b \cosh (2 (c+d x))-b \cosh (4 (c+d x))}+\text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-b c-b d x-2 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+12 a c \text {$\#$1}^2-5 b c \text {$\#$1}^2+12 a d x \text {$\#$1}^2-5 b d x \text {$\#$1}^2+24 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-10 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-12 a c \text {$\#$1}^4+5 b c \text {$\#$1}^4-12 a d x \text {$\#$1}^4+5 b d x \text {$\#$1}^4-24 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+10 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+b c \text {$\#$1}^6+b d x \text {$\#$1}^6+2 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{32 a (a-b) d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]/(a - b*Sinh[c + d*x]^4)^2,x]

[Out]

((32*Cosh[c + d*x]*(2*a + b - b*Cosh[2*(c + d*x)]))/(8*a - 3*b + 4*b*Cosh[2*(c + d*x)] - b*Cosh[4*(c + d*x)])
+ RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-(b*c) - b*d*x - 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] + 12*a*c*#1^2 - 5*b*c*#1^2 + 12*a*d*x*
#1^2 - 5*b*d*x*#1^2 + 24*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 - Sinh[(c + d*x)/2]*#1]*#
1^2 - 12*a*c*#1^4 + 5*b*c*#1^4 - 12*a*d*x*#1^4 + 5*b*d*x*#1^4 - 24*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] - Sinh[(c + d*x)/2] + Cosh
[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + b*c*#1^6 + b*d*x*#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) & ])/(32*a*(a - b)*d)

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Maple [A]
time = 5.01, size = 339, normalized size = 1.53

method	result	size
derivativedivides	\(\frac {\frac {-\frac {\left (-2 b +a \right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a \left (a -b \right )}+\frac {\left (3 a -8 b \right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a \left (a -b \right )}-\frac {\left (3 a +2 b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a \left (a -b \right )}+\frac {2}{4 a -4 b}}{a \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {-\frac {\left (-\sqrt {a b}+3 a -2 b \right ) \arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {-\sqrt {a b}\, a -a b}}+\frac {\left (\sqrt {a b}+3 a -2 b \right ) \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {\sqrt {a b}\, a -a b}}}{2 a -2 b}}{d}\)	\(339\)
default	\(\frac {\frac {-\frac {\left (-2 b +a \right ) \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a \left (a -b \right )}+\frac {\left (3 a -8 b \right ) \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a \left (a -b \right )}-\frac {\left (3 a +2 b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a \left (a -b \right )}+\frac {2}{4 a -4 b}}{a \left (\tanh ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {-\frac {\left (-\sqrt {a b}+3 a -2 b \right ) \arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {-\sqrt {a b}\, a -a b}}+\frac {\left (\sqrt {a b}+3 a -2 b \right ) \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {\sqrt {a b}\, a -a b}}}{2 a -2 b}}{d}\)	\(339\)
risch	\(\frac {{\mathrm e}^{d x +c} \left (-b \,{\mathrm e}^{6 d x +6 c}+4 a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}+b \,{\mathrm e}^{2 d x +2 c}-b \right )}{2 a d \left (a -b \right ) \left (-b \,{\mathrm e}^{8 d x +8 c}+4 b \,{\mathrm e}^{6 d x +6 c}+16 a \,{\mathrm e}^{4 d x +4 c}-6 b \,{\mathrm e}^{4 d x +4 c}+4 b \,{\mathrm e}^{2 d x +2 c}-b \right )}+\left (\munderset {\textit {\_R} =\RootOf \left (\left (65536 a^{9} b \,d^{4}-196608 a^{8} b^{2} d^{4}+196608 a^{7} b^{3} d^{4}-65536 a^{6} b^{4} d^{4}\right ) \textit {\_Z}^{4}+\left (7680 a^{5} b \,d^{2}-7680 b^{2} d^{2} a^{4}+2048 a^{3} d^{2} b^{3}\right ) \textit {\_Z}^{2}-81 a^{2}+72 a b -16 b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (\frac {32768 a^{7} d^{3} b}{81 a^{2}-81 a b +20 b^{2}}-\frac {114688 a^{6} d^{3} b^{2}}{81 a^{2}-81 a b +20 b^{2}}+\frac {147456 a^{5} d^{3} b^{3}}{81 a^{2}-81 a b +20 b^{2}}-\frac {81920 a^{4} b^{4} d^{3}}{81 a^{2}-81 a b +20 b^{2}}+\frac {16384 a^{3} b^{5} d^{3}}{81 a^{2}-81 a b +20 b^{2}}\right ) \textit {\_R}^{3}+\left (\frac {864 a^{4} d}{81 a^{2}-81 a b +20 b^{2}}+\frac {1152 a^{3} d b}{81 a^{2}-81 a b +20 b^{2}}-\frac {2720 a^{2} d \,b^{2}}{81 a^{2}-81 a b +20 b^{2}}+\frac {1472 a \,b^{3} d}{81 a^{2}-81 a b +20 b^{2}}-\frac {256 b^{4} d}{81 a^{2}-81 a b +20 b^{2}}\right ) \textit {\_R} \right ) {\mathrm e}^{d x +c}+1\right )\right )\)	\(541\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)/(a-b*sinh(d*x+c)^4)^2,x,method=_RETURNVERBOSE)

[Out]

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

[Out]

-1/2*((4*a*e^(5*c) + b*e^(5*c))*e^(5*d*x) + (4*a*e^(3*c) + b*e^(3*c))*e^(3*d*x) - b*e^(7*d*x + 7*c) - b*e^(d*x
+ c))/(a^2*b*d - a*b^2*d + (a^2*b*d*e^(8*c) - a*b^2*d*e^(8*c))*e^(8*d*x) - 4*(a^2*b*d*e^(6*c) - a*b^2*d*e^(6*
c))*e^(6*d*x) - 2*(8*a^3*d*e^(4*c) - 11*a^2*b*d*e^(4*c) + 3*a*b^2*d*e^(4*c))*e^(4*d*x) - 4*(a^2*b*d*e^(2*c) -
a*b^2*d*e^(2*c))*e^(2*d*x)) + 1/2*integrate(-((12*a*e^(5*c) - 5*b*e^(5*c))*e^(5*d*x) - (12*a*e^(3*c) - 5*b*e^(
3*c))*e^(3*d*x) - b*e^(7*d*x + 7*c) + b*e^(d*x + c))/(a^2*b - a*b^2 + (a^2*b*e^(8*c) - a*b^2*e^(8*c))*e^(8*d*x
) - 4*(a^2*b*e^(6*c) - a*b^2*e^(6*c))*e^(6*d*x) - 2*(8*a^3*e^(4*c) - 11*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c))*e^(4*
d*x) - 4*(a^2*b*e^(2*c) - a*b^2*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. 6018 vs. \(2 (173) = 346\).
time = 0.54, size = 6018, normalized size = 27.23 \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)^4)^2,x, algorithm="fricas")

[Out]

1/16*(8*b*cosh(d*x + c)^7 + 56*b*cosh(d*x + c)*sinh(d*x + c)^6 + 8*b*sinh(d*x + c)^7 - 8*(4*a + b)*cosh(d*x +
c)^5 + 8*(21*b*cosh(d*x + c)^2 - 4*a - b)*sinh(d*x + c)^5 + 40*(7*b*cosh(d*x + c)^3 - (4*a + b)*cosh(d*x + c))
*sinh(d*x + c)^4 - 8*(4*a + b)*cosh(d*x + c)^3 + 8*(35*b*cosh(d*x + c)^4 - 10*(4*a + b)*cosh(d*x + c)^2 - 4*a
- b)*sinh(d*x + c)^3 + 8*(21*b*cosh(d*x + c)^5 - 10*(4*a + b)*cosh(d*x + c)^3 - 3*(4*a + b)*cosh(d*x + c))*sin
h(d*x + c)^2 + ((a^2*b - a*b^2)*d*cosh(d*x + c)^8 + 8*(a^2*b - a*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^2*b
- a*b^2)*d*sinh(d*x + c)^8 - 4*(a^2*b - a*b^2)*d*cosh(d*x + c)^6 + 4*(7*(a^2*b - a*b^2)*d*cosh(d*x + c)^2 - (
a^2*b - a*b^2)*d)*sinh(d*x + c)^6 - 2*(8*a^3 - 11*a^2*b + 3*a*b^2)*d*cosh(d*x + c)^4 + 8*(7*(a^2*b - a*b^2)*d*
cosh(d*x + c)^3 - 3*(a^2*b - a*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^2*b - a*b^2)*d*cosh(d*x + c)^4
- 30*(a^2*b - a*b^2)*d*cosh(d*x + c)^2 - (8*a^3 - 11*a^2*b + 3*a*b^2)*d)*sinh(d*x + c)^4 - 4*(a^2*b - a*b^2)*
d*cosh(d*x + c)^2 + 8*(7*(a^2*b - a*b^2)*d*cosh(d*x + c)^5 - 10*(a^2*b - a*b^2)*d*cosh(d*x + c)^3 - (8*a^3 - 1
1*a^2*b + 3*a*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^2*b - a*b^2)*d*cosh(d*x + c)^6 - 15*(a^2*b - a*b
^2)*d*cosh(d*x + c)^4 - 3*(8*a^3 - 11*a^2*b + 3*a*b^2)*d*cosh(d*x + c)^2 - (a^2*b - a*b^2)*d)*sinh(d*x + c)^2
+ (a^2*b - a*b^2)*d + 8*((a^2*b - a*b^2)*d*cosh(d*x + c)^7 - 3*(a^2*b - a*b^2)*d*cosh(d*x + c)^5 - (8*a^3 - 11
*a^2*b + 3*a*b^2)*d*cosh(d*x + c)^3 - (a^2*b - a*b^2)*d*cosh(d*x + c))*sinh(d*x + c))*sqrt(-((a^6 - 3*a^5*b +
3*a^4*b^2 - a^3*b^3)*d^2*sqrt((81*a^2 - 90*a*b + 25*b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^
5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) + 15*a^2 - 15*a*b + 4*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))*log
((81*a^2 - 81*a*b + 20*b^2)*cosh(d*x + c)^2 + 2*(81*a^2 - 81*a*b + 20*b^2)*cosh(d*x + c)*sinh(d*x + c) + (81*a
^2 - 81*a*b + 20*b^2)*sinh(d*x + c)^2 + 81*a^2 - 81*a*b + 20*b^2 + 2*((27*a^4 - 24*a^3*b + 5*a^2*b^2)*d*cosh(d
*x + c) + (27*a^4 - 24*a^3*b + 5*a^2*b^2)*d*sinh(d*x + c) - 2*((2*a^7*b - 7*a^6*b^2 + 9*a^5*b^3 - 5*a^4*b^4 +
a^3*b^5)*d^3*cosh(d*x + c) + (2*a^7*b - 7*a^6*b^2 + 9*a^5*b^3 - 5*a^4*b^4 + a^3*b^5)*d^3*sinh(d*x + c))*sqrt((
81*a^2 - 90*a*b + 25*b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^
4)))*sqrt(-((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((81*a^2 - 90*a*b + 25*b^2)/((a^9*b - 6*a^8*b^2 + 15
*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) + 15*a^2 - 15*a*b + 4*b^2)/((a^6 - 3*a^5*b + 3
*a^4*b^2 - a^3*b^3)*d^2))) - ((a^2*b - a*b^2)*d*cosh(d*x + c)^8 + 8*(a^2*b - a*b^2)*d*cosh(d*x + c)*sinh(d*x +
c)^7 + (a^2*b - a*b^2)*d*sinh(d*x + c)^8 - 4*(a^2*b - a*b^2)*d*cosh(d*x + c)^6 + 4*(7*(a^2*b - a*b^2)*d*cosh(
d*x + c)^2 - (a^2*b - a*b^2)*d)*sinh(d*x + c)^6 - 2*(8*a^3 - 11*a^2*b + 3*a*b^2)*d*cosh(d*x + c)^4 + 8*(7*(a^2
*b - a*b^2)*d*cosh(d*x + c)^3 - 3*(a^2*b - a*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^2*b - a*b^2)*d*c
osh(d*x + c)^4 - 30*(a^2*b - a*b^2)*d*cosh(d*x + c)^2 - (8*a^3 - 11*a^2*b + 3*a*b^2)*d)*sinh(d*x + c)^4 - 4*(a
^2*b - a*b^2)*d*cosh(d*x + c)^2 + 8*(7*(a^2*b - a*b^2)*d*cosh(d*x + c)^5 - 10*(a^2*b - a*b^2)*d*cosh(d*x + c)^
3 - (8*a^3 - 11*a^2*b + 3*a*b^2)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^2*b - a*b^2)*d*cosh(d*x + c)^6 - 1
5*(a^2*b - a*b^2)*d*cosh(d*x + c)^4 - 3*(8*a^3 - 11*a^2*b + 3*a*b^2)*d*cosh(d*x + c)^2 - (a^2*b - a*b^2)*d)*si
nh(d*x + c)^2 + (a^2*b - a*b^2)*d + 8*((a^2*b - a*b^2)*d*cosh(d*x + c)^7 - 3*(a^2*b - a*b^2)*d*cosh(d*x + c)^5
- (8*a^3 - 11*a^2*b + 3*a*b^2)*d*cosh(d*x + c)^3 - (a^2*b - a*b^2)*d*cosh(d*x + c))*sinh(d*x + c))*sqrt(-((a^
6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((81*a^2 - 90*a*b + 25*b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a
^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) + 15*a^2 - 15*a*b + 4*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*
b^3)*d^2))*log((81*a^2 - 81*a*b + 20*b^2)*cosh(d*x + c)^2 + 2*(81*a^2 - 81*a*b + 20*b^2)*cosh(d*x + c)*sinh(d*
x + c) + (81*a^2 - 81*a*b + 20*b^2)*sinh(d*x + c)^2 + 81*a^2 - 81*a*b + 20*b^2 - 2*((27*a^4 - 24*a^3*b + 5*a^2
*b^2)*d*cosh(d*x + c) + (27*a^4 - 24*a^3*b + 5*a^2*b^2)*d*sinh(d*x + c) - 2*((2*a^7*b - 7*a^6*b^2 + 9*a^5*b^3
- 5*a^4*b^4 + a^3*b^5)*d^3*cosh(d*x + c) + (2*a^7*b - 7*a^6*b^2 + 9*a^5*b^3 - 5*a^4*b^4 + a^3*b^5)*d^3*sinh(d*
x + c))*sqrt((81*a^2 - 90*a*b + 25*b^2)/((a^9*b - 6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6
+ a^3*b^7)*d^4)))*sqrt(-((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((81*a^2 - 90*a*b + 25*b^2)/((a^9*b -
6*a^8*b^2 + 15*a^7*b^3 - 20*a^6*b^4 + 15*a^5*b^5 - 6*a^4*b^6 + a^3*b^7)*d^4)) + 15*a^2 - 15*a*b + 4*b^2)/((a^6
- 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))) + ((a^2*b - a*b^2)*d*cosh(d*x + c)^8 + 8*(a^2*b - a*b^2)*d*cosh(d*x +
c)*sinh(d*x + c)^7 + (a^2*b - a*b^2)*d*sinh(d*x + c)^8 - 4*(a^2*b - a*b^2)*d*cosh(d*x + c)^6 + 4*(7*(a^2*b -
a*b^2)*d*cosh(d*x + c)^2 - (a^2*b - a*b^2)*d)*sinh(d*x + c)^6 - 2*(8*a^3 - 11*a^2*b + 3*a*b^2)*d*cosh(d*x + c)
^4 + 8*(7*(a^2*b - a*b^2)*d*cosh(d*x + c)^3 - 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)/(a-b*sinh(d*x+c)**4)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1054 vs. \(2 (173) = 346\).
time = 0.60, size = 1054, normalized size = 4.77 \begin {gather*} -\frac {\frac {{\left ({\left (4 \, \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^{2} - a b\right )}^{2} {\left | b \right |} - {\left (12 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{4} b - \sqrt {-b^{2} + \sqrt {a b} b} a^{3} b^{2} - 16 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b^{3} + 5 \, \sqrt {-b^{2} + \sqrt {a b} b} a b^{4}\right )} {\left | -a^{2} + a b \right |} {\left | b \right |} + {\left (12 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{5} b - 17 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{4} b^{2} - 12 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{3} b^{3} + 27 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b^{4} - 10 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b^{5}\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 - a b^{2} + \sqrt {{\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} {\left (a^{2} b - a b^{2}\right )} + {\left (a^{2} b - a b^{2}\right )}^{2}}}{a^{2} b - a b^{2}}}}\right )}{{\left (4 \, a^{6} b^{3} - 7 \, a^{5} b^{4} - 3 \, a^{4} b^{5} + 11 \, a^{3} b^{6} - 5 \, a^{2} b^{7}\right )} {\left | -a^{2} + a b \right |}} + \frac {{\left ({\left (4 \, \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^{2} - a b\right )}^{2} {\left | b \right |} - {\left (12 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{4} b - \sqrt {-b^{2} - \sqrt {a b} b} a^{3} b^{2} - 16 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b^{3} + 5 \, \sqrt {-b^{2} - \sqrt {a b} b} a b^{4}\right )} {\left | -a^{2} + a b \right |} {\left | b \right |} + {\left (12 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{5} b - 17 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{4} b^{2} - 12 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{3} b^{3} + 27 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b^{4} - 10 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b^{5}\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 - a b^{2} - \sqrt {{\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} {\left (a^{2} b - a b^{2}\right )} + {\left (a^{2} b - a b^{2}\right )}^{2}}}{a^{2} b - a b^{2}}}}\right )}{{\left (4 \, a^{6} b^{3} - 7 \, a^{5} b^{4} - 3 \, a^{4} b^{5} + 11 \, a^{3} b^{6} - 5 \, a^{2} b^{7}\right )} {\left | -a^{2} + a b \right |}} - \frac {4 \, {\left (b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 4 \, a {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 4 \, b {\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 )} {\left (a^{2} - a b\right )}}}{8 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)/(a-b*sinh(d*x+c)^4)^2,x, algorithm="giac")

[Out]

-1/8*(((4*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^2 - a*b)^2*abs
(b) - (12*sqrt(-b^2 + sqrt(a*b)*b)*a^4*b - sqrt(-b^2 + sqrt(a*b)*b)*a^3*b^2 - 16*sqrt(-b^2 + sqrt(a*b)*b)*a^2*
b^3 + 5*sqrt(-b^2 + sqrt(a*b)*b)*a*b^4)*abs(-a^2 + a*b)*abs(b) + (12*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^5*b
- 17*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^4*b^2 - 12*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a^3*b^3 + 27*sqrt(a*b)
*sqrt(-b^2 + sqrt(a*b)*b)*a^2*b^4 - 10*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a*b^5)*abs(b))*arctan(1/2*(e^(d*x +
c) + e^(-d*x - c))/sqrt(-(a^2*b - a*b^2 + sqrt((a^3 - 2*a^2*b + a*b^2)*(a^2*b - a*b^2) + (a^2*b - a*b^2)^2))/(
a^2*b - a*b^2)))/((4*a^6*b^3 - 7*a^5*b^4 - 3*a^4*b^5 + 11*a^3*b^6 - 5*a^2*b^7)*abs(-a^2 + a*b)) + ((4*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^2 - a*b)^2*abs(b) - (12*sqrt(-b^
2 - sqrt(a*b)*b)*a^4*b - sqrt(-b^2 - sqrt(a*b)*b)*a^3*b^2 - 16*sqrt(-b^2 - sqrt(a*b)*b)*a^2*b^3 + 5*sqrt(-b^2
- sqrt(a*b)*b)*a*b^4)*abs(-a^2 + a*b)*abs(b) + (12*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^5*b - 17*sqrt(a*b)*sqr
t(-b^2 - sqrt(a*b)*b)*a^4*b^2 - 12*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a^3*b^3 + 27*sqrt(a*b)*sqrt(-b^2 - sqrt(
a*b)*b)*a^2*b^4 - 10*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a*b^5)*abs(b))*arctan(1/2*(e^(d*x + c) + e^(-d*x - c))
/sqrt(-(a^2*b - a*b^2 - sqrt((a^3 - 2*a^2*b + a*b^2)*(a^2*b - a*b^2) + (a^2*b - a*b^2)^2))/(a^2*b - a*b^2)))/(
(4*a^6*b^3 - 7*a^5*b^4 - 3*a^4*b^5 + 11*a^3*b^6 - 5*a^2*b^7)*abs(-a^2 + a*b)) - 4*(b*(e^(d*x + c) + e^(-d*x -
c))^3 - 4*a*(e^(d*x + c) + e^(-d*x - c)) - 4*b*(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)*(a^2 - a*b)))/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 )}{{\left (a-b\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}^2} \,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)^4)^2,x)

[Out]

int(sinh(c + d*x)/(a - b*sinh(c + d*x)^4)^2, x)

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