3.3.31 \(\int \frac {\sinh ^3(c+d x)}{a-b \sinh ^4(c+d x)} \, dx\) [231]

Optimal. Leaf size=115 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^{3/4} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^{3/4} d} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3294, 1180, 211, 214} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 b^{3/4} d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{3/4} d \sqrt {\sqrt {a}-\sqrt {b}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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)}{a-b \sinh ^4(c+d x)} \, dx &=-\frac {\text {Subst}\left (\int \frac {1-x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{2 d}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{2 d}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^{3/4} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^{3/4} d}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.13, size = 365, normalized size = 3.17 \begin {gather*} -\frac {\text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-c-d x-2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+3 c \text {$\#$1}^2+3 d x \text {$\#$1}^2+6 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-3 c \text {$\#$1}^4-3 d x \text {$\#$1}^4-6 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+c \text {$\#$1}^6+d x \text {$\#$1}^6+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{8 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-c - d*x - 2*Log[-Cosh[(c + d*x)/2]
- Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 3*c*#1^2 + 3*d*x*#1^2 + 6*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*c*#1^4 - 3*d*x*#1^4 - 6*Lo
g[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 + c*#1^6 + d*x*#1
^6 + 2*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) & ]/d

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Maple [A]
time = 1.41, size = 148, normalized size = 1.29

method	result	size
risch	\(\munderset {\textit {\_R} =\RootOf \left (-1+\left (256 a \,b^{3} d^{4}-256 b^{4} d^{4}\right ) \textit {\_Z}^{4}+32 b^{2} d^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (128 a \,b^{2} d^{3}-128 b^{3} d^{3}\right ) \textit {\_R}^{3}+16 b d \textit {\_R} \right ) {\mathrm e}^{d x +c}+1\right )\)	\(89\)
derivativedivides	\(\frac {8 a \left (-\frac {\sqrt {a b}\, \arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{16 a b \sqrt {-\sqrt {a b}\, a -a b}}-\frac {\sqrt {a b}\, \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{16 a b \sqrt {\sqrt {a b}\, a -a b}}\right )}{d}\)	\(148\)
default	\(\frac {8 a \left (-\frac {\sqrt {a b}\, \arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{16 a b \sqrt {-\sqrt {a b}\, a -a b}}-\frac {\sqrt {a b}\, \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{16 a b \sqrt {\sqrt {a b}\, a -a b}}\right )}{d}\)	\(148\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/d*a*(-1/16*(a*b)^(1/2)/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))-1/16*(a*b)^(1/2)/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),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 975 vs. \(2 (79) = 158\).
time = 0.41, size = 975, normalized size = 8.48 \begin {gather*} \frac {1}{4} \, \sqrt {-\frac {{\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + 1}{{\left (a b - b^{2}\right )} d^{2}}} \log \left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 2 \, {\left (b d \cosh \left (d x + c\right ) + b d \sinh \left (d x + c\right ) - {\left ({\left (a b^{2} - b^{3}\right )} d^{3} \cosh \left (d x + c\right ) + {\left (a b^{2} - b^{3}\right )} d^{3} \sinh \left (d x + c\right )\right )} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}}\right )} \sqrt {-\frac {{\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + 1}{{\left (a b - b^{2}\right )} d^{2}}} + 1\right ) - \frac {1}{4} \, \sqrt {-\frac {{\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + 1}{{\left (a b - b^{2}\right )} d^{2}}} \log \left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 2 \, {\left (b d \cosh \left (d x + c\right ) + b d \sinh \left (d x + c\right ) - {\left ({\left (a b^{2} - b^{3}\right )} d^{3} \cosh \left (d x + c\right ) + {\left (a b^{2} - b^{3}\right )} d^{3} \sinh \left (d x + c\right )\right )} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}}\right )} \sqrt {-\frac {{\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + 1}{{\left (a b - b^{2}\right )} d^{2}}} + 1\right ) + \frac {1}{4} \, \sqrt {\frac {{\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} - 1}{{\left (a b - b^{2}\right )} d^{2}}} \log \left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 2 \, {\left (b d \cosh \left (d x + c\right ) + b d \sinh \left (d x + c\right ) + {\left ({\left (a b^{2} - b^{3}\right )} d^{3} \cosh \left (d x + c\right ) + {\left (a b^{2} - b^{3}\right )} d^{3} \sinh \left (d x + c\right )\right )} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}}\right )} \sqrt {\frac {{\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} - 1}{{\left (a b - b^{2}\right )} d^{2}}} + 1\right ) - \frac {1}{4} \, \sqrt {\frac {{\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} - 1}{{\left (a b - b^{2}\right )} d^{2}}} \log \left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 2 \, {\left (b d \cosh \left (d x + c\right ) + b d \sinh \left (d x + c\right ) + {\left ({\left (a b^{2} - b^{3}\right )} d^{3} \cosh \left (d x + c\right ) + {\left (a b^{2} - b^{3}\right )} d^{3} \sinh \left (d x + c\right )\right )} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}}\right )} \sqrt {\frac {{\left (a b - b^{2}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} - 1}{{\left (a b - b^{2}\right )} d^{2}}} + 1\right ) \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),x, algorithm="fricas")

[Out]

1/4*sqrt(-((a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + 1)/((a*b - b^2)*d^2))*log(cosh(d*x + c)^2
+ 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 2*(b*d*cosh(d*x + c) + b*d*sinh(d*x + c) - ((a*b^2 - b^3)
*d^3*cosh(d*x + c) + (a*b^2 - b^3)*d^3*sinh(d*x + c))*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)))*sqrt(-((a*b - b
^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + 1)/((a*b - b^2)*d^2)) + 1) - 1/4*sqrt(-((a*b - b^2)*d^2*sqrt
(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) + 1)/((a*b - b^2)*d^2))*log(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c
) + sinh(d*x + c)^2 - 2*(b*d*cosh(d*x + c) + b*d*sinh(d*x + c) - ((a*b^2 - b^3)*d^3*cosh(d*x + c) + (a*b^2 - b
^3)*d^3*sinh(d*x + c))*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)))*sqrt(-((a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*
b^4 + b^5)*d^4)) + 1)/((a*b - b^2)*d^2)) + 1) + 1/4*sqrt(((a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^
4)) - 1)/((a*b - b^2)*d^2))*log(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 2*(b*d*cos
h(d*x + c) + b*d*sinh(d*x + c) + ((a*b^2 - b^3)*d^3*cosh(d*x + c) + (a*b^2 - b^3)*d^3*sinh(d*x + c))*sqrt(a/((
a^2*b^3 - 2*a*b^4 + b^5)*d^4)))*sqrt(((a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - 1)/((a*b - b^2
)*d^2)) + 1) - 1/4*sqrt(((a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - 1)/((a*b - b^2)*d^2))*log(c
osh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 2*(b*d*cosh(d*x + c) + b*d*sinh(d*x + c) +
((a*b^2 - b^3)*d^3*cosh(d*x + c) + (a*b^2 - b^3)*d^3*sinh(d*x + c))*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)))*s
qrt(((a*b - b^2)*d^2*sqrt(a/((a^2*b^3 - 2*a*b^4 + b^5)*d^4)) - 1)/((a*b - b^2)*d^2)) + 1)

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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),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (79) = 158\).
time = 0.54, size = 314, normalized size = 2.73 \begin {gather*} -\frac {\frac {{\left (4 \, \sqrt {-b^{2} - \sqrt {a b} b} a b + 5 \, \sqrt {-b^{2} - \sqrt {a b} b} b^{2} - 4 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a - 5 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} b\right )} {\left | b \right |} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {b + \sqrt {{\left (a - b\right )} b + b^{2}}}{b}}}\right )}{4 \, a^{2} b^{3} + a b^{4} - 5 \, b^{5}} + \frac {{\left (4 \, \sqrt {-b^{2} + \sqrt {a b} b} a b + 5 \, \sqrt {-b^{2} + \sqrt {a b} b} b^{2} + 4 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a + 5 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} b\right )} {\left | b \right |} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {b - \sqrt {{\left (a - b\right )} b + b^{2}}}{b}}}\right )}{4 \, a^{2} b^{3} + a b^{4} - 5 \, b^{5}}}{2 \, 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),x, algorithm="giac")

[Out]

-1/2*((4*sqrt(-b^2 - sqrt(a*b)*b)*a*b + 5*sqrt(-b^2 - sqrt(a*b)*b)*b^2 - 4*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*
a - 5*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*b)*abs(b)*arctan(1/2*(e^(d*x + c) + e^(-d*x - c))/sqrt(-(b + sqrt((a
- b)*b + b^2))/b))/(4*a^2*b^3 + a*b^4 - 5*b^5) + (4*sqrt(-b^2 + sqrt(a*b)*b)*a*b + 5*sqrt(-b^2 + sqrt(a*b)*b)*
b^2 + 4*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a + 5*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*b)*abs(b)*arctan(1/2*(e^(d
*x + c) + e^(-d*x - c))/sqrt(-(b - sqrt((a - b)*b + b^2))/b))/(4*a^2*b^3 + a*b^4 - 5*b^5))/d

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Mupad [B]
time = 6.16, size = 975, normalized size = 8.48 \begin {gather*} \ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^4\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^7\,{\left (a-b\right )}^2}-\frac {8388608\,a^4\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (a+b\right )\,\sqrt {-\frac {b^2-\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{b^7\,\left (a-b\right )}\right )\,\sqrt {-\frac {b^2-\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{4}+\frac {2097152\,a^4\,d\,{\mathrm {e}}^{c+d\,x}}{b^8\,\left (a-b\right )}\right )\,\sqrt {-\frac {b^2-\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a^4\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^9\,{\left (a-b\right )}^2}\right )\,\sqrt {\frac {b^2-\sqrt {a\,b^3}}{16\,\left (b^4\,d^2-a\,b^3\,d^2\right )}}-\ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^4\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^7\,{\left (a-b\right )}^2}+\frac {8388608\,a^4\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (a+b\right )\,\sqrt {-\frac {b^2-\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{b^7\,\left (a-b\right )}\right )\,\sqrt {-\frac {b^2-\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{4}-\frac {2097152\,a^4\,d\,{\mathrm {e}}^{c+d\,x}}{b^8\,\left (a-b\right )}\right )\,\sqrt {-\frac {b^2-\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a^4\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^9\,{\left (a-b\right )}^2}\right )\,\sqrt {\frac {b^2-\sqrt {a\,b^3}}{16\,\left (b^4\,d^2-a\,b^3\,d^2\right )}}+\ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^4\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^7\,{\left (a-b\right )}^2}-\frac {8388608\,a^4\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (a+b\right )\,\sqrt {-\frac {b^2+\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{b^7\,\left (a-b\right )}\right )\,\sqrt {-\frac {b^2+\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{4}+\frac {2097152\,a^4\,d\,{\mathrm {e}}^{c+d\,x}}{b^8\,\left (a-b\right )}\right )\,\sqrt {-\frac {b^2+\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a^4\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^9\,{\left (a-b\right )}^2}\right )\,\sqrt {\frac {b^2+\sqrt {a\,b^3}}{16\,\left (b^4\,d^2-a\,b^3\,d^2\right )}}-\ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^4\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^7\,{\left (a-b\right )}^2}+\frac {8388608\,a^4\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (a+b\right )\,\sqrt {-\frac {b^2+\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{b^7\,\left (a-b\right )}\right )\,\sqrt {-\frac {b^2+\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{4}-\frac {2097152\,a^4\,d\,{\mathrm {e}}^{c+d\,x}}{b^8\,\left (a-b\right )}\right )\,\sqrt {-\frac {b^2+\sqrt {a\,b^3}}{b^3\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a^4\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^9\,{\left (a-b\right )}^2}\right )\,\sqrt {\frac {b^2+\sqrt {a\,b^3}}{16\,\left (b^4\,d^2-a\,b^3\,d^2\right )}} \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),x)

[Out]

log((((((4194304*a^4*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b))/(b^7*(a - b)^2) - (8388608*a^4*d^3*exp(c + d*x)*(a
+ b)*(-(b^2 - (a*b^3)^(1/2))/(b^3*d^2*(a - b)))^(1/2))/(b^7*(a - b)))*(-(b^2 - (a*b^3)^(1/2))/(b^3*d^2*(a - b)
))^(1/2))/4 + (2097152*a^4*d*exp(c + d*x))/(b^8*(a - b)))*(-(b^2 - (a*b^3)^(1/2))/(b^3*d^2*(a - b)))^(1/2))/4
- (262144*a^4*(exp(2*c + 2*d*x) + 1)*(a + b))/(b^9*(a - b)^2))*((b^2 - (a*b^3)^(1/2))/(16*(b^4*d^2 - a*b^3*d^2
)))^(1/2) - log((((((4194304*a^4*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b))/(b^7*(a - b)^2) + (8388608*a^4*d^3*exp(
c + d*x)*(a + b)*(-(b^2 - (a*b^3)^(1/2))/(b^3*d^2*(a - b)))^(1/2))/(b^7*(a - b)))*(-(b^2 - (a*b^3)^(1/2))/(b^3
*d^2*(a - b)))^(1/2))/4 - (2097152*a^4*d*exp(c + d*x))/(b^8*(a - b)))*(-(b^2 - (a*b^3)^(1/2))/(b^3*d^2*(a - b)
))^(1/2))/4 - (262144*a^4*(exp(2*c + 2*d*x) + 1)*(a + b))/(b^9*(a - b)^2))*((b^2 - (a*b^3)^(1/2))/(16*(b^4*d^2
- a*b^3*d^2)))^(1/2) + log((((((4194304*a^4*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b))/(b^7*(a - b)^2) - (8388608*
a^4*d^3*exp(c + d*x)*(a + b)*(-(b^2 + (a*b^3)^(1/2))/(b^3*d^2*(a - b)))^(1/2))/(b^7*(a - b)))*(-(b^2 + (a*b^3)
^(1/2))/(b^3*d^2*(a - b)))^(1/2))/4 + (2097152*a^4*d*exp(c + d*x))/(b^8*(a - b)))*(-(b^2 + (a*b^3)^(1/2))/(b^3
*d^2*(a - b)))^(1/2))/4 - (262144*a^4*(exp(2*c + 2*d*x) + 1)*(a + b))/(b^9*(a - b)^2))*((b^2 + (a*b^3)^(1/2))/
(16*(b^4*d^2 - a*b^3*d^2)))^(1/2) - log((((((4194304*a^4*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b))/(b^7*(a - b)^2)
+ (8388608*a^4*d^3*exp(c + d*x)*(a + b)*(-(b^2 + (a*b^3)^(1/2))/(b^3*d^2*(a - b)))^(1/2))/(b^7*(a - b)))*(-(b
^2 + (a*b^3)^(1/2))/(b^3*d^2*(a - b)))^(1/2))/4 - (2097152*a^4*d*exp(c + d*x))/(b^8*(a - b)))*(-(b^2 + (a*b^3)
^(1/2))/(b^3*d^2*(a - b)))^(1/2))/4 - (262144*a^4*(exp(2*c + 2*d*x) + 1)*(a + b))/(b^9*(a - b)^2))*((b^2 + (a*
b^3)^(1/2))/(16*(b^4*d^2 - a*b^3*d^2)))^(1/2)

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