3.3.30 \(\int \frac {\sinh ^5(c+d x)}{a-b \sinh ^4(c+d x)} \, dx\) [230]

Optimal. Leaf size=139 \[ \frac {\sqrt {a} \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^{5/4} d}+\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^{5/4} d}-\frac {\cosh (c+d x)}{b d} \]

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3294, 1184, 1107, 211, 214} \begin {gather*} \frac {\sqrt {a} \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{5/4} d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 b^{5/4} d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\cosh (c+d x)}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^5/(a - b*Sinh[c + d*x]^4),x]

[Out]

(Sqrt[a]*ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^(5/4)*d) + (Sqr
t[a]*ArcTanh[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^(5/4)*d) - Cosh[c
+ d*x]/(b*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 1107

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/
2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && PosQ[b^2 - 4*a*c]

Rule 1184

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x
^2)^q/(a + b*x^2 + c*x^4), 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] && IntegerQ[q]

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 ^5(c+d x)}{a-b \sinh ^4(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {1}{b}+\frac {a}{b \left (a-b+2 b x^2-b x^4\right )}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\cosh (c+d x)}{b d}+\frac {a \text {Subst}\left (\int \frac {1}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{b d}\\ &=-\frac {\cosh (c+d x)}{b d}-\frac {\sqrt {a} \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{2 \sqrt {b} d}+\frac {\sqrt {a} \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{2 \sqrt {b} d}\\ &=\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^{5/4} d}+\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^{5/4} d}-\frac {\cosh (c+d x)}{b 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 = 0.20, size = 235, normalized size = 1.69 \begin {gather*} -\frac {2 \cosh (c+d x)+a \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-c \text {$\#$1}-d x \text {$\#$1}-2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}+c \text {$\#$1}^3+d x \text {$\#$1}^3+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^3}{-b-8 a \text {$\#$1}^2+3 b \text {$\#$1}^2-3 b \text {$\#$1}^4+b \text {$\#$1}^6}\&\right ]}{2 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^5/(a - b*Sinh[c + d*x]^4),x]

[Out]

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

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Maple [A]
time = 1.62, size = 174, normalized size = 1.25

method	result	size
risch	\(-\frac {{\mathrm e}^{d x +c}}{2 b d}-\frac {{\mathrm e}^{-d x -c}}{2 b d}+\left (\munderset {\textit {\_R} =\RootOf \left (\left (256 a \,b^{5} d^{4}-256 b^{6} d^{4}\right ) \textit {\_Z}^{4}+32 a \,d^{2} \textit {\_Z}^{2} b^{3}-a^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (\frac {128 b^{4} d^{3}}{a}-\frac {128 b^{5} d^{3}}{a^{2}}\right ) \textit {\_R}^{3}+\left (8 b d +\frac {8 b^{2} d}{a}\right ) \textit {\_R} \right ) {\mathrm e}^{d x +c}+1\right )\right )\)	\(142\)
derivativedivides	\(\frac {\frac {2 a^{2} \left (\frac {\arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {\sqrt {a b}\, a -a b}}-\frac {\arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {-\sqrt {a b}\, a -a b}}\right )}{b}-\frac {1}{b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {1}{b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\)	\(174\)
default	\(\frac {\frac {2 a^{2} \left (\frac {\arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {\sqrt {a b}\, a -a b}}-\frac {\arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{4 a \sqrt {-\sqrt {a b}\, a -a b}}\right )}{b}-\frac {1}{b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {1}{b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\)	\(174\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^5/(a-b*sinh(d*x+c)^4),x,method=_RETURNVERBOSE)

[Out]

1/d*(2*a^2/b*(1/4/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/(-(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/b/(tanh(1/2*d*x+1/2*c)+1)+1/b/(tanh(1/2*d*x+1/2*c)-1))

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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)^5/(a-b*sinh(d*x+c)^4),x, algorithm="maxima")

[Out]

-1/2*(e^(2*d*x + 2*c) + 1)*e^(-d*x - c)/(b*d) - 1/32*integrate(256*(a*e^(5*d*x + 5*c) - a*e^(3*d*x + 3*c))/(b^
2*e^(8*d*x + 8*c) - 4*b^2*e^(6*d*x + 6*c) - 4*b^2*e^(2*d*x + 2*c) + b^2 - 2*(8*a*b*e^(4*c) - 3*b^2*e^(4*c))*e^
(4*d*x)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1247 vs. \(2 (99) = 198\).
time = 0.43, size = 1247, normalized size = 8.97 \begin {gather*} \frac {{\left (b d \cosh \left (d x + c\right ) + b d \sinh \left (d x + c\right )\right )} \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (a^{2} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} \sinh \left (d x + c\right )^{2} + a^{2} + 2 \, {\left (a^{2} b d \cosh \left (d x + c\right ) + a^{2} b d \sinh \left (d x + c\right ) - {\left ({\left (a b^{4} - b^{5}\right )} d^{3} \cosh \left (d x + c\right ) + {\left (a b^{4} - b^{5}\right )} d^{3} \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}}\right )} \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}}\right ) - {\left (b d \cosh \left (d x + c\right ) + b d \sinh \left (d x + c\right )\right )} \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (a^{2} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} \sinh \left (d x + c\right )^{2} + a^{2} - 2 \, {\left (a^{2} b d \cosh \left (d x + c\right ) + a^{2} b d \sinh \left (d x + c\right ) - {\left ({\left (a b^{4} - b^{5}\right )} d^{3} \cosh \left (d x + c\right ) + {\left (a b^{4} - b^{5}\right )} d^{3} \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}}\right )} \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}}\right ) + {\left (b d \cosh \left (d x + c\right ) + b d \sinh \left (d x + c\right )\right )} \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (a^{2} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} \sinh \left (d x + c\right )^{2} + a^{2} + 2 \, {\left (a^{2} b d \cosh \left (d x + c\right ) + a^{2} b d \sinh \left (d x + c\right ) + {\left ({\left (a b^{4} - b^{5}\right )} d^{3} \cosh \left (d x + c\right ) + {\left (a b^{4} - b^{5}\right )} d^{3} \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}}\right )} \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}}\right ) - {\left (b d \cosh \left (d x + c\right ) + b d \sinh \left (d x + c\right )\right )} \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (a^{2} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} \sinh \left (d x + c\right )^{2} + a^{2} - 2 \, {\left (a^{2} b d \cosh \left (d x + c\right ) + a^{2} b d \sinh \left (d x + c\right ) + {\left ({\left (a b^{4} - b^{5}\right )} d^{3} \cosh \left (d x + c\right ) + {\left (a b^{4} - b^{5}\right )} d^{3} \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}}\right )} \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a^{3}}{{\left (a^{2} b^{5} - 2 \, a b^{6} + b^{7}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}}\right ) - 2 \, \cosh \left (d x + c\right )^{2} - 4 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) - 2 \, \sinh \left (d x + c\right )^{2} - 2}{4 \, {\left (b d \cosh \left (d x + c\right ) + b d \sinh \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^5/(a-b*sinh(d*x+c)^4),x, algorithm="fricas")

[Out]

1/4*((b*d*cosh(d*x + c) + b*d*sinh(d*x + c))*sqrt(-((a*b^2 - b^3)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)
) + a)/((a*b^2 - b^3)*d^2))*log(a^2*cosh(d*x + c)^2 + 2*a^2*cosh(d*x + c)*sinh(d*x + c) + a^2*sinh(d*x + c)^2
+ a^2 + 2*(a^2*b*d*cosh(d*x + c) + a^2*b*d*sinh(d*x + c) - ((a*b^4 - b^5)*d^3*cosh(d*x + c) + (a*b^4 - b^5)*d^
3*sinh(d*x + c))*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)))*sqrt(-((a*b^2 - b^3)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*
b^6 + b^7)*d^4)) + a)/((a*b^2 - b^3)*d^2))) - (b*d*cosh(d*x + c) + b*d*sinh(d*x + c))*sqrt(-((a*b^2 - b^3)*d^2
*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a)/((a*b^2 - b^3)*d^2))*log(a^2*cosh(d*x + c)^2 + 2*a^2*cosh(d*x
+ c)*sinh(d*x + c) + a^2*sinh(d*x + c)^2 + a^2 - 2*(a^2*b*d*cosh(d*x + c) + a^2*b*d*sinh(d*x + c) - ((a*b^4 -
b^5)*d^3*cosh(d*x + c) + (a*b^4 - b^5)*d^3*sinh(d*x + c))*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)))*sqrt(-((a
*b^2 - b^3)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) + a)/((a*b^2 - b^3)*d^2))) + (b*d*cosh(d*x + c) + b*
d*sinh(d*x + c))*sqrt(((a*b^2 - b^3)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a)/((a*b^2 - b^3)*d^2))*l
og(a^2*cosh(d*x + c)^2 + 2*a^2*cosh(d*x + c)*sinh(d*x + c) + a^2*sinh(d*x + c)^2 + a^2 + 2*(a^2*b*d*cosh(d*x +
c) + a^2*b*d*sinh(d*x + c) + ((a*b^4 - b^5)*d^3*cosh(d*x + c) + (a*b^4 - b^5)*d^3*sinh(d*x + c))*sqrt(a^3/((a
^2*b^5 - 2*a*b^6 + b^7)*d^4)))*sqrt(((a*b^2 - b^3)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)) - a)/((a*b^2
- b^3)*d^2))) - (b*d*cosh(d*x + c) + b*d*sinh(d*x + c))*sqrt(((a*b^2 - b^3)*d^2*sqrt(a^3/((a^2*b^5 - 2*a*b^6 +
b^7)*d^4)) - a)/((a*b^2 - b^3)*d^2))*log(a^2*cosh(d*x + c)^2 + 2*a^2*cosh(d*x + c)*sinh(d*x + c) + a^2*sinh(d
*x + c)^2 + a^2 - 2*(a^2*b*d*cosh(d*x + c) + a^2*b*d*sinh(d*x + c) + ((a*b^4 - b^5)*d^3*cosh(d*x + c) + (a*b^4
- b^5)*d^3*sinh(d*x + c))*sqrt(a^3/((a^2*b^5 - 2*a*b^6 + b^7)*d^4)))*sqrt(((a*b^2 - b^3)*d^2*sqrt(a^3/((a^2*b
^5 - 2*a*b^6 + b^7)*d^4)) - a)/((a*b^2 - b^3)*d^2))) - 2*cosh(d*x + c)^2 - 4*cosh(d*x + c)*sinh(d*x + c) - 2*s
inh(d*x + c)^2 - 2)/(b*d*cosh(d*x + c) + b*d*sinh(d*x + c))

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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)**5/(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. 525 vs. \(2 (99) = 198\).
time = 0.58, size = 525, normalized size = 3.78 \begin {gather*} \frac {\frac {2 \, {\left (4 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b^{2} + 5 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} b^{3} + {\left (4 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b + 5 \, \sqrt {-b^{2} + \sqrt {a b} b} a b^{2}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {b^{2} - \sqrt {b^{4} + {\left (a b - b^{2}\right )} b^{2}}}{b^{2}}}}\right )}{4 \, a^{2} b^{4} + a b^{5} - 5 \, b^{6}} - \frac {{\left (4 \, \sqrt {a b} \sqrt {b^{2} + \sqrt {a b} b} a b^{2} - 3 \, \sqrt {a b} \sqrt {b^{2} + \sqrt {a b} b} b^{3} - {\left (4 \, \sqrt {b^{2} + \sqrt {a b} b} a^{2} b - 3 \, \sqrt {b^{2} + \sqrt {a b} b} a b^{2}\right )} {\left | b \right |}\right )} \log \left (2 \, \sqrt {\frac {b^{2} + \sqrt {b^{4} + {\left (a b - b^{2}\right )} b^{2}}}{b^{2}}} + e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}{4 \, a^{2} b^{4} - 7 \, a b^{5} + 3 \, b^{6}} + \frac {{\left (4 \, \sqrt {a b} \sqrt {b^{2} + \sqrt {a b} b} a b^{2} - 3 \, \sqrt {a b} \sqrt {b^{2} + \sqrt {a b} b} b^{3} - {\left (4 \, \sqrt {b^{2} + \sqrt {a b} b} a^{2} b - 3 \, \sqrt {b^{2} + \sqrt {a b} b} a b^{2}\right )} {\left | b \right |}\right )} \log \left ({\left | -2 \, \sqrt {\frac {b^{2} + \sqrt {b^{4} + {\left (a b - b^{2}\right )} b^{2}}}{b^{2}}} + e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} \right |}\right )}{4 \, a^{2} b^{4} - 7 \, a b^{5} + 3 \, b^{6}} - \frac {2 \, {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{b}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^5/(a-b*sinh(d*x+c)^4),x, algorithm="giac")

[Out]

1/4*(2*(4*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a*b^2 + 5*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*b^3 + (4*sqrt(-b^2 +
sqrt(a*b)*b)*a^2*b + 5*sqrt(-b^2 + sqrt(a*b)*b)*a*b^2)*abs(b))*arctan(1/2*(e^(d*x + c) + e^(-d*x - c))/sqrt(-
(b^2 - sqrt(b^4 + (a*b - b^2)*b^2))/b^2))/(4*a^2*b^4 + a*b^5 - 5*b^6) - (4*sqrt(a*b)*sqrt(b^2 + sqrt(a*b)*b)*a
*b^2 - 3*sqrt(a*b)*sqrt(b^2 + sqrt(a*b)*b)*b^3 - (4*sqrt(b^2 + sqrt(a*b)*b)*a^2*b - 3*sqrt(b^2 + sqrt(a*b)*b)*
a*b^2)*abs(b))*log(2*sqrt((b^2 + sqrt(b^4 + (a*b - b^2)*b^2))/b^2) + e^(d*x + c) + e^(-d*x - c))/(4*a^2*b^4 -
7*a*b^5 + 3*b^6) + (4*sqrt(a*b)*sqrt(b^2 + sqrt(a*b)*b)*a*b^2 - 3*sqrt(a*b)*sqrt(b^2 + sqrt(a*b)*b)*b^3 - (4*s
qrt(b^2 + sqrt(a*b)*b)*a^2*b - 3*sqrt(b^2 + sqrt(a*b)*b)*a*b^2)*abs(b))*log(abs(-2*sqrt((b^2 + sqrt(b^4 + (a*b
- b^2)*b^2))/b^2) + e^(d*x + c) + e^(-d*x - c)))/(4*a^2*b^4 - 7*a*b^5 + 3*b^6) - 2*(e^(d*x + c) + e^(-d*x - c
))/b)/d

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Mupad [B]
time = 7.85, size = 1046, normalized size = 7.53 \begin {gather*} \ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^6\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^9\,{\left (a-b\right )}^2}+\frac {16777216\,a^6\,d^3\,{\mathrm {e}}^{c+d\,x}\,\sqrt {-\frac {\sqrt {a^3\,b^5}+a\,b^3}{b^5\,d^2\,\left (a-b\right )}}}{b^8\,\left (a-b\right )}\right )\,\sqrt {-\frac {\sqrt {a^3\,b^5}+a\,b^3}{b^5\,d^2\,\left (a-b\right )}}}{4}-\frac {2097152\,a^7\,d\,{\mathrm {e}}^{c+d\,x}}{b^{11}\,\left (a-b\right )}\right )\,\sqrt {-\frac {\sqrt {a^3\,b^5}+a\,b^3}{b^5\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a^7\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^{12}\,{\left (a-b\right )}^2}\right )\,\sqrt {\frac {\sqrt {a^3\,b^5}+a\,b^3}{16\,\left (b^6\,d^2-a\,b^5\,d^2\right )}}-\ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^6\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^9\,{\left (a-b\right )}^2}-\frac {16777216\,a^6\,d^3\,{\mathrm {e}}^{c+d\,x}\,\sqrt {-\frac {\sqrt {a^3\,b^5}+a\,b^3}{b^5\,d^2\,\left (a-b\right )}}}{b^8\,\left (a-b\right )}\right )\,\sqrt {-\frac {\sqrt {a^3\,b^5}+a\,b^3}{b^5\,d^2\,\left (a-b\right )}}}{4}+\frac {2097152\,a^7\,d\,{\mathrm {e}}^{c+d\,x}}{b^{11}\,\left (a-b\right )}\right )\,\sqrt {-\frac {\sqrt {a^3\,b^5}+a\,b^3}{b^5\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a^7\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^{12}\,{\left (a-b\right )}^2}\right )\,\sqrt {\frac {\sqrt {a^3\,b^5}+a\,b^3}{16\,\left (b^6\,d^2-a\,b^5\,d^2\right )}}-\ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^6\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^9\,{\left (a-b\right )}^2}-\frac {16777216\,a^6\,d^3\,{\mathrm {e}}^{c+d\,x}\,\sqrt {\frac {\sqrt {a^3\,b^5}-a\,b^3}{b^5\,d^2\,\left (a-b\right )}}}{b^8\,\left (a-b\right )}\right )\,\sqrt {\frac {\sqrt {a^3\,b^5}-a\,b^3}{b^5\,d^2\,\left (a-b\right )}}}{4}+\frac {2097152\,a^7\,d\,{\mathrm {e}}^{c+d\,x}}{b^{11}\,\left (a-b\right )}\right )\,\sqrt {\frac {\sqrt {a^3\,b^5}-a\,b^3}{b^5\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a^7\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^{12}\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {\sqrt {a^3\,b^5}-a\,b^3}{16\,\left (b^6\,d^2-a\,b^5\,d^2\right )}}+\ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^6\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^9\,{\left (a-b\right )}^2}+\frac {16777216\,a^6\,d^3\,{\mathrm {e}}^{c+d\,x}\,\sqrt {\frac {\sqrt {a^3\,b^5}-a\,b^3}{b^5\,d^2\,\left (a-b\right )}}}{b^8\,\left (a-b\right )}\right )\,\sqrt {\frac {\sqrt {a^3\,b^5}-a\,b^3}{b^5\,d^2\,\left (a-b\right )}}}{4}-\frac {2097152\,a^7\,d\,{\mathrm {e}}^{c+d\,x}}{b^{11}\,\left (a-b\right )}\right )\,\sqrt {\frac {\sqrt {a^3\,b^5}-a\,b^3}{b^5\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a^7\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^{12}\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {\sqrt {a^3\,b^5}-a\,b^3}{16\,\left (b^6\,d^2-a\,b^5\,d^2\right )}}-\frac {{\mathrm {e}}^{c+d\,x}}{2\,b\,d}-\frac {{\mathrm {e}}^{-c-d\,x}}{2\,b\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^5/(a - b*sinh(c + d*x)^4),x)

[Out]

log((((((4194304*a^6*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b))/(b^9*(a - b)^2) + (16777216*a^6*d^3*exp(c + d*x)*(-
((a^3*b^5)^(1/2) + a*b^3)/(b^5*d^2*(a - b)))^(1/2))/(b^8*(a - b)))*(-((a^3*b^5)^(1/2) + a*b^3)/(b^5*d^2*(a - b
)))^(1/2))/4 - (2097152*a^7*d*exp(c + d*x))/(b^11*(a - b)))*(-((a^3*b^5)^(1/2) + a*b^3)/(b^5*d^2*(a - b)))^(1/
2))/4 - (262144*a^7*(exp(2*c + 2*d*x) + 1)*(a + b))/(b^12*(a - b)^2))*(((a^3*b^5)^(1/2) + a*b^3)/(16*(b^6*d^2
- a*b^5*d^2)))^(1/2) - log((((((4194304*a^6*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b))/(b^9*(a - b)^2) - (16777216*
a^6*d^3*exp(c + d*x)*(-((a^3*b^5)^(1/2) + a*b^3)/(b^5*d^2*(a - b)))^(1/2))/(b^8*(a - b)))*(-((a^3*b^5)^(1/2) +
a*b^3)/(b^5*d^2*(a - b)))^(1/2))/4 + (2097152*a^7*d*exp(c + d*x))/(b^11*(a - b)))*(-((a^3*b^5)^(1/2) + a*b^3)
/(b^5*d^2*(a - b)))^(1/2))/4 - (262144*a^7*(exp(2*c + 2*d*x) + 1)*(a + b))/(b^12*(a - b)^2))*(((a^3*b^5)^(1/2)
+ a*b^3)/(16*(b^6*d^2 - a*b^5*d^2)))^(1/2) - log((((((4194304*a^6*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b))/(b^9*
(a - b)^2) - (16777216*a^6*d^3*exp(c + d*x)*(((a^3*b^5)^(1/2) - a*b^3)/(b^5*d^2*(a - b)))^(1/2))/(b^8*(a - b))
)*(((a^3*b^5)^(1/2) - a*b^3)/(b^5*d^2*(a - b)))^(1/2))/4 + (2097152*a^7*d*exp(c + d*x))/(b^11*(a - b)))*(((a^3
*b^5)^(1/2) - a*b^3)/(b^5*d^2*(a - b)))^(1/2))/4 - (262144*a^7*(exp(2*c + 2*d*x) + 1)*(a + b))/(b^12*(a - b)^2
))*(-((a^3*b^5)^(1/2) - a*b^3)/(16*(b^6*d^2 - a*b^5*d^2)))^(1/2) + log((((((4194304*a^6*d^2*(exp(2*c + 2*d*x)
+ 1)*(3*a + b))/(b^9*(a - b)^2) + (16777216*a^6*d^3*exp(c + d*x)*(((a^3*b^5)^(1/2) - a*b^3)/(b^5*d^2*(a - b)))
^(1/2))/(b^8*(a - b)))*(((a^3*b^5)^(1/2) - a*b^3)/(b^5*d^2*(a - b)))^(1/2))/4 - (2097152*a^7*d*exp(c + d*x))/(
b^11*(a - b)))*(((a^3*b^5)^(1/2) - a*b^3)/(b^5*d^2*(a - b)))^(1/2))/4 - (262144*a^7*(exp(2*c + 2*d*x) + 1)*(a
+ b))/(b^12*(a - b)^2))*(-((a^3*b^5)^(1/2) - a*b^3)/(16*(b^6*d^2 - a*b^5*d^2)))^(1/2) - exp(c + d*x)/(2*b*d) -
exp(- c - d*x)/(2*b*d)

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