3.4.17 \(\int \frac {\cosh ^6(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) [317]

Optimal. Leaf size=121 \[ \frac {\left (8 a^2-20 a b+15 b^2\right ) x}{8 b^3}-\frac {(a-b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^3 d}-\frac {(4 a-7 b) \cosh (c+d x) \sinh (c+d x)}{8 b^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 b d} \]

[Out]

1/8*(8*a^2-20*a*b+15*b^2)*x/b^3-1/8*(4*a-7*b)*cosh(d*x+c)*sinh(d*x+c)/b^2/d+1/4*cosh(d*x+c)^3*sinh(d*x+c)/b/d-
(a-b)^(5/2)*arctanh((a-b)^(1/2)*tanh(d*x+c)/a^(1/2))/b^3/d/a^(1/2)

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Rubi [A]
time = 0.14, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3270, 425, 541, 536, 212, 214} \begin {gather*} \frac {x \left (8 a^2-20 a b+15 b^2\right )}{8 b^3}-\frac {(a-b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^3 d}-\frac {(4 a-7 b) \sinh (c+d x) \cosh (c+d x)}{8 b^2 d}+\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]^6/(a + b*Sinh[c + d*x]^2),x]

[Out]

((8*a^2 - 20*a*b + 15*b^2)*x)/(8*b^3) - ((a - b)^(5/2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*
b^3*d) - ((4*a - 7*b)*Cosh[c + d*x]*Sinh[c + d*x])/(8*b^2*d) + (Cosh[c + d*x]^3*Sinh[c + d*x])/(4*b*d)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 3270

Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, T
an[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\cosh ^6(c+d x)}{a+b \sinh ^2(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^3 \left (a-(a-b) x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 b d}+\frac {\text {Subst}\left (\int \frac {-a+4 b-3 (a-b) x^2}{\left (1-x^2\right )^2 \left (a+(-a+b) x^2\right )} \, dx,x,\tanh (c+d x)\right )}{4 b d}\\ &=-\frac {(4 a-7 b) \cosh (c+d x) \sinh (c+d x)}{8 b^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 b d}+\frac {\text {Subst}\left (\int \frac {4 a^2-9 a b+8 b^2+(4 a-7 b) (a-b) x^2}{\left (1-x^2\right ) \left (a+(-a+b) x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 b^2 d}\\ &=-\frac {(4 a-7 b) \cosh (c+d x) \sinh (c+d x)}{8 b^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 b d}-\frac {(a-b)^3 \text {Subst}\left (\int \frac {1}{a+(-a+b) x^2} \, dx,x,\tanh (c+d x)\right )}{b^3 d}+\frac {\left (8 a^2-20 a b+15 b^2\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 b^3 d}\\ &=\frac {\left (8 a^2-20 a b+15 b^2\right ) x}{8 b^3}-\frac {(a-b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^3 d}-\frac {(4 a-7 b) \cosh (c+d x) \sinh (c+d x)}{8 b^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 b d}\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 106, normalized size = 0.88 \begin {gather*} \frac {-32 (a-b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )+\sqrt {a} \left (4 \left (8 a^2-20 a b+15 b^2\right ) (c+d x)-8 (a-2 b) b \sinh (2 (c+d x))+b^2 \sinh (4 (c+d x))\right )}{32 \sqrt {a} b^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]^6/(a + b*Sinh[c + d*x]^2),x]

[Out]

(-32*(a - b)^(5/2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]] + Sqrt[a]*(4*(8*a^2 - 20*a*b + 15*b^2)*(c + d*
x) - 8*(a - 2*b)*b*Sinh[2*(c + d*x)] + b^2*Sinh[4*(c + d*x)]))/(32*Sqrt[a]*b^3*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(437\) vs. \(2(107)=214\).
time = 1.82, size = 438, normalized size = 3.62

method	result	size
risch	\(\frac {x \,a^{2}}{b^{3}}-\frac {5 a x}{2 b^{2}}+\frac {15 x}{8 b}+\frac {{\mathrm e}^{4 d x +4 c}}{64 b d}+\frac {{\mathrm e}^{2 d x +2 c}}{4 b d}-\frac {a \,{\mathrm e}^{2 d x +2 c}}{8 b^{2} d}-\frac {{\mathrm e}^{-2 d x -2 c}}{4 b d}+\frac {a \,{\mathrm e}^{-2 d x -2 c}}{8 b^{2} d}-\frac {{\mathrm e}^{-4 d x -4 c}}{64 b d}+\frac {a \sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {a \left (a -b \right )}+2 a -b}{b}\right )}{2 d \,b^{3}}-\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {a \left (a -b \right )}+2 a -b}{b}\right )}{d \,b^{2}}+\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {a \left (a -b \right )}+2 a -b}{b}\right )}{2 a d b}-\frac {a \sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {a \left (a -b \right )}-2 a +b}{b}\right )}{2 d \,b^{3}}+\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {a \left (a -b \right )}-2 a +b}{b}\right )}{d \,b^{2}}-\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {a \left (a -b \right )}-2 a +b}{b}\right )}{2 a d b}\)	\(431\)
derivativedivides	\(\frac {\frac {2 a \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{b^{3}}-\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-9 b +4 a}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {11 b -4 a}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (8 a^{2}-20 a b +15 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 b^{3}}+\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-11 b +4 a}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-9 b +4 a}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-8 a^{2}+20 a b -15 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 b^{3}}}{d}\)	\(438\)
default	\(\frac {\frac {2 a \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{b^{3}}-\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-9 b +4 a}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {11 b -4 a}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\left (8 a^{2}-20 a b +15 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 b^{3}}+\frac {1}{4 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {-11 b +4 a}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-9 b +4 a}{8 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-8 a^{2}+20 a b -15 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 b^{3}}}{d}\)	\(438\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^6/(a+b*sinh(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

1/d*(2/b^3*a*(a^3-3*a^2*b+3*a*b^2-b^3)*(-1/2*((-b*(a-b))^(1/2)-b)/a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)+a-2*
b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))+1/2*((-b*(a-b))^(1/2)+b)/a/(-b
*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)
^(1/2)))-1/4/b/(tanh(1/2*d*x+1/2*c)+1)^4+1/2/b/(tanh(1/2*d*x+1/2*c)+1)^3-1/8*(-9*b+4*a)/b^2/(tanh(1/2*d*x+1/2*
c)+1)-1/8*(11*b-4*a)/b^2/(tanh(1/2*d*x+1/2*c)+1)^2+1/8*(8*a^2-20*a*b+15*b^2)/b^3*ln(tanh(1/2*d*x+1/2*c)+1)+1/4
/b/(tanh(1/2*d*x+1/2*c)-1)^4+1/2/b/(tanh(1/2*d*x+1/2*c)-1)^3-1/8*(-11*b+4*a)/b^2/(tanh(1/2*d*x+1/2*c)-1)^2-1/8
*(-9*b+4*a)/b^2/(tanh(1/2*d*x+1/2*c)-1)+1/8/b^3*(-8*a^2+20*a*b-15*b^2)*ln(tanh(1/2*d*x+1/2*c)-1))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^6/(a+b*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for mor
e details)Is

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 776 vs. \(2 (107) = 214\).
time = 0.45, size = 1817, normalized size = 15.02 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^6/(a+b*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

[1/64*(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 + 8*(8*a^2 - 20*a*b + 1
5*b^2)*d*x*cosh(d*x + c)^4 - 8*(a*b - 2*b^2)*cosh(d*x + c)^6 + 4*(7*b^2*cosh(d*x + c)^2 - 2*a*b + 4*b^2)*sinh(
d*x + c)^6 + 8*(7*b^2*cosh(d*x + c)^3 - 6*(a*b - 2*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*b^2*cosh(d*x +
c)^4 + 4*(8*a^2 - 20*a*b + 15*b^2)*d*x - 60*(a*b - 2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*b^2*cosh(d*x
+ c)^5 + 4*(8*a^2 - 20*a*b + 15*b^2)*d*x*cosh(d*x + c) - 20*(a*b - 2*b^2)*cosh(d*x + c)^3)*sinh(d*x + c)^3 +
8*(a*b - 2*b^2)*cosh(d*x + c)^2 + 4*(7*b^2*cosh(d*x + c)^6 + 12*(8*a^2 - 20*a*b + 15*b^2)*d*x*cosh(d*x + c)^2
- 30*(a*b - 2*b^2)*cosh(d*x + c)^4 + 2*a*b - 4*b^2)*sinh(d*x + c)^2 + 32*((a^2 - 2*a*b + b^2)*cosh(d*x + c)^4
+ 4*(a^2 - 2*a*b + b^2)*cosh(d*x + c)^3*sinh(d*x + c) + 6*(a^2 - 2*a*b + b^2)*cosh(d*x + c)^2*sinh(d*x + c)^2
+ 4*(a^2 - 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 - 2*a*b + b^2)*sinh(d*x + c)^4)*sqrt((a - b)/a)*l
og((b^2*cosh(d*x + c)^4 + 4*b^2*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*sinh(d*x + c)^4 + 2*(2*a*b - b^2)*cosh(d*x
+ c)^2 + 2*(3*b^2*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)^2 + 8*a^2 - 8*a*b + b^2 + 4*(b^2*cosh(d*x + c)
^3 + (2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c) + 4*(a*b*cosh(d*x + c)^2 + 2*a*b*cosh(d*x + c)*sinh(d*x + c) +
a*b*sinh(d*x + c)^2 + 2*a^2 - a*b)*sqrt((a - b)/a))/(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*cos
h(d*x + c)^3 + (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + b)) - b^2 + 8*(b^2*cosh(d*x + c)^7 + 4*(8*a^2 - 20*a*b
+ 15*b^2)*d*x*cosh(d*x + c)^3 - 6*(a*b - 2*b^2)*cosh(d*x + c)^5 + 2*(a*b - 2*b^2)*cosh(d*x + c))*sinh(d*x + c
))/(b^3*d*cosh(d*x + c)^4 + 4*b^3*d*cosh(d*x + c)^3*sinh(d*x + c) + 6*b^3*d*cosh(d*x + c)^2*sinh(d*x + c)^2 +
4*b^3*d*cosh(d*x + c)*sinh(d*x + c)^3 + b^3*d*sinh(d*x + c)^4), 1/64*(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 + 8*(8*a^2 - 20*a*b + 15*b^2)*d*x*cosh(d*x + c)^4 - 8*(a*b - 2*b^2)*co
sh(d*x + c)^6 + 4*(7*b^2*cosh(d*x + c)^2 - 2*a*b + 4*b^2)*sinh(d*x + c)^6 + 8*(7*b^2*cosh(d*x + c)^3 - 6*(a*b
- 2*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*b^2*cosh(d*x + c)^4 + 4*(8*a^2 - 20*a*b + 15*b^2)*d*x - 60*(a*
b - 2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*b^2*cosh(d*x + c)^5 + 4*(8*a^2 - 20*a*b + 15*b^2)*d*x*cosh(
d*x + c) - 20*(a*b - 2*b^2)*cosh(d*x + c)^3)*sinh(d*x + c)^3 + 8*(a*b - 2*b^2)*cosh(d*x + c)^2 + 4*(7*b^2*cosh
(d*x + c)^6 + 12*(8*a^2 - 20*a*b + 15*b^2)*d*x*cosh(d*x + c)^2 - 30*(a*b - 2*b^2)*cosh(d*x + c)^4 + 2*a*b - 4*
b^2)*sinh(d*x + c)^2 + 64*((a^2 - 2*a*b + b^2)*cosh(d*x + c)^4 + 4*(a^2 - 2*a*b + b^2)*cosh(d*x + c)^3*sinh(d*
x + c) + 6*(a^2 - 2*a*b + b^2)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^2 - 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x
+ c)^3 + (a^2 - 2*a*b + b^2)*sinh(d*x + c)^4)*sqrt(-(a - b)/a)*arctan(-1/2*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x +
c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + 2*a - b)*sqrt(-(a - b)/a)/(a - b)) - b^2 + 8*(b^2*cosh(d*x + c)^7 + 4*
(8*a^2 - 20*a*b + 15*b^2)*d*x*cosh(d*x + c)^3 - 6*(a*b - 2*b^2)*cosh(d*x + c)^5 + 2*(a*b - 2*b^2)*cosh(d*x + c
))*sinh(d*x + c))/(b^3*d*cosh(d*x + c)^4 + 4*b^3*d*cosh(d*x + c)^3*sinh(d*x + c) + 6*b^3*d*cosh(d*x + c)^2*sin
h(d*x + c)^2 + 4*b^3*d*cosh(d*x + c)*sinh(d*x + c)^3 + b^3*d*sinh(d*x + c)^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(cosh(d*x+c)**6/(a+b*sinh(d*x+c)**2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (107) = 214\).
time = 2.42, size = 226, normalized size = 1.87 \begin {gather*} \frac {\frac {8 \, {\left (8 \, a^{2} - 20 \, a b + 15 \, b^{2}\right )} {\left (d x + c\right )}}{b^{3}} + \frac {b e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a e^{\left (2 \, d x + 2 \, c\right )} + 16 \, b e^{\left (2 \, d x + 2 \, c\right )}}{b^{2}} - \frac {{\left (48 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 120 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 90 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 16 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{b^{3}} - \frac {64 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt {-a^{2} + a b}}\right )}{\sqrt {-a^{2} + a b} b^{3}}}{64 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^6/(a+b*sinh(d*x+c)^2),x, algorithm="giac")

[Out]

1/64*(8*(8*a^2 - 20*a*b + 15*b^2)*(d*x + c)/b^3 + (b*e^(4*d*x + 4*c) - 8*a*e^(2*d*x + 2*c) + 16*b*e^(2*d*x + 2
*c))/b^2 - (48*a^2*e^(4*d*x + 4*c) - 120*a*b*e^(4*d*x + 4*c) + 90*b^2*e^(4*d*x + 4*c) - 8*a*b*e^(2*d*x + 2*c)
+ 16*b^2*e^(2*d*x + 2*c) + b^2)*e^(-4*d*x - 4*c)/b^3 - 64*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*arctan(1/2*(b*e^(2*d
*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/(sqrt(-a^2 + a*b)*b^3))/d

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Mupad [B]
time = 1.37, size = 264, normalized size = 2.18 \begin {gather*} \frac {x\,\left (8\,a^2-20\,a\,b+15\,b^2\right )}{8\,b^3}-\frac {{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,b\,d}+\frac {{\mathrm {e}}^{4\,c+4\,d\,x}}{64\,b\,d}+\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (a-2\,b\right )}{8\,b^2\,d}-\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a-2\,b\right )}{8\,b^2\,d}+\frac {\ln \left (\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,{\left (a-b\right )}^3}{b^4}-\frac {2\,{\left (a-b\right )}^{5/2}\,\left (b+2\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}-b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{\sqrt {a}\,b^4}\right )\,{\left (a-b\right )}^{5/2}}{2\,\sqrt {a}\,b^3\,d}-\frac {\ln \left (\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,{\left (a-b\right )}^3}{b^4}+\frac {2\,{\left (a-b\right )}^{5/2}\,\left (b+2\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}-b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{\sqrt {a}\,b^4}\right )\,{\left (a-b\right )}^{5/2}}{2\,\sqrt {a}\,b^3\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)^6/(a + b*sinh(c + d*x)^2),x)

[Out]

(x*(8*a^2 - 20*a*b + 15*b^2))/(8*b^3) - exp(- 4*c - 4*d*x)/(64*b*d) + exp(4*c + 4*d*x)/(64*b*d) + (exp(- 2*c -
2*d*x)*(a - 2*b))/(8*b^2*d) - (exp(2*c + 2*d*x)*(a - 2*b))/(8*b^2*d) + (log((4*exp(2*c + 2*d*x)*(a - b)^3)/b^
4 - (2*(a - b)^(5/2)*(b + 2*a*exp(2*c + 2*d*x) - b*exp(2*c + 2*d*x)))/(a^(1/2)*b^4))*(a - b)^(5/2))/(2*a^(1/2)
*b^3*d) - (log((4*exp(2*c + 2*d*x)*(a - b)^3)/b^4 + (2*(a - b)^(5/2)*(b + 2*a*exp(2*c + 2*d*x) - b*exp(2*c + 2
*d*x)))/(a^(1/2)*b^4))*(a - b)^(5/2))/(2*a^(1/2)*b^3*d)

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