3.4.29 \(\int \frac {\cosh ^6(c+d x)}{(a+b \sinh ^2(c+d x))^2} \, dx\) [329]

Optimal. Leaf size=158 \[ -\frac {(4 a-5 b) x}{2 b^3}+\frac {(a-b)^{3/2} (4 a+b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^3 d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 b d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac {(a-b) (2 a-b) \tanh (c+d x)}{2 a b^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )} \]

[Out]

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

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Rubi [A]
time = 0.19, antiderivative size = 158, 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 {(4 a+b) (a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} b^3 d}-\frac {x (4 a-5 b)}{2 b^3}+\frac {(2 a-b) (a-b) \tanh (c+d x)}{2 a b^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac {\sinh (c+d x) \cosh (c+d x)}{2 b d \left (a-(a-b) \tanh ^2(c+d x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((4*a - 5*b)*x)/b^3 + ((a - b)^(3/2)*(4*a + b)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b
^3*d) + (Cosh[c + d*x]*Sinh[c + d*x])/(2*b*d*(a - (a - b)*Tanh[c + d*x]^2)) + ((a - b)*(2*a - b)*Tanh[c + d*x]
)/(2*a*b^2*d*(a - (a - b)*Tanh[c + d*x]^2))

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

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Mathematica [A]
time = 0.42, size = 118, normalized size = 0.75 \begin {gather*} \frac {2 (-4 a+5 b) (c+d x)+\frac {2 (a-b)^{3/2} (4 a+b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{3/2}}+b \sinh (2 (c+d x))+\frac {2 (a-b)^2 b \sinh (2 (c+d x))}{a (2 a-b+b \cosh (2 (c+d x)))}}{4 b^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(424\) vs. \(2(142)=284\).
time = 2.00, size = 425, normalized size = 2.69

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (\frac {-\frac {b \left (a^{2}-2 a b +b^{2}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {b \left (a^{2}-2 a b +b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\left (4 a^{3}-7 a^{2} b +2 a \,b^{2}+b^{3}\right ) \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2}\right )}{b^{3}}+\frac {1}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (4 a -5 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{3}}-\frac {1}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (-4 a +5 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{3}}}{d}\)	\(425\)
default	\(\frac {-\frac {2 \left (\frac {-\frac {b \left (a^{2}-2 a b +b^{2}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {b \left (a^{2}-2 a b +b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\left (4 a^{3}-7 a^{2} b +2 a \,b^{2}+b^{3}\right ) \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2}\right )}{b^{3}}+\frac {1}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (4 a -5 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{3}}-\frac {1}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (-4 a +5 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{3}}}{d}\)	\(425\)
risch	\(-\frac {2 a x}{b^{3}}+\frac {5 x}{2 b^{2}}+\frac {{\mathrm e}^{2 d x +2 c}}{8 b^{2} d}-\frac {{\mathrm e}^{-2 d x -2 c}}{8 b^{2} d}-\frac {2 a^{3} {\mathrm e}^{2 d x +2 c}-5 a^{2} b \,{\mathrm e}^{2 d x +2 c}+4 a \,b^{2} {\mathrm e}^{2 d x +2 c}-b^{3} {\mathrm e}^{2 d x +2 c}+a^{2} b -2 a \,b^{2}+b^{3}}{b^{3} a d \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )}+\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {a \left (a -b \right )}-2 a +b}{b}\right )}{d \,b^{3}}-\frac {3 \sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {a \left (a -b \right )}-2 a +b}{b}\right )}{4 a d \,b^{2}}-\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {a \left (a -b \right )}-2 a +b}{b}\right )}{4 a^{2} d b}-\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {a \left (a -b \right )}+2 a -b}{b}\right )}{d \,b^{3}}+\frac {3 \sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {a \left (a -b \right )}+2 a -b}{b}\right )}{4 a d \,b^{2}}+\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {a \left (a -b \right )}+2 a -b}{b}\right )}{4 a^{2} d b}\)	\(480\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-2/b^3*((-1/2*b*(a^2-2*a*b+b^2)/a*tanh(1/2*d*x+1/2*c)^3-1/2*b*(a^2-2*a*b+b^2)/a*tanh(1/2*d*x+1/2*c))/(a*t
anh(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)+1/2*(4*a^3-7*a^2*b+2*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))))+1/2/b^2/(tanh(1/2*d*x+1/2*c)-1)
^2+1/2/b^2/(tanh(1/2*d*x+1/2*c)-1)+1/2*(4*a-5*b)/b^3*ln(tanh(1/2*d*x+1/2*c)-1)-1/2/b^2/(tanh(1/2*d*x+1/2*c)+1)
^2+1/2/b^2/(tanh(1/2*d*x+1/2*c)+1)+1/2/b^3*(-4*a+5*b)*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)^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. 1682 vs. \(2 (144) = 288\).
time = 0.42, size = 3629, normalized size = 22.97 \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)^2,x, algorithm="fricas")

[Out]

[1/8*(a*b^2*cosh(d*x + c)^8 + 8*a*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + a*b^2*sinh(d*x + c)^8 + 2*(2*a^2*b - a*b
^2 - 2*(4*a^2*b - 5*a*b^2)*d*x)*cosh(d*x + c)^6 + 2*(14*a*b^2*cosh(d*x + c)^2 + 2*a^2*b - a*b^2 - 2*(4*a^2*b -
5*a*b^2)*d*x)*sinh(d*x + c)^6 + 4*(14*a*b^2*cosh(d*x + c)^3 + 3*(2*a^2*b - a*b^2 - 2*(4*a^2*b - 5*a*b^2)*d*x)
*cosh(d*x + c))*sinh(d*x + c)^5 - 8*(2*a^3 - 5*a^2*b + 4*a*b^2 - b^3 + (8*a^3 - 14*a^2*b + 5*a*b^2)*d*x)*cosh(
d*x + c)^4 + 2*(35*a*b^2*cosh(d*x + c)^4 - 8*a^3 + 20*a^2*b - 16*a*b^2 + 4*b^3 - 4*(8*a^3 - 14*a^2*b + 5*a*b^2
)*d*x + 15*(2*a^2*b - a*b^2 - 2*(4*a^2*b - 5*a*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*a*b^2*cosh(d*
x + c)^5 + 5*(2*a^2*b - a*b^2 - 2*(4*a^2*b - 5*a*b^2)*d*x)*cosh(d*x + c)^3 - 4*(2*a^3 - 5*a^2*b + 4*a*b^2 - b^
3 + (8*a^3 - 14*a^2*b + 5*a*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 - a*b^2 - 2*(6*a^2*b - 9*a*b^2 + 4*b^3 +
2*(4*a^2*b - 5*a*b^2)*d*x)*cosh(d*x + c)^2 + 2*(14*a*b^2*cosh(d*x + c)^6 + 15*(2*a^2*b - a*b^2 - 2*(4*a^2*b -
5*a*b^2)*d*x)*cosh(d*x + c)^4 - 6*a^2*b + 9*a*b^2 - 4*b^3 - 2*(4*a^2*b - 5*a*b^2)*d*x - 24*(2*a^3 - 5*a^2*b +
4*a*b^2 - b^3 + (8*a^3 - 14*a^2*b + 5*a*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 2*((4*a^2*b - 3*a*b^2 - b
^3)*cosh(d*x + c)^6 + 6*(4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c)*sinh(d*x + c)^5 + (4*a^2*b - 3*a*b^2 - b^3)*si
nh(d*x + c)^6 + 2*(8*a^3 - 10*a^2*b + a*b^2 + b^3)*cosh(d*x + c)^4 + (16*a^3 - 20*a^2*b + 2*a*b^2 + 2*b^3 + 15
*(4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c)^3 +
2*(8*a^3 - 10*a^2*b + a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + (4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c)^2
+ (15*(4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c)^4 + 4*a^2*b - 3*a*b^2 - b^3 + 12*(8*a^3 - 10*a^2*b + a*b^2 + b^
3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(3*(4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c)^5 + 4*(8*a^3 - 10*a^2*b + a
*b^2 + b^3)*cosh(d*x + c)^3 + (4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt((a - b)/a)*log((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*si
nh(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*cosh(d*x +
c)^3 + (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + b)) + 4*(2*a*b^2*cosh(d*x + c)^7 + 3*(2*a^2*b - a*b^2 - 2*(4*
a^2*b - 5*a*b^2)*d*x)*cosh(d*x + c)^5 - 8*(2*a^3 - 5*a^2*b + 4*a*b^2 - b^3 + (8*a^3 - 14*a^2*b + 5*a*b^2)*d*x)
*cosh(d*x + c)^3 - (6*a^2*b - 9*a*b^2 + 4*b^3 + 2*(4*a^2*b - 5*a*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c))/(a*b^
4*d*cosh(d*x + c)^6 + 6*a*b^4*d*cosh(d*x + c)*sinh(d*x + c)^5 + a*b^4*d*sinh(d*x + c)^6 + a*b^4*d*cosh(d*x + c
)^2 + 2*(2*a^2*b^3 - a*b^4)*d*cosh(d*x + c)^4 + (15*a*b^4*d*cosh(d*x + c)^2 + 2*(2*a^2*b^3 - a*b^4)*d)*sinh(d*
x + c)^4 + 4*(5*a*b^4*d*cosh(d*x + c)^3 + 2*(2*a^2*b^3 - a*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (15*a*b^4*d
*cosh(d*x + c)^4 + a*b^4*d + 12*(2*a^2*b^3 - a*b^4)*d*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(3*a*b^4*d*cosh(d*x
+ c)^5 + a*b^4*d*cosh(d*x + c) + 4*(2*a^2*b^3 - a*b^4)*d*cosh(d*x + c)^3)*sinh(d*x + c)), 1/8*(a*b^2*cosh(d*x
+ c)^8 + 8*a*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + a*b^2*sinh(d*x + c)^8 + 2*(2*a^2*b - a*b^2 - 2*(4*a^2*b - 5*
a*b^2)*d*x)*cosh(d*x + c)^6 + 2*(14*a*b^2*cosh(d*x + c)^2 + 2*a^2*b - a*b^2 - 2*(4*a^2*b - 5*a*b^2)*d*x)*sinh(
d*x + c)^6 + 4*(14*a*b^2*cosh(d*x + c)^3 + 3*(2*a^2*b - a*b^2 - 2*(4*a^2*b - 5*a*b^2)*d*x)*cosh(d*x + c))*sinh
(d*x + c)^5 - 8*(2*a^3 - 5*a^2*b + 4*a*b^2 - b^3 + (8*a^3 - 14*a^2*b + 5*a*b^2)*d*x)*cosh(d*x + c)^4 + 2*(35*a
*b^2*cosh(d*x + c)^4 - 8*a^3 + 20*a^2*b - 16*a*b^2 + 4*b^3 - 4*(8*a^3 - 14*a^2*b + 5*a*b^2)*d*x + 15*(2*a^2*b
- a*b^2 - 2*(4*a^2*b - 5*a*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*a*b^2*cosh(d*x + c)^5 + 5*(2*a^2*
b - a*b^2 - 2*(4*a^2*b - 5*a*b^2)*d*x)*cosh(d*x + c)^3 - 4*(2*a^3 - 5*a^2*b + 4*a*b^2 - b^3 + (8*a^3 - 14*a^2*
b + 5*a*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c)^3 - a*b^2 - 2*(6*a^2*b - 9*a*b^2 + 4*b^3 + 2*(4*a^2*b - 5*a*b^2
)*d*x)*cosh(d*x + c)^2 + 2*(14*a*b^2*cosh(d*x + c)^6 + 15*(2*a^2*b - a*b^2 - 2*(4*a^2*b - 5*a*b^2)*d*x)*cosh(d
*x + c)^4 - 6*a^2*b + 9*a*b^2 - 4*b^3 - 2*(4*a^2*b - 5*a*b^2)*d*x - 24*(2*a^3 - 5*a^2*b + 4*a*b^2 - b^3 + (8*a
^3 - 14*a^2*b + 5*a*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 4*((4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c)^6
+ 6*(4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c)*sinh(d*x + c)^5 + (4*a^2*b - 3*a*b^2 - b^3)*sinh(d*x + c)^6 + 2*(8
*a^3 - 10*a^2*b + a*b^2 + b^3)*cosh(d*x + c)^4 + (16*a^3 - 20*a^2*b + 2*a*b^2 + 2*b^3 + 15*(4*a^2*b - 3*a*b^2
- b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(4*a^2*b - 3*a*b^2 - b^3)*cosh(d*x + c)^3 + 2*(8*a^3 - 10*a^2*b
+ a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^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(cosh(d*x+c)**6/(a+b*sinh(d*x+c)**2)**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (144) = 288\).
time = 1.65, size = 305, normalized size = 1.93 \begin {gather*} -\frac {\frac {12 \, {\left (d x + c\right )} {\left (4 \, a - 5 \, b\right )}}{b^{3}} - \frac {3 \, e^{\left (2 \, d x + 2 \, c\right )}}{b^{2}} - \frac {12 \, {\left (4 \, a^{3} - 7 \, a^{2} b + 2 \, 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} a b^{3}} - \frac {8 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 10 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 16 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 64 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 79 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 24 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 28 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 44 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 24 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 3 \, a b^{2}}{{\left (b e^{\left (6 \, d x + 6 \, c\right )} + 4 \, a e^{\left (4 \, d x + 4 \, c\right )} - 2 \, b e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )}\right )} a b^{3}}}{24 \, 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)^2,x, algorithm="giac")

[Out]

-1/24*(12*(d*x + c)*(4*a - 5*b)/b^3 - 3*e^(2*d*x + 2*c)/b^2 - 12*(4*a^3 - 7*a^2*b + 2*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)*a*b^3) - (8*a^2*b*e^(6*d*x + 6*c) - 10*a*b^2
*e^(6*d*x + 6*c) - 16*a^3*e^(4*d*x + 4*c) + 64*a^2*b*e^(4*d*x + 4*c) - 79*a*b^2*e^(4*d*x + 4*c) + 24*b^3*e^(4*
d*x + 4*c) - 28*a^2*b*e^(2*d*x + 2*c) + 44*a*b^2*e^(2*d*x + 2*c) - 24*b^3*e^(2*d*x + 2*c) - 3*a*b^2)/((b*e^(6*
d*x + 6*c) + 4*a*e^(4*d*x + 4*c) - 2*b*e^(4*d*x + 4*c) + b*e^(2*d*x + 2*c))*a*b^3))/d

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6}{{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \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)^2,x)

[Out]

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

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