3.4.39 \(\int \frac {\cosh ^6(c+d x)}{(a+b \sinh ^2(c+d x))^3} \, dx\) [339]

Optimal. Leaf size=160 \[ \frac {x}{b^3}-\frac {\sqrt {a-b} \left (8 a^2+4 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} b^3 d}-\frac {(a-b) \tanh (c+d x)}{4 a b d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}-\frac {(a-b) (4 a+3 b) \tanh (c+d x)}{8 a^2 b^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )} \]

[Out]

x/b^3-1/8*(8*a^2+4*a*b+3*b^2)*arctanh((a-b)^(1/2)*tanh(d*x+c)/a^(1/2))*(a-b)^(1/2)/a^(5/2)/b^3/d-1/4*(a-b)*tan
h(d*x+c)/a/b/d/(a-(a-b)*tanh(d*x+c)^2)^2-1/8*(a-b)*(4*a+3*b)*tanh(d*x+c)/a^2/b^2/d/(a-(a-b)*tanh(d*x+c)^2)

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Rubi [A]
time = 0.16, antiderivative size = 160, 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 {(a-b) (4 a+3 b) \tanh (c+d x)}{8 a^2 b^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )}-\frac {\sqrt {a-b} \left (8 a^2+4 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} b^3 d}-\frac {(a-b) \tanh (c+d x)}{4 a b d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac {x}{b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]^6/(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

x/b^3 - (Sqrt[a - b]*(8*a^2 + 4*a*b + 3*b^2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*b^3*d) -
((a - b)*Tanh[c + d*x])/(4*a*b*d*(a - (a - b)*Tanh[c + d*x]^2)^2) - ((a - b)*(4*a + 3*b)*Tanh[c + d*x])/(8*a^
2*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 )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a-(a-b) x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {(a-b) \tanh (c+d x)}{4 a b d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-a-3 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 )}{4 a b d}\\ &=-\frac {(a-b) \tanh (c+d x)}{4 a b d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}-\frac {(a-b) (4 a+3 b) \tanh (c+d x)}{8 a^2 b^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {4 a^2+a b+3 b^2+(a-b) (4 a+3 b) x^2}{\left (1-x^2\right ) \left (a+(-a+b) x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 a^2 b^2 d}\\ &=-\frac {(a-b) \tanh (c+d x)}{4 a b d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}-\frac {(a-b) (4 a+3 b) \tanh (c+d x)}{8 a^2 b^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{b^3 d}-\frac {\left ((a-b) \left (8 a^2+4 a b+3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+(-a+b) x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^2 b^3 d}\\ &=\frac {x}{b^3}-\frac {\sqrt {a-b} \left (8 a^2+4 a b+3 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} b^3 d}-\frac {(a-b) \tanh (c+d x)}{4 a b d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}-\frac {(a-b) (4 a+3 b) \tanh (c+d x)}{8 a^2 b^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [A]
time = 1.33, size = 164, normalized size = 1.02 \begin {gather*} \frac {8 (c+d x)-\frac {\left (8 a^3-4 a^2 b-a b^2-3 b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{5/2} \sqrt {a-b}}+\frac {4 (a-b)^2 b \sinh (2 (c+d x))}{a (2 a-b+b \cosh (2 (c+d x)))^2}+\frac {3 b \left (-2 a^2+a b+b^2\right ) \sinh (2 (c+d x))}{a^2 (2 a-b+b \cosh (2 (c+d x)))}}{8 b^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]^6/(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(428\) vs. \(2(146)=292\).
time = 2.05, size = 429, normalized size = 2.68

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 \left (-\frac {b \left (4 a^{2}+a b -5 b^{2}\right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {\left (4 a^{3}-23 a^{2} b +7 a \,b^{2}+12 b^{3}\right ) b \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}+\frac {\left (4 a^{3}-23 a^{2} b +7 a \,b^{2}+12 b^{3}\right ) b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}-\frac {b \left (4 a^{2}+a b -5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {\left (8 a^{3}-4 a^{2} b -a \,b^{2}-3 b^{3}\right ) \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{4 a}}{b^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{3}}}{d}\)	\(429\)
default	\(\frac {\frac {\frac {2 \left (-\frac {b \left (4 a^{2}+a b -5 b^{2}\right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {\left (4 a^{3}-23 a^{2} b +7 a \,b^{2}+12 b^{3}\right ) b \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}+\frac {\left (4 a^{3}-23 a^{2} b +7 a \,b^{2}+12 b^{3}\right ) b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}-\frac {b \left (4 a^{2}+a b -5 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}+\frac {\left (8 a^{3}-4 a^{2} b -a \,b^{2}-3 b^{3}\right ) \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{4 a}}{b^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{3}}}{d}\)	\(429\)
risch	\(\frac {x}{b^{3}}+\frac {16 a^{3} b \,{\mathrm e}^{6 d x +6 c}-20 a^{2} b^{2} {\mathrm e}^{6 d x +6 c}+a \,b^{3} {\mathrm e}^{6 d x +6 c}+3 b^{4} {\mathrm e}^{6 d x +6 c}+48 a^{4} {\mathrm e}^{4 d x +4 c}-72 a^{3} b \,{\mathrm e}^{4 d x +4 c}+18 a^{2} b^{2} {\mathrm e}^{4 d x +4 c}+15 a \,b^{3} {\mathrm e}^{4 d x +4 c}-9 b^{4} {\mathrm e}^{4 d x +4 c}+32 a^{3} b \,{\mathrm e}^{2 d x +2 c}-28 a^{2} b^{2} {\mathrm e}^{2 d x +2 c}-13 a \,b^{3} {\mathrm e}^{2 d x +2 c}+9 b^{4} {\mathrm e}^{2 d x +2 c}+6 a^{2} b^{2}-3 a \,b^{3}-3 b^{4}}{4 b^{3} a^{2} d \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )^{2}}+\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {a \left (a -b \right )}+2 a -b}{b}\right )}{2 a d \,b^{3}}+\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {a \left (a -b \right )}+2 a -b}{b}\right )}{4 a^{2} d \,b^{2}}+\frac {3 \sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {a \left (a -b \right )}+2 a -b}{b}\right )}{16 a^{3} d b}-\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {a \left (a -b \right )}-2 a +b}{b}\right )}{2 a d \,b^{3}}-\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {a \left (a -b \right )}-2 a +b}{b}\right )}{4 a^{2} d \,b^{2}}-\frac {3 \sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {a \left (a -b \right )}-2 a +b}{b}\right )}{16 a^{3} d b}\)	\(588\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^6/(a+b*sinh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(2/b^3*((-1/8*b*(4*a^2+a*b-5*b^2)/a*tanh(1/2*d*x+1/2*c)^7+1/8*(4*a^3-23*a^2*b+7*a*b^2+12*b^3)/a^2*b*tanh(1
/2*d*x+1/2*c)^5+1/8*(4*a^3-23*a^2*b+7*a*b^2+12*b^3)/a^2*b*tanh(1/2*d*x+1/2*c)^3-1/8*b*(4*a^2+a*b-5*b^2)/a*tanh
(1/2*d*x+1/2*c))/(a*tanh(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)^2+1/8/a*(8*a^
3-4*a^2*b-a*b^2-3*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)*arct
anh(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/b^3*
ln(tanh(1/2*d*x+1/2*c)-1)+1/b^3*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)^3,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. 2623 vs. \(2 (148) = 296\).
time = 0.46, size = 5511, normalized size = 34.44 \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)^3,x, algorithm="fricas")

[Out]

[1/16*(16*a^2*b^2*d*x*cosh(d*x + c)^8 + 128*a^2*b^2*d*x*cosh(d*x + c)*sinh(d*x + c)^7 + 16*a^2*b^2*d*x*sinh(d*
x + c)^8 + 4*(16*a^3*b - 20*a^2*b^2 + a*b^3 + 3*b^4 + 16*(2*a^3*b - a^2*b^2)*d*x)*cosh(d*x + c)^6 + 4*(112*a^2
*b^2*d*x*cosh(d*x + c)^2 + 16*a^3*b - 20*a^2*b^2 + a*b^3 + 3*b^4 + 16*(2*a^3*b - a^2*b^2)*d*x)*sinh(d*x + c)^6
+ 16*a^2*b^2*d*x + 8*(112*a^2*b^2*d*x*cosh(d*x + c)^3 + 3*(16*a^3*b - 20*a^2*b^2 + a*b^3 + 3*b^4 + 16*(2*a^3*
b - a^2*b^2)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 + 4*(48*a^4 - 72*a^3*b + 18*a^2*b^2 + 15*a*b^3 - 9*b^4 + 8*(8
*a^4 - 8*a^3*b + 3*a^2*b^2)*d*x)*cosh(d*x + c)^4 + 4*(280*a^2*b^2*d*x*cosh(d*x + c)^4 + 48*a^4 - 72*a^3*b + 18
*a^2*b^2 + 15*a*b^3 - 9*b^4 + 8*(8*a^4 - 8*a^3*b + 3*a^2*b^2)*d*x + 15*(16*a^3*b - 20*a^2*b^2 + a*b^3 + 3*b^4
+ 16*(2*a^3*b - a^2*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 24*a^2*b^2 - 12*a*b^3 - 12*b^4 + 16*(56*a^2*b
^2*d*x*cosh(d*x + c)^5 + 5*(16*a^3*b - 20*a^2*b^2 + a*b^3 + 3*b^4 + 16*(2*a^3*b - a^2*b^2)*d*x)*cosh(d*x + c)^
3 + (48*a^4 - 72*a^3*b + 18*a^2*b^2 + 15*a*b^3 - 9*b^4 + 8*(8*a^4 - 8*a^3*b + 3*a^2*b^2)*d*x)*cosh(d*x + c))*s
inh(d*x + c)^3 + 4*(32*a^3*b - 28*a^2*b^2 - 13*a*b^3 + 9*b^4 + 16*(2*a^3*b - a^2*b^2)*d*x)*cosh(d*x + c)^2 + 4
*(112*a^2*b^2*d*x*cosh(d*x + c)^6 + 15*(16*a^3*b - 20*a^2*b^2 + a*b^3 + 3*b^4 + 16*(2*a^3*b - a^2*b^2)*d*x)*co
sh(d*x + c)^4 + 32*a^3*b - 28*a^2*b^2 - 13*a*b^3 + 9*b^4 + 16*(2*a^3*b - a^2*b^2)*d*x + 6*(48*a^4 - 72*a^3*b +
18*a^2*b^2 + 15*a*b^3 - 9*b^4 + 8*(8*a^4 - 8*a^3*b + 3*a^2*b^2)*d*x)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((8*a
^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c)^8 + 8*(8*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (8
*a^2*b^2 + 4*a*b^3 + 3*b^4)*sinh(d*x + c)^8 + 4*(16*a^3*b + 2*a*b^3 - 3*b^4)*cosh(d*x + c)^6 + 4*(16*a^3*b + 2
*a*b^3 - 3*b^4 + 7*(8*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(8*a^2*b^2 + 4*a*b^3
+ 3*b^4)*cosh(d*x + c)^3 + 3*(16*a^3*b + 2*a*b^3 - 3*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(64*a^4 - 32*a^3*
b + 16*a^2*b^2 - 12*a*b^3 + 9*b^4)*cosh(d*x + c)^4 + 2*(35*(8*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c)^4 + 64*
a^4 - 32*a^3*b + 16*a^2*b^2 - 12*a*b^3 + 9*b^4 + 30*(16*a^3*b + 2*a*b^3 - 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c
)^4 + 8*a^2*b^2 + 4*a*b^3 + 3*b^4 + 8*(7*(8*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c)^5 + 10*(16*a^3*b + 2*a*b^
3 - 3*b^4)*cosh(d*x + c)^3 + (64*a^4 - 32*a^3*b + 16*a^2*b^2 - 12*a*b^3 + 9*b^4)*cosh(d*x + c))*sinh(d*x + c)^
3 + 4*(16*a^3*b + 2*a*b^3 - 3*b^4)*cosh(d*x + c)^2 + 4*(7*(8*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c)^6 + 15*(
16*a^3*b + 2*a*b^3 - 3*b^4)*cosh(d*x + c)^4 + 16*a^3*b + 2*a*b^3 - 3*b^4 + 3*(64*a^4 - 32*a^3*b + 16*a^2*b^2 -
12*a*b^3 + 9*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((8*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c)^7 + 3*(16
*a^3*b + 2*a*b^3 - 3*b^4)*cosh(d*x + c)^5 + (64*a^4 - 32*a^3*b + 16*a^2*b^2 - 12*a*b^3 + 9*b^4)*cosh(d*x + c)^
3 + (16*a^3*b + 2*a*b^3 - 3*b^4)*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*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*cosh(d*x + c)^3 + (2*a - b)*cosh(
d*x + c))*sinh(d*x + c) + b)) + 8*(16*a^2*b^2*d*x*cosh(d*x + c)^7 + 3*(16*a^3*b - 20*a^2*b^2 + a*b^3 + 3*b^4 +
16*(2*a^3*b - a^2*b^2)*d*x)*cosh(d*x + c)^5 + 2*(48*a^4 - 72*a^3*b + 18*a^2*b^2 + 15*a*b^3 - 9*b^4 + 8*(8*a^4
- 8*a^3*b + 3*a^2*b^2)*d*x)*cosh(d*x + c)^3 + (32*a^3*b - 28*a^2*b^2 - 13*a*b^3 + 9*b^4 + 16*(2*a^3*b - a^2*b
^2)*d*x)*cosh(d*x + c))*sinh(d*x + c))/(a^2*b^5*d*cosh(d*x + c)^8 + 8*a^2*b^5*d*cosh(d*x + c)*sinh(d*x + c)^7
+ a^2*b^5*d*sinh(d*x + c)^8 + a^2*b^5*d + 4*(2*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^6 + 4*(7*a^2*b^5*d*cosh(d*x
+ c)^2 + (2*a^3*b^4 - a^2*b^5)*d)*sinh(d*x + c)^6 + 2*(8*a^4*b^3 - 8*a^3*b^4 + 3*a^2*b^5)*d*cosh(d*x + c)^4 +
8*(7*a^2*b^5*d*cosh(d*x + c)^3 + 3*(2*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*a^2*b^5*d*co
sh(d*x + c)^4 + 30*(2*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^2 + (8*a^4*b^3 - 8*a^3*b^4 + 3*a^2*b^5)*d)*sinh(d*x +
c)^4 + 4*(2*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^2 + 8*(7*a^2*b^5*d*cosh(d*x + c)^5 + 10*(2*a^3*b^4 - a^2*b^5)*
d*cosh(d*x + c)^3 + (8*a^4*b^3 - 8*a^3*b^4 + 3*a^2*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*a^2*b^5*d*cosh
(d*x + c)^6 + 15*(2*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c)^4 + 3*(8*a^4*b^3 - 8*a^3*b^4 + 3*a^2*b^5)*d*cosh(d*x +
c)^2 + (2*a^3*b^4 - a^2*b^5)*d)*sinh(d*x + c)^2 + 8*(a^2*b^5*d*cosh(d*x + c)^7 + 3*(2*a^3*b^4 - a^2*b^5)*d*cos
h(d*x + c)^5 + (8*a^4*b^3 - 8*a^3*b^4 + 3*a^2*b^5)*d*cosh(d*x + c)^3 + (2*a^3*b^4 - a^2*b^5)*d*cosh(d*x + c))*
sinh(d*x + c)), 1/8*(8*a^2*b^2*d*x*cosh(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(cosh(d*x+c)**6/(a+b*sinh(d*x+c)**2)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 353 vs. \(2 (148) = 296\).
time = 4.08, size = 353, normalized size = 2.21 \begin {gather*} \frac {\frac {8 \, {\left (d x + c\right )}}{b^{3}} - \frac {{\left (8 \, a^{3} - 4 \, a^{2} b - a b^{2} - 3 \, 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^{2} b^{3}} + \frac {2 \, {\left (16 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} - 20 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + a b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 3 \, b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 48 \, a^{4} e^{\left (4 \, d x + 4 \, c\right )} - 72 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + 18 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 15 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 9 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 32 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} - 28 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 13 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 6 \, a^{2} b^{2} - 3 \, a b^{3} - 3 \, b^{4}\right )}}{{\left (b e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + b\right )}^{2} a^{2} b^{3}}}{8 \, 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)^3,x, algorithm="giac")

[Out]

1/8*(8*(d*x + c)/b^3 - (8*a^3 - 4*a^2*b - a*b^2 - 3*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^2*b^3) + 2*(16*a^3*b*e^(6*d*x + 6*c) - 20*a^2*b^2*e^(6*d*x + 6*c) + a*b^3*e^(6*d*x +
6*c) + 3*b^4*e^(6*d*x + 6*c) + 48*a^4*e^(4*d*x + 4*c) - 72*a^3*b*e^(4*d*x + 4*c) + 18*a^2*b^2*e^(4*d*x + 4*c)
+ 15*a*b^3*e^(4*d*x + 4*c) - 9*b^4*e^(4*d*x + 4*c) + 32*a^3*b*e^(2*d*x + 2*c) - 28*a^2*b^2*e^(2*d*x + 2*c) -
13*a*b^3*e^(2*d*x + 2*c) + 9*b^4*e^(2*d*x + 2*c) + 6*a^2*b^2 - 3*a*b^3 - 3*b^4)/((b*e^(4*d*x + 4*c) + 4*a*e^(2
*d*x + 2*c) - 2*b*e^(2*d*x + 2*c) + b)^2*a^2*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 )}^3} \,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)^3,x)

[Out]

int(cosh(c + d*x)^6/(a + b*sinh(c + d*x)^2)^3, x)

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