Integrand size = 10, antiderivative size = 175 \[ \int \frac {1}{a+b \sinh ^6(x)} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {\sqrt [3]{a}-\sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{b}}}+\frac {\text {arctanh}\left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {\text {arctanh}\left (\frac {\sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}} \] Output:
1/3*arctanh((a^(1/3)-b^(1/3))^(1/2)*tanh(x)/a^(1/6))/a^(5/6)/(a^(1/3)-b^(1 /3))^(1/2)+1/3*arctanh((a^(1/3)+(-1)^(1/3)*b^(1/3))^(1/2)*tanh(x)/a^(1/6)) /a^(5/6)/(a^(1/3)+(-1)^(1/3)*b^(1/3))^(1/2)+1/3*arctanh((a^(1/3)-(-1)^(2/3 )*b^(1/3))^(1/2)*tanh(x)/a^(1/6))/a^(5/6)/(a^(1/3)-(-1)^(2/3)*b^(1/3))^(1/ 2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 5.04 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.77 \[ \int \frac {1}{a+b \sinh ^6(x)} \, dx=\frac {16}{3} \text {RootSum}\left [b-6 b \text {$\#$1}+15 b \text {$\#$1}^2+64 a \text {$\#$1}^3-20 b \text {$\#$1}^3+15 b \text {$\#$1}^4-6 b \text {$\#$1}^5+b \text {$\#$1}^6\&,\frac {x \text {$\#$1}^2+\log (-\cosh (x)-\sinh (x)+\cosh (x) \text {$\#$1}-\sinh (x) \text {$\#$1}) \text {$\#$1}^2}{-b+5 b \text {$\#$1}+32 a \text {$\#$1}^2-10 b \text {$\#$1}^2+10 b \text {$\#$1}^3-5 b \text {$\#$1}^4+b \text {$\#$1}^5}\&\right ] \] Input:
Integrate[(a + b*Sinh[x]^6)^(-1),x]
Output:
(16*RootSum[b - 6*b*#1 + 15*b*#1^2 + 64*a*#1^3 - 20*b*#1^3 + 15*b*#1^4 - 6 *b*#1^5 + b*#1^6 & , (x*#1^2 + Log[-Cosh[x] - Sinh[x] + Cosh[x]*#1 - Sinh[ x]*#1]*#1^2)/(-b + 5*b*#1 + 32*a*#1^2 - 10*b*#1^2 + 10*b*#1^3 - 5*b*#1^4 + b*#1^5) & ])/3
Time = 0.56 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3690, 3042, 3660, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{a+b \sinh ^6(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{a-b \sin (i x)^6}dx\) |
\(\Big \downarrow \) 3690 |
\(\displaystyle \frac {\int \frac {1}{\frac {\sqrt [3]{b} \sinh ^2(x)}{\sqrt [3]{a}}+1}dx}{3 a}+\frac {\int \frac {1}{1-\frac {\sqrt [3]{-1} \sqrt [3]{b} \sinh ^2(x)}{\sqrt [3]{a}}}dx}{3 a}+\frac {\int \frac {1}{\frac {(-1)^{2/3} \sqrt [3]{b} \sinh ^2(x)}{\sqrt [3]{a}}+1}dx}{3 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {1}{1-\frac {\sqrt [3]{b} \sin (i x)^2}{\sqrt [3]{a}}}dx}{3 a}+\frac {\int \frac {1}{\frac {\sqrt [3]{-1} \sqrt [3]{b} \sin (i x)^2}{\sqrt [3]{a}}+1}dx}{3 a}+\frac {\int \frac {1}{1-\frac {(-1)^{2/3} \sqrt [3]{b} \sin (i x)^2}{\sqrt [3]{a}}}dx}{3 a}\) |
\(\Big \downarrow \) 3660 |
\(\displaystyle \frac {\int \frac {1}{1-\left (1-\frac {\sqrt [3]{b}}{\sqrt [3]{a}}\right ) \tanh ^2(x)}d\tanh (x)}{3 a}+\frac {\int \frac {1}{1-\left (\frac {\sqrt [3]{-1} \sqrt [3]{b}}{\sqrt [3]{a}}+1\right ) \tanh ^2(x)}d\tanh (x)}{3 a}+\frac {\int \frac {1}{1-\left (1-\frac {(-1)^{2/3} \sqrt [3]{b}}{\sqrt [3]{a}}\right ) \tanh ^2(x)}d\tanh (x)}{3 a}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\text {arctanh}\left (\frac {\sqrt {\sqrt [3]{a}-\sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-\sqrt [3]{b}}}+\frac {\text {arctanh}\left (\frac {\sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b}}}+\frac {\text {arctanh}\left (\frac {\sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}} \tanh (x)}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b}}}\) |
Input:
Int[(a + b*Sinh[x]^6)^(-1),x]
Output:
ArcTanh[(Sqrt[a^(1/3) - b^(1/3)]*Tanh[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) - b^(1/3)]) + ArcTanh[(Sqrt[a^(1/3) + (-1)^(1/3)*b^(1/3)]*Tanh[x])/a^(1/6 )]/(3*a^(5/6)*Sqrt[a^(1/3) + (-1)^(1/3)*b^(1/3)]) + ArcTanh[(Sqrt[a^(1/3) - (-1)^(2/3)*b^(1/3)]*Tanh[x])/a^(1/6)]/(3*a^(5/6)*Sqrt[a^(1/3) - (-1)^(2/ 3)*b^(1/3)])
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[1/(a + (a + b)*ff^2*x^ 2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(-1), x_Symbol] :> Module[{ k}, Simp[2/(a*n) Sum[Int[1/(1 - Sin[e + f*x]^2/((-1)^(4*(k/n))*Rt[-a/b, n /2])), x], {k, 1, n/2}], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[n/2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.17 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{12}-6 a \,\textit {\_Z}^{10}+15 a \,\textit {\_Z}^{8}+\left (-20 a +64 b \right ) \textit {\_Z}^{6}+15 a \,\textit {\_Z}^{4}-6 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (-\textit {\_R}^{10}+5 \textit {\_R}^{8}-10 \textit {\_R}^{6}+10 \textit {\_R}^{4}-5 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{11} a -5 \textit {\_R}^{9} a +10 \textit {\_R}^{7} a -10 \textit {\_R}^{5} a +32 \textit {\_R}^{5} b +5 \textit {\_R}^{3} a -\textit {\_R} a}\right )}{6}\) | \(128\) |
risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (-1+\left (46656 a^{6}-46656 a^{5} b \right ) \textit {\_Z}^{6}-3888 a^{4} \textit {\_Z}^{4}+108 a^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 x}+\left (-\frac {15552 a^{6}}{b}+15552 a^{5}\right ) \textit {\_R}^{5}+\left (\frac {2592 a^{5}}{b}-2592 a^{4}\right ) \textit {\_R}^{4}+\left (\frac {864 a^{4}}{b}+432 a^{3}\right ) \textit {\_R}^{3}+\left (-\frac {144 a^{3}}{b}-72 a^{2}\right ) \textit {\_R}^{2}+\left (-\frac {12 a^{2}}{b}+12 a \right ) \textit {\_R} +\frac {2 a}{b}-1\right )\) | \(140\) |
Input:
int(1/(a+b*sinh(x)^6),x,method=_RETURNVERBOSE)
Output:
1/6*sum((-_R^10+5*_R^8-10*_R^6+10*_R^4-5*_R^2+1)/(_R^11*a-5*_R^9*a+10*_R^7 *a-10*_R^5*a+32*_R^5*b+5*_R^3*a-_R*a)*ln(tanh(1/2*x)-_R),_R=RootOf(a*_Z^12 -6*a*_Z^10+15*a*_Z^8+(-20*a+64*b)*_Z^6+15*a*_Z^4-6*a*_Z^2+a))
Result contains complex when optimal does not.
Time = 1.23 (sec) , antiderivative size = 16401, normalized size of antiderivative = 93.72 \[ \int \frac {1}{a+b \sinh ^6(x)} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+b*sinh(x)^6),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {1}{a+b \sinh ^6(x)} \, dx=\int \frac {1}{a + b \sinh ^{6}{\left (x \right )}}\, dx \] Input:
integrate(1/(a+b*sinh(x)**6),x)
Output:
Integral(1/(a + b*sinh(x)**6), x)
\[ \int \frac {1}{a+b \sinh ^6(x)} \, dx=\int { \frac {1}{b \sinh \left (x\right )^{6} + a} \,d x } \] Input:
integrate(1/(a+b*sinh(x)^6),x, algorithm="maxima")
Output:
integrate(1/(b*sinh(x)^6 + a), x)
\[ \int \frac {1}{a+b \sinh ^6(x)} \, dx=\int { \frac {1}{b \sinh \left (x\right )^{6} + a} \,d x } \] Input:
integrate(1/(a+b*sinh(x)^6),x, algorithm="giac")
Output:
sage0*x
Time = 67.05 (sec) , antiderivative size = 857, normalized size of antiderivative = 4.90 \[ \int \frac {1}{a+b \sinh ^6(x)} \, dx =\text {Too large to display} \] Input:
int(1/(a + b*sinh(x)^6),x)
Output:
symsum(log(root(46656*a^5*b*d^6 - 46656*a^6*d^6 + 3888*a^4*d^4 - 108*a^2*d ^2 + 1, d, k)*(root(46656*a^5*b*d^6 - 46656*a^6*d^6 + 3888*a^4*d^4 - 108*a ^2*d^2 + 1, d, k)*(root(46656*a^5*b*d^6 - 46656*a^6*d^6 + 3888*a^4*d^4 - 1 08*a^2*d^2 + 1, d, k)*(root(46656*a^5*b*d^6 - 46656*a^6*d^6 + 3888*a^4*d^4 - 108*a^2*d^2 + 1, d, k)*((1459166279268040704*(327680*a^7*exp(2*x) + 298 496*a^6*b - 65536*a^7 + 158*a^2*b^5 - 91315*a^3*b^4 + 348176*a^4*b^3 - 489 952*a^5*b^2 - 196*a^2*b^5*exp(2*x) + 274019*a^3*b^4*exp(2*x) - 1132876*a^4 *b^3*exp(2*x) + 1770440*a^5*b^2*exp(2*x) - 1239040*a^6*b*exp(2*x)))/(b^10* (a - b)^3) + (17509995351216488448*root(46656*a^5*b*d^6 - 46656*a^6*d^6 + 3888*a^4*d^4 - 108*a^2*d^2 + 1, d, k)*(262144*a^7*exp(2*x) + 203520*a^6*b - 65536*a^7 - 453*a^3*b^4 + 72022*a^4*b^3 - 209472*a^5*b^2 + 630*a^3*b^4*e xp(2*x) - 254512*a^4*b^3*exp(2*x) + 767508*a^5*b^2*exp(2*x) - 775680*a^6*b *exp(2*x)))/(b^10*(a - b)^2)) - (486388759756013568*(655360*a^5*exp(2*x) + 9*a*b^4 + 370176*a^4*b - 196608*a^5 - 24408*a^2*b^3 - 149088*a^3*b^2 + 63 676*a^2*b^3*exp(2*x) + 526248*a^3*b^2*exp(2*x) - 10*a*b^4*exp(2*x) - 12451 84*a^4*b*exp(2*x)))/(b^10*(a - b)^2)) - (40532396646334464*(655360*a^5*exp (2*x) + b^5*exp(2*x) + 24677*a*b^4 + 773120*a^4*b - 262144*a^5 - b^5 + 198 071*a^2*b^3 - 733696*a^3*b^2 - 477713*a^2*b^3*exp(2*x) + 1770640*a^3*b^2*e xp(2*x) - 53861*a*b^4*exp(2*x) - 1894400*a^4*b*exp(2*x)))/(b^10*(a - b)^3) ) + (13510798882111488*(655360*a^3*exp(2*x) - 11382*b^3*exp(2*x) - 1444...
\[ \int \frac {1}{a+b \sinh ^6(x)} \, dx=\int \frac {1}{\sinh \left (x \right )^{6} b +a}d x \] Input:
int(1/(a+b*sinh(x)^6),x)
Output:
int(1/(sinh(x)**6*b + a),x)