3.2.76 \(\int \frac {\sinh (c+d x)}{a+b \sinh ^3(c+d x)} \, dx\) [176]

3.2.76.1 Optimal result
3.2.76.2 Mathematica [C] (verified)
3.2.76.3 Rubi [A] (verified)
3.2.76.4 Maple [C] (verified)
3.2.76.5 Fricas [B] (verification not implemented)
3.2.76.6 Sympy [F]
3.2.76.7 Maxima [F]
3.2.76.8 Giac [F]
3.2.76.9 Mupad [B] (verification not implemented)

3.2.76.1 Optimal result

Integrand size = 21, antiderivative size = 290 \[ \int \frac {\sinh (c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\frac {2 \arctan \left (\frac {\sqrt [6]{-1} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 \sqrt [3]{a} \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}} \sqrt [3]{b} d}-\frac {2 \sqrt [3]{-1} \arctan \left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 \sqrt [3]{a} \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}} \sqrt [3]{b} d}+\frac {2 \text {arctanh}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 \sqrt [3]{a} \sqrt {a^{2/3}+b^{2/3}} \sqrt [3]{b} d} \]

output
-2/3*(-1)^(1/3)*arctan((-1)^(1/6)*((-1)^(5/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*d 
*x+1/2*c))/((-1)^(1/3)*a^(2/3)-b^(2/3))^(1/2))/a^(1/3)/b^(1/3)/d/((-1)^(1/ 
3)*a^(2/3)-b^(2/3))^(1/2)+2/3*arctanh((b^(1/3)-a^(1/3)*tanh(1/2*d*x+1/2*c) 
)/(a^(2/3)+b^(2/3))^(1/2))/a^(1/3)/b^(1/3)/d/(a^(2/3)+b^(2/3))^(1/2)+2/3*a 
rctan((-1)^(1/6)*((-1)^(1/6)*b^(1/3)+I*a^(1/3)*tanh(1/2*d*x+1/2*c))/((-1)^ 
(1/3)*a^(2/3)-(-1)^(2/3)*b^(2/3))^(1/2))/a^(1/3)/b^(1/3)/d/((-1)^(1/3)*a^( 
2/3)-(-1)^(2/3)*b^(2/3))^(1/2)
 
3.2.76.2 Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 11.06 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.69 \[ \int \frac {\sinh (c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\frac {\text {RootSum}\left [-b+3 b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 b \text {$\#$1}^4+b \text {$\#$1}^6\&,\frac {-c-d x-2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+c \text {$\#$1}^2+d x \text {$\#$1}^2+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2}{b+4 a \text {$\#$1}-2 b \text {$\#$1}^2+b \text {$\#$1}^4}\&\right ]}{3 d} \]

input
Integrate[Sinh[c + d*x]/(a + b*Sinh[c + d*x]^3),x]
 
output
RootSum[-b + 3*b*#1^2 + 8*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (-c - d*x - 2*Log 
[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + 
 d*x)/2]*#1] + c*#1^2 + d*x*#1^2 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d* 
x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2)/(b + 4*a*#1 - 2 
*b*#1^2 + b*#1^4) & ]/(3*d)
 
3.2.76.3 Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 26, 3699, 2009}

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.

\(\displaystyle \int \frac {\sinh (c+d x)}{a+b \sinh ^3(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {i \sin (i c+i d x)}{a+i b \sin (i c+i d x)^3}dx\)

\(\Big \downarrow \) 26

\(\displaystyle -i \int \frac {\sin (i c+i d x)}{i b \sin (i c+i d x)^3+a}dx\)

\(\Big \downarrow \) 3699

\(\displaystyle -i \int \left (\frac {\sqrt [3]{-1}}{3 \sqrt [3]{a} \left (\sqrt [6]{-1} \sqrt [3]{a}-i \sqrt [3]{b} \sinh (c+d x)\right ) \sqrt [3]{b}}-\frac {(-1)^{2/3}}{3 \sqrt [3]{a} \left (\sqrt [6]{-1} \sqrt [3]{b} \sinh (c+d x)+\sqrt [6]{-1} \sqrt [3]{a}\right ) \sqrt [3]{b}}-\frac {1}{3 \sqrt [3]{a} \left ((-1)^{5/6} \sqrt [3]{b} \sinh (c+d x)+\sqrt [6]{-1} \sqrt [3]{a}\right ) \sqrt [3]{b}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -i \left (\frac {2 i \arctan \left (\frac {\sqrt [6]{-1} \left (\sqrt [6]{-1} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 \sqrt [3]{a} \sqrt [3]{b} d \sqrt {\sqrt [3]{-1} a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac {2 (-1)^{5/6} \arctan \left (\frac {\sqrt [6]{-1} \left ((-1)^{5/6} \sqrt [3]{b}+i \sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}\right )}{3 \sqrt [3]{a} \sqrt [3]{b} d \sqrt {\sqrt [3]{-1} a^{2/3}-b^{2/3}}}+\frac {2 i \text {arctanh}\left (\frac {\sqrt [3]{b}-\sqrt [3]{a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+b^{2/3}}}\right )}{3 \sqrt [3]{a} \sqrt [3]{b} d \sqrt {a^{2/3}+b^{2/3}}}\right )\)

input
Int[Sinh[c + d*x]/(a + b*Sinh[c + d*x]^3),x]
 
output
(-I)*((((2*I)/3)*ArcTan[((-1)^(1/6)*((-1)^(1/6)*b^(1/3) + I*a^(1/3)*Tanh[( 
c + d*x)/2]))/Sqrt[(-1)^(1/3)*a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(a^(1/3)*Sqr 
t[(-1)^(1/3)*a^(2/3) - (-1)^(2/3)*b^(2/3)]*b^(1/3)*d) - (2*(-1)^(5/6)*ArcT 
an[((-1)^(1/6)*((-1)^(5/6)*b^(1/3) + I*a^(1/3)*Tanh[(c + d*x)/2]))/Sqrt[(- 
1)^(1/3)*a^(2/3) - b^(2/3)]])/(3*a^(1/3)*Sqrt[(-1)^(1/3)*a^(2/3) - b^(2/3) 
]*b^(1/3)*d) + (((2*I)/3)*ArcTanh[(b^(1/3) - a^(1/3)*Tanh[(c + d*x)/2])/Sq 
rt[a^(2/3) + b^(2/3)]])/(a^(1/3)*Sqrt[a^(2/3) + b^(2/3)]*b^(1/3)*d))
 

3.2.76.3.1 Defintions of rubi rules used

rule 26
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2009
Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042
Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3699
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_ 
))^(p_.), x_Symbol] :> Int[ExpandTrig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n) 
^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4] || Gt 
Q[p, 0] || (EqQ[p, -1] && IntegerQ[n]))
 
3.2.76.4 Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.39 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.28

method result size
derivativedivides \(\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (\textit {\_R}^{3}-\textit {\_R} \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 d}\) \(82\)
default \(\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}-3 a \,\textit {\_Z}^{4}-8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}-a \right )}{\sum }\frac {\left (\textit {\_R}^{3}-\textit {\_R} \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a -2 \textit {\_R}^{3} a -4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 d}\) \(82\)
risch \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (-1+\left (729 a^{4} b^{2} d^{6}+729 a^{2} b^{4} d^{6}\right ) \textit {\_Z}^{6}+243 a^{2} b^{2} d^{4} \textit {\_Z}^{4}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{d x +c}+\left (\frac {243 d^{5} b^{2} a^{5}}{a^{2}-b^{2}}+\frac {243 d^{5} b^{4} a^{3}}{a^{2}-b^{2}}\right ) \textit {\_R}^{5}+\left (\frac {81 d^{4} b \,a^{5}}{a^{2}-b^{2}}+\frac {81 d^{4} b^{3} a^{3}}{a^{2}-b^{2}}\right ) \textit {\_R}^{4}+\left (\frac {54 d^{3} b^{2} a^{3}}{a^{2}-b^{2}}-\frac {27 d^{3} b^{4} a}{a^{2}-b^{2}}\right ) \textit {\_R}^{3}+\left (\frac {18 d^{2} b \,a^{3}}{a^{2}-b^{2}}-\frac {9 d^{2} b^{3} a}{a^{2}-b^{2}}\right ) \textit {\_R}^{2}+\left (\frac {3 d \,a^{3}}{a^{2}-b^{2}}-\frac {6 d \,b^{2} a}{a^{2}-b^{2}}\right ) \textit {\_R} -\frac {a b}{a^{2}-b^{2}}\right )\) \(299\)

input
int(sinh(d*x+c)/(a+b*sinh(d*x+c)^3),x,method=_RETURNVERBOSE)
 
output
2/3/d*sum((_R^3-_R)/(_R^5*a-2*_R^3*a-4*_R^2*b+_R*a)*ln(tanh(1/2*d*x+1/2*c) 
-_R),_R=RootOf(_Z^6*a-3*_Z^4*a-8*_Z^3*b+3*_Z^2*a-a))
 
3.2.76.5 Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 18312 vs. \(2 (197) = 394\).

Time = 1.02 (sec) , antiderivative size = 18312, normalized size of antiderivative = 63.14 \[ \int \frac {\sinh (c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\text {Too large to display} \]

input
integrate(sinh(d*x+c)/(a+b*sinh(d*x+c)^3),x, algorithm="fricas")
 
output
Too large to include
 
3.2.76.6 Sympy [F]

\[ \int \frac {\sinh (c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\int \frac {\sinh {\left (c + d x \right )}}{a + b \sinh ^{3}{\left (c + d x \right )}}\, dx \]

input
integrate(sinh(d*x+c)/(a+b*sinh(d*x+c)**3),x)
 
output
Integral(sinh(c + d*x)/(a + b*sinh(c + d*x)**3), x)
 
3.2.76.7 Maxima [F]

\[ \int \frac {\sinh (c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\int { \frac {\sinh \left (d x + c\right )}{b \sinh \left (d x + c\right )^{3} + a} \,d x } \]

input
integrate(sinh(d*x+c)/(a+b*sinh(d*x+c)^3),x, algorithm="maxima")
 
output
integrate(sinh(d*x + c)/(b*sinh(d*x + c)^3 + a), x)
 
3.2.76.8 Giac [F]

\[ \int \frac {\sinh (c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\int { \frac {\sinh \left (d x + c\right )}{b \sinh \left (d x + c\right )^{3} + a} \,d x } \]

input
integrate(sinh(d*x+c)/(a+b*sinh(d*x+c)^3),x, algorithm="giac")
 
output
integrate(sinh(d*x + c)/(b*sinh(d*x + c)^3 + a), x)
 
3.2.76.9 Mupad [B] (verification not implemented)

Time = 23.24 (sec) , antiderivative size = 857, normalized size of antiderivative = 2.96 \[ \int \frac {\sinh (c+d x)}{a+b \sinh ^3(c+d x)} \, dx=\sum _{k=1}^6\ln \left (\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )\,\left (\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )\,\left (\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )\,\left (\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )\,\left (\frac {\left (4\,a^4\,b\,d^4+a^5\,d^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\,16+a^3\,b^2\,d^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\,11\right )\,663552}{b^6}-\frac {\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )\,\left (-a^4\,b\,d^5+a^5\,d^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\,4+a^3\,b^2\,d^5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\,5\right )\,1990656}{b^5}\right )+\frac {\left (8\,a^4\,d^3+a^2\,b^2\,d^3-a^3\,b\,d^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\,25\right )\,221184}{b^6}\right )-\frac {a^2\,d^2\,\left (b-a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\,7\right )\,294912}{b^6}\right )+\frac {a^2\,d\,\left (b-a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\,2\right )\,196608}{b^7}\right )-\frac {a\,\left (8\,a-b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^{\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right )}\right )\,8192}{b^7}\right )\,\mathrm {root}\left (729\,a^4\,b^2\,d^6\,z^6+729\,a^2\,b^4\,d^6\,z^6+243\,a^2\,b^2\,d^4\,z^4-1,z,k\right ) \]

input
int(sinh(c + d*x)/(a + b*sinh(c + d*x)^3),x)
 
output
symsum(log(root(729*a^4*b^2*d^6*z^6 + 729*a^2*b^4*d^6*z^6 + 243*a^2*b^2*d^ 
4*z^4 - 1, z, k)*(root(729*a^4*b^2*d^6*z^6 + 729*a^2*b^4*d^6*z^6 + 243*a^2 
*b^2*d^4*z^4 - 1, z, k)*(root(729*a^4*b^2*d^6*z^6 + 729*a^2*b^4*d^6*z^6 + 
243*a^2*b^2*d^4*z^4 - 1, z, k)*(root(729*a^4*b^2*d^6*z^6 + 729*a^2*b^4*d^6 
*z^6 + 243*a^2*b^2*d^4*z^4 - 1, z, k)*((663552*(4*a^4*b*d^4 + 16*a^5*d^4*e 
xp(d*x)*exp(root(729*a^4*b^2*d^6*z^6 + 729*a^2*b^4*d^6*z^6 + 243*a^2*b^2*d 
^4*z^4 - 1, z, k)) + 11*a^3*b^2*d^4*exp(d*x)*exp(root(729*a^4*b^2*d^6*z^6 
+ 729*a^2*b^4*d^6*z^6 + 243*a^2*b^2*d^4*z^4 - 1, z, k))))/b^6 - (1990656*r 
oot(729*a^4*b^2*d^6*z^6 + 729*a^2*b^4*d^6*z^6 + 243*a^2*b^2*d^4*z^4 - 1, z 
, k)*(4*a^5*d^5*exp(d*x)*exp(root(729*a^4*b^2*d^6*z^6 + 729*a^2*b^4*d^6*z^ 
6 + 243*a^2*b^2*d^4*z^4 - 1, z, k)) - a^4*b*d^5 + 5*a^3*b^2*d^5*exp(d*x)*e 
xp(root(729*a^4*b^2*d^6*z^6 + 729*a^2*b^4*d^6*z^6 + 243*a^2*b^2*d^4*z^4 - 
1, z, k))))/b^5) + (221184*(8*a^4*d^3 + a^2*b^2*d^3 - 25*a^3*b*d^3*exp(d*x 
)*exp(root(729*a^4*b^2*d^6*z^6 + 729*a^2*b^4*d^6*z^6 + 243*a^2*b^2*d^4*z^4 
 - 1, z, k))))/b^6) - (294912*a^2*d^2*(b - 7*a*exp(d*x)*exp(root(729*a^4*b 
^2*d^6*z^6 + 729*a^2*b^4*d^6*z^6 + 243*a^2*b^2*d^4*z^4 - 1, z, k))))/b^6) 
+ (196608*a^2*d*(b - 2*a*exp(d*x)*exp(root(729*a^4*b^2*d^6*z^6 + 729*a^2*b 
^4*d^6*z^6 + 243*a^2*b^2*d^4*z^4 - 1, z, k))))/b^7) - (8192*a*(8*a - b*exp 
(d*x)*exp(root(729*a^4*b^2*d^6*z^6 + 729*a^2*b^4*d^6*z^6 + 243*a^2*b^2*d^4 
*z^4 - 1, z, k))))/b^7)*root(729*a^4*b^2*d^6*z^6 + 729*a^2*b^4*d^6*z^6 ...