Integrand size = 7, antiderivative size = 87 \[ \int \sinh (x) \tanh (6 x) \, dx=-\frac {\arctan \left (\sqrt {2} \sinh (x)\right )}{3 \sqrt {2}}-\frac {\arctan \left (2 \sqrt {2-\sqrt {3}} \sinh (x)\right )}{6 \sqrt {2-\sqrt {3}}}-\frac {\arctan \left (2 \sqrt {2+\sqrt {3}} \sinh (x)\right )}{6 \sqrt {2+\sqrt {3}}}+\sinh (x) \] Output:
-1/6*arctan(sinh(x)*2^(1/2))*2^(1/2)-1/6*arctan(2*(1/2*6^(1/2)-1/2*2^(1/2) )*sinh(x))/(1/2*6^(1/2)-1/2*2^(1/2))-1/6*arctan(2*(1/2*6^(1/2)+1/2*2^(1/2) )*sinh(x))/(1/2*6^(1/2)+1/2*2^(1/2))+sinh(x)
Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int \sinh (x) \tanh (6 x) \, dx=-\frac {\arctan \left (\sqrt {2} \sinh (x)\right )}{3 \sqrt {2}}-\frac {1}{6} \sqrt {2-\sqrt {3}} \arctan \left (\frac {2 \sinh (x)}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{6} \sqrt {2+\sqrt {3}} \arctan \left (\frac {2 \sinh (x)}{\sqrt {2+\sqrt {3}}}\right )+\sinh (x) \] Input:
Integrate[Sinh[x]*Tanh[6*x],x]
Output:
-1/3*ArcTan[Sqrt[2]*Sinh[x]]/Sqrt[2] - (Sqrt[2 - Sqrt[3]]*ArcTan[(2*Sinh[x ])/Sqrt[2 - Sqrt[3]]])/6 - (Sqrt[2 + Sqrt[3]]*ArcTan[(2*Sinh[x])/Sqrt[2 + Sqrt[3]]])/6 + Sinh[x]
Time = 0.40 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {3042, 25, 4878, 27, 2460, 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.
\(\displaystyle \int \sinh (x) \tanh (6 x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\sin (i x) \tan (6 i x)dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \sin (i x) \tan (6 i x)dx\) |
\(\Big \downarrow \) 4878 |
\(\displaystyle -\int -\frac {2 \sinh ^2(x) \left (16 \sinh ^4(x)+16 \sinh ^2(x)+3\right )}{32 \sinh ^6(x)+48 \sinh ^4(x)+18 \sinh ^2(x)+1}d\sinh (x)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {\sinh ^2(x) \left (16 \sinh ^4(x)+16 \sinh ^2(x)+3\right )}{32 \sinh ^6(x)+48 \sinh ^4(x)+18 \sinh ^2(x)+1}d\sinh (x)\) |
\(\Big \downarrow \) 2460 |
\(\displaystyle 2 \int \left (\frac {-8 \sinh ^2(x)-1}{3 \left (16 \sinh ^4(x)+16 \sinh ^2(x)+1\right )}-\frac {1}{6 \left (2 \sinh ^2(x)+1\right )}+\frac {1}{2}\right )d\sinh (x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {\arctan \left (\sqrt {2} \sinh (x)\right )}{6 \sqrt {2}}-\frac {1}{12} \sqrt {2-\sqrt {3}} \arctan \left (\frac {2 \sinh (x)}{\sqrt {2-\sqrt {3}}}\right )-\frac {1}{12} \sqrt {2+\sqrt {3}} \arctan \left (\frac {2 \sinh (x)}{\sqrt {2+\sqrt {3}}}\right )+\frac {\sinh (x)}{2}\right )\) |
Input:
Int[Sinh[x]*Tanh[6*x],x]
Output:
2*(-1/6*ArcTan[Sqrt[2]*Sinh[x]]/Sqrt[2] - (Sqrt[2 - Sqrt[3]]*ArcTan[(2*Sin h[x])/Sqrt[2 - Sqrt[3]]])/12 - (Sqrt[2 + Sqrt[3]]*ArcTan[(2*Sinh[x])/Sqrt[ 2 + Sqrt[3]]])/12 + Sinh[x]/2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px /. x -> Sqrt[x]]}, Int[ExpandIntegrand[u*(Qx /. x -> x^2)^p, x], x] /; !SumQ[NonfreeFactors[Q x, x]]] /; PolyQ[Px, x^2] && GtQ[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0] && RationalFunctionQ[u, x]
Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, Simp[With[{d = FreeFa ctors[Sin[v], x]}, d/Coefficient[v, x, 1] Subst[Int[SubstFor[1, Sin[v]/d, u/Cos[v], x], x], x, Sin[v]/d]], x] /; !FalseQ[v] && FunctionOfQ[NonfreeF actors[Sin[v], x], u/Cos[v], x]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.97
method | result | size |
risch | \(\frac {{\mathrm e}^{x}}{2}-\frac {{\mathrm e}^{-x}}{2}+\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-i \sqrt {2}\, {\mathrm e}^{x}-1\right )}{12}-\frac {i \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+i \sqrt {2}\, {\mathrm e}^{x}-1\right )}{12}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (20736 \textit {\_Z}^{4}+576 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (-12 \textit {\_R} \,{\mathrm e}^{x}+{\mathrm e}^{2 x}-1\right )\right )\) | \(84\) |
Input:
int(sinh(x)*tanh(6*x),x,method=_RETURNVERBOSE)
Output:
1/2*exp(x)-1/2*exp(-x)+1/12*I*2^(1/2)*ln(exp(2*x)-I*2^(1/2)*exp(x)-1)-1/12 *I*2^(1/2)*ln(exp(2*x)+I*2^(1/2)*exp(x)-1)+sum(_R*ln(-12*_R*exp(x)+exp(2*x )-1),_R=RootOf(20736*_Z^4+576*_Z^2+1))
Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (63) = 126\).
Time = 0.13 (sec) , antiderivative size = 375, normalized size of antiderivative = 4.31 \[ \int \sinh (x) \tanh (6 x) \, dx =\text {Too large to display} \] Input:
integrate(sinh(x)*tanh(6*x),x, algorithm="fricas")
Output:
1/6*(sqrt(sqrt(3) + 2)*(cosh(x) + sinh(x))*arctan(((sqrt(3) - 2)*cosh(x)^3 + 3*(sqrt(3) - 2)*cosh(x)*sinh(x)^2 + (sqrt(3) - 2)*sinh(x)^3 - (sqrt(3) - 1)*cosh(x) + (3*(sqrt(3) - 2)*cosh(x)^2 - sqrt(3) + 1)*sinh(x))*sqrt(sqr t(3) + 2)) + sqrt(sqrt(3) + 2)*(cosh(x) + sinh(x))*arctan(((sqrt(3) - 2)*c osh(x) + (sqrt(3) - 2)*sinh(x))*sqrt(sqrt(3) + 2)) - sqrt(-sqrt(3) + 2)*(c osh(x) + sinh(x))*arctan(((sqrt(3) + 2)*cosh(x)^3 + 3*(sqrt(3) + 2)*cosh(x )*sinh(x)^2 + (sqrt(3) + 2)*sinh(x)^3 - (sqrt(3) + 1)*cosh(x) + (3*(sqrt(3 ) + 2)*cosh(x)^2 - sqrt(3) - 1)*sinh(x))*sqrt(-sqrt(3) + 2)) - sqrt(-sqrt( 3) + 2)*(cosh(x) + sinh(x))*arctan(((sqrt(3) + 2)*cosh(x) + (sqrt(3) + 2)* sinh(x))*sqrt(-sqrt(3) + 2)) - (sqrt(2)*cosh(x) + sqrt(2)*sinh(x))*arctan( 1/2*sqrt(2)*cosh(x) + 1/2*sqrt(2)*sinh(x)) + (sqrt(2)*cosh(x) + sqrt(2)*si nh(x))*arctan(-1/2*(sqrt(2)*cosh(x)^2 + 2*sqrt(2)*cosh(x)*sinh(x) + sqrt(2 )*sinh(x)^2 + sqrt(2))/(cosh(x) - sinh(x))) + 3*cosh(x)^2 + 6*cosh(x)*sinh (x) + 3*sinh(x)^2 - 3)/(cosh(x) + sinh(x))
\[ \int \sinh (x) \tanh (6 x) \, dx=\int \sinh {\left (x \right )} \tanh {\left (6 x \right )}\, dx \] Input:
integrate(sinh(x)*tanh(6*x),x)
Output:
Integral(sinh(x)*tanh(6*x), x)
\[ \int \sinh (x) \tanh (6 x) \, dx=\int { \sinh \left (x\right ) \tanh \left (6 \, x\right ) \,d x } \] Input:
integrate(sinh(x)*tanh(6*x),x, algorithm="maxima")
Output:
1/2*(e^(2*x) - 1)*e^(-x) - 1/6*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*e^x )) - 1/6*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*e^x)) - 1/2*integrate(2/ 3*(2*e^(7*x) - e^(5*x) - e^(3*x) + 2*e^x)/(e^(8*x) - e^(4*x) + 1), x)
Time = 0.12 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.15 \[ \int \sinh (x) \tanh (6 x) \, dx=-\frac {1}{12} \, {\left (\sqrt {6} + \sqrt {2}\right )} \arctan \left (-\frac {2 \, {\left (e^{\left (-x\right )} - e^{x}\right )}}{\sqrt {6} + \sqrt {2}}\right ) - \frac {1}{12} \, {\left (\sqrt {6} - \sqrt {2}\right )} \arctan \left (-\frac {2 \, {\left (e^{\left (-x\right )} - e^{x}\right )}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{12} \, \sqrt {2} {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} - \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \] Input:
integrate(sinh(x)*tanh(6*x),x, algorithm="giac")
Output:
-1/12*(sqrt(6) + sqrt(2))*arctan(-2*(e^(-x) - e^x)/(sqrt(6) + sqrt(2))) - 1/12*(sqrt(6) - sqrt(2))*arctan(-2*(e^(-x) - e^x)/(sqrt(6) - sqrt(2))) - 1 /12*sqrt(2)*(pi + 2*arctan(1/2*sqrt(2)*(e^(2*x) - 1)*e^(-x))) - 1/2*e^(-x) + 1/2*e^x
Time = 2.23 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.13 \[ \int \sinh (x) \tanh (6 x) \, dx=\frac {{\mathrm {e}}^x}{2}-\frac {{\mathrm {e}}^{-x}}{2}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^{2\,x}-1\right )}{2}\right )}{6}-2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^{2\,x}-1\right )}{12\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}}{144}}}\right )\,\sqrt {\frac {1}{72}-\frac {\sqrt {3}}{144}}-2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^{2\,x}-1\right )}{12\,\sqrt {\frac {\sqrt {3}}{144}+\frac {1}{72}}}\right )\,\sqrt {\frac {\sqrt {3}}{144}+\frac {1}{72}} \] Input:
int(tanh(6*x)*sinh(x),x)
Output:
exp(x)/2 - exp(-x)/2 - (2^(1/2)*atan((2^(1/2)*exp(-x)*(exp(2*x) - 1))/2))/ 6 - 2*atan((exp(-x)*(exp(2*x) - 1))/(12*(1/72 - 3^(1/2)/144)^(1/2)))*(1/72 - 3^(1/2)/144)^(1/2) - 2*atan((exp(-x)*(exp(2*x) - 1))/(12*(3^(1/2)/144 + 1/72)^(1/2)))*(3^(1/2)/144 + 1/72)^(1/2)
Time = 0.26 (sec) , antiderivative size = 254, normalized size of antiderivative = 2.92 \[ \int \sinh (x) \tanh (6 x) \, dx=\frac {-2 e^{x} \sqrt {-\sqrt {3}+2}\, \mathit {atan} \left (\frac {4 e^{x}-\sqrt {6}-\sqrt {2}}{2 \sqrt {-\sqrt {3}+2}}\right )-2 e^{x} \sqrt {-\sqrt {3}+2}\, \mathit {atan} \left (\frac {4 e^{x}+\sqrt {6}+\sqrt {2}}{2 \sqrt {-\sqrt {3}+2}}\right )-2 e^{x} \sqrt {2}\, \mathit {atan} \left (\frac {2 e^{x}-\sqrt {2}}{\sqrt {2}}\right )-2 e^{x} \sqrt {2}\, \mathit {atan} \left (\frac {2 e^{x}+\sqrt {2}}{\sqrt {2}}\right )+e^{x} \sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 e^{x}}{\sqrt {6}+\sqrt {2}}\right )+e^{x} \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}-4 e^{x}}{\sqrt {6}+\sqrt {2}}\right )-e^{x} \sqrt {6}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 e^{x}}{\sqrt {6}+\sqrt {2}}\right )-e^{x} \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {-\sqrt {3}+2}+4 e^{x}}{\sqrt {6}+\sqrt {2}}\right )+6 e^{2 x}-6}{12 e^{x}} \] Input:
int(sinh(x)*tanh(6*x),x)
Output:
( - 2*e**x*sqrt( - sqrt(3) + 2)*atan((4*e**x - sqrt(6) - sqrt(2))/(2*sqrt( - sqrt(3) + 2))) - 2*e**x*sqrt( - sqrt(3) + 2)*atan((4*e**x + sqrt(6) + s qrt(2))/(2*sqrt( - sqrt(3) + 2))) - 2*e**x*sqrt(2)*atan((2*e**x - sqrt(2)) /sqrt(2)) - 2*e**x*sqrt(2)*atan((2*e**x + sqrt(2))/sqrt(2)) + e**x*sqrt(6) *atan((2*sqrt( - sqrt(3) + 2) - 4*e**x)/(sqrt(6) + sqrt(2))) + e**x*sqrt(2 )*atan((2*sqrt( - sqrt(3) + 2) - 4*e**x)/(sqrt(6) + sqrt(2))) - e**x*sqrt( 6)*atan((2*sqrt( - sqrt(3) + 2) + 4*e**x)/(sqrt(6) + sqrt(2))) - e**x*sqrt (2)*atan((2*sqrt( - sqrt(3) + 2) + 4*e**x)/(sqrt(6) + sqrt(2))) + 6*e**(2* x) - 6)/(12*e**x)