Optimal. Leaf size=157 \[ -\frac {\cos (3) \text {Ci}(3-3 \coth (a+b x))}{8 b}-\frac {3 \cos (1) \text {Ci}(1-\coth (a+b x))}{8 b}+\frac {3 \cos (1) \text {Ci}(\coth (a+b x)+1)}{8 b}+\frac {\cos (3) \text {Ci}(3 \coth (a+b x)+3)}{8 b}-\frac {\sin (3) \text {Si}(3-3 \coth (a+b x))}{8 b}-\frac {3 \sin (1) \text {Si}(1-\coth (a+b x))}{8 b}+\frac {3 \sin (1) \text {Si}(\coth (a+b x)+1)}{8 b}+\frac {\sin (3) \text {Si}(3 \coth (a+b x)+3)}{8 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.37, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 5, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {6725, 3312, 3303, 3299, 3302} \[ -\frac {\cos (3) \text {CosIntegral}(3-3 \coth (a+b x))}{8 b}-\frac {3 \cos (1) \text {CosIntegral}(1-\coth (a+b x))}{8 b}+\frac {3 \cos (1) \text {CosIntegral}(\coth (a+b x)+1)}{8 b}+\frac {\cos (3) \text {CosIntegral}(3 \coth (a+b x)+3)}{8 b}-\frac {\sin (3) \text {Si}(3-3 \coth (a+b x))}{8 b}-\frac {3 \sin (1) \text {Si}(1-\coth (a+b x))}{8 b}+\frac {3 \sin (1) \text {Si}(\coth (a+b x)+1)}{8 b}+\frac {\sin (3) \text {Si}(3 \coth (a+b x)+3)}{8 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3299
Rule 3302
Rule 3303
Rule 3312
Rule 6725
Rubi steps
\begin {align*} \int \cos ^3(\coth (a+b x)) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cos ^3(x)}{1-x^2} \, dx,x,\coth (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {\cos ^3(x)}{2 (-1+x)}+\frac {\cos ^3(x)}{2 (1+x)}\right ) \, dx,x,\coth (a+b x)\right )}{b}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\cos ^3(x)}{-1+x} \, dx,x,\coth (a+b x)\right )}{2 b}+\frac {\operatorname {Subst}\left (\int \frac {\cos ^3(x)}{1+x} \, dx,x,\coth (a+b x)\right )}{2 b}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {3 \cos (x)}{4 (-1+x)}+\frac {\cos (3 x)}{4 (-1+x)}\right ) \, dx,x,\coth (a+b x)\right )}{2 b}+\frac {\operatorname {Subst}\left (\int \left (\frac {3 \cos (x)}{4 (1+x)}+\frac {\cos (3 x)}{4 (1+x)}\right ) \, dx,x,\coth (a+b x)\right )}{2 b}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\cos (3 x)}{-1+x} \, dx,x,\coth (a+b x)\right )}{8 b}+\frac {\operatorname {Subst}\left (\int \frac {\cos (3 x)}{1+x} \, dx,x,\coth (a+b x)\right )}{8 b}-\frac {3 \operatorname {Subst}\left (\int \frac {\cos (x)}{-1+x} \, dx,x,\coth (a+b x)\right )}{8 b}+\frac {3 \operatorname {Subst}\left (\int \frac {\cos (x)}{1+x} \, dx,x,\coth (a+b x)\right )}{8 b}\\ &=-\frac {(3 \cos (1)) \operatorname {Subst}\left (\int \frac {\cos (1-x)}{-1+x} \, dx,x,\coth (a+b x)\right )}{8 b}+\frac {(3 \cos (1)) \operatorname {Subst}\left (\int \frac {\cos (1+x)}{1+x} \, dx,x,\coth (a+b x)\right )}{8 b}-\frac {\cos (3) \operatorname {Subst}\left (\int \frac {\cos (3-3 x)}{-1+x} \, dx,x,\coth (a+b x)\right )}{8 b}+\frac {\cos (3) \operatorname {Subst}\left (\int \frac {\cos (3+3 x)}{1+x} \, dx,x,\coth (a+b x)\right )}{8 b}-\frac {(3 \sin (1)) \operatorname {Subst}\left (\int \frac {\sin (1-x)}{-1+x} \, dx,x,\coth (a+b x)\right )}{8 b}+\frac {(3 \sin (1)) \operatorname {Subst}\left (\int \frac {\sin (1+x)}{1+x} \, dx,x,\coth (a+b x)\right )}{8 b}-\frac {\sin (3) \operatorname {Subst}\left (\int \frac {\sin (3-3 x)}{-1+x} \, dx,x,\coth (a+b x)\right )}{8 b}+\frac {\sin (3) \operatorname {Subst}\left (\int \frac {\sin (3+3 x)}{1+x} \, dx,x,\coth (a+b x)\right )}{8 b}\\ &=-\frac {\cos (3) \text {Ci}(3-3 \coth (a+b x))}{8 b}-\frac {3 \cos (1) \text {Ci}(1-\coth (a+b x))}{8 b}+\frac {3 \cos (1) \text {Ci}(1+\coth (a+b x))}{8 b}+\frac {\cos (3) \text {Ci}(3+3 \coth (a+b x))}{8 b}-\frac {\sin (3) \text {Si}(3-3 \coth (a+b x))}{8 b}-\frac {3 \sin (1) \text {Si}(1-\coth (a+b x))}{8 b}+\frac {3 \sin (1) \text {Si}(1+\coth (a+b x))}{8 b}+\frac {\sin (3) \text {Si}(3+3 \coth (a+b x))}{8 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.26, size = 124, normalized size = 0.79 \[ \frac {-2 \cos (3) \text {Ci}(3-3 \coth (a+b x))-6 \cos (1) \text {Ci}(1-\coth (a+b x))+6 \cos (1) \text {Ci}(\coth (a+b x)+1)+2 \cos (3) \text {Ci}(3 \coth (a+b x)+3)-2 \sin (3) \text {Si}(3-3 \coth (a+b x))-6 \sin (1) \text {Si}(1-\coth (a+b x))+6 \sin (1) \text {Si}(\coth (a+b x)+1)+2 \sin (3) \text {Si}(3 \coth (a+b x)+3)}{16 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.48, size = 298, normalized size = 1.90 \[ \frac {\cos \relax (3) \operatorname {Ci}\left (\frac {6 \, e^{\left (2 \, b x + 2 \, a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right ) + 3 \, \cos \relax (1) \operatorname {Ci}\left (\frac {2 \, e^{\left (2 \, b x + 2 \, a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right ) + 3 \, \cos \relax (1) \operatorname {Ci}\left (-\frac {2 \, e^{\left (2 \, b x + 2 \, a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right ) + \cos \relax (3) \operatorname {Ci}\left (-\frac {6 \, e^{\left (2 \, b x + 2 \, a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right ) - \cos \relax (3) \operatorname {Ci}\left (\frac {6}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right ) - 3 \, \cos \relax (1) \operatorname {Ci}\left (\frac {2}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right ) - 3 \, \cos \relax (1) \operatorname {Ci}\left (-\frac {2}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right ) - \cos \relax (3) \operatorname {Ci}\left (-\frac {6}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right ) + 2 \, \sin \relax (3) \operatorname {Si}\left (\frac {6 \, e^{\left (2 \, b x + 2 \, a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right ) + 6 \, \sin \relax (1) \operatorname {Si}\left (\frac {2 \, e^{\left (2 \, b x + 2 \, a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right ) + 2 \, \sin \relax (3) \operatorname {Si}\left (\frac {6}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right ) + 6 \, \sin \relax (1) \operatorname {Si}\left (\frac {2}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right )}{16 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos \left (\coth \left (b x + a\right )\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.14, size = 118, normalized size = 0.75 \[ \frac {\frac {\Si \left (3+3 \coth \left (b x +a \right )\right ) \sin \relax (3)}{8}+\frac {\Ci \left (3+3 \coth \left (b x +a \right )\right ) \cos \relax (3)}{8}+\frac {\Si \left (-3+3 \coth \left (b x +a \right )\right ) \sin \relax (3)}{8}-\frac {\Ci \left (-3+3 \coth \left (b x +a \right )\right ) \cos \relax (3)}{8}+\frac {3 \Si \left (1+\coth \left (b x +a \right )\right ) \sin \relax (1)}{8}+\frac {3 \Ci \left (1+\coth \left (b x +a \right )\right ) \cos \relax (1)}{8}+\frac {3 \Si \left (-1+\coth \left (b x +a \right )\right ) \sin \relax (1)}{8}-\frac {3 \Ci \left (-1+\coth \left (b x +a \right )\right ) \cos \relax (1)}{8}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos \left (\coth \left (b x + a\right )\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\cos \left (\mathrm {coth}\left (a+b\,x\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^{3}{\left (\coth {\left (a + b x \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________