Integrand size = 9, antiderivative size = 157 \[ \int \cos ^3(\tanh (a+b x)) \, dx=-\frac {\cos (3) \operatorname {CosIntegral}(3-3 \tanh (a+b x))}{8 b}-\frac {3 \cos (1) \operatorname {CosIntegral}(1-\tanh (a+b x))}{8 b}+\frac {3 \cos (1) \operatorname {CosIntegral}(1+\tanh (a+b x))}{8 b}+\frac {\cos (3) \operatorname {CosIntegral}(3+3 \tanh (a+b x))}{8 b}-\frac {\sin (3) \text {Si}(3-3 \tanh (a+b x))}{8 b}-\frac {3 \sin (1) \text {Si}(1-\tanh (a+b x))}{8 b}+\frac {3 \sin (1) \text {Si}(1+\tanh (a+b x))}{8 b}+\frac {\sin (3) \text {Si}(3+3 \tanh (a+b x))}{8 b} \]
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Time = 0.28 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {6857, 3393, 3384, 3380, 3383} \[ \int \cos ^3(\tanh (a+b x)) \, dx=-\frac {\cos (3) \operatorname {CosIntegral}(3-3 \tanh (a+b x))}{8 b}-\frac {3 \cos (1) \operatorname {CosIntegral}(1-\tanh (a+b x))}{8 b}+\frac {3 \cos (1) \operatorname {CosIntegral}(\tanh (a+b x)+1)}{8 b}+\frac {\cos (3) \operatorname {CosIntegral}(3 \tanh (a+b x)+3)}{8 b}-\frac {\sin (3) \text {Si}(3-3 \tanh (a+b x))}{8 b}-\frac {3 \sin (1) \text {Si}(1-\tanh (a+b x))}{8 b}+\frac {3 \sin (1) \text {Si}(\tanh (a+b x)+1)}{8 b}+\frac {\sin (3) \text {Si}(3 \tanh (a+b x)+3)}{8 b} \]
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Rule 3380
Rule 3383
Rule 3384
Rule 3393
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cos ^3(x)}{1-x^2} \, dx,x,\tanh (a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {\cos ^3(x)}{2 (-1+x)}+\frac {\cos ^3(x)}{2 (1+x)}\right ) \, dx,x,\tanh (a+b x)\right )}{b} \\ & = -\frac {\text {Subst}\left (\int \frac {\cos ^3(x)}{-1+x} \, dx,x,\tanh (a+b x)\right )}{2 b}+\frac {\text {Subst}\left (\int \frac {\cos ^3(x)}{1+x} \, dx,x,\tanh (a+b x)\right )}{2 b} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {3 \cos (x)}{4 (-1+x)}+\frac {\cos (3 x)}{4 (-1+x)}\right ) \, dx,x,\tanh (a+b x)\right )}{2 b}+\frac {\text {Subst}\left (\int \left (\frac {3 \cos (x)}{4 (1+x)}+\frac {\cos (3 x)}{4 (1+x)}\right ) \, dx,x,\tanh (a+b x)\right )}{2 b} \\ & = -\frac {\text {Subst}\left (\int \frac {\cos (3 x)}{-1+x} \, dx,x,\tanh (a+b x)\right )}{8 b}+\frac {\text {Subst}\left (\int \frac {\cos (3 x)}{1+x} \, dx,x,\tanh (a+b x)\right )}{8 b}-\frac {3 \text {Subst}\left (\int \frac {\cos (x)}{-1+x} \, dx,x,\tanh (a+b x)\right )}{8 b}+\frac {3 \text {Subst}\left (\int \frac {\cos (x)}{1+x} \, dx,x,\tanh (a+b x)\right )}{8 b} \\ & = -\frac {(3 \cos (1)) \text {Subst}\left (\int \frac {\cos (1-x)}{-1+x} \, dx,x,\tanh (a+b x)\right )}{8 b}+\frac {(3 \cos (1)) \text {Subst}\left (\int \frac {\cos (1+x)}{1+x} \, dx,x,\tanh (a+b x)\right )}{8 b}-\frac {\cos (3) \text {Subst}\left (\int \frac {\cos (3-3 x)}{-1+x} \, dx,x,\tanh (a+b x)\right )}{8 b}+\frac {\cos (3) \text {Subst}\left (\int \frac {\cos (3+3 x)}{1+x} \, dx,x,\tanh (a+b x)\right )}{8 b}-\frac {(3 \sin (1)) \text {Subst}\left (\int \frac {\sin (1-x)}{-1+x} \, dx,x,\tanh (a+b x)\right )}{8 b}+\frac {(3 \sin (1)) \text {Subst}\left (\int \frac {\sin (1+x)}{1+x} \, dx,x,\tanh (a+b x)\right )}{8 b}-\frac {\sin (3) \text {Subst}\left (\int \frac {\sin (3-3 x)}{-1+x} \, dx,x,\tanh (a+b x)\right )}{8 b}+\frac {\sin (3) \text {Subst}\left (\int \frac {\sin (3+3 x)}{1+x} \, dx,x,\tanh (a+b x)\right )}{8 b} \\ & = -\frac {\cos (3) \operatorname {CosIntegral}(3-3 \tanh (a+b x))}{8 b}-\frac {3 \cos (1) \operatorname {CosIntegral}(1-\tanh (a+b x))}{8 b}+\frac {3 \cos (1) \operatorname {CosIntegral}(1+\tanh (a+b x))}{8 b}+\frac {\cos (3) \operatorname {CosIntegral}(3+3 \tanh (a+b x))}{8 b}-\frac {\sin (3) \text {Si}(3-3 \tanh (a+b x))}{8 b}-\frac {3 \sin (1) \text {Si}(1-\tanh (a+b x))}{8 b}+\frac {3 \sin (1) \text {Si}(1+\tanh (a+b x))}{8 b}+\frac {\sin (3) \text {Si}(3+3 \tanh (a+b x))}{8 b} \\ \end{align*}
Time = 0.71 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.79 \[ \int \cos ^3(\tanh (a+b x)) \, dx=\frac {-2 \cos (3) \operatorname {CosIntegral}(3-3 \tanh (a+b x))-6 \cos (1) \operatorname {CosIntegral}(1-\tanh (a+b x))+6 \cos (1) \operatorname {CosIntegral}(1+\tanh (a+b x))+2 \cos (3) \operatorname {CosIntegral}(3+3 \tanh (a+b x))-2 \sin (3) \text {Si}(3-3 \tanh (a+b x))-6 \sin (1) \text {Si}(1-\tanh (a+b x))+6 \sin (1) \text {Si}(1+\tanh (a+b x))+2 \sin (3) \text {Si}(3+3 \tanh (a+b x))}{16 b} \]
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Time = 0.81 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {Si}\left (-3+3 \tanh \left (b x +a \right )\right ) \sin \left (3\right )}{8}-\frac {\operatorname {Ci}\left (-3+3 \tanh \left (b x +a \right )\right ) \cos \left (3\right )}{8}+\frac {\operatorname {Si}\left (3+3 \tanh \left (b x +a \right )\right ) \sin \left (3\right )}{8}+\frac {\operatorname {Ci}\left (3+3 \tanh \left (b x +a \right )\right ) \cos \left (3\right )}{8}+\frac {3 \,\operatorname {Si}\left (-1+\tanh \left (b x +a \right )\right ) \sin \left (1\right )}{8}-\frac {3 \,\operatorname {Ci}\left (-1+\tanh \left (b x +a \right )\right ) \cos \left (1\right )}{8}+\frac {3 \,\operatorname {Si}\left (1+\tanh \left (b x +a \right )\right ) \sin \left (1\right )}{8}+\frac {3 \,\operatorname {Ci}\left (1+\tanh \left (b x +a \right )\right ) \cos \left (1\right )}{8}}{b}\) | \(118\) |
default | \(\frac {\frac {\operatorname {Si}\left (-3+3 \tanh \left (b x +a \right )\right ) \sin \left (3\right )}{8}-\frac {\operatorname {Ci}\left (-3+3 \tanh \left (b x +a \right )\right ) \cos \left (3\right )}{8}+\frac {\operatorname {Si}\left (3+3 \tanh \left (b x +a \right )\right ) \sin \left (3\right )}{8}+\frac {\operatorname {Ci}\left (3+3 \tanh \left (b x +a \right )\right ) \cos \left (3\right )}{8}+\frac {3 \,\operatorname {Si}\left (-1+\tanh \left (b x +a \right )\right ) \sin \left (1\right )}{8}-\frac {3 \,\operatorname {Ci}\left (-1+\tanh \left (b x +a \right )\right ) \cos \left (1\right )}{8}+\frac {3 \,\operatorname {Si}\left (1+\tanh \left (b x +a \right )\right ) \sin \left (1\right )}{8}+\frac {3 \,\operatorname {Ci}\left (1+\tanh \left (b x +a \right )\right ) \cos \left (1\right )}{8}}{b}\) | \(118\) |
risch | \(-\frac {{\mathrm e}^{3 i} \operatorname {Ei}_{1}\left (-\frac {6 i}{1+{\mathrm e}^{2 b x +2 a}}+6 i\right )}{16 b}+\frac {{\mathrm e}^{3 i} \operatorname {Ei}_{1}\left (\frac {6 i}{1+{\mathrm e}^{2 b x +2 a}}\right )}{16 b}+\frac {{\mathrm e}^{-3 i} \operatorname {Ei}_{1}\left (-\frac {6 i}{1+{\mathrm e}^{2 b x +2 a}}\right )}{16 b}-\frac {{\mathrm e}^{-3 i} \operatorname {Ei}_{1}\left (\frac {6 i}{1+{\mathrm e}^{2 b x +2 a}}-6 i\right )}{16 b}-\frac {3 \,{\mathrm e}^{-i} \operatorname {Ei}_{1}\left (\frac {2 i}{1+{\mathrm e}^{2 b x +2 a}}-2 i\right )}{16 b}+\frac {3 \,{\mathrm e}^{i} \operatorname {Ei}_{1}\left (\frac {2 i}{1+{\mathrm e}^{2 b x +2 a}}\right )}{16 b}-\frac {3 \,{\mathrm e}^{i} \operatorname {Ei}_{1}\left (-\frac {2 i}{1+{\mathrm e}^{2 b x +2 a}}+2 i\right )}{16 b}+\frac {3 \,{\mathrm e}^{-i} \operatorname {Ei}_{1}\left (-\frac {2 i}{1+{\mathrm e}^{2 b x +2 a}}\right )}{16 b}\) | \(222\) |
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 698, normalized size of antiderivative = 4.45 \[ \int \cos ^3(\tanh (a+b x)) \, dx=\text {Too large to display} \]
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\[ \int \cos ^3(\tanh (a+b x)) \, dx=\int \cos ^{3}{\left (\tanh {\left (a + b x \right )} \right )}\, dx \]
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\[ \int \cos ^3(\tanh (a+b x)) \, dx=\int { \cos \left (\tanh \left (b x + a\right )\right )^{3} \,d x } \]
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\[ \int \cos ^3(\tanh (a+b x)) \, dx=\int { \cos \left (\tanh \left (b x + a\right )\right )^{3} \,d x } \]
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Timed out. \[ \int \cos ^3(\tanh (a+b x)) \, dx=\int {\cos \left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3 \,d x \]
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