Optimal. Leaf size=311 \[ \frac{e^{c (a+b x)} \sqrt{\tanh ^2(a c+b c x)} \coth (a c+b c x)}{b c}+\frac{25 e^{c (a+b x)} \sqrt{\tanh ^2(a c+b c x)} \coth (a c+b c x)}{4 b c \left (e^{2 c (a+b x)}+1\right )}-\frac{55 e^{c (a+b x)} \sqrt{\tanh ^2(a c+b c x)} \coth (a c+b c x)}{6 b c \left (e^{2 c (a+b x)}+1\right )^2}+\frac{26 e^{c (a+b x)} \sqrt{\tanh ^2(a c+b c x)} \coth (a c+b c x)}{3 b c \left (e^{2 c (a+b x)}+1\right )^3}-\frac{4 e^{c (a+b x)} \sqrt{\tanh ^2(a c+b c x)} \coth (a c+b c x)}{b c \left (e^{2 c (a+b x)}+1\right )^4}-\frac{15 \tan ^{-1}\left (e^{c (a+b x)}\right ) \sqrt{\tanh ^2(a c+b c x)} \coth (a c+b c x)}{4 b c} \]
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Rubi [A] time = 0.902018, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {6720, 2282, 390, 1814, 1157, 385, 203} \[ \frac{e^{c (a+b x)} \sqrt{\tanh ^2(a c+b c x)} \coth (a c+b c x)}{b c}+\frac{25 e^{c (a+b x)} \sqrt{\tanh ^2(a c+b c x)} \coth (a c+b c x)}{4 b c \left (e^{2 c (a+b x)}+1\right )}-\frac{55 e^{c (a+b x)} \sqrt{\tanh ^2(a c+b c x)} \coth (a c+b c x)}{6 b c \left (e^{2 c (a+b x)}+1\right )^2}+\frac{26 e^{c (a+b x)} \sqrt{\tanh ^2(a c+b c x)} \coth (a c+b c x)}{3 b c \left (e^{2 c (a+b x)}+1\right )^3}-\frac{4 e^{c (a+b x)} \sqrt{\tanh ^2(a c+b c x)} \coth (a c+b c x)}{b c \left (e^{2 c (a+b x)}+1\right )^4}-\frac{15 \tan ^{-1}\left (e^{c (a+b x)}\right ) \sqrt{\tanh ^2(a c+b c x)} \coth (a c+b c x)}{4 b c} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 2282
Rule 390
Rule 1814
Rule 1157
Rule 385
Rule 203
Rubi steps
\begin{align*} \int e^{c (a+b x)} \tanh ^2(a c+b c x)^{5/2} \, dx &=\left (\coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}\right ) \int e^{c (a+b x)} \tanh ^5(a c+b c x) \, dx\\ &=\frac{\left (\coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}\right ) \operatorname{Subst}\left (\int \frac{\left (-1+x^2\right )^5}{\left (1+x^2\right )^5} \, dx,x,e^{c (a+b x)}\right )}{b c}\\ &=\frac{\left (\coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}\right ) \operatorname{Subst}\left (\int \left (1-\frac{2 \left (1+10 x^4+5 x^8\right )}{\left (1+x^2\right )^5}\right ) \, dx,x,e^{c (a+b x)}\right )}{b c}\\ &=\frac{e^{c (a+b x)} \coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}}{b c}-\frac{\left (2 \coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}\right ) \operatorname{Subst}\left (\int \frac{1+10 x^4+5 x^8}{\left (1+x^2\right )^5} \, dx,x,e^{c (a+b x)}\right )}{b c}\\ &=\frac{e^{c (a+b x)} \coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}}{b c}-\frac{4 e^{c (a+b x)} \coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}}{b c \left (1+e^{2 c (a+b x)}\right )^4}+\frac{\left (\coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}\right ) \operatorname{Subst}\left (\int \frac{8-120 x^2+40 x^4-40 x^6}{\left (1+x^2\right )^4} \, dx,x,e^{c (a+b x)}\right )}{4 b c}\\ &=\frac{e^{c (a+b x)} \coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}}{b c}-\frac{4 e^{c (a+b x)} \coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}}{b c \left (1+e^{2 c (a+b x)}\right )^4}+\frac{26 e^{c (a+b x)} \coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}}{3 b c \left (1+e^{2 c (a+b x)}\right )^3}-\frac{\left (\coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}\right ) \operatorname{Subst}\left (\int \frac{160-480 x^2+240 x^4}{\left (1+x^2\right )^3} \, dx,x,e^{c (a+b x)}\right )}{24 b c}\\ &=\frac{e^{c (a+b x)} \coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}}{b c}-\frac{4 e^{c (a+b x)} \coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}}{b c \left (1+e^{2 c (a+b x)}\right )^4}+\frac{26 e^{c (a+b x)} \coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}}{3 b c \left (1+e^{2 c (a+b x)}\right )^3}-\frac{55 e^{c (a+b x)} \coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}}{6 b c \left (1+e^{2 c (a+b x)}\right )^2}+\frac{\left (\coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}\right ) \operatorname{Subst}\left (\int \frac{240-960 x^2}{\left (1+x^2\right )^2} \, dx,x,e^{c (a+b x)}\right )}{96 b c}\\ &=\frac{e^{c (a+b x)} \coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}}{b c}-\frac{4 e^{c (a+b x)} \coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}}{b c \left (1+e^{2 c (a+b x)}\right )^4}+\frac{26 e^{c (a+b x)} \coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}}{3 b c \left (1+e^{2 c (a+b x)}\right )^3}-\frac{55 e^{c (a+b x)} \coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}}{6 b c \left (1+e^{2 c (a+b x)}\right )^2}+\frac{25 e^{c (a+b x)} \coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}}{4 b c \left (1+e^{2 c (a+b x)}\right )}-\frac{\left (15 \coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,e^{c (a+b x)}\right )}{4 b c}\\ &=\frac{e^{c (a+b x)} \coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}}{b c}-\frac{4 e^{c (a+b x)} \coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}}{b c \left (1+e^{2 c (a+b x)}\right )^4}+\frac{26 e^{c (a+b x)} \coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}}{3 b c \left (1+e^{2 c (a+b x)}\right )^3}-\frac{55 e^{c (a+b x)} \coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}}{6 b c \left (1+e^{2 c (a+b x)}\right )^2}+\frac{25 e^{c (a+b x)} \coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}}{4 b c \left (1+e^{2 c (a+b x)}\right )}-\frac{15 \tan ^{-1}\left (e^{c (a+b x)}\right ) \coth (a c+b c x) \sqrt{\tanh ^2(a c+b c x)}}{4 b c}\\ \end{align*}
Mathematica [A] time = 0.217265, size = 133, normalized size = 0.43 \[ \frac{\left (e^{c (a+b x)} \left (157 e^{2 c (a+b x)}+187 e^{4 c (a+b x)}+123 e^{6 c (a+b x)}+12 e^{8 c (a+b x)}+33\right )-45 \left (e^{2 c (a+b x)}+1\right )^4 \tan ^{-1}\left (e^{c (a+b x)}\right )\right ) \sqrt{\tanh ^2(c (a+b x))} \coth (c (a+b x))}{12 b c \left (e^{2 c (a+b x)}+1\right )^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.277, size = 324, normalized size = 1. \begin{align*}{\frac{ \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ){{\rm e}^{c \left ( bx+a \right ) }}}{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) cb}\sqrt{{\frac{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) ^{2}}{ \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) ^{2}}}}}+{\frac{{{\rm e}^{c \left ( bx+a \right ) }} \left ( 75\,{{\rm e}^{6\,c \left ( bx+a \right ) }}+115\,{{\rm e}^{4\,c \left ( bx+a \right ) }}+109\,{{\rm e}^{2\,c \left ( bx+a \right ) }}+21 \right ) }{ \left ( 12\,{{\rm e}^{2\,c \left ( bx+a \right ) }}-12 \right ) \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) ^{3}cb}\sqrt{{\frac{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) ^{2}}{ \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) ^{2}}}}}+{\frac{{\frac{15\,i}{8}} \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) \ln \left ({{\rm e}^{c \left ( bx+a \right ) }}-i \right ) }{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) cb}\sqrt{{\frac{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) ^{2}}{ \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) ^{2}}}}}-{\frac{{\frac{15\,i}{8}} \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) \ln \left ({{\rm e}^{c \left ( bx+a \right ) }}+i \right ) }{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) cb}\sqrt{{\frac{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) ^{2}}{ \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.70385, size = 196, normalized size = 0.63 \begin{align*} -\frac{15 \, \arctan \left (e^{\left (b c x + a c\right )}\right )}{4 \, b c} + \frac{12 \, e^{\left (9 \, b c x + 9 \, a c\right )} + 123 \, e^{\left (7 \, b c x + 7 \, a c\right )} + 187 \, e^{\left (5 \, b c x + 5 \, a c\right )} + 157 \, e^{\left (3 \, b c x + 3 \, a c\right )} + 33 \, e^{\left (b c x + a c\right )}}{12 \, b c{\left (e^{\left (8 \, b c x + 8 \, a c\right )} + 4 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.1185, size = 3186, normalized size = 10.24 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33277, size = 248, normalized size = 0.8 \begin{align*} -\frac{45 \, \arctan \left (e^{\left (b c x + a c\right )}\right ) \mathrm{sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) - 12 \, e^{\left (b c x + a c\right )} \mathrm{sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) - \frac{75 \, e^{\left (7 \, b c x + 7 \, a c\right )} \mathrm{sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) + 115 \, e^{\left (5 \, b c x + 5 \, a c\right )} \mathrm{sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) + 109 \, e^{\left (3 \, b c x + 3 \, a c\right )} \mathrm{sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) + 21 \, e^{\left (b c x + a c\right )} \mathrm{sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}{{\left (e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}^{4}}}{12 \, b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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