Optimal. Leaf size=72 \[ \frac{a}{2 b d (a+b) \left (a+b \tanh ^2(c+d x)\right )}+\frac{\log \left (a+b \tanh ^2(c+d x)\right )}{2 d (a+b)^2}+\frac{\log (\cosh (c+d x))}{d (a+b)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.118215, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3670, 446, 77} \[ \frac{a}{2 b d (a+b) \left (a+b \tanh ^2(c+d x)\right )}+\frac{\log \left (a+b \tanh ^2(c+d x)\right )}{2 d (a+b)^2}+\frac{\log (\cosh (c+d x))}{d (a+b)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3670
Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{\tanh ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{\left (1-x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{(1-x) (a+b x)^2} \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{(a+b)^2 (-1+x)}-\frac{a}{(a+b) (a+b x)^2}+\frac{b}{(a+b)^2 (a+b x)}\right ) \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=\frac{\log (\cosh (c+d x))}{(a+b)^2 d}+\frac{\log \left (a+b \tanh ^2(c+d x)\right )}{2 (a+b)^2 d}+\frac{a}{2 b (a+b) d \left (a+b \tanh ^2(c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.438799, size = 57, normalized size = 0.79 \[ \frac{\frac{a (a+b)}{b \left (a+b \tanh ^2(c+d x)\right )}+\log \left (a+b \tanh ^2(c+d x)\right )+2 \log (\cosh (c+d x))}{2 d (a+b)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.024, size = 118, normalized size = 1.6 \begin{align*} -{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ) }{2\,d \left ( a+b \right ) ^{2}}}+{\frac{{a}^{2}}{2\,d \left ( a+b \right ) ^{2}b \left ( a+b \left ( \tanh \left ( dx+c \right ) \right ) ^{2} \right ) }}+{\frac{a}{2\,d \left ( a+b \right ) ^{2} \left ( a+b \left ( \tanh \left ( dx+c \right ) \right ) ^{2} \right ) }}+{\frac{\ln \left ( a+b \left ( \tanh \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d \left ( a+b \right ) ^{2}}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) }{2\,d \left ( a+b \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.08869, size = 230, normalized size = 3.19 \begin{align*} \frac{2 \, a e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3} + 2 \,{\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} +{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} + \frac{d x + c}{{\left (a^{2} + 2 \, a b + b^{2}\right )} d} + \frac{\log \left (2 \,{\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} +{\left (a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )} + a + b\right )}{2 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.7972, size = 1611, normalized size = 22.38 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.23256, size = 208, normalized size = 2.89 \begin{align*} \frac{\log \left ({\left | a{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + b{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 2 \, a - 2 \, b \right |}\right )}{2 \,{\left (a^{2} d + 2 \, a b d + b^{2} d\right )}} - \frac{e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} - 2}{2 \,{\left (a d + b d\right )}{\left (a{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + b{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 2 \, a - 2 \, b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]