Optimal. Leaf size=85 \[ \frac{3 \left (a^2-b^2\right ) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{1}{8} x \left (3 a^2-2 a b+3 b^2\right )+\frac{(a+b) \sinh (c+d x) \cosh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )}{4 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0868324, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3675, 413, 385, 206} \[ \frac{3 \left (a^2-b^2\right ) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{1}{8} x \left (3 a^2-2 a b+3 b^2\right )+\frac{(a+b) \sinh (c+d x) \cosh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3675
Rule 413
Rule 385
Rule 206
Rubi steps
\begin{align*} \int \cosh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^2}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{(a+b) \cosh ^3(c+d x) \sinh (c+d x) \left (a+b \tanh ^2(c+d x)\right )}{4 d}-\frac{\operatorname{Subst}\left (\int \frac{-a (3 a-b)-(a-3 b) b x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 d}\\ &=\frac{3 \left (a^2-b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{(a+b) \cosh ^3(c+d x) \sinh (c+d x) \left (a+b \tanh ^2(c+d x)\right )}{4 d}+\frac{\left (3 a^2-2 a b+3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac{1}{8} \left (3 a^2-2 a b+3 b^2\right ) x+\frac{3 \left (a^2-b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{(a+b) \cosh ^3(c+d x) \sinh (c+d x) \left (a+b \tanh ^2(c+d x)\right )}{4 d}\\ \end{align*}
Mathematica [A] time = 0.292658, size = 63, normalized size = 0.74 \[ \frac{4 \left (3 a^2-2 a b+3 b^2\right ) (c+d x)+8 \left (a^2-b^2\right ) \sinh (2 (c+d x))+(a+b)^2 \sinh (4 (c+d x))}{32 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.04, size = 124, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ({b}^{2} \left ( \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{4}}-{\frac{3\,\sinh \left ( dx+c \right ) }{8}} \right ) \cosh \left ( dx+c \right ) +{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +2\,ab \left ( 1/4\,\sinh \left ( dx+c \right ) \left ( \cosh \left ( dx+c \right ) \right ) ^{3}-1/8\,\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) -1/8\,dx-c/8 \right ) +{a}^{2} \left ( \left ({\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{3\,\cosh \left ( dx+c \right ) }{8}} \right ) \sinh \left ( dx+c \right ) +{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.03492, size = 231, normalized size = 2.72 \begin{align*} \frac{1}{64} \, a^{2}{\left (24 \, x + \frac{e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac{8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac{8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac{e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + \frac{1}{64} \, b^{2}{\left (24 \, x + \frac{e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac{8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac{e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac{1}{32} \, a b{\left (\frac{8 \,{\left (d x + c\right )}}{d} - \frac{e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac{e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.95999, size = 234, normalized size = 2.75 \begin{align*} \frac{{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} +{\left (3 \, a^{2} - 2 \, a b + 3 \, b^{2}\right )} d x +{\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{3} + 4 \,{\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{8 \, d} \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 = 1.93304, size = 255, normalized size = 3. \begin{align*} \frac{8 \,{\left (3 \, a^{2} - 2 \, a b + 3 \, b^{2}\right )} d x -{\left (18 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 12 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 18 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 8 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{2} + 2 \, a b + b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )} +{\left (a^{2} e^{\left (4 \, d x + 12 \, c\right )} + 2 \, a b e^{\left (4 \, d x + 12 \, c\right )} + b^{2} e^{\left (4 \, d x + 12 \, c\right )} + 8 \, a^{2} e^{\left (2 \, d x + 10 \, c\right )} - 8 \, b^{2} e^{\left (2 \, d x + 10 \, c\right )}\right )} e^{\left (-8 \, c\right )}}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]