Optimal. Leaf size=137 \[ \frac {3}{8} b x \left (8 a^2-4 a b+b^2\right )-\frac {a (2 a+b) (4 a+b) \coth (c+d x)}{8 d}+\frac {b \cosh ^4(c+d x) \coth (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{4 d}+\frac {b \cosh ^2(c+d x) \coth (c+d x) \left (a (4 a+b)-(4 a-3 b) (a-b) \tanh ^2(c+d x)\right )}{8 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3187, 468, 577, 453, 206} \[ \frac {3}{8} b x \left (8 a^2-4 a b+b^2\right )-\frac {a (2 a+b) (4 a+b) \coth (c+d x)}{8 d}+\frac {b \cosh ^4(c+d x) \coth (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{4 d}+\frac {b \cosh ^2(c+d x) \coth (c+d x) \left (a (4 a+b)-(4 a-3 b) (a-b) \tanh ^2(c+d x)\right )}{8 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 453
Rule 468
Rule 577
Rule 3187
Rubi steps
\begin {align*} \int \text {csch}^2(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a-(a-b) x^2\right )^3}{x^2 \left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b \cosh ^4(c+d x) \coth (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{4 d}+\frac {\operatorname {Subst}\left (\int \frac {\left (a (4 a+b)-(4 a-3 b) (a-b) x^2\right ) \left (a+(-a+b) x^2\right )}{x^2 \left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 d}\\ &=\frac {b \cosh ^4(c+d x) \coth (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{4 d}+\frac {b \cosh ^2(c+d x) \coth (c+d x) \left (a (4 a+b)-(4 a-3 b) (a-b) \tanh ^2(c+d x)\right )}{8 d}+\frac {\operatorname {Subst}\left (\int \frac {a (2 a+b) (4 a+b)-(4 a-3 b) (a-b) (2 a-b) x^2}{x^2 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=-\frac {a (2 a+b) (4 a+b) \coth (c+d x)}{8 d}+\frac {b \cosh ^4(c+d x) \coth (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{4 d}+\frac {b \cosh ^2(c+d x) \coth (c+d x) \left (a (4 a+b)-(4 a-3 b) (a-b) \tanh ^2(c+d x)\right )}{8 d}+\frac {\left (3 b \left (8 a^2-4 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac {3}{8} b \left (8 a^2-4 a b+b^2\right ) x-\frac {a (2 a+b) (4 a+b) \coth (c+d x)}{8 d}+\frac {b \cosh ^4(c+d x) \coth (c+d x) \left (a-(a-b) \tanh ^2(c+d x)\right )^2}{4 d}+\frac {b \cosh ^2(c+d x) \coth (c+d x) \left (a (4 a+b)-(4 a-3 b) (a-b) \tanh ^2(c+d x)\right )}{8 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.88, size = 113, normalized size = 0.82 \[ \frac {\sinh ^6(c+d x) \left (a \text {csch}^2(c+d x)+b\right )^3 \left (-32 a^3 \coth (c+d x)+12 b \left (8 a^2-4 a b+b^2\right ) (c+d x)+8 b^2 (3 a-b) \sinh (2 (c+d x))+b^3 \sinh (4 (c+d x))\right )}{4 d (2 a+b \cosh (2 (c+d x))-b)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.70, size = 169, normalized size = 1.23 \[ \frac {b^{3} \cosh \left (d x + c\right )^{5} + 5 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 3 \, {\left (8 \, a b^{2} - 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + {\left (10 \, b^{3} \cosh \left (d x + c\right )^{3} + 9 \, {\left (8 \, a b^{2} - 3 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 8 \, {\left (8 \, a^{3} + 3 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right ) + 8 \, {\left (8 \, a^{3} + 3 \, {\left (8 \, a^{2} b - 4 \, a b^{2} + b^{3}\right )} d x\right )} \sinh \left (d x + c\right )}{64 \, d \sinh \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 177, normalized size = 1.29 \[ \frac {b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 24 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 8 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 24 \, {\left (8 \, a^{2} b - 4 \, a b^{2} + b^{3}\right )} {\left (d x + c\right )} - \frac {128 \, a^{3}}{e^{\left (2 \, d x + 2 \, c\right )} - 1} - {\left (144 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 72 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 18 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 24 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 8 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 94, normalized size = 0.69 \[ \frac {-a^{3} \coth \left (d x +c \right )+3 a^{2} b \left (d x +c \right )+3 a \,b^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+b^{3} \left (\left (\frac {\left (\sinh ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 130, normalized size = 0.95 \[ \frac {1}{64} \, b^{3} {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac {3}{8} \, a b^{2} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + 3 \, a^{2} b x + \frac {2 \, a^{3}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.76, size = 121, normalized size = 0.88 \[ \frac {3\,b\,x\,\left (8\,a^2-4\,a\,b+b^2\right )}{8}-\frac {2\,a^3}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {b^3\,{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,d}+\frac {b^3\,{\mathrm {e}}^{4\,c+4\,d\,x}}{64\,d}-\frac {b^2\,{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (3\,a-b\right )}{8\,d}+\frac {b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (3\,a-b\right )}{8\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________