Optimal. Leaf size=88 \[ -\frac {a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {a b \sinh (c+d x) \cosh (c+d x)}{d}-a b x+\frac {b^2 \cosh ^5(c+d x)}{5 d}-\frac {2 b^2 \cosh ^3(c+d x)}{3 d}+\frac {b^2 \cosh (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3220, 3770, 2635, 8, 2633} \[ -\frac {a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {a b \sinh (c+d x) \cosh (c+d x)}{d}-a b x+\frac {b^2 \cosh ^5(c+d x)}{5 d}-\frac {2 b^2 \cosh ^3(c+d x)}{3 d}+\frac {b^2 \cosh (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2633
Rule 2635
Rule 3220
Rule 3770
Rubi steps
\begin {align*} \int \text {csch}(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx &=i \int \left (-i a^2 \text {csch}(c+d x)-2 i a b \sinh ^2(c+d x)-i b^2 \sinh ^5(c+d x)\right ) \, dx\\ &=a^2 \int \text {csch}(c+d x) \, dx+(2 a b) \int \sinh ^2(c+d x) \, dx+b^2 \int \sinh ^5(c+d x) \, dx\\ &=-\frac {a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {a b \cosh (c+d x) \sinh (c+d x)}{d}-(a b) \int 1 \, dx+\frac {b^2 \operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-a b x-\frac {a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {b^2 \cosh (c+d x)}{d}-\frac {2 b^2 \cosh ^3(c+d x)}{3 d}+\frac {b^2 \cosh ^5(c+d x)}{5 d}+\frac {a b \cosh (c+d x) \sinh (c+d x)}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.31, size = 96, normalized size = 1.09 \[ \frac {120 a \left (b \sinh (2 (c+d x))-2 \left (-a \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+a \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+b c+b d x\right )\right )+150 b^2 \cosh (c+d x)-25 b^2 \cosh (3 (c+d x))+3 b^2 \cosh (5 (c+d x))}{240 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.49, size = 1052, normalized size = 11.95 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 154, normalized size = 1.75 \[ -\frac {480 \, {\left (d x + c\right )} a b - 3 \, b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 25 \, b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 120 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 150 \, b^{2} e^{\left (d x + c\right )} + 480 \, a^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) - 480 \, a^{2} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - {\left (150 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 120 \, a b e^{\left (3 \, d x + 3 \, c\right )} - 25 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{2}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.12, size = 76, normalized size = 0.86 \[ \frac {-2 a^{2} \arctanh \left ({\mathrm e}^{d x +c}\right )+2 a b \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+b^{2} \left (\frac {8}{15}+\frac {\left (\sinh ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sinh ^{2}\left (d x +c \right )\right )}{15}\right ) \cosh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 140, normalized size = 1.59 \[ -\frac {1}{4} \, a b {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + \frac {1}{480} \, b^{2} {\left (\frac {3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac {25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {150 \, e^{\left (d x + c\right )}}{d} + \frac {150 \, e^{\left (-d x - c\right )}}{d} - \frac {25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} + \frac {a^{2} \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.21, size = 177, normalized size = 2.01 \[ \frac {5\,b^2\,{\mathrm {e}}^{c+d\,x}}{16\,d}-\frac {2\,\mathrm {atan}\left (\frac {a^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^4}}\right )\,\sqrt {a^4}}{\sqrt {-d^2}}-a\,b\,x+\frac {5\,b^2\,{\mathrm {e}}^{-c-d\,x}}{16\,d}-\frac {5\,b^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{96\,d}-\frac {5\,b^2\,{\mathrm {e}}^{3\,c+3\,d\,x}}{96\,d}+\frac {b^2\,{\mathrm {e}}^{-5\,c-5\,d\,x}}{160\,d}+\frac {b^2\,{\mathrm {e}}^{5\,c+5\,d\,x}}{160\,d}-\frac {a\,b\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{4\,d}+\frac {a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{4\,d} \]
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]
________________________________________________________________________________________