Optimal. Leaf size=83 \[ -\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {b \left (3 a^2-3 a b+b^2\right ) \cosh (c+d x)}{d}+\frac {b^2 (3 a-2 b) \cosh ^3(c+d x)}{3 d}+\frac {b^3 \cosh ^5(c+d x)}{5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3186, 390, 206} \[ \frac {b \left (3 a^2-3 a b+b^2\right ) \cosh (c+d x)}{d}-\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {b^2 (3 a-2 b) \cosh ^3(c+d x)}{3 d}+\frac {b^3 \cosh ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 390
Rule 3186
Rubi steps
\begin {align*} \int \text {csch}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (a-b+b x^2\right )^3}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (-b \left (3 a^2-3 a b+b^2\right )-(3 a-2 b) b^2 x^2-b^3 x^4+\frac {a^3}{1-x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {b \left (3 a^2-3 a b+b^2\right ) \cosh (c+d x)}{d}+\frac {(3 a-2 b) b^2 \cosh ^3(c+d x)}{3 d}+\frac {b^3 \cosh ^5(c+d x)}{5 d}-\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {b \left (3 a^2-3 a b+b^2\right ) \cosh (c+d x)}{d}+\frac {(3 a-2 b) b^2 \cosh ^3(c+d x)}{3 d}+\frac {b^3 \cosh ^5(c+d x)}{5 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.22, size = 83, normalized size = 1.00 \[ \frac {3 \left (80 a^3 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+b^3 \cosh (5 (c+d x))\right )+30 b \left (24 a^2-18 a b+5 b^2\right ) \cosh (c+d x)+5 b^2 (12 a-5 b) \cosh (3 (c+d x))}{240 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.56, size = 1128, normalized size = 13.59 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.21, size = 202, normalized size = 2.43 \[ \frac {3 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 60 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 25 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 720 \, a^{2} b e^{\left (d x + c\right )} - 540 \, a b^{2} e^{\left (d x + c\right )} + 150 \, b^{3} e^{\left (d x + c\right )} - 480 \, a^{3} \log \left (e^{\left (d x + c\right )} + 1\right ) + 480 \, a^{3} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) + {\left (720 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 540 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 150 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 60 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 25 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{3}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 86, normalized size = 1.04 \[ \frac {-2 a^{3} \arctanh \left ({\mathrm e}^{d x +c}\right )+3 a^{2} b \cosh \left (d x +c \right )+3 a \,b^{2} \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\left (d x +c \right )\right )}{3}\right ) \cosh \left (d x +c \right )+b^{3} \left (\frac {8}{15}+\frac {\left (\sinh ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sinh ^{2}\left (d x +c \right )\right )}{15}\right ) \cosh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.80, size = 193, normalized size = 2.33 \[ \frac {1}{480} \, b^{3} {\left (\frac {3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac {25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {150 \, e^{\left (d x + c\right )}}{d} + \frac {150 \, e^{\left (-d x - c\right )}}{d} - \frac {25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} + \frac {1}{8} \, a b^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {3}{2} \, a^{2} b {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {a^{3} \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.30, size = 184, normalized size = 2.22 \[ \frac {{\mathrm {e}}^{c+d\,x}\,\left (24\,a^2\,b-18\,a\,b^2+5\,b^3\right )}{16\,d}-\frac {2\,\mathrm {atan}\left (\frac {a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^6}}\right )\,\sqrt {a^6}}{\sqrt {-d^2}}+\frac {{\mathrm {e}}^{-c-d\,x}\,\left (24\,a^2\,b-18\,a\,b^2+5\,b^3\right )}{16\,d}+\frac {b^3\,{\mathrm {e}}^{-5\,c-5\,d\,x}}{160\,d}+\frac {b^3\,{\mathrm {e}}^{5\,c+5\,d\,x}}{160\,d}+\frac {b^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}\,\left (12\,a-5\,b\right )}{96\,d}+\frac {b^2\,{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (12\,a-5\,b\right )}{96\,d} \]
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]
________________________________________________________________________________________