Optimal. Leaf size=189 \[ -\frac {a^3 \coth (c+d x) \text {csch}^9(c+d x)}{10 d}+\frac {9 a^3 \coth (c+d x) \text {csch}^7(c+d x)}{80 d}+\frac {3 a \left (21 a^2+80 a b+128 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{256 d}-\frac {3 a \left (21 a^2+80 a b+128 b^2\right ) \coth (c+d x) \text {csch}(c+d x)}{256 d}-\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^5(c+d x)}{160 d}+\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^3(c+d x)}{128 d}+\frac {b^3 \cosh (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.37, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3215, 1157, 1814, 388, 206} \[ \frac {3 a \left (21 a^2+80 a b+128 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{256 d}-\frac {3 a \left (21 a^2+80 a b+128 b^2\right ) \coth (c+d x) \text {csch}(c+d x)}{256 d}-\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^5(c+d x)}{160 d}+\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^3(c+d x)}{128 d}-\frac {a^3 \coth (c+d x) \text {csch}^9(c+d x)}{10 d}+\frac {9 a^3 \coth (c+d x) \text {csch}^7(c+d x)}{80 d}+\frac {b^3 \cosh (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 388
Rule 1157
Rule 1814
Rule 3215
Rubi steps
\begin {align*} \int \text {csch}^{11}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b-2 b x^2+b x^4\right )^3}{\left (1-x^2\right )^6} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a^3 \coth (c+d x) \text {csch}^9(c+d x)}{10 d}-\frac {\operatorname {Subst}\left (\int \frac {-9 a^3-30 a^2 b-30 a b^2-10 b^3+10 b \left (3 a^2+9 a b+5 b^2\right ) x^2-10 b^2 (9 a+10 b) x^4+10 b^2 (3 a+10 b) x^6-50 b^3 x^8+10 b^3 x^{10}}{\left (1-x^2\right )^5} \, dx,x,\cosh (c+d x)\right )}{10 d}\\ &=\frac {9 a^3 \coth (c+d x) \text {csch}^7(c+d x)}{80 d}-\frac {a^3 \coth (c+d x) \text {csch}^9(c+d x)}{10 d}+\frac {\operatorname {Subst}\left (\int \frac {63 a^3+240 a^2 b+240 a b^2+80 b^3-160 b^2 (3 a+2 b) x^2+240 b^2 (a+2 b) x^4-320 b^3 x^6+80 b^3 x^8}{\left (1-x^2\right )^4} \, dx,x,\cosh (c+d x)\right )}{80 d}\\ &=-\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^5(c+d x)}{160 d}+\frac {9 a^3 \coth (c+d x) \text {csch}^7(c+d x)}{80 d}-\frac {a^3 \coth (c+d x) \text {csch}^9(c+d x)}{10 d}-\frac {\operatorname {Subst}\left (\int \frac {-15 \left (21 a^3+80 a^2 b+96 a b^2+32 b^3\right )+1440 b^2 (a+b) x^2-1440 b^3 x^4+480 b^3 x^6}{\left (1-x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{480 d}\\ &=\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^3(c+d x)}{128 d}-\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^5(c+d x)}{160 d}+\frac {9 a^3 \coth (c+d x) \text {csch}^7(c+d x)}{80 d}-\frac {a^3 \coth (c+d x) \text {csch}^9(c+d x)}{10 d}+\frac {\operatorname {Subst}\left (\int \frac {15 \left (63 a^3+240 a^2 b+384 a b^2+128 b^3\right )-3840 b^3 x^2+1920 b^3 x^4}{\left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{1920 d}\\ &=-\frac {3 a \left (21 a^2+80 a b+128 b^2\right ) \coth (c+d x) \text {csch}(c+d x)}{256 d}+\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^3(c+d x)}{128 d}-\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^5(c+d x)}{160 d}+\frac {9 a^3 \coth (c+d x) \text {csch}^7(c+d x)}{80 d}-\frac {a^3 \coth (c+d x) \text {csch}^9(c+d x)}{10 d}-\frac {\operatorname {Subst}\left (\int \frac {-15 \left (63 a^3+240 a^2 b+384 a b^2+256 b^3\right )+3840 b^3 x^2}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{3840 d}\\ &=\frac {b^3 \cosh (c+d x)}{d}-\frac {3 a \left (21 a^2+80 a b+128 b^2\right ) \coth (c+d x) \text {csch}(c+d x)}{256 d}+\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^3(c+d x)}{128 d}-\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^5(c+d x)}{160 d}+\frac {9 a^3 \coth (c+d x) \text {csch}^7(c+d x)}{80 d}-\frac {a^3 \coth (c+d x) \text {csch}^9(c+d x)}{10 d}+\frac {\left (3 a \left (21 a^2+80 a b+128 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{256 d}\\ &=\frac {3 a \left (21 a^2+80 a b+128 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{256 d}+\frac {b^3 \cosh (c+d x)}{d}-\frac {3 a \left (21 a^2+80 a b+128 b^2\right ) \coth (c+d x) \text {csch}(c+d x)}{256 d}+\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^3(c+d x)}{128 d}-\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^5(c+d x)}{160 d}+\frac {9 a^3 \coth (c+d x) \text {csch}^7(c+d x)}{80 d}-\frac {a^3 \coth (c+d x) \text {csch}^9(c+d x)}{10 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.46, size = 265, normalized size = 1.40 \[ \frac {b^3 \cosh (c+d x)}{d}-\frac {a \left (60 \left (21 a^2+80 a b+128 b^2\right ) \text {csch}^2\left (\frac {1}{2} (c+d x)\right )+60 \left (21 a^2+80 a b+128 b^2\right ) \text {sech}^2\left (\frac {1}{2} (c+d x)\right )+240 \left (21 a^2+80 a b+128 b^2\right ) \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+2 a^2 \text {csch}^{10}\left (\frac {1}{2} (c+d x)\right )-15 a^2 \text {csch}^8\left (\frac {1}{2} (c+d x)\right )+2 a^2 \text {sech}^{10}\left (\frac {1}{2} (c+d x)\right )+15 a^2 \text {sech}^8\left (\frac {1}{2} (c+d x)\right )+10 a (7 a+16 b) \text {csch}^6\left (\frac {1}{2} (c+d x)\right )-40 a (7 a+24 b) \text {csch}^4\left (\frac {1}{2} (c+d x)\right )+10 a (7 a+16 b) \text {sech}^6\left (\frac {1}{2} (c+d x)\right )+40 a (7 a+24 b) \text {sech}^4\left (\frac {1}{2} (c+d x)\right )\right )}{20480 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [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]
________________________________________________________________________________________
giac [B] time = 0.53, size = 477, normalized size = 2.52 \[ \frac {1280 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 15 \, {\left (21 \, a^{3} + 80 \, a^{2} b + 128 \, a b^{2}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - 15 \, {\left (21 \, a^{3} + 80 \, a^{2} b + 128 \, a b^{2}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {4 \, {\left (315 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{9} + 1200 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{9} + 1920 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{9} - 5880 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} - 22400 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} - 30720 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} + 43008 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 163840 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 184320 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 151680 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 542720 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 491520 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 247040 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 675840 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 491520 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4\right )}^{5}}}{2560 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.16, size = 166, normalized size = 0.88 \[ \frac {a^{3} \left (\left (-\frac {\mathrm {csch}\left (d x +c \right )^{9}}{10}+\frac {9 \mathrm {csch}\left (d x +c \right )^{7}}{80}-\frac {21 \mathrm {csch}\left (d x +c \right )^{5}}{160}+\frac {21 \mathrm {csch}\left (d x +c \right )^{3}}{128}-\frac {63 \,\mathrm {csch}\left (d x +c \right )}{256}\right ) \coth \left (d x +c \right )+\frac {63 \arctanh \left ({\mathrm e}^{d x +c}\right )}{128}\right )+3 a^{2} b \left (\left (-\frac {\mathrm {csch}\left (d x +c \right )^{5}}{6}+\frac {5 \mathrm {csch}\left (d x +c \right )^{3}}{24}-\frac {5 \,\mathrm {csch}\left (d x +c \right )}{16}\right ) \coth \left (d x +c \right )+\frac {5 \arctanh \left ({\mathrm e}^{d x +c}\right )}{8}\right )+3 a \,b^{2} \left (-\frac {\mathrm {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\arctanh \left ({\mathrm e}^{d x +c}\right )\right )+b^{3} \cosh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.34, size = 573, normalized size = 3.03 \[ \frac {1}{2} \, b^{3} {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {1}{1280} \, a^{3} {\left (\frac {315 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {315 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (315 \, e^{\left (-d x - c\right )} - 3045 \, e^{\left (-3 \, d x - 3 \, c\right )} + 13188 \, e^{\left (-5 \, d x - 5 \, c\right )} - 33660 \, e^{\left (-7 \, d x - 7 \, c\right )} + 55970 \, e^{\left (-9 \, d x - 9 \, c\right )} + 55970 \, e^{\left (-11 \, d x - 11 \, c\right )} - 33660 \, e^{\left (-13 \, d x - 13 \, c\right )} + 13188 \, e^{\left (-15 \, d x - 15 \, c\right )} - 3045 \, e^{\left (-17 \, d x - 17 \, c\right )} + 315 \, e^{\left (-19 \, d x - 19 \, c\right )}\right )}}{d {\left (10 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} + 120 \, e^{\left (-6 \, d x - 6 \, c\right )} - 210 \, e^{\left (-8 \, d x - 8 \, c\right )} + 252 \, e^{\left (-10 \, d x - 10 \, c\right )} - 210 \, e^{\left (-12 \, d x - 12 \, c\right )} + 120 \, e^{\left (-14 \, d x - 14 \, c\right )} - 45 \, e^{\left (-16 \, d x - 16 \, c\right )} + 10 \, e^{\left (-18 \, d x - 18 \, c\right )} - e^{\left (-20 \, d x - 20 \, c\right )} - 1\right )}}\right )} + \frac {1}{16} \, a^{2} b {\left (\frac {15 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {15 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (15 \, e^{\left (-d x - c\right )} - 85 \, e^{\left (-3 \, d x - 3 \, c\right )} + 198 \, e^{\left (-5 \, d x - 5 \, c\right )} + 198 \, e^{\left (-7 \, d x - 7 \, c\right )} - 85 \, e^{\left (-9 \, d x - 9 \, c\right )} + 15 \, e^{\left (-11 \, d x - 11 \, c\right )}\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} - 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} - 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} - e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}}\right )} + \frac {3}{2} \, a b^{2} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.11, size = 1194, normalized size = 6.32 \[ \text {result too large to display} \]
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]
________________________________________________________________________________________