Optimal. Leaf size=171 \[ -\frac {a \left (35 a^2+144 a b+384 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{128 d}-\frac {b^3 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^3(c+d x)}{3 d}+\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}(c+d x)}{128 d}-\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}^3(c+d x)}{192 d}+\frac {7 a^3 \coth (c+d x) \text {csch}^5(c+d x)}{48 d}-\frac {a^3 \coth (c+d x) \text {csch}^7(c+d x)}{8 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.23, antiderivative size = 171, 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 = {3294, 1171,
1828, 1167, 212} \begin {gather*} -\frac {a^3 \coth (c+d x) \text {csch}^7(c+d x)}{8 d}+\frac {7 a^3 \coth (c+d x) \text {csch}^5(c+d x)}{48 d}-\frac {a \left (35 a^2+144 a b+384 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{128 d}-\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}^3(c+d x)}{192 d}+\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}(c+d x)}{128 d}+\frac {b^3 \cosh ^3(c+d x)}{3 d}-\frac {b^3 \cosh (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 1167
Rule 1171
Rule 1828
Rule 3294
Rubi steps
\begin {align*} \int \text {csch}^9(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (a+b-2 b x^2+b x^4\right )^3}{\left (1-x^2\right )^5} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a^3 \coth (c+d x) \text {csch}^7(c+d x)}{8 d}+\frac {\text {Subst}\left (\int \frac {-(a+2 b) \left (7 a^2+10 a b+4 b^2\right )+8 b \left (3 a^2+9 a b+5 b^2\right ) x^2-8 b^2 (9 a+10 b) x^4+8 b^2 (3 a+10 b) x^6-40 b^3 x^8+8 b^3 x^{10}}{\left (1-x^2\right )^4} \, dx,x,\cosh (c+d x)\right )}{8 d}\\ &=\frac {7 a^3 \coth (c+d x) \text {csch}^5(c+d x)}{48 d}-\frac {a^3 \coth (c+d x) \text {csch}^7(c+d x)}{8 d}-\frac {\text {Subst}\left (\int \frac {35 a^3+144 a^2 b+144 a b^2+48 b^3-96 b^2 (3 a+2 b) x^2+144 b^2 (a+2 b) x^4-192 b^3 x^6+48 b^3 x^8}{\left (1-x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{48 d}\\ &=-\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}^3(c+d x)}{192 d}+\frac {7 a^3 \coth (c+d x) \text {csch}^5(c+d x)}{48 d}-\frac {a^3 \coth (c+d x) \text {csch}^7(c+d x)}{8 d}+\frac {\text {Subst}\left (\int \frac {-3 \left (35 a^3+144 a^2 b+192 a b^2+64 b^3\right )+576 b^2 (a+b) x^2-576 b^3 x^4+192 b^3 x^6}{\left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{192 d}\\ &=\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}(c+d x)}{128 d}-\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}^3(c+d x)}{192 d}+\frac {7 a^3 \coth (c+d x) \text {csch}^5(c+d x)}{48 d}-\frac {a^3 \coth (c+d x) \text {csch}^7(c+d x)}{8 d}-\frac {\text {Subst}\left (\int \frac {3 \left (35 a^3+144 a^2 b+384 a b^2+128 b^3\right )-768 b^3 x^2+384 b^3 x^4}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{384 d}\\ &=\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}(c+d x)}{128 d}-\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}^3(c+d x)}{192 d}+\frac {7 a^3 \coth (c+d x) \text {csch}^5(c+d x)}{48 d}-\frac {a^3 \coth (c+d x) \text {csch}^7(c+d x)}{8 d}-\frac {\text {Subst}\left (\int \left (384 b^3-384 b^3 x^2+\frac {3 \left (35 a^3+144 a^2 b+384 a b^2\right )}{1-x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{384 d}\\ &=-\frac {b^3 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^3(c+d x)}{3 d}+\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}(c+d x)}{128 d}-\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}^3(c+d x)}{192 d}+\frac {7 a^3 \coth (c+d x) \text {csch}^5(c+d x)}{48 d}-\frac {a^3 \coth (c+d x) \text {csch}^7(c+d x)}{8 d}-\frac {\left (a \left (35 a^2+144 a b+384 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{128 d}\\ &=-\frac {a \left (35 a^2+144 a b+384 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{128 d}-\frac {b^3 \cosh (c+d x)}{d}+\frac {b^3 \cosh ^3(c+d x)}{3 d}+\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}(c+d x)}{128 d}-\frac {a^2 (35 a+144 b) \coth (c+d x) \text {csch}^3(c+d x)}{192 d}+\frac {7 a^3 \coth (c+d x) \text {csch}^5(c+d x)}{48 d}-\frac {a^3 \coth (c+d x) \text {csch}^7(c+d x)}{8 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.27, size = 219, normalized size = 1.28 \begin {gather*} \frac {-4608 b^3 \cosh (c+d x)+512 b^3 \cosh (3 (c+d x))+a \left (12 a (35 a+144 b) \text {csch}^2\left (\frac {1}{2} (c+d x)\right )-18 a (5 a+16 b) \text {csch}^4\left (\frac {1}{2} (c+d x)\right )+20 a^2 \text {csch}^6\left (\frac {1}{2} (c+d x)\right )-3 a^2 \text {csch}^8\left (\frac {1}{2} (c+d x)\right )+48 \left (35 a^2+144 a b+384 b^2\right ) \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+12 a (35 a+144 b) \text {sech}^2\left (\frac {1}{2} (c+d x)\right )+18 a (5 a+16 b) \text {sech}^4\left (\frac {1}{2} (c+d x)\right )+20 a^2 \text {sech}^6\left (\frac {1}{2} (c+d x)\right )+3 a^2 \text {sech}^8\left (\frac {1}{2} (c+d x)\right )\right )}{6144 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(374\) vs.
\(2(159)=318\).
time = 1.52, size = 375, normalized size = 2.19
method | result | size |
risch | \(\frac {{\mathrm e}^{3 d x +3 c} b^{3}}{24 d}-\frac {3 b^{3} {\mathrm e}^{d x +c}}{8 d}-\frac {3 b^{3} {\mathrm e}^{-d x -c}}{8 d}+\frac {b^{3} {\mathrm e}^{-3 d x -3 c}}{24 d}+\frac {a^{2} {\mathrm e}^{d x +c} \left (105 a \,{\mathrm e}^{14 d x +14 c}+432 b \,{\mathrm e}^{14 d x +14 c}-805 a \,{\mathrm e}^{12 d x +12 c}-3312 b \,{\mathrm e}^{12 d x +12 c}+2681 a \,{\mathrm e}^{10 d x +10 c}+7344 b \,{\mathrm e}^{10 d x +10 c}-5053 a \,{\mathrm e}^{8 d x +8 c}-4464 b \,{\mathrm e}^{8 d x +8 c}-5053 a \,{\mathrm e}^{6 d x +6 c}-4464 b \,{\mathrm e}^{6 d x +6 c}+2681 a \,{\mathrm e}^{4 d x +4 c}+7344 b \,{\mathrm e}^{4 d x +4 c}-805 a \,{\mathrm e}^{2 d x +2 c}-3312 b \,{\mathrm e}^{2 d x +2 c}+105 a +432 b \right )}{192 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{8}}+\frac {35 a^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{128 d}+\frac {9 a^{2} b \ln \left ({\mathrm e}^{d x +c}-1\right )}{8 d}+\frac {3 a \ln \left ({\mathrm e}^{d x +c}-1\right ) b^{2}}{d}-\frac {35 a^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{128 d}-\frac {9 a^{2} b \ln \left ({\mathrm e}^{d x +c}+1\right )}{8 d}-\frac {3 a \ln \left ({\mathrm e}^{d x +c}+1\right ) b^{2}}{d}\) | \(375\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 463 vs.
\(2 (159) = 318\).
time = 0.30, size = 463, normalized size = 2.71 \begin {gather*} \frac {1}{24} \, b^{3} {\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 {1}{384} \, a^{3} {\left (\frac {105 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {105 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (105 \, e^{\left (-d x - c\right )} - 805 \, e^{\left (-3 \, d x - 3 \, c\right )} + 2681 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5053 \, e^{\left (-7 \, d x - 7 \, c\right )} - 5053 \, e^{\left (-9 \, d x - 9 \, c\right )} + 2681 \, e^{\left (-11 \, d x - 11 \, c\right )} - 805 \, e^{\left (-13 \, d x - 13 \, c\right )} + 105 \, e^{\left (-15 \, d x - 15 \, c\right )}\right )}}{d {\left (8 \, e^{\left (-2 \, d x - 2 \, c\right )} - 28 \, e^{\left (-4 \, d x - 4 \, c\right )} + 56 \, e^{\left (-6 \, d x - 6 \, c\right )} - 70 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} - 28 \, e^{\left (-12 \, d x - 12 \, c\right )} + 8 \, e^{\left (-14 \, d x - 14 \, c\right )} - e^{\left (-16 \, d x - 16 \, c\right )} - 1\right )}}\right )} - \frac {3}{8} \, a^{2} b {\left (\frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (3 \, e^{\left (-d x - c\right )} - 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} + 3 \, e^{\left (-7 \, d x - 7 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} - 3 \, 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}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 10848 vs.
\(2 (159) = 318\).
time = 0.48, size = 10848, normalized size = 63.44 \begin {gather*} \text {too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 335 vs.
\(2 (159) = 318\).
time = 0.60, size = 335, normalized size = 1.96 \begin {gather*} \frac {32 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 384 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 3 \, {\left (35 \, a^{3} + 144 \, a^{2} b + 384 \, a b^{2}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) + 3 \, {\left (35 \, a^{3} + 144 \, a^{2} b + 384 \, a b^{2}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) + \frac {4 \, {\left (105 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} + 432 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} - 1540 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 6336 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 8176 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 29952 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 17856 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 46080 \, a^{2} b {\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 )}^{4}}}{768 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.12, size = 759, normalized size = 4.44 \begin {gather*} \frac {b^3\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}-\frac {3\,b^3\,{\mathrm {e}}^{-c-d\,x}}{8\,d}-\frac {3\,b^3\,{\mathrm {e}}^{c+d\,x}}{8\,d}+\frac {b^3\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (35\,a^3\,\sqrt {-d^2}+384\,a\,b^2\,\sqrt {-d^2}+144\,a^2\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {1225\,a^6+10080\,a^5\,b+47616\,a^4\,b^2+110592\,a^3\,b^3+147456\,a^2\,b^4}}\right )\,\sqrt {1225\,a^6+10080\,a^5\,b+47616\,a^4\,b^2+110592\,a^3\,b^3+147456\,a^2\,b^4}}{64\,\sqrt {-d^2}}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (35\,a^3+144\,b\,a^2\right )}{64\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^3+48\,b\,a^2\right )}{4\,d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (35\,a^3+144\,b\,a^2\right )}{96\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (432\,a^2\,b-7\,a^3\right )}{24\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {170\,a^3\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1\right )}-\frac {404\,a^3\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (15\,{\mathrm {e}}^{4\,c+4\,d\,x}-6\,{\mathrm {e}}^{2\,c+2\,d\,x}-20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}-6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1\right )}-\frac {112\,a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (7\,{\mathrm {e}}^{2\,c+2\,d\,x}-21\,{\mathrm {e}}^{4\,c+4\,d\,x}+35\,{\mathrm {e}}^{6\,c+6\,d\,x}-35\,{\mathrm {e}}^{8\,c+8\,d\,x}+21\,{\mathrm {e}}^{10\,c+10\,d\,x}-7\,{\mathrm {e}}^{12\,c+12\,d\,x}+{\mathrm {e}}^{14\,c+14\,d\,x}-1\right )}-\frac {32\,a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (28\,{\mathrm {e}}^{4\,c+4\,d\,x}-8\,{\mathrm {e}}^{2\,c+2\,d\,x}-56\,{\mathrm {e}}^{6\,c+6\,d\,x}+70\,{\mathrm {e}}^{8\,c+8\,d\,x}-56\,{\mathrm {e}}^{10\,c+10\,d\,x}+28\,{\mathrm {e}}^{12\,c+12\,d\,x}-8\,{\mathrm {e}}^{14\,c+14\,d\,x}+{\mathrm {e}}^{16\,c+16\,d\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________