Optimal. Leaf size=136 \[ -\frac {\sqrt [4]{b} \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a \sqrt {\sqrt {a}+\sqrt {b}} d} \]
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Rubi [A]
time = 0.14, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3294, 1184,
213, 1180, 211, 214} \begin {gather*} -\frac {\sqrt [4]{b} \text {ArcTan}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 213
Rule 214
Rule 1180
Rule 1184
Rule 3294
Rubi steps
\begin {align*} \int \frac {\text {csch}(c+d x)}{a-b \sinh ^4(c+d x)} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a-b+2 b x^2-b x^4\right )} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \left (-\frac {1}{a \left (-1+x^2\right )}+\frac {b-b x^2}{a \left (a-b+2 b x^2-b x^4\right )}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\cosh (c+d x)\right )}{a d}-\frac {\text {Subst}\left (\int \frac {b-b x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{a d}\\ &=-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {b \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{2 a d}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{2 a d}\\ &=-\frac {\sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {\sqrt [4]{b} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a \sqrt {\sqrt {a}+\sqrt {b}} d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.18, size = 385, normalized size = 2.83 \begin {gather*} \frac {8 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )-b \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-c-d x-2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+3 c \text {$\#$1}^2+3 d x \text {$\#$1}^2+6 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-3 c \text {$\#$1}^4-3 d x \text {$\#$1}^4-6 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+c \text {$\#$1}^6+d x \text {$\#$1}^6+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{8 a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.98, size = 164, normalized size = 1.21
method | result | size |
risch | \(\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{d a}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{d a}+2 \left (\munderset {\textit {\_R} =\RootOf \left (\left (4096 a^{5} d^{4}-4096 a^{4} b \,d^{4}\right ) \textit {\_Z}^{4}+128 a^{2} b \,d^{2} \textit {\_Z}^{2}-b \right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 d x +2 c}+\left (\left (\frac {1024 d^{3} a^{4}}{b}-1024 a^{3} d^{3}\right ) \textit {\_R}^{3}+32 a d \textit {\_R} \right ) {\mathrm e}^{d x +c}+1\right )\right )\) | \(130\) |
derivativedivides | \(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+8 b \left (-\frac {\sqrt {a b}\, \arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{16 a b \sqrt {-\sqrt {a b}\, a -a b}}-\frac {\sqrt {a b}\, \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{16 a b \sqrt {\sqrt {a b}\, a -a b}}\right )}{d}\) | \(164\) |
default | \(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+8 b \left (-\frac {\sqrt {a b}\, \arctan \left (\frac {-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}+2 a}{4 \sqrt {-\sqrt {a b}\, a -a b}}\right )}{16 a b \sqrt {-\sqrt {a b}\, a -a b}}-\frac {\sqrt {a b}\, \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \sqrt {a b}-2 a}{4 \sqrt {\sqrt {a b}\, a -a b}}\right )}{16 a b \sqrt {\sqrt {a b}\, a -a b}}\right )}{d}\) | \(164\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1067 vs.
\(2 (100) = 200\).
time = 0.43, size = 1067, normalized size = 7.85 \begin {gather*} \frac {a d \sqrt {-\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} \log \left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + 2 \, {\left (a b d \cosh \left (d x + c\right ) + a b d \sinh \left (d x + c\right ) - {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \cosh \left (d x + c\right ) + {\left (a^{4} - a^{3} b\right )} d^{3} \sinh \left (d x + c\right )\right )} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}}\right )} \sqrt {-\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} + b\right ) - a d \sqrt {-\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} \log \left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} - 2 \, {\left (a b d \cosh \left (d x + c\right ) + a b d \sinh \left (d x + c\right ) - {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \cosh \left (d x + c\right ) + {\left (a^{4} - a^{3} b\right )} d^{3} \sinh \left (d x + c\right )\right )} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}}\right )} \sqrt {-\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} + b\right ) + a d \sqrt {\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} \log \left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + 2 \, {\left (a b d \cosh \left (d x + c\right ) + a b d \sinh \left (d x + c\right ) + {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \cosh \left (d x + c\right ) + {\left (a^{4} - a^{3} b\right )} d^{3} \sinh \left (d x + c\right )\right )} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}}\right )} \sqrt {\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} + b\right ) - a d \sqrt {\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} \log \left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} - 2 \, {\left (a b d \cosh \left (d x + c\right ) + a b d \sinh \left (d x + c\right ) + {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \cosh \left (d x + c\right ) + {\left (a^{4} - a^{3} b\right )} d^{3} \sinh \left (d x + c\right )\right )} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}}\right )} \sqrt {\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} + b\right ) - 4 \, \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + 4 \, \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right )}{4 \, a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 419 vs.
\(2 (100) = 200\).
time = 0.47, size = 419, normalized size = 3.08 \begin {gather*} \frac {\frac {{\left ({\left (4 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a + 5 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} b\right )} {\left | a \right |} {\left | b \right |} - {\left (4 \, \sqrt {-b^{2} - \sqrt {a b} b} a^{2} b + 5 \, \sqrt {-b^{2} - \sqrt {a b} b} a b^{2}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a b + \sqrt {a^{2} b^{2} + {\left (a^{2} - a b\right )} a b}}{a b}}}\right )}{4 \, a^{4} b^{2} + a^{3} b^{3} - 5 \, a^{2} b^{4}} - \frac {{\left ({\left (4 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a + 5 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} b\right )} {\left | a \right |} {\left | b \right |} + {\left (4 \, \sqrt {-b^{2} + \sqrt {a b} b} a^{2} b + 5 \, \sqrt {-b^{2} + \sqrt {a b} b} a b^{2}\right )} {\left | b \right |}\right )} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {a b - \sqrt {a^{2} b^{2} + {\left (a^{2} - a b\right )} a b}}{a b}}}\right )}{4 \, a^{4} b^{2} + a^{3} b^{3} - 5 \, a^{2} b^{4}} - \frac {\log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right )}{a} + \frac {\log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right )}{a}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.19, size = 1243, normalized size = 9.14 \begin {gather*} \ln \left (\frac {\left (\frac {\left (\frac {4294967296\,a\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (-49\,a^2+26\,a\,b+15\,b^2\right )}{b^6\,{\left (a-b\right )}^3}+\frac {8589934592\,a^2\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (16\,a^2+3\,a\,b-15\,b^2\right )\,\sqrt {-\frac {a^2\,b+\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{b^7\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a^2\,b+\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{4}-\frac {2147483648\,d\,{\mathrm {e}}^{c+d\,x}\,\left (17\,a-15\,b\right )}{b^6\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a^2\,b+\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{4}+\frac {268435456\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (16\,a^2+3\,a\,b-15\,b^2\right )}{a\,b^6\,{\left (a-b\right )}^3}\right )\,\sqrt {-\frac {a^2\,b+\sqrt {a^5\,b}}{16\,\left (a^5\,d^2-a^4\,b\,d^2\right )}}-\ln \left (\frac {\left (\frac {\left (\frac {4294967296\,a\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (-49\,a^2+26\,a\,b+15\,b^2\right )}{b^6\,{\left (a-b\right )}^3}-\frac {8589934592\,a^2\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (16\,a^2+3\,a\,b-15\,b^2\right )\,\sqrt {-\frac {a^2\,b+\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{b^7\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a^2\,b+\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{4}+\frac {2147483648\,d\,{\mathrm {e}}^{c+d\,x}\,\left (17\,a-15\,b\right )}{b^6\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a^2\,b+\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{4}+\frac {268435456\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (16\,a^2+3\,a\,b-15\,b^2\right )}{a\,b^6\,{\left (a-b\right )}^3}\right )\,\sqrt {-\frac {a^2\,b+\sqrt {a^5\,b}}{16\,\left (a^5\,d^2-a^4\,b\,d^2\right )}}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (65536\,a^2\,\sqrt {-a^2\,d^2}+50625\,b^2\,\sqrt {-a^2\,d^2}-115200\,a\,b\,\sqrt {-a^2\,d^2}\right )}{65536\,d\,a^3-115200\,d\,a^2\,b+50625\,d\,a\,b^2}\right )}{\sqrt {-a^2\,d^2}}-\ln \left (\frac {\left (\frac {\left (\frac {4294967296\,a\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (-49\,a^2+26\,a\,b+15\,b^2\right )}{b^6\,{\left (a-b\right )}^3}-\frac {8589934592\,a^2\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (16\,a^2+3\,a\,b-15\,b^2\right )\,\sqrt {-\frac {a^2\,b-\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{b^7\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a^2\,b-\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{4}+\frac {2147483648\,d\,{\mathrm {e}}^{c+d\,x}\,\left (17\,a-15\,b\right )}{b^6\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a^2\,b-\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{4}+\frac {268435456\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (16\,a^2+3\,a\,b-15\,b^2\right )}{a\,b^6\,{\left (a-b\right )}^3}\right )\,\sqrt {-\frac {a^2\,b-\sqrt {a^5\,b}}{16\,\left (a^5\,d^2-a^4\,b\,d^2\right )}}+\ln \left (\frac {\left (\frac {\left (\frac {4294967296\,a\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (-49\,a^2+26\,a\,b+15\,b^2\right )}{b^6\,{\left (a-b\right )}^3}+\frac {8589934592\,a^2\,d^3\,{\mathrm {e}}^{c+d\,x}\,\left (16\,a^2+3\,a\,b-15\,b^2\right )\,\sqrt {-\frac {a^2\,b-\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{b^7\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a^2\,b-\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{4}-\frac {2147483648\,d\,{\mathrm {e}}^{c+d\,x}\,\left (17\,a-15\,b\right )}{b^6\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a^2\,b-\sqrt {a^5\,b}}{a^4\,d^2\,\left (a-b\right )}}}{4}+\frac {268435456\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (16\,a^2+3\,a\,b-15\,b^2\right )}{a\,b^6\,{\left (a-b\right )}^3}\right )\,\sqrt {-\frac {a^2\,b-\sqrt {a^5\,b}}{16\,\left (a^5\,d^2-a^4\,b\,d^2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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