Optimal. Leaf size=487 \[ \frac {\left (\sqrt {a}+\sqrt {a+b}\right ) \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}-\sqrt {2} (a+b)^{3/4} \cot (x)}{\sqrt [4]{a} \sqrt {a+b+\sqrt {a} \sqrt {a+b}}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {a+b+\sqrt {a} \sqrt {a+b}}}-\frac {\left (\sqrt {a}+\sqrt {a+b}\right ) \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}+\sqrt {2} (a+b)^{3/4} \cot (x)}{\sqrt [4]{a} \sqrt {a+b+\sqrt {a} \sqrt {a+b}}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {a+b+\sqrt {a} \sqrt {a+b}}}-\frac {\left (\sqrt {a}-\sqrt {a+b}\right ) \log \left (\sqrt {a} \sqrt [4]{a+b}-\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} \cot (x)+(a+b)^{3/4} \cot ^2(x)\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}+\frac {\left (\sqrt {a}-\sqrt {a+b}\right ) \log \left (\sqrt {a} \sqrt [4]{a+b}+\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} \cot (x)+(a+b)^{3/4} \cot ^2(x)\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}} \]
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Rubi [A]
time = 0.76, antiderivative size = 487, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3288, 1183,
648, 632, 210, 642} \begin {gather*} \frac {\left (\sqrt {a+b}+\sqrt {a}\right ) \text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}-\sqrt {2} (a+b)^{3/4} \cot (x)}{\sqrt [4]{a} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}-\frac {\left (\sqrt {a+b}+\sqrt {a}\right ) \text {ArcTan}\left (\frac {\sqrt {2} (a+b)^{3/4} \cot (x)+\sqrt [4]{a} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}{\sqrt [4]{a} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}-\frac {\left (\sqrt {a}-\sqrt {a+b}\right ) \log \left ((a+b)^{3/4} \cot ^2(x)-\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \cot (x)+\sqrt {a} \sqrt [4]{a+b}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}+\frac {\left (\sqrt {a}-\sqrt {a+b}\right ) \log \left ((a+b)^{3/4} \cot ^2(x)+\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \cot (x)+\sqrt {a} \sqrt [4]{a+b}\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1183
Rule 3288
Rubi steps
\begin {align*} \int \frac {1}{a+b \cos ^4(x)} \, dx &=-\text {Subst}\left (\int \frac {1+x^2}{a+2 a x^2+(a+b) x^4} \, dx,x,\cot (x)\right )\\ &=-\frac {\sqrt [4]{a+b} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}{(a+b)^{3/4}}-\left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) x}{\frac {\sqrt {a}}{\sqrt {a+b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} x}{(a+b)^{3/4}}+x^2} \, dx,x,\cot (x)\right )}{2 \sqrt {2} a^{3/4} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}-\frac {\sqrt [4]{a+b} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}{(a+b)^{3/4}}+\left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) x}{\frac {\sqrt {a}}{\sqrt {a+b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} x}{(a+b)^{3/4}}+x^2} \, dx,x,\cot (x)\right )}{2 \sqrt {2} a^{3/4} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}\\ &=-\frac {\left (1+\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {a+b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} x}{(a+b)^{3/4}}+x^2} \, dx,x,\cot (x)\right )}{4 (a+b)}-\frac {\left (1+\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {a+b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} x}{(a+b)^{3/4}}+x^2} \, dx,x,\cot (x)\right )}{4 (a+b)}-\frac {\left (\sqrt [4]{a+b} \left (-1+\frac {\sqrt {a}}{\sqrt {a+b}}\right )\right ) \text {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}{(a+b)^{3/4}}+2 x}{\frac {\sqrt {a}}{\sqrt {a+b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} x}{(a+b)^{3/4}}+x^2} \, dx,x,\cot (x)\right )}{4 \sqrt {2} a^{3/4} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}+\frac {\left (\sqrt {a}-\sqrt {a+b}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}{(a+b)^{3/4}}+2 x}{\frac {\sqrt {a}}{\sqrt {a+b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} x}{(a+b)^{3/4}}+x^2} \, dx,x,\cot (x)\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}\\ &=\frac {\sqrt [4]{a+b} \left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \log \left (\sqrt {a} \sqrt [4]{a+b}-\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} \cot (x)+(a+b)^{3/4} \cot ^2(x)\right )}{4 \sqrt {2} a^{3/4} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}+\frac {\left (\sqrt {a}-\sqrt {a+b}\right ) \log \left (\sqrt {a} \sqrt [4]{a+b}+\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} \cot (x)+(a+b)^{3/4} \cot ^2(x)\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}+\frac {\left (1+\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \text {Subst}\left (\int \frac {1}{-\frac {2 \sqrt {a} \left (a+b+\sqrt {a} \sqrt {a+b}\right )}{(a+b)^{3/2}}-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}{(a+b)^{3/4}}+2 \cot (x)\right )}{2 (a+b)}+\frac {\left (1+\frac {\sqrt {a+b}}{\sqrt {a}}\right ) \text {Subst}\left (\int \frac {1}{-\frac {2 \sqrt {a} \left (a+b+\sqrt {a} \sqrt {a+b}\right )}{(a+b)^{3/2}}-x^2} \, dx,x,\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}{(a+b)^{3/4}}+2 \cot (x)\right )}{2 (a+b)}\\ &=\frac {\left (\sqrt {a}+\sqrt {a+b}\right ) \tan ^{-1}\left (\frac {(a+b)^{3/4} \left (\frac {\sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}{(a+b)^{3/4}}-\sqrt {2} \cot (x)\right )}{\sqrt [4]{a} \sqrt {a+b+\sqrt {a} \sqrt {a+b}}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {a+b+\sqrt {a} \sqrt {a+b}}}-\frac {\left (\sqrt {a}+\sqrt {a+b}\right ) \tan ^{-1}\left (\frac {(a+b)^{3/4} \left (\frac {\sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}{(a+b)^{3/4}}+\sqrt {2} \cot (x)\right )}{\sqrt [4]{a} \sqrt {a+b+\sqrt {a} \sqrt {a+b}}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {a+b+\sqrt {a} \sqrt {a+b}}}+\frac {\sqrt [4]{a+b} \left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \log \left (\sqrt {a} \sqrt [4]{a+b}-\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} \cot (x)+(a+b)^{3/4} \cot ^2(x)\right )}{4 \sqrt {2} a^{3/4} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}+\frac {\left (\sqrt {a}-\sqrt {a+b}\right ) \log \left (\sqrt {a} \sqrt [4]{a+b}+\sqrt {2} \sqrt [4]{a} \sqrt {a+b-\sqrt {a} \sqrt {a+b}} \cot (x)+(a+b)^{3/4} \cot ^2(x)\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{a+b} \sqrt {a+b-\sqrt {a} \sqrt {a+b}}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.26, size = 121, normalized size = 0.25 \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {a+i \sqrt {a} \sqrt {b}}}\right )}{2 \sqrt {a} \sqrt {a+i \sqrt {a} \sqrt {b}}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {-a+i \sqrt {a} \sqrt {b}}}\right )}{2 \sqrt {a} \sqrt {-a+i \sqrt {a} \sqrt {b}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1232\) vs.
\(2(343)=686\).
time = 0.21, size = 1233, normalized size = 2.53
method | result | size |
risch | \(\munderset {\textit {\_R} =\RootOf \left (1+\left (256 a^{4}+256 a^{3} b \right ) \textit {\_Z}^{4}+32 a^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (\frac {128 i a^{4}}{b}+128 i a^{3}\right ) \textit {\_R}^{3}+\left (-\frac {32 a^{3}}{b}-32 a^{2}\right ) \textit {\_R}^{2}+\left (\frac {8 i a^{2}}{b}-8 i a \right ) \textit {\_R} -\frac {2 a}{b}+1\right )\) | \(101\) |
default | \(\frac {\frac {\left (-a^{\frac {5}{2}} \sqrt {2 \sqrt {a^{2}+a b}-2 a}-a^{\frac {3}{2}} \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}-a^{\frac {3}{2}} \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, b -\sqrt {a}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}\, b +\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}\, a +\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}\, b +\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a^{2}+\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a b \right ) \ln \left (\sqrt {a}\, \left (\tan ^{2}\left (x \right )\right )-\tan \left (x \right ) \sqrt {2 \sqrt {\left (a +b \right ) a}-2 a}+\sqrt {a +b}\right )}{2 \sqrt {a}}+\frac {2 \left (2 a^{2} b +2 b^{2} a +\frac {\left (-a^{\frac {5}{2}} \sqrt {2 \sqrt {a^{2}+a b}-2 a}-a^{\frac {3}{2}} \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}-a^{\frac {3}{2}} \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, b -\sqrt {a}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}\, b +\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}\, a +\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}\, b +\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a^{2}+\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a b \right ) \sqrt {2 \sqrt {\left (a +b \right ) a}-2 a}}{2 \sqrt {a}}\right ) \arctan \left (\frac {2 \sqrt {a}\, \tan \left (x \right )-\sqrt {2 \sqrt {\left (a +b \right ) a}-2 a}}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {\left (a +b \right ) a}+2 a}}\right )}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {\left (a +b \right ) a}+2 a}}}{4 a b \left (a +b \right )^{\frac {3}{2}}}+\frac {\frac {\left (a^{\frac {5}{2}} \sqrt {2 \sqrt {a^{2}+a b}-2 a}+a^{\frac {3}{2}} \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}+a^{\frac {3}{2}} \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, b +\sqrt {a}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}\, b -\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}\, a -\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}\, b -\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a^{2}-\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a b \right ) \ln \left (\sqrt {a}\, \left (\tan ^{2}\left (x \right )\right )+\tan \left (x \right ) \sqrt {2 \sqrt {\left (a +b \right ) a}-2 a}+\sqrt {a +b}\right )}{2 \sqrt {a}}+\frac {2 \left (2 a^{2} b +2 b^{2} a -\frac {\left (a^{\frac {5}{2}} \sqrt {2 \sqrt {a^{2}+a b}-2 a}+a^{\frac {3}{2}} \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}+a^{\frac {3}{2}} \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, b +\sqrt {a}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}\, b -\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}\, a -\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, \sqrt {a^{2}+a b}\, b -\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a^{2}-\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+a b}-2 a}\, a b \right ) \sqrt {2 \sqrt {\left (a +b \right ) a}-2 a}}{2 \sqrt {a}}\right ) \arctan \left (\frac {2 \sqrt {a}\, \tan \left (x \right )+\sqrt {2 \sqrt {\left (a +b \right ) a}-2 a}}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {\left (a +b \right ) a}+2 a}}\right )}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {\left (a +b \right ) a}+2 a}}}{4 a b \left (a +b \right )^{\frac {3}{2}}}\) | \(1233\) |
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. 809 vs.
\(2 (344) = 688\).
time = 0.53, size = 809, normalized size = 1.66 \begin {gather*} -\frac {1}{8} \, \sqrt {-\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} \log \left (b \cos \left (x\right )^{2} + 2 \, {\left (a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{4} + a^{3} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} \cos \left (x\right ) \sin \left (x\right )\right )} \sqrt {-\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} - {\left (a^{3} + a^{2} b - 2 \, {\left (a^{3} + a^{2} b\right )} \cos \left (x\right )^{2}\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}}\right ) + \frac {1}{8} \, \sqrt {-\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} \log \left (b \cos \left (x\right )^{2} - 2 \, {\left (a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{4} + a^{3} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} \cos \left (x\right ) \sin \left (x\right )\right )} \sqrt {-\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} + a b}} - {\left (a^{3} + a^{2} b - 2 \, {\left (a^{3} + a^{2} b\right )} \cos \left (x\right )^{2}\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}}\right ) + \frac {1}{8} \, \sqrt {\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} \log \left (-b \cos \left (x\right )^{2} + 2 \, {\left (a b \cos \left (x\right ) \sin \left (x\right ) - {\left (a^{4} + a^{3} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} \cos \left (x\right ) \sin \left (x\right )\right )} \sqrt {\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} - {\left (a^{3} + a^{2} b - 2 \, {\left (a^{3} + a^{2} b\right )} \cos \left (x\right )^{2}\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}}\right ) - \frac {1}{8} \, \sqrt {\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} \log \left (-b \cos \left (x\right )^{2} - 2 \, {\left (a b \cos \left (x\right ) \sin \left (x\right ) - {\left (a^{4} + a^{3} b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} \cos \left (x\right ) \sin \left (x\right )\right )} \sqrt {\frac {{\left (a^{2} + a b\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} + a b}} - {\left (a^{3} + a^{2} b - 2 \, {\left (a^{3} + a^{2} b\right )} \cos \left (x\right )^{2}\right )} \sqrt {-\frac {b}{a^{5} + 2 \, a^{4} b + a^{3} b^{2}}}\right ) \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 [A]
time = 0.43, size = 307, normalized size = 0.63 \begin {gather*} \frac {{\left (3 \, \sqrt {a^{2} + \sqrt {-a b} a} a^{2} + 4 \, \sqrt {a^{2} + \sqrt {-a b} a} a b - 3 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} a - 4 \, \sqrt {a^{2} + \sqrt {-a b} a} \sqrt {-a b} b\right )} {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2 \, \tan \left (x\right )}{\sqrt {\frac {4 \, a + \sqrt {-16 \, {\left (a + b\right )} a + 16 \, a^{2}}}{a}}}\right )\right )} {\left | a \right |}}{2 \, {\left (3 \, a^{5} + 7 \, a^{4} b + 4 \, a^{3} b^{2}\right )}} + \frac {{\left (3 \, \sqrt {a^{2} - \sqrt {-a b} a} a^{2} + 4 \, \sqrt {a^{2} - \sqrt {-a b} a} a b - 3 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} a - 4 \, \sqrt {a^{2} - \sqrt {-a b} a} \sqrt {-a b} b\right )} {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2 \, \tan \left (x\right )}{\sqrt {\frac {4 \, a - \sqrt {-16 \, {\left (a + b\right )} a + 16 \, a^{2}}}{a}}}\right )\right )} {\left | a \right |}}{2 \, {\left (3 \, a^{5} + 7 \, a^{4} b + 4 \, a^{3} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.66, size = 926, normalized size = 1.90 \begin {gather*} -2\,\mathrm {atanh}\left (\frac {8\,a^6\,b\,\mathrm {tan}\left (x\right )\,\sqrt {-\frac {a^2}{16\,\left (a^4+b\,a^3\right )}-\frac {\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}}}{\frac {2\,a^9\,b}{a^4+b\,a^3}-2\,a^4\,b^2-2\,a^5\,b+\frac {2\,a^8\,b^2}{a^4+b\,a^3}+\frac {2\,a^7\,b\,\sqrt {-a^3\,b}}{a^4+b\,a^3}+\frac {2\,a^6\,b^2\,\sqrt {-a^3\,b}}{a^4+b\,a^3}}-\frac {8\,a^2\,b\,\mathrm {tan}\left (x\right )\,\sqrt {-\frac {a^2}{16\,\left (a^4+b\,a^3\right )}-\frac {\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}}}{\frac {2\,a^5\,b}{a^4+b\,a^3}-2\,a\,b+\frac {2\,a^3\,b\,\sqrt {-a^3\,b}}{a^4+b\,a^3}}+\frac {8\,a^4\,b\,\mathrm {tan}\left (x\right )\,\sqrt {-a^3\,b}\,\sqrt {-\frac {a^2}{16\,\left (a^4+b\,a^3\right )}-\frac {\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}}}{\frac {2\,a^9\,b}{a^4+b\,a^3}-2\,a^4\,b^2-2\,a^5\,b+\frac {2\,a^8\,b^2}{a^4+b\,a^3}+\frac {2\,a^7\,b\,\sqrt {-a^3\,b}}{a^4+b\,a^3}+\frac {2\,a^6\,b^2\,\sqrt {-a^3\,b}}{a^4+b\,a^3}}\right )\,\sqrt {-\frac {a^2+\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}}-2\,\mathrm {atanh}\left (\frac {8\,a^2\,b\,\mathrm {tan}\left (x\right )\,\sqrt {\frac {\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}-\frac {a^2}{16\,\left (a^4+b\,a^3\right )}}}{2\,a\,b-\frac {2\,a^5\,b}{a^4+b\,a^3}+\frac {2\,a^3\,b\,\sqrt {-a^3\,b}}{a^4+b\,a^3}}-\frac {8\,a^6\,b\,\mathrm {tan}\left (x\right )\,\sqrt {\frac {\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}-\frac {a^2}{16\,\left (a^4+b\,a^3\right )}}}{2\,a^5\,b+2\,a^4\,b^2-\frac {2\,a^9\,b}{a^4+b\,a^3}-\frac {2\,a^8\,b^2}{a^4+b\,a^3}+\frac {2\,a^7\,b\,\sqrt {-a^3\,b}}{a^4+b\,a^3}+\frac {2\,a^6\,b^2\,\sqrt {-a^3\,b}}{a^4+b\,a^3}}+\frac {8\,a^4\,b\,\mathrm {tan}\left (x\right )\,\sqrt {-a^3\,b}\,\sqrt {\frac {\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}-\frac {a^2}{16\,\left (a^4+b\,a^3\right )}}}{2\,a^5\,b+2\,a^4\,b^2-\frac {2\,a^9\,b}{a^4+b\,a^3}-\frac {2\,a^8\,b^2}{a^4+b\,a^3}+\frac {2\,a^7\,b\,\sqrt {-a^3\,b}}{a^4+b\,a^3}+\frac {2\,a^6\,b^2\,\sqrt {-a^3\,b}}{a^4+b\,a^3}}\right )\,\sqrt {-\frac {a^2-\sqrt {-a^3\,b}}{16\,\left (a^4+b\,a^3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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