Optimal. Leaf size=84 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2}}+\frac {b}{a (a-b) \sqrt {a+b \cot ^2(x)}} \]
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Rubi [A]
time = 0.09, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3751, 457, 87,
162, 65, 214} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {b}{a (a-b) \sqrt {a+b \cot ^2(x)}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 87
Rule 162
Rule 214
Rule 457
Rule 3751
Rubi steps
\begin {align*} \int \frac {\tan (x)}{\left (a+b \cot ^2(x)\right )^{3/2}} \, dx &=-\text {Subst}\left (\int \frac {1}{x \left (1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\cot (x)\right )\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{x (1+x) (a+b x)^{3/2}} \, dx,x,\cot ^2(x)\right )\right )\\ &=\frac {b}{a (a-b) \sqrt {a+b \cot ^2(x)}}-\frac {\text {Subst}\left (\int \frac {a-b-b x}{x (1+x) \sqrt {a+b x}} \, dx,x,\cot ^2(x)\right )}{2 a (a-b)}\\ &=\frac {b}{a (a-b) \sqrt {a+b \cot ^2(x)}}-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\cot ^2(x)\right )}{2 a}+\frac {\text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,\cot ^2(x)\right )}{2 (a-b)}\\ &=\frac {b}{a (a-b) \sqrt {a+b \cot ^2(x)}}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cot ^2(x)}\right )}{a b}+\frac {\text {Subst}\left (\int \frac {1}{1-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cot ^2(x)}\right )}{(a-b) b}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2}}+\frac {b}{a (a-b) \sqrt {a+b \cot ^2(x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.06, size = 75, normalized size = 0.89 \begin {gather*} \frac {a \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {a+b \cot ^2(x)}{a-b}\right )+(-a+b) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1+\frac {b \cot ^2(x)}{a}\right )}{a (a-b) \sqrt {a+b \cot ^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.66, size = 962, normalized size = 11.45
method | result | size |
default | \(-\frac {\left (\left (\cos ^{2}\left (x \right )\right ) a -b \left (\cos ^{2}\left (x \right )\right )-a \right ) \left (\sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sqrt {a}\, \sqrt {a -b}-a \cos \left (x \right )+b \cos \left (x \right )-\sqrt {a}\, \sqrt {a -b}+a}{\left (\cos \left (x \right )+1\right ) b}}\, \sqrt {-\frac {2 \left (\cos \left (x \right ) \sqrt {a}\, \sqrt {a -b}+a \cos \left (x \right )-b \cos \left (x \right )-\sqrt {a}\, \sqrt {a -b}-a \right )}{\left (\cos \left (x \right )+1\right ) b}}\, \EllipticF \left (\frac {\left (-1+\cos \left (x \right )\right ) \sqrt {\frac {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}{b}}}{\sin \left (x \right )}, \sqrt {\frac {8 a^{\frac {3}{2}} \sqrt {a -b}-4 \sqrt {a}\, \sqrt {a -b}\, b +8 a^{2}-8 a b +b^{2}}{b^{2}}}\right ) b \sin \left (x \right )+2 \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sqrt {a}\, \sqrt {a -b}-a \cos \left (x \right )+b \cos \left (x \right )-\sqrt {a}\, \sqrt {a -b}+a}{\left (\cos \left (x \right )+1\right ) b}}\, \sqrt {-\frac {2 \left (\cos \left (x \right ) \sqrt {a}\, \sqrt {a -b}+a \cos \left (x \right )-b \cos \left (x \right )-\sqrt {a}\, \sqrt {a -b}-a \right )}{\left (\cos \left (x \right )+1\right ) b}}\, \EllipticPi \left (\frac {\left (-1+\cos \left (x \right )\right ) \sqrt {\frac {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}{b}}}{\sin \left (x \right )}, \frac {b}{2 \sqrt {a}\, \sqrt {a -b}-2 a +b}, \frac {\sqrt {-\frac {2 \sqrt {a}\, \sqrt {a -b}+2 a -b}{b}}}{\sqrt {\frac {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}{b}}}\right ) a \sin \left (x \right )-2 \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sqrt {a}\, \sqrt {a -b}-a \cos \left (x \right )+b \cos \left (x \right )-\sqrt {a}\, \sqrt {a -b}+a}{\left (\cos \left (x \right )+1\right ) b}}\, \sqrt {-\frac {2 \left (\cos \left (x \right ) \sqrt {a}\, \sqrt {a -b}+a \cos \left (x \right )-b \cos \left (x \right )-\sqrt {a}\, \sqrt {a -b}-a \right )}{\left (\cos \left (x \right )+1\right ) b}}\, \EllipticPi \left (\frac {\left (-1+\cos \left (x \right )\right ) \sqrt {\frac {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}{b}}}{\sin \left (x \right )}, \frac {b}{2 \sqrt {a}\, \sqrt {a -b}-2 a +b}, \frac {\sqrt {-\frac {2 \sqrt {a}\, \sqrt {a -b}+2 a -b}{b}}}{\sqrt {\frac {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}{b}}}\right ) b \sin \left (x \right )-2 \sqrt {2}\, \sqrt {\frac {\cos \left (x \right ) \sqrt {a}\, \sqrt {a -b}-a \cos \left (x \right )+b \cos \left (x \right )-\sqrt {a}\, \sqrt {a -b}+a}{\left (\cos \left (x \right )+1\right ) b}}\, \sqrt {-\frac {2 \left (\cos \left (x \right ) \sqrt {a}\, \sqrt {a -b}+a \cos \left (x \right )-b \cos \left (x \right )-\sqrt {a}\, \sqrt {a -b}-a \right )}{\left (\cos \left (x \right )+1\right ) b}}\, \EllipticPi \left (\frac {\left (-1+\cos \left (x \right )\right ) \sqrt {\frac {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}{b}}}{\sin \left (x \right )}, -\frac {b}{2 \sqrt {a}\, \sqrt {a -b}-2 a +b}, \frac {\sqrt {-\frac {2 \sqrt {a}\, \sqrt {a -b}+2 a -b}{b}}}{\sqrt {\frac {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}{b}}}\right ) a \sin \left (x \right )+\sqrt {\frac {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}{b}}\, b \cos \left (x \right )-\sqrt {\frac {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}{b}}\, b \right )}{\left (-1+\cos \left (x \right )\right ) \sin \left (x \right )^{2} \left (\frac {\left (\cos ^{2}\left (x \right )\right ) a -b \left (\cos ^{2}\left (x \right )\right )-a}{\cos ^{2}\left (x \right )-1}\right )^{\frac {3}{2}} \sqrt {\frac {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}{b}}\, \left (a -b \right ) a}\) | \(962\) |
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. 201 vs.
\(2 (70) = 140\).
time = 2.89, size = 863, normalized size = 10.27 \begin {gather*} \left [\frac {2 \, {\left (a^{2} b - a b^{2}\right )} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )^{2} + {\left (a^{2} b - 2 \, a b^{2} + b^{3} + {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \tan \left (x\right )^{2}\right )} \sqrt {a} \log \left (2 \, a \tan \left (x\right )^{2} + 2 \, \sqrt {a} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )^{2} + b\right ) - {\left (a^{3} \tan \left (x\right )^{2} + a^{2} b\right )} \sqrt {a - b} \log \left (\frac {{\left (2 \, a - b\right )} \tan \left (x\right )^{2} + 2 \, \sqrt {a - b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2} + 1}\right )}{2 \, {\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} \tan \left (x\right )^{2}\right )}}, \frac {2 \, {\left (a^{2} b - a b^{2}\right )} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )^{2} - 2 \, {\left (a^{3} \tan \left (x\right )^{2} + a^{2} b\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {-a + b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{a - b}\right ) + {\left (a^{2} b - 2 \, a b^{2} + b^{3} + {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \tan \left (x\right )^{2}\right )} \sqrt {a} \log \left (2 \, a \tan \left (x\right )^{2} + 2 \, \sqrt {a} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )^{2} + b\right )}{2 \, {\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} \tan \left (x\right )^{2}\right )}}, \frac {2 \, {\left (a^{2} b - a b^{2}\right )} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )^{2} - 2 \, {\left (a^{2} b - 2 \, a b^{2} + b^{3} + {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \tan \left (x\right )^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{a}\right ) - {\left (a^{3} \tan \left (x\right )^{2} + a^{2} b\right )} \sqrt {a - b} \log \left (\frac {{\left (2 \, a - b\right )} \tan \left (x\right )^{2} + 2 \, \sqrt {a - b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2} + 1}\right )}{2 \, {\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} \tan \left (x\right )^{2}\right )}}, \frac {{\left (a^{2} b - a b^{2}\right )} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}} \tan \left (x\right )^{2} - {\left (a^{2} b - 2 \, a b^{2} + b^{3} + {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \tan \left (x\right )^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{a}\right ) - {\left (a^{3} \tan \left (x\right )^{2} + a^{2} b\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {-a + b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{a - b}\right )}{a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} \tan \left (x\right )^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tan {\left (x \right )}}{\left (a + b \cot ^{2}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \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. 340 vs.
\(2 (70) = 140\).
time = 0.49, size = 340, normalized size = 4.05 \begin {gather*} -\frac {{\left (2 \, \sqrt {a - b} a^{2} \arctan \left (-\frac {a - b}{\sqrt {-a^{2} + a b}}\right ) - 4 \, \sqrt {a - b} a b \arctan \left (-\frac {a - b}{\sqrt {-a^{2} + a b}}\right ) + 2 \, \sqrt {a - b} b^{2} \arctan \left (-\frac {a - b}{\sqrt {-a^{2} + a b}}\right ) + \sqrt {-a^{2} + a b} \sqrt {a - b} a \log \left (b\right )\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{2 \, {\left (\sqrt {-a^{2} + a b} a^{3} - 2 \, \sqrt {-a^{2} + a b} a^{2} b + \sqrt {-a^{2} + a b} a b^{2}\right )}} + \frac {\frac {\sqrt {a - b} \log \left ({\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2}\right )}{a^{2} - 2 \, a b + b^{2}} + \frac {2 \, b \sin \left (x\right )}{\sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b} {\left (a^{2} - a b\right )}} + \frac {2 \, \sqrt {a - b} \arctan \left (\frac {{\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2} - 2 \, a + b}{2 \, \sqrt {-a^{2} + a b}}\right )}{\sqrt {-a^{2} + a b} a}}{2 \, \mathrm {sgn}\left (\sin \left (x\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.48, size = 1451, normalized size = 17.27 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {2\,a^2\,b^8\,\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3+24\,a^5\,b^4-38\,a^4\,b^5+30\,a^3\,b^6-12\,a^2\,b^7+2\,a\,b^8\right )}-\frac {12\,a^3\,b^7\,\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3+24\,a^5\,b^4-38\,a^4\,b^5+30\,a^3\,b^6-12\,a^2\,b^7+2\,a\,b^8\right )}+\frac {30\,a^4\,b^6\,\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3+24\,a^5\,b^4-38\,a^4\,b^5+30\,a^3\,b^6-12\,a^2\,b^7+2\,a\,b^8\right )}-\frac {38\,a^5\,b^5\,\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3+24\,a^5\,b^4-38\,a^4\,b^5+30\,a^3\,b^6-12\,a^2\,b^7+2\,a\,b^8\right )}+\frac {24\,a^6\,b^4\,\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3+24\,a^5\,b^4-38\,a^4\,b^5+30\,a^3\,b^6-12\,a^2\,b^7+2\,a\,b^8\right )}-\frac {6\,a^7\,b^3\,\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3+24\,a^5\,b^4-38\,a^4\,b^5+30\,a^3\,b^6-12\,a^2\,b^7+2\,a\,b^8\right )}\right )}{\sqrt {a^3}}-\frac {b}{\left (a\,b-a^2\right )\,\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}}-\frac {\mathrm {atan}\left (\frac {\frac {\sqrt {{\left (a-b\right )}^3}\,\left (\frac {\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}\,\left (-4\,a^8\,b^2+16\,a^7\,b^3-26\,a^6\,b^4+22\,a^5\,b^5-10\,a^4\,b^6+2\,a^3\,b^7\right )}{2}+\frac {\sqrt {{\left (a-b\right )}^3}\,\left (12\,a^5\,b^7-2\,a^4\,b^8-28\,a^6\,b^6+32\,a^7\,b^5-18\,a^8\,b^4+4\,a^9\,b^3+\frac {\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}\,\sqrt {{\left (a-b\right )}^3}\,\left (16\,a^{11}\,b^2-88\,a^{10}\,b^3+200\,a^9\,b^4-240\,a^8\,b^5+160\,a^7\,b^6-56\,a^6\,b^7+8\,a^5\,b^8\right )}{4\,{\left (a-b\right )}^3}\right )}{2\,{\left (a-b\right )}^3}\right )\,1{}\mathrm {i}}{{\left (a-b\right )}^3}+\frac {\sqrt {{\left (a-b\right )}^3}\,\left (\frac {\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}\,\left (-4\,a^8\,b^2+16\,a^7\,b^3-26\,a^6\,b^4+22\,a^5\,b^5-10\,a^4\,b^6+2\,a^3\,b^7\right )}{2}+\frac {\sqrt {{\left (a-b\right )}^3}\,\left (2\,a^4\,b^8-12\,a^5\,b^7+28\,a^6\,b^6-32\,a^7\,b^5+18\,a^8\,b^4-4\,a^9\,b^3+\frac {\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}\,\sqrt {{\left (a-b\right )}^3}\,\left (16\,a^{11}\,b^2-88\,a^{10}\,b^3+200\,a^9\,b^4-240\,a^8\,b^5+160\,a^7\,b^6-56\,a^6\,b^7+8\,a^5\,b^8\right )}{4\,{\left (a-b\right )}^3}\right )}{2\,{\left (a-b\right )}^3}\right )\,1{}\mathrm {i}}{{\left (a-b\right )}^3}}{2\,a^3\,b^6-6\,a^4\,b^5+6\,a^5\,b^4-2\,a^6\,b^3-\frac {\sqrt {{\left (a-b\right )}^3}\,\left (\frac {\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}\,\left (-4\,a^8\,b^2+16\,a^7\,b^3-26\,a^6\,b^4+22\,a^5\,b^5-10\,a^4\,b^6+2\,a^3\,b^7\right )}{2}+\frac {\sqrt {{\left (a-b\right )}^3}\,\left (12\,a^5\,b^7-2\,a^4\,b^8-28\,a^6\,b^6+32\,a^7\,b^5-18\,a^8\,b^4+4\,a^9\,b^3+\frac {\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}\,\sqrt {{\left (a-b\right )}^3}\,\left (16\,a^{11}\,b^2-88\,a^{10}\,b^3+200\,a^9\,b^4-240\,a^8\,b^5+160\,a^7\,b^6-56\,a^6\,b^7+8\,a^5\,b^8\right )}{4\,{\left (a-b\right )}^3}\right )}{2\,{\left (a-b\right )}^3}\right )}{{\left (a-b\right )}^3}+\frac {\sqrt {{\left (a-b\right )}^3}\,\left (\frac {\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}\,\left (-4\,a^8\,b^2+16\,a^7\,b^3-26\,a^6\,b^4+22\,a^5\,b^5-10\,a^4\,b^6+2\,a^3\,b^7\right )}{2}+\frac {\sqrt {{\left (a-b\right )}^3}\,\left (2\,a^4\,b^8-12\,a^5\,b^7+28\,a^6\,b^6-32\,a^7\,b^5+18\,a^8\,b^4-4\,a^9\,b^3+\frac {\sqrt {a+\frac {b}{{\mathrm {tan}\left (x\right )}^2}}\,\sqrt {{\left (a-b\right )}^3}\,\left (16\,a^{11}\,b^2-88\,a^{10}\,b^3+200\,a^9\,b^4-240\,a^8\,b^5+160\,a^7\,b^6-56\,a^6\,b^7+8\,a^5\,b^8\right )}{4\,{\left (a-b\right )}^3}\right )}{2\,{\left (a-b\right )}^3}\right )}{{\left (a-b\right )}^3}}\right )\,\sqrt {{\left (a-b\right )}^3}\,1{}\mathrm {i}}{{\left (a-b\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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