Optimal. Leaf size=149 \[ \frac {b \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{7/4} \sqrt {\sqrt {a}-\sqrt {b}} d}+\frac {b \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{7/4} \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {\cot (c+d x)}{a d}-\frac {\cot ^3(c+d x)}{3 a d} \]
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Rubi [A]
time = 0.13, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3296, 1301,
1180, 211} \begin {gather*} \frac {b \text {ArcTan}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{7/4} d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {b \text {ArcTan}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{7/4} d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\cot ^3(c+d x)}{3 a d}-\frac {\cot (c+d x)}{a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 1180
Rule 1301
Rule 3296
Rubi steps
\begin {align*} \int \frac {\csc ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^4 \left (a+2 a x^2+(a-b) x^4\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{a x^4}+\frac {1}{a x^2}+\frac {b \left (1+x^2\right )}{a \left (a+2 a x^2+(a-b) x^4\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {\cot (c+d x)}{a d}-\frac {\cot ^3(c+d x)}{3 a d}+\frac {b \text {Subst}\left (\int \frac {1+x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{a d}\\ &=-\frac {\cot (c+d x)}{a d}-\frac {\cot ^3(c+d x)}{3 a d}+\frac {\left (\left (\sqrt {a}+\sqrt {b}\right ) b\right ) \text {Subst}\left (\int \frac {1}{a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 a^{3/2} d}+\frac {\left (\left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) b\right ) \text {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 a d}\\ &=\frac {b \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{7/4} \sqrt {\sqrt {a}-\sqrt {b}} d}+\frac {b \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{7/4} \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {\cot (c+d x)}{a d}-\frac {\cot ^3(c+d x)}{3 a d}\\ \end {align*}
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Mathematica [A]
time = 1.08, size = 165, normalized size = 1.11 \begin {gather*} \frac {\frac {3 b \tan ^{-1}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}}}-\frac {3 b \tanh ^{-1}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}}}-4 \sqrt {a} \cot (c+d x)-2 \sqrt {a} \cot (c+d x) \csc ^2(c+d x)}{6 a^{3/2} d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.54, size = 177, normalized size = 1.19
method | result | size |
derivativedivides | \(\frac {-\frac {1}{3 a \tan \left (d x +c \right )^{3}}-\frac {1}{a \tan \left (d x +c \right )}+\frac {b \left (a -b \right ) \left (\frac {\left (\sqrt {a b}-b \right ) \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}+b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{a}}{d}\) | \(177\) |
default | \(\frac {-\frac {1}{3 a \tan \left (d x +c \right )^{3}}-\frac {1}{a \tan \left (d x +c \right )}+\frac {b \left (a -b \right ) \left (\frac {\left (\sqrt {a b}-b \right ) \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}+b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{a}}{d}\) | \(177\) |
risch | \(\frac {4 i \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{3 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+16 \left (\munderset {\textit {\_R} =\RootOf \left (\left (16777216 a^{8} d^{4}-16777216 a^{7} b \,d^{4}\right ) \textit {\_Z}^{4}+8192 a^{4} b^{2} d^{2} \textit {\_Z}^{2}+b^{4}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {524288 i d^{3} a^{7}}{b^{4}}-\frac {524288 i a^{6} d^{3}}{b^{3}}\right ) \textit {\_R}^{3}+\left (-\frac {8192 d^{2} a^{5}}{b^{3}}+\frac {8192 a^{4} d^{2}}{b^{2}}\right ) \textit {\_R}^{2}+\left (\frac {128 i d \,a^{3}}{b^{2}}+\frac {128 i a^{2} d}{b}\right ) \textit {\_R} -\frac {2 a}{b}-1\right )\right )\) | \(182\) |
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. 1365 vs.
\(2 (111) = 222\).
time = 0.58, size = 1365, normalized size = 9.16 \begin {gather*} -\frac {3 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sqrt {-\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{5}}{{\left (a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right )} d^{4}}} + b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}} \log \left (\frac {1}{4} \, b^{4} \cos \left (d x + c\right )^{2} - \frac {1}{4} \, b^{4} - \frac {1}{4} \, {\left (2 \, {\left (a^{5} b - a^{4} b^{2}\right )} d^{2} \cos \left (d x + c\right )^{2} - {\left (a^{5} b - a^{4} b^{2}\right )} d^{2}\right )} \sqrt {\frac {b^{5}}{{\left (a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right )} d^{4}}} + \frac {1}{2} \, {\left (a^{2} b^{3} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - {\left (a^{7} - a^{6} b\right )} d^{3} \sqrt {\frac {b^{5}}{{\left (a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right )} d^{4}}} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \sqrt {-\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{5}}{{\left (a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right )} d^{4}}} + b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}}\right ) \sin \left (d x + c\right ) - 3 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sqrt {-\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{5}}{{\left (a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right )} d^{4}}} + b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}} \log \left (\frac {1}{4} \, b^{4} \cos \left (d x + c\right )^{2} - \frac {1}{4} \, b^{4} - \frac {1}{4} \, {\left (2 \, {\left (a^{5} b - a^{4} b^{2}\right )} d^{2} \cos \left (d x + c\right )^{2} - {\left (a^{5} b - a^{4} b^{2}\right )} d^{2}\right )} \sqrt {\frac {b^{5}}{{\left (a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right )} d^{4}}} - \frac {1}{2} \, {\left (a^{2} b^{3} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - {\left (a^{7} - a^{6} b\right )} d^{3} \sqrt {\frac {b^{5}}{{\left (a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right )} d^{4}}} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \sqrt {-\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{5}}{{\left (a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right )} d^{4}}} + b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}}\right ) \sin \left (d x + c\right ) - 3 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sqrt {\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{5}}{{\left (a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right )} d^{4}}} - b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}} \log \left (-\frac {1}{4} \, b^{4} \cos \left (d x + c\right )^{2} + \frac {1}{4} \, b^{4} - \frac {1}{4} \, {\left (2 \, {\left (a^{5} b - a^{4} b^{2}\right )} d^{2} \cos \left (d x + c\right )^{2} - {\left (a^{5} b - a^{4} b^{2}\right )} d^{2}\right )} \sqrt {\frac {b^{5}}{{\left (a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right )} d^{4}}} + \frac {1}{2} \, {\left (a^{2} b^{3} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{7} - a^{6} b\right )} d^{3} \sqrt {\frac {b^{5}}{{\left (a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right )} d^{4}}} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \sqrt {\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{5}}{{\left (a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right )} d^{4}}} - b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}}\right ) \sin \left (d x + c\right ) + 3 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sqrt {\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{5}}{{\left (a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right )} d^{4}}} - b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}} \log \left (-\frac {1}{4} \, b^{4} \cos \left (d x + c\right )^{2} + \frac {1}{4} \, b^{4} - \frac {1}{4} \, {\left (2 \, {\left (a^{5} b - a^{4} b^{2}\right )} d^{2} \cos \left (d x + c\right )^{2} - {\left (a^{5} b - a^{4} b^{2}\right )} d^{2}\right )} \sqrt {\frac {b^{5}}{{\left (a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right )} d^{4}}} - \frac {1}{2} \, {\left (a^{2} b^{3} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{7} - a^{6} b\right )} d^{3} \sqrt {\frac {b^{5}}{{\left (a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right )} d^{4}}} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \sqrt {\frac {{\left (a^{4} - a^{3} b\right )} d^{2} \sqrt {\frac {b^{5}}{{\left (a^{9} - 2 \, a^{8} b + a^{7} b^{2}\right )} d^{4}}} - b^{2}}{{\left (a^{4} - a^{3} b\right )} d^{2}}}\right ) \sin \left (d x + c\right ) + 16 \, \cos \left (d x + c\right )^{3} - 24 \, \cos \left (d x + c\right )}{24 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )} \sin \left (d x + c\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 {\csc ^{4}{\left (c + d x \right )}}{a - b \sin ^{4}{\left (c + d x \right )}}\, 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. 938 vs.
\(2 (111) = 222\).
time = 0.87, size = 938, normalized size = 6.30 \begin {gather*} -\frac {\frac {3 \, {\left ({\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{2} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{3}\right )} a^{2} {\left | a - b \right |} - {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{4} b - 9 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{3} b^{2} + 5 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{2} b^{3} + \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a b^{4}\right )} {\left | a - b \right |} {\left | a \right |} - {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{4} b - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} b^{2} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{3}\right )} {\left | a - b \right |}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a^{2} + \sqrt {a^{4} - {\left (a^{2} - a b\right )} a^{2}}}{a^{2} - a b}}}\right )\right )}}{{\left (3 \, a^{8} - 15 \, a^{7} b + 26 \, a^{6} b^{2} - 18 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right )} {\left | a \right |}} - \frac {3 \, {\left ({\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{2} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{3}\right )} a^{2} {\left | a - b \right |} + {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{4} b - 9 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{3} b^{2} + 5 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{2} b^{3} + \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a b^{4}\right )} {\left | a - b \right |} {\left | a \right |} - {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{4} b - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} b^{2} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{3}\right )} {\left | a - b \right |}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a^{2} - \sqrt {a^{4} - {\left (a^{2} - a b\right )} a^{2}}}{a^{2} - a b}}}\right )\right )}}{{\left (3 \, a^{8} - 15 \, a^{7} b + 26 \, a^{6} b^{2} - 18 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right )} {\left | a \right |}} + \frac {2 \, {\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )}}{a \tan \left (d x + c\right )^{3}}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 15.55, size = 1670, normalized size = 11.21 \begin {gather*} -\frac {\frac {1}{3\,a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2}{a}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^3}+\frac {\mathrm {atan}\left (\frac {\left (\sqrt {\frac {\sqrt {a^7\,b^5}+a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}\,\left (16\,a^5\,b^4-32\,a^6\,b^3+16\,a^7\,b^2+\mathrm {tan}\left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^7\,b^5}+a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}\,\left (64\,a^9\,b-128\,a^8\,b^2+64\,a^7\,b^3\right )\right )-\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^3\,b^5-4\,a^5\,b^3\right )\right )\,\sqrt {\frac {\sqrt {a^7\,b^5}+a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}\,1{}\mathrm {i}-\left (\sqrt {\frac {\sqrt {a^7\,b^5}+a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}\,\left (16\,a^5\,b^4-32\,a^6\,b^3+16\,a^7\,b^2-\mathrm {tan}\left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^7\,b^5}+a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}\,\left (64\,a^9\,b-128\,a^8\,b^2+64\,a^7\,b^3\right )\right )+\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^3\,b^5-4\,a^5\,b^3\right )\right )\,\sqrt {\frac {\sqrt {a^7\,b^5}+a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}\,1{}\mathrm {i}}{\left (\sqrt {\frac {\sqrt {a^7\,b^5}+a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}\,\left (16\,a^5\,b^4-32\,a^6\,b^3+16\,a^7\,b^2+\mathrm {tan}\left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^7\,b^5}+a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}\,\left (64\,a^9\,b-128\,a^8\,b^2+64\,a^7\,b^3\right )\right )-\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^3\,b^5-4\,a^5\,b^3\right )\right )\,\sqrt {\frac {\sqrt {a^7\,b^5}+a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}+\left (\sqrt {\frac {\sqrt {a^7\,b^5}+a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}\,\left (16\,a^5\,b^4-32\,a^6\,b^3+16\,a^7\,b^2-\mathrm {tan}\left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^7\,b^5}+a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}\,\left (64\,a^9\,b-128\,a^8\,b^2+64\,a^7\,b^3\right )\right )+\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^3\,b^5-4\,a^5\,b^3\right )\right )\,\sqrt {\frac {\sqrt {a^7\,b^5}+a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}-2\,a^2\,b^5+2\,a^3\,b^4}\right )\,\sqrt {\frac {\sqrt {a^7\,b^5}+a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}\,2{}\mathrm {i}}{d}+\frac {\mathrm {atan}\left (\frac {\left (\sqrt {-\frac {\sqrt {a^7\,b^5}-a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}\,\left (16\,a^5\,b^4-32\,a^6\,b^3+16\,a^7\,b^2+\mathrm {tan}\left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^7\,b^5}-a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}\,\left (64\,a^9\,b-128\,a^8\,b^2+64\,a^7\,b^3\right )\right )-\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^3\,b^5-4\,a^5\,b^3\right )\right )\,\sqrt {-\frac {\sqrt {a^7\,b^5}-a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}\,1{}\mathrm {i}-\left (\sqrt {-\frac {\sqrt {a^7\,b^5}-a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}\,\left (16\,a^5\,b^4-32\,a^6\,b^3+16\,a^7\,b^2-\mathrm {tan}\left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^7\,b^5}-a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}\,\left (64\,a^9\,b-128\,a^8\,b^2+64\,a^7\,b^3\right )\right )+\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^3\,b^5-4\,a^5\,b^3\right )\right )\,\sqrt {-\frac {\sqrt {a^7\,b^5}-a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}\,1{}\mathrm {i}}{\left (\sqrt {-\frac {\sqrt {a^7\,b^5}-a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}\,\left (16\,a^5\,b^4-32\,a^6\,b^3+16\,a^7\,b^2+\mathrm {tan}\left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^7\,b^5}-a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}\,\left (64\,a^9\,b-128\,a^8\,b^2+64\,a^7\,b^3\right )\right )-\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^3\,b^5-4\,a^5\,b^3\right )\right )\,\sqrt {-\frac {\sqrt {a^7\,b^5}-a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}+\left (\sqrt {-\frac {\sqrt {a^7\,b^5}-a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}\,\left (16\,a^5\,b^4-32\,a^6\,b^3+16\,a^7\,b^2-\mathrm {tan}\left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^7\,b^5}-a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}\,\left (64\,a^9\,b-128\,a^8\,b^2+64\,a^7\,b^3\right )\right )+\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^3\,b^5-4\,a^5\,b^3\right )\right )\,\sqrt {-\frac {\sqrt {a^7\,b^5}-a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}-2\,a^2\,b^5+2\,a^3\,b^4}\right )\,\sqrt {-\frac {\sqrt {a^7\,b^5}-a^4\,b^2}{16\,\left (a^7\,b-a^8\right )}}\,2{}\mathrm {i}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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