Optimal. Leaf size=344 \[ \frac {2 (-1)^{2/3} b^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{7/3} \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}} d}-\frac {2 b^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a^{7/3} \sqrt {a^{2/3}-b^{2/3}} d}+\frac {2 \sqrt [3]{-1} b^{5/3} \tan ^{-1}\left (\frac {(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{7/3} \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}} d}-\frac {3 \tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac {b \cot (c+d x)}{a^2 d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d} \]
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Rubi [A]
time = 0.37, antiderivative size = 344, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3299, 3852, 8,
3853, 3855, 2739, 632, 210} \begin {gather*} \frac {2 (-1)^{2/3} b^{5/3} \text {ArcTan}\left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{7/3} d \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac {2 b^{5/3} \text {ArcTan}\left (\frac {\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a^{7/3} d \sqrt {a^{2/3}-b^{2/3}}}+\frac {2 \sqrt [3]{-1} b^{5/3} \text {ArcTan}\left (\frac {\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+(-1)^{2/3} \sqrt [3]{b}}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{7/3} d \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}+\frac {b \cot (c+d x)}{a^2 d}-\frac {3 \tanh ^{-1}(\cos (c+d x))}{8 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 210
Rule 632
Rule 2739
Rule 3299
Rule 3852
Rule 3853
Rule 3855
Rubi steps
\begin {align*} \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx &=\int \left (-\frac {b \csc ^2(c+d x)}{a^2}+\frac {\csc ^5(c+d x)}{a}+\frac {b^2 \sin (c+d x)}{a^2 \left (a+b \sin ^3(c+d x)\right )}\right ) \, dx\\ &=\frac {\int \csc ^5(c+d x) \, dx}{a}-\frac {b \int \csc ^2(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {\sin (c+d x)}{a+b \sin ^3(c+d x)} \, dx}{a^2}\\ &=-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac {3 \int \csc ^3(c+d x) \, dx}{4 a}+\frac {b^2 \int \left (-\frac {1}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}-\frac {(-1)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} \sin (c+d x)\right )}+\frac {\sqrt [3]{-1}}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} \sin (c+d x)\right )}\right ) \, dx}{a^2}+\frac {b \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^2 d}\\ &=\frac {b \cot (c+d x)}{a^2 d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac {3 \int \csc (c+d x) \, dx}{8 a}-\frac {b^{5/3} \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)} \, dx}{3 a^{7/3}}+\frac {\left (\sqrt [3]{-1} b^{5/3}\right ) \int \frac {1}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} \sin (c+d x)} \, dx}{3 a^{7/3}}-\frac {\left ((-1)^{2/3} b^{5/3}\right ) \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} \sin (c+d x)} \, dx}{3 a^{7/3}}\\ &=-\frac {3 \tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac {b \cot (c+d x)}{a^2 d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {\left (2 b^{5/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+2 \sqrt [3]{b} x+\sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^{7/3} d}+\frac {\left (2 \sqrt [3]{-1} b^{5/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+2 (-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^{7/3} d}-\frac {\left (2 (-1)^{2/3} b^{5/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-2 \sqrt [3]{-1} \sqrt [3]{b} x+\sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^{7/3} d}\\ &=-\frac {3 \tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac {b \cot (c+d x)}{a^2 d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac {\left (4 b^{5/3}\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/3}-b^{2/3}\right )-x^2} \, dx,x,2 \sqrt [3]{b}+2 \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^{7/3} d}-\frac {\left (4 \sqrt [3]{-1} b^{5/3}\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/3}+\sqrt [3]{-1} b^{2/3}\right )-x^2} \, dx,x,2 (-1)^{2/3} \sqrt [3]{b}+2 \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^{7/3} d}+\frac {\left (4 (-1)^{2/3} b^{5/3}\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/3}-(-1)^{2/3} b^{2/3}\right )-x^2} \, dx,x,-2 \sqrt [3]{-1} \sqrt [3]{b}+2 \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^{7/3} d}\\ &=\frac {2 (-1)^{2/3} b^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{7/3} \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}} d}-\frac {2 b^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a^{7/3} \sqrt {a^{2/3}-b^{2/3}} d}+\frac {2 \sqrt [3]{-1} b^{5/3} \tan ^{-1}\left (\frac {(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{7/3} \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}} d}-\frac {3 \tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac {b \cot (c+d x)}{a^2 d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a 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 = 1.41, size = 290, normalized size = 0.84 \begin {gather*} \frac {-64 b^2 \text {RootSum}\left [-b+3 b \text {$\#$1}^2-8 i a \text {$\#$1}^3-3 b \text {$\#$1}^4+b \text {$\#$1}^6\&,\frac {-2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2}{b-4 i a \text {$\#$1}-2 b \text {$\#$1}^2+b \text {$\#$1}^4}\&\right ]+3 \left (32 b \cot \left (\frac {1}{2} (c+d x)\right )-6 a \csc ^2\left (\frac {1}{2} (c+d x)\right )-a \csc ^4\left (\frac {1}{2} (c+d x)\right )-24 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+24 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+6 a \sec ^2\left (\frac {1}{2} (c+d x)\right )+a \sec ^4\left (\frac {1}{2} (c+d x)\right )-32 b \tan \left (\frac {1}{2} (c+d x)\right )\right )}{192 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.07, size = 196, normalized size = 0.57
method | result | size |
derivativedivides | \(\frac {\frac {\frac {a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+2 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a^{2}}+\frac {2 b^{2} \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{3}+\textit {\_R} \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a^{2}}-\frac {1}{64 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{8 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {b}{2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(196\) |
default | \(\frac {\frac {\frac {a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+2 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a^{2}}+\frac {2 b^{2} \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{3}+\textit {\_R} \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a^{2}}-\frac {1}{64 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{8 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {b}{2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(196\) |
risch | \(\frac {3 a \,{\mathrm e}^{7 i \left (d x +c \right )}-11 a \,{\mathrm e}^{5 i \left (d x +c \right )}+8 i b \,{\mathrm e}^{6 i \left (d x +c \right )}-11 a \,{\mathrm e}^{3 i \left (d x +c \right )}-24 i b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 a \,{\mathrm e}^{i \left (d x +c \right )}+24 i b \,{\mathrm e}^{2 i \left (d x +c \right )}-8 i b}{4 a^{2} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+32 i \left (\munderset {\textit {\_R} =\RootOf \left (\left (782757789696 a^{16} d^{6}-782757789696 a^{14} b^{2} d^{6}\right ) \textit {\_Z}^{6}-254803968 a^{10} b^{4} d^{4} \textit {\_Z}^{4}-b^{10}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\left (\frac {8153726976 i d^{5} a^{15}}{a^{2} b^{8}+b^{10}}-\frac {8153726976 i d^{5} a^{13} b^{2}}{a^{2} b^{8}+b^{10}}\right ) \textit {\_R}^{5}+\left (\frac {84934656 i d^{4} a^{13} b}{a^{2} b^{8}+b^{10}}-\frac {84934656 i d^{4} a^{11} b^{3}}{a^{2} b^{8}+b^{10}}\right ) \textit {\_R}^{4}+\left (-\frac {1769472 i d^{3} a^{9} b^{4}}{a^{2} b^{8}+b^{10}}-\frac {884736 i d^{3} a^{7} b^{6}}{a^{2} b^{8}+b^{10}}\right ) \textit {\_R}^{3}+\left (-\frac {18432 i d^{2} a^{7} b^{5}}{a^{2} b^{8}+b^{10}}-\frac {9216 i d^{2} a^{5} b^{7}}{a^{2} b^{8}+b^{10}}\right ) \textit {\_R}^{2}+\left (-\frac {96 i d \,a^{5} b^{6}}{a^{2} b^{8}+b^{10}}-\frac {192 i d \,a^{3} b^{8}}{a^{2} b^{8}+b^{10}}\right ) \textit {\_R} -\frac {i b^{9} a}{a^{2} b^{8}+b^{10}}\right )\right )-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 a d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 a d}\) | \(503\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 104.91, size = 21564, normalized size = 62.69 \begin {gather*} \text {too large to display} \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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 14.60, size = 1560, normalized size = 4.53 \begin {gather*} \frac {\sum _{k=1}^6\ln \left (\frac {-3072\,a^3\,b^{11}+262144\,b^{14}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )\,a^4\,b^{11}\,155648-{\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )}^2\,a^5\,b^{11}\,393216+{\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )}^2\,a^7\,b^9\,774144-{\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )}^3\,a^8\,b^9\,2064384+{\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )}^3\,a^{10}\,b^7\,2073600-{\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )}^4\,a^{11}\,b^7\,9510912+{\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )}^4\,a^{13}\,b^5\,2737152+{\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )}^5\,a^{12}\,b^7\,10616832-{\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )}^5\,a^{14}\,b^5\,10285056+{\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )}^5\,a^{16}\,b^3\,3732480+{\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )}^6\,a^{15}\,b^5\,7962624-{\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )}^6\,a^{17}\,b^3\,9953280+98304\,a^2\,b^{12}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )\,a^3\,b^{12}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,262144+\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )\,a^5\,b^{10}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,165888-{\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )}^2\,a^6\,b^{10}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1327104+{\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )}^2\,a^8\,b^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,165888+{\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )}^3\,a^7\,b^{10}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,2359296-{\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )}^3\,a^9\,b^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,7077888+{\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )}^3\,a^{11}\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,82944+{\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )}^4\,a^{10}\,b^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,81395712-{\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )}^4\,a^{12}\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1714176+{\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )}^5\,a^{13}\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,27869184-{\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )}^5\,a^{15}\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,23141376-{\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )}^6\,a^{14}\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,254803968+{\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )}^6\,a^{16}\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,252813312}{a^9}\right )\,\mathrm {root}\left (729\,a^{14}\,b^2\,z^6-729\,a^{16}\,z^6-243\,a^{10}\,b^4\,z^4-b^{10},z,k\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,a\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,a\,d}+\frac {3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,a\,d}+\frac {b\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^2\,d}-\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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