Optimal. Leaf size=17 \[ \text {Int}\left (\frac {1}{\left (a+b \sin ^3(c+d x)\right )^2},x\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {1}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx &=\int \frac {1}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx\\ \end {align*}
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Mathematica [A] Result contains complex when optimal does not.
time = 0.37, size = 502, normalized size = 29.53 \begin {gather*} \frac {\frac {i \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\&,\frac {2 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+4 i a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+2 a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}-24 a^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+12 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+12 i a^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-6 i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-4 i a b \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-2 a b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+2 b^2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-i b^2 \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{b \text {$\#$1}-4 i a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\&\right ]}{a^2-b^2}-\frac {12 b \cos (c+d x) (-3 a+a \cos (2 (c+d x))+2 b \sin (c+d x))}{(a-b) (a+b) (4 a+3 b \sin (c+d x)-b \sin (3 (c+d x)))}}{18 a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A] Leaf count of result is larger than twice the leaf count of optimal. \(348\) vs.
\(2(16)=32\).
time = 3.13, size = 349, normalized size = 20.53
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {2 b^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a \left (a^{2}-b^{2}\right )}-\frac {2 b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-b^{2}\right )}+\frac {8 b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a \left (a^{2}-b^{2}\right )}+\frac {8 b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-b^{2}\right )}-\frac {2 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a \left (a^{2}-b^{2}\right )}+\frac {2 b}{3 a^{2}-3 b^{2}}}{a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (3 a^{2}-2 b^{2}\right ) \textit {\_R}^{4}-2 a b \,\textit {\_R}^{3}+6 a^{2} \textit {\_R}^{2}-2 a \textit {\_R} b +3 a^{2}-2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}}{9 a \left (a^{2}-b^{2}\right )}}{d}\) | \(349\) |
| default | \(\frac {\frac {\frac {2 b^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a \left (a^{2}-b^{2}\right )}-\frac {2 b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-b^{2}\right )}+\frac {8 b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a \left (a^{2}-b^{2}\right )}+\frac {8 b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-b^{2}\right )}-\frac {2 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a \left (a^{2}-b^{2}\right )}+\frac {2 b}{3 a^{2}-3 b^{2}}}{a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (3 a^{2}-2 b^{2}\right ) \textit {\_R}^{4}-2 a b \,\textit {\_R}^{3}+6 a^{2} \textit {\_R}^{2}-2 a \textit {\_R} b +3 a^{2}-2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}}{9 a \left (a^{2}-b^{2}\right )}}{d}\) | \(349\) |
| risch | \(\text {Expression too large to display}\) | \(2453\) |
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 [A] Result contains complex when optimal does not.
time = 8.35, size = 70185, normalized size = 4128.53 \begin {gather*} \text {too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \sin ^{3}{\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
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 [A]
time = 17.89, size = 1567, normalized size = 92.18 \begin {gather*} \frac {\sum _{k=1}^6\ln \left (-\frac {8192\,\left (80\,b^6-270\,a^2\,b^4\right )}{243\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}-\mathrm {root}\left (1594323\,a^{14}\,b^2\,d^6-1594323\,a^{12}\,b^4\,d^6+531441\,a^{10}\,b^6\,d^6-531441\,a^{16}\,d^6-59049\,a^{10}\,b^2\,d^4+59049\,a^8\,b^4\,d^4-177147\,a^{12}\,d^4+8019\,a^6\,b^2\,d^2-19683\,a^8\,d^2+432\,a^2\,b^2-64\,b^4-729\,a^4,d,k\right )\,\left (\frac {8192\,\left (-2187\,a^5\,b^3+648\,a^3\,b^5+144\,a\,b^7\right )}{243\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}-\mathrm {root}\left (1594323\,a^{14}\,b^2\,d^6-1594323\,a^{12}\,b^4\,d^6+531441\,a^{10}\,b^6\,d^6-531441\,a^{16}\,d^6-59049\,a^{10}\,b^2\,d^4+59049\,a^8\,b^4\,d^4-177147\,a^{12}\,d^4+8019\,a^6\,b^2\,d^2-19683\,a^8\,d^2+432\,a^2\,b^2-64\,b^4-729\,a^4,d,k\right )\,\left (\mathrm {root}\left (1594323\,a^{14}\,b^2\,d^6-1594323\,a^{12}\,b^4\,d^6+531441\,a^{10}\,b^6\,d^6-531441\,a^{16}\,d^6-59049\,a^{10}\,b^2\,d^4+59049\,a^8\,b^4\,d^4-177147\,a^{12}\,d^4+8019\,a^6\,b^2\,d^2-19683\,a^8\,d^2+432\,a^2\,b^2-64\,b^4-729\,a^4,d,k\right )\,\left (-\mathrm {root}\left (1594323\,a^{14}\,b^2\,d^6-1594323\,a^{12}\,b^4\,d^6+531441\,a^{10}\,b^6\,d^6-531441\,a^{16}\,d^6-59049\,a^{10}\,b^2\,d^4+59049\,a^8\,b^4\,d^4-177147\,a^{12}\,d^4+8019\,a^6\,b^2\,d^2-19683\,a^8\,d^2+432\,a^2\,b^2-64\,b^4-729\,a^4,d,k\right )\,\left (\mathrm {root}\left (1594323\,a^{14}\,b^2\,d^6-1594323\,a^{12}\,b^4\,d^6+531441\,a^{10}\,b^6\,d^6-531441\,a^{16}\,d^6-59049\,a^{10}\,b^2\,d^4+59049\,a^8\,b^4\,d^4-177147\,a^{12}\,d^4+8019\,a^6\,b^2\,d^2-19683\,a^8\,d^2+432\,a^2\,b^2-64\,b^4-729\,a^4,d,k\right )\,\left (\frac {8192\,\left (-177147\,a^{13}\,b^3+590490\,a^{11}\,b^5-649539\,a^9\,b^7+236196\,a^7\,b^9\right )}{243\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}+\frac {8192\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6561\,a^{12}\,b^4-13122\,a^{10}\,b^6+6561\,a^8\,b^8\right )}{27\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}\right )+\frac {8192\,\left (72171\,a^{10}\,b^4-85293\,a^8\,b^6+13122\,a^6\,b^8\right )}{243\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}+\frac {8192\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (-8748\,a^{11}\,b^3+37908\,a^9\,b^5-40824\,a^7\,b^7+11664\,a^5\,b^9\right )}{27\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}\right )+\frac {8192\,\left (39366\,a^9\,b^3+26973\,a^7\,b^5-20412\,a^5\,b^7\right )}{243\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}+\frac {8192\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3078\,a^6\,b^6-8181\,a^8\,b^4\right )}{27\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}\right )-\frac {8192\,\left (11664\,a^6\,b^4-11340\,a^4\,b^6+2592\,a^2\,b^8\right )}{243\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}+\frac {8192\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (1944\,a^7\,b^3+1260\,a^5\,b^5-720\,a^3\,b^7\right )}{27\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}\right )+\frac {8192\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (1053\,a^4\,b^4-688\,a^2\,b^6+128\,b^8\right )}{27\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}\right )-\frac {8192\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (32\,a\,b^5-108\,a^3\,b^3\right )}{27\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}\right )\,\mathrm {root}\left (1594323\,a^{14}\,b^2\,d^6-1594323\,a^{12}\,b^4\,d^6+531441\,a^{10}\,b^6\,d^6-531441\,a^{16}\,d^6-59049\,a^{10}\,b^2\,d^4+59049\,a^8\,b^4\,d^4-177147\,a^{12}\,d^4+8019\,a^6\,b^2\,d^2-19683\,a^8\,d^2+432\,a^2\,b^2-64\,b^4-729\,a^4,d,k\right )}{d}+\frac {\frac {2\,b}{3\,\left (a^2-b^2\right )}+\frac {8\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3\,\left (a^2-b^2\right )}-\frac {2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3\,\left (a^2-b^2\right )}-\frac {2\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3\,a\,\left (a^2-b^2\right )}+\frac {8\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3\,a\,\left (a^2-b^2\right )}+\frac {2\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3\,a\,\left (a^2-b^2\right )}}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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