Integrand size = 23, antiderivative size = 240 \[ \int \frac {\sin ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {2 \arctan \left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 \sqrt {a^{2/3}-b^{2/3}} b^{2/3} d}-\frac {2 \text {arctanh}\left (\frac {\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-(-1)^{2/3} a^{2/3}+b^{2/3}}}\right )}{3 \sqrt {-(-1)^{2/3} a^{2/3}+b^{2/3}} b^{2/3} d}-\frac {2 \text {arctanh}\left (\frac {\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}}}\right )}{3 \sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}} b^{2/3} d} \]
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Time = 0.18 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3299, 2739, 632, 210, 212} \[ \int \frac {\sin ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {2 \arctan \left (\frac {\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 b^{2/3} d \sqrt {a^{2/3}-b^{2/3}}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^{2/3}-(-1)^{2/3} a^{2/3}}}\right )}{3 b^{2/3} d \sqrt {b^{2/3}-(-1)^{2/3} a^{2/3}}}-\frac {2 \text {arctanh}\left (\frac {(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}}}\right )}{3 b^{2/3} d \sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}}} \]
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Rule 210
Rule 212
Rule 632
Rule 2739
Rule 3299
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}+\frac {1}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}+\frac {1}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}\right ) \, dx \\ & = \frac {\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)} \, dx}{3 b^{2/3}}+\frac {\int \frac {1}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)} \, dx}{3 b^{2/3}}+\frac {\int \frac {1}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)} \, dx}{3 b^{2/3}} \\ & = \frac {2 \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 b^{2/3} d}+\frac {2 \text {Subst}\left (\int \frac {1}{-\sqrt [3]{-1} \sqrt [3]{a}+2 \sqrt [3]{b} x-\sqrt [3]{-1} \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^{2/3} d}+\frac {2 \text {Subst}\left (\int \frac {1}{(-1)^{2/3} \sqrt [3]{a}+2 \sqrt [3]{b} x+(-1)^{2/3} \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^{2/3} d} \\ & = -\frac {4 \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 b^{2/3} d}-\frac {4 \text {Subst}\left (\int \frac {1}{-4 \left ((-1)^{2/3} a^{2/3}-b^{2/3}\right )-x^2} \, dx,x,2 \sqrt [3]{b}-2 \sqrt [3]{-1} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^{2/3} d}-\frac {4 \text {Subst}\left (\int \frac {1}{4 \left (\sqrt [3]{-1} a^{2/3}+b^{2/3}\right )-x^2} \, dx,x,2 \sqrt [3]{b}+2 (-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^{2/3} d} \\ & = \frac {2 \arctan \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 \sqrt {a^{2/3}-b^{2/3}} b^{2/3} d}-\frac {2 \text {arctanh}\left (\frac {\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-(-1)^{2/3} a^{2/3}+b^{2/3}}}\right )}{3 \sqrt {-(-1)^{2/3} a^{2/3}+b^{2/3}} b^{2/3} d}-\frac {2 \text {arctanh}\left (\frac {\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}}}\right )}{3 \sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}} b^{2/3} d} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 11.04 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.96 \[ \int \frac {\sin ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {i \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\&,\frac {2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-4 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+2 i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{b \text {$\#$1}-4 i a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\&\right ]}{6 d} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.81 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.32
method | result | size |
derivativedivides | \(\frac {4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 d}\) | \(76\) |
default | \(\frac {4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 d}\) | \(76\) |
risch | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4096+\left (729 a^{2} b^{4} d^{6}-729 b^{6} d^{6}\right ) \textit {\_Z}^{6}+3888 b^{4} d^{4} \textit {\_Z}^{4}-6912 b^{2} d^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\left (-\frac {243}{1024} b^{3} d^{5} a^{2}+\frac {243}{1024} b^{5} d^{5}\right ) \textit {\_R}^{5}+\left (-\frac {81 i a \,b^{3} d^{4}}{256}+\frac {81 i b^{5} d^{4}}{256 a}\right ) \textit {\_R}^{4}-\frac {81 b^{3} d^{3} \textit {\_R}^{3}}{64}+\left (-\frac {9 i a b \,d^{2}}{16}-\frac {9 i d^{2} b^{3}}{8 a}\right ) \textit {\_R}^{2}+\frac {9 b d \textit {\_R}}{4}+\frac {i b}{a}\right )\right )}{4}\) | \(167\) |
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Result contains complex when optimal does not.
Time = 1.18 (sec) , antiderivative size = 25253, normalized size of antiderivative = 105.22 \[ \int \frac {\sin ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sin ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int \frac {\sin ^{2}{\left (c + d x \right )}}{a + b \sin ^{3}{\left (c + d x \right )}}\, dx \]
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\[ \int \frac {\sin ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int { \frac {\sin \left (d x + c\right )^{2}}{b \sin \left (d x + c\right )^{3} + a} \,d x } \]
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\[ \int \frac {\sin ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int { \frac {\sin \left (d x + c\right )^{2}}{b \sin \left (d x + c\right )^{3} + a} \,d x } \]
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Time = 14.79 (sec) , antiderivative size = 590, normalized size of antiderivative = 2.46 \[ \int \frac {\sin ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {\sum _{k=1}^6\ln \left (-\frac {8192\,a^4\,\left (-729\,a^2\,b^3-81\,a^2\,b^2\,\mathrm {root}\left (d^6-27\,b^2\,d^4+243\,b^4\,d^2+729\,b^4\,\left (a^2-b^2\right ),d,k\right )+243\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^4+324\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^3\,\mathrm {root}\left (d^6-27\,b^2\,d^4+243\,b^4\,d^2+729\,b^4\,\left (a^2-b^2\right ),d,k\right )+162\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^2\,{\mathrm {root}\left (d^6-27\,b^2\,d^4+243\,b^4\,d^2+729\,b^4\,\left (a^2-b^2\right ),d,k\right )}^2+36\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b\,{\mathrm {root}\left (d^6-27\,b^2\,d^4+243\,b^4\,d^2+729\,b^4\,\left (a^2-b^2\right ),d,k\right )}^3+3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,{\mathrm {root}\left (d^6-27\,b^2\,d^4+243\,b^4\,d^2+729\,b^4\,\left (a^2-b^2\right ),d,k\right )}^4+972\,b^5+324\,b^4\,\mathrm {root}\left (d^6-27\,b^2\,d^4+243\,b^4\,d^2+729\,b^4\,\left (a^2-b^2\right ),d,k\right )-216\,b^3\,{\mathrm {root}\left (d^6-27\,b^2\,d^4+243\,b^4\,d^2+729\,b^4\,\left (a^2-b^2\right ),d,k\right )}^2-72\,b^2\,{\mathrm {root}\left (d^6-27\,b^2\,d^4+243\,b^4\,d^2+729\,b^4\,\left (a^2-b^2\right ),d,k\right )}^3+12\,b\,{\mathrm {root}\left (d^6-27\,b^2\,d^4+243\,b^4\,d^2+729\,b^4\,\left (a^2-b^2\right ),d,k\right )}^4+4\,{\mathrm {root}\left (d^6-27\,b^2\,d^4+243\,b^4\,d^2+729\,b^4\,\left (a^2-b^2\right ),d,k\right )}^5\right )}{{\mathrm {root}\left (d^6-27\,b^2\,d^4+243\,b^4\,d^2+729\,b^4\,\left (a^2-b^2\right ),d,k\right )}^5}\right )\,\mathrm {root}\left (729\,a^2\,b^4\,d^6-729\,b^6\,d^6+243\,b^4\,d^4-27\,b^2\,d^2+1,d,k\right )}{d} \]
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