Optimal. Leaf size=93 \[ -\frac {2 b \left (2 a^2-b^2\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right )^{3/2}}-\frac {\tanh ^{-1}(\cos (x))}{a^2}-\frac {b^2 \cos (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))} \]
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Rubi [A]
time = 0.14, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {2881, 3080,
3855, 2739, 632, 210} \begin {gather*} -\frac {2 b \left (2 a^2-b^2\right ) \text {ArcTan}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right )^{3/2}}-\frac {b^2 \cos (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}-\frac {\tanh ^{-1}(\cos (x))}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 2881
Rule 3080
Rule 3855
Rubi steps
\begin {align*} \int \frac {\csc (x)}{(a+b \sin (x))^2} \, dx &=-\frac {b^2 \cos (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}+\frac {\int \frac {\csc (x) \left (a^2-b^2-a b \sin (x)\right )}{a+b \sin (x)} \, dx}{a \left (a^2-b^2\right )}\\ &=-\frac {b^2 \cos (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}+\frac {\int \csc (x) \, dx}{a^2}-\frac {\left (b \left (2 a^2-b^2\right )\right ) \int \frac {1}{a+b \sin (x)} \, dx}{a^2 \left (a^2-b^2\right )}\\ &=-\frac {\tanh ^{-1}(\cos (x))}{a^2}-\frac {b^2 \cos (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}-\frac {\left (2 b \left (2 a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{a^2 \left (a^2-b^2\right )}\\ &=-\frac {\tanh ^{-1}(\cos (x))}{a^2}-\frac {b^2 \cos (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}+\frac {\left (4 b \left (2 a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {x}{2}\right )\right )}{a^2 \left (a^2-b^2\right )}\\ &=-\frac {2 b \left (2 a^2-b^2\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right )^{3/2}}-\frac {\tanh ^{-1}(\cos (x))}{a^2}-\frac {b^2 \cos (x)}{a \left (a^2-b^2\right ) (a+b \sin (x))}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 99, normalized size = 1.06 \begin {gather*} \frac {\frac {2 b \left (-2 a^2+b^2\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-\log \left (\cos \left (\frac {x}{2}\right )\right )+\log \left (\sin \left (\frac {x}{2}\right )\right )-\frac {a b^2 \cos (x)}{(a-b) (a+b) (a+b \sin (x))}}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 123, normalized size = 1.32
method | result | size |
default | \(\frac {\ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{2}}-\frac {4 b \left (\frac {\frac {b^{2} \tan \left (\frac {x}{2}\right )}{2 a^{2}-2 b^{2}}+\frac {a b}{2 a^{2}-2 b^{2}}}{a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 b \tan \left (\frac {x}{2}\right )+a}+\frac {\left (2 a^{2}-b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{a^{2}}\) | \(123\) |
risch | \(\frac {2 i b \left (-i a \,{\mathrm e}^{i x}+b \right )}{a \left (-a^{2}+b^{2}\right ) \left (b \,{\mathrm e}^{2 i x}-b +2 i a \,{\mathrm e}^{i x}\right )}+\frac {2 i b \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right )}-\frac {i b^{3} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) a^{2}}-\frac {2 i b \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right )}+\frac {i b^{3} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) a^{2}}+\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{a^{2}}-\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{a^{2}}\) | \(380\) |
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: ValueError} \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. 224 vs.
\(2 (87) = 174\).
time = 0.69, size = 511, normalized size = 5.49 \begin {gather*} \left [-\frac {{\left (2 \, a^{3} b - a b^{3} + {\left (2 \, a^{2} b^{2} - b^{4}\right )} \sin \left (x\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (x\right ) \sin \left (x\right ) + b \cos \left (x\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2}}\right ) + 2 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (x\right ) + {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4} + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (x\right )\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4} + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (x\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right )}{2 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4} + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \sin \left (x\right )\right )}}, \frac {2 \, {\left (2 \, a^{3} b - a b^{3} + {\left (2 \, a^{2} b^{2} - b^{4}\right )} \sin \left (x\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (x\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (x\right )}\right ) - 2 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (x\right ) - {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4} + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (x\right )\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4} + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (x\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right )}{2 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4} + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \sin \left (x\right )\right )}}\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 {\left (x \right )}}{\left (a + b \sin {\left (x \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 134, normalized size = 1.44 \begin {gather*} -\frac {2 \, {\left (2 \, a^{2} b - b^{3}\right )} {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{4} - a^{2} b^{2}\right )} \sqrt {a^{2} - b^{2}}} - \frac {2 \, {\left (b^{3} \tan \left (\frac {1}{2} \, x\right ) + a b^{2}\right )}}{{\left (a^{4} - a^{2} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, x\right ) + a\right )}} + \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.82, size = 1356, normalized size = 14.58 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{a^2}-\frac {\frac {2\,b^2}{a\,\left (a^2-b^2\right )}+\frac {2\,b^3\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^2\,\left (a^2-b^2\right )}}{a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )+a}-\frac {b\,\mathrm {atan}\left (\frac {\frac {b\,\left (2\,a^2-b^2\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (\frac {2\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a^6-8\,a^4\,b^2+11\,a^2\,b^4-4\,b^6\right )}{a^5-2\,a^3\,b^2+a\,b^4}-\frac {2\,\left (3\,a^4\,b-2\,a^2\,b^3\right )}{a^4-a^2\,b^2}+\frac {b\,\left (\frac {2\,\left (a^6\,b-a^4\,b^3\right )}{a^4-a^2\,b^2}-\frac {2\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (3\,a^8-10\,a^6\,b^2+11\,a^4\,b^4-4\,a^2\,b^6\right )}{a^5-2\,a^3\,b^2+a\,b^4}\right )\,\left (2\,a^2-b^2\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}}{a^8-3\,a^6\,b^2+3\,a^4\,b^4-a^2\,b^6}\right )\,1{}\mathrm {i}}{a^8-3\,a^6\,b^2+3\,a^4\,b^4-a^2\,b^6}-\frac {b\,\left (2\,a^2-b^2\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (\frac {2\,\left (3\,a^4\,b-2\,a^2\,b^3\right )}{a^4-a^2\,b^2}-\frac {2\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a^6-8\,a^4\,b^2+11\,a^2\,b^4-4\,b^6\right )}{a^5-2\,a^3\,b^2+a\,b^4}+\frac {b\,\left (\frac {2\,\left (a^6\,b-a^4\,b^3\right )}{a^4-a^2\,b^2}-\frac {2\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (3\,a^8-10\,a^6\,b^2+11\,a^4\,b^4-4\,a^2\,b^6\right )}{a^5-2\,a^3\,b^2+a\,b^4}\right )\,\left (2\,a^2-b^2\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}}{a^8-3\,a^6\,b^2+3\,a^4\,b^4-a^2\,b^6}\right )\,1{}\mathrm {i}}{a^8-3\,a^6\,b^2+3\,a^4\,b^4-a^2\,b^6}}{\frac {4\,\left (2\,a^2\,b-b^3\right )}{a^4-a^2\,b^2}+\frac {4\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (2\,b^4-4\,a^2\,b^2\right )}{a^5-2\,a^3\,b^2+a\,b^4}+\frac {b\,\left (2\,a^2-b^2\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (\frac {2\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a^6-8\,a^4\,b^2+11\,a^2\,b^4-4\,b^6\right )}{a^5-2\,a^3\,b^2+a\,b^4}-\frac {2\,\left (3\,a^4\,b-2\,a^2\,b^3\right )}{a^4-a^2\,b^2}+\frac {b\,\left (\frac {2\,\left (a^6\,b-a^4\,b^3\right )}{a^4-a^2\,b^2}-\frac {2\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (3\,a^8-10\,a^6\,b^2+11\,a^4\,b^4-4\,a^2\,b^6\right )}{a^5-2\,a^3\,b^2+a\,b^4}\right )\,\left (2\,a^2-b^2\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}}{a^8-3\,a^6\,b^2+3\,a^4\,b^4-a^2\,b^6}\right )}{a^8-3\,a^6\,b^2+3\,a^4\,b^4-a^2\,b^6}+\frac {b\,\left (2\,a^2-b^2\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (\frac {2\,\left (3\,a^4\,b-2\,a^2\,b^3\right )}{a^4-a^2\,b^2}-\frac {2\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a^6-8\,a^4\,b^2+11\,a^2\,b^4-4\,b^6\right )}{a^5-2\,a^3\,b^2+a\,b^4}+\frac {b\,\left (\frac {2\,\left (a^6\,b-a^4\,b^3\right )}{a^4-a^2\,b^2}-\frac {2\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (3\,a^8-10\,a^6\,b^2+11\,a^4\,b^4-4\,a^2\,b^6\right )}{a^5-2\,a^3\,b^2+a\,b^4}\right )\,\left (2\,a^2-b^2\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}}{a^8-3\,a^6\,b^2+3\,a^4\,b^4-a^2\,b^6}\right )}{a^8-3\,a^6\,b^2+3\,a^4\,b^4-a^2\,b^6}}\right )\,\left (2\,a^2-b^2\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,2{}\mathrm {i}}{a^8-3\,a^6\,b^2+3\,a^4\,b^4-a^2\,b^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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