Optimal. Leaf size=144 \[ \frac {b \sin ^2(x) \cos (x)}{3 a^2}+\frac {2 b^5 \tanh ^{-1}\left (\frac {a+b \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \sqrt {a^2-b^2}}+\frac {b \left (2 a^2+3 b^2\right ) \cos (x)}{3 a^4}-\frac {\left (3 a^2+4 b^2\right ) \sin (x) \cos (x)}{8 a^3}+\frac {x \left (3 a^4+4 a^2 b^2+8 b^4\right )}{8 a^5}-\frac {\sin ^3(x) \cos (x)}{4 a} \]
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Rubi [A] time = 0.59, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3853, 4104, 3919, 3831, 2660, 618, 206} \[ \frac {x \left (4 a^2 b^2+3 a^4+8 b^4\right )}{8 a^5}+\frac {b \left (2 a^2+3 b^2\right ) \cos (x)}{3 a^4}-\frac {\left (3 a^2+4 b^2\right ) \sin (x) \cos (x)}{8 a^3}+\frac {2 b^5 \tanh ^{-1}\left (\frac {a+b \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \sqrt {a^2-b^2}}+\frac {b \sin ^2(x) \cos (x)}{3 a^2}-\frac {\sin ^3(x) \cos (x)}{4 a} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 2660
Rule 3831
Rule 3853
Rule 3919
Rule 4104
Rubi steps
\begin {align*} \int \frac {\sin ^4(x)}{a+b \csc (x)} \, dx &=-\frac {\cos (x) \sin ^3(x)}{4 a}+\frac {\int \frac {\left (-4 b+3 a \csc (x)+3 b \csc ^2(x)\right ) \sin ^3(x)}{a+b \csc (x)} \, dx}{4 a}\\ &=\frac {b \cos (x) \sin ^2(x)}{3 a^2}-\frac {\cos (x) \sin ^3(x)}{4 a}-\frac {\int \frac {\left (-3 \left (3 a^2+4 b^2\right )-a b \csc (x)+8 b^2 \csc ^2(x)\right ) \sin ^2(x)}{a+b \csc (x)} \, dx}{12 a^2}\\ &=-\frac {\left (3 a^2+4 b^2\right ) \cos (x) \sin (x)}{8 a^3}+\frac {b \cos (x) \sin ^2(x)}{3 a^2}-\frac {\cos (x) \sin ^3(x)}{4 a}+\frac {\int \frac {\left (-8 b \left (2 a^2+3 b^2\right )+a \left (9 a^2-4 b^2\right ) \csc (x)+3 b \left (3 a^2+4 b^2\right ) \csc ^2(x)\right ) \sin (x)}{a+b \csc (x)} \, dx}{24 a^3}\\ &=\frac {b \left (2 a^2+3 b^2\right ) \cos (x)}{3 a^4}-\frac {\left (3 a^2+4 b^2\right ) \cos (x) \sin (x)}{8 a^3}+\frac {b \cos (x) \sin ^2(x)}{3 a^2}-\frac {\cos (x) \sin ^3(x)}{4 a}-\frac {\int \frac {-3 \left (3 a^4+4 a^2 b^2+8 b^4\right )-3 a b \left (3 a^2+4 b^2\right ) \csc (x)}{a+b \csc (x)} \, dx}{24 a^4}\\ &=\frac {\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}+\frac {b \left (2 a^2+3 b^2\right ) \cos (x)}{3 a^4}-\frac {\left (3 a^2+4 b^2\right ) \cos (x) \sin (x)}{8 a^3}+\frac {b \cos (x) \sin ^2(x)}{3 a^2}-\frac {\cos (x) \sin ^3(x)}{4 a}-\frac {b^5 \int \frac {\csc (x)}{a+b \csc (x)} \, dx}{a^5}\\ &=\frac {\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}+\frac {b \left (2 a^2+3 b^2\right ) \cos (x)}{3 a^4}-\frac {\left (3 a^2+4 b^2\right ) \cos (x) \sin (x)}{8 a^3}+\frac {b \cos (x) \sin ^2(x)}{3 a^2}-\frac {\cos (x) \sin ^3(x)}{4 a}-\frac {b^4 \int \frac {1}{1+\frac {a \sin (x)}{b}} \, dx}{a^5}\\ &=\frac {\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}+\frac {b \left (2 a^2+3 b^2\right ) \cos (x)}{3 a^4}-\frac {\left (3 a^2+4 b^2\right ) \cos (x) \sin (x)}{8 a^3}+\frac {b \cos (x) \sin ^2(x)}{3 a^2}-\frac {\cos (x) \sin ^3(x)}{4 a}-\frac {\left (2 b^4\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {2 a x}{b}+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{a^5}\\ &=\frac {\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}+\frac {b \left (2 a^2+3 b^2\right ) \cos (x)}{3 a^4}-\frac {\left (3 a^2+4 b^2\right ) \cos (x) \sin (x)}{8 a^3}+\frac {b \cos (x) \sin ^2(x)}{3 a^2}-\frac {\cos (x) \sin ^3(x)}{4 a}+\frac {\left (4 b^4\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (1-\frac {a^2}{b^2}\right )-x^2} \, dx,x,\frac {2 a}{b}+2 \tan \left (\frac {x}{2}\right )\right )}{a^5}\\ &=\frac {\left (3 a^4+4 a^2 b^2+8 b^4\right ) x}{8 a^5}+\frac {2 b^5 \tanh ^{-1}\left (\frac {b \left (\frac {a}{b}+\tan \left (\frac {x}{2}\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \sqrt {a^2-b^2}}+\frac {b \left (2 a^2+3 b^2\right ) \cos (x)}{3 a^4}-\frac {\left (3 a^2+4 b^2\right ) \cos (x) \sin (x)}{8 a^3}+\frac {b \cos (x) \sin ^2(x)}{3 a^2}-\frac {\cos (x) \sin ^3(x)}{4 a}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 129, normalized size = 0.90 \[ \frac {36 a^4 x-24 a^4 \sin (2 x)+3 a^4 \sin (4 x)-8 a^3 b \cos (3 x)+48 a^2 b^2 x-24 a^2 b^2 \sin (2 x)+24 a b \left (3 a^2+4 b^2\right ) \cos (x)-\frac {192 b^5 \tan ^{-1}\left (\frac {a+b \tan \left (\frac {x}{2}\right )}{\sqrt {b^2-a^2}}\right )}{\sqrt {b^2-a^2}}+96 b^4 x}{96 a^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 410, normalized size = 2.85 \[ \left [\frac {12 \, \sqrt {a^{2} - b^{2}} b^{5} \log \left (\frac {{\left (a^{2} - 2 \, b^{2}\right )} \cos \relax (x)^{2} + 2 \, a b \sin \relax (x) + a^{2} + b^{2} + 2 \, {\left (b \cos \relax (x) \sin \relax (x) + a \cos \relax (x)\right )} \sqrt {a^{2} - b^{2}}}{a^{2} \cos \relax (x)^{2} - 2 \, a b \sin \relax (x) - a^{2} - b^{2}}\right ) - 8 \, {\left (a^{5} b - a^{3} b^{3}\right )} \cos \relax (x)^{3} + 3 \, {\left (3 \, a^{6} + a^{4} b^{2} + 4 \, a^{2} b^{4} - 8 \, b^{6}\right )} x + 24 \, {\left (a^{5} b - a b^{5}\right )} \cos \relax (x) + 3 \, {\left (2 \, {\left (a^{6} - a^{4} b^{2}\right )} \cos \relax (x)^{3} - {\left (5 \, a^{6} - a^{4} b^{2} - 4 \, a^{2} b^{4}\right )} \cos \relax (x)\right )} \sin \relax (x)}{24 \, {\left (a^{7} - a^{5} b^{2}\right )}}, \frac {24 \, \sqrt {-a^{2} + b^{2}} b^{5} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \sin \relax (x) + a\right )}}{{\left (a^{2} - b^{2}\right )} \cos \relax (x)}\right ) - 8 \, {\left (a^{5} b - a^{3} b^{3}\right )} \cos \relax (x)^{3} + 3 \, {\left (3 \, a^{6} + a^{4} b^{2} + 4 \, a^{2} b^{4} - 8 \, b^{6}\right )} x + 24 \, {\left (a^{5} b - a b^{5}\right )} \cos \relax (x) + 3 \, {\left (2 \, {\left (a^{6} - a^{4} b^{2}\right )} \cos \relax (x)^{3} - {\left (5 \, a^{6} - a^{4} b^{2} - 4 \, a^{2} b^{4}\right )} \cos \relax (x)\right )} \sin \relax (x)}{24 \, {\left (a^{7} - a^{5} b^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.66, size = 252, normalized size = 1.75 \[ -\frac {2 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (b) + \arctan \left (\frac {b \tan \left (\frac {1}{2} \, x\right ) + a}{\sqrt {-a^{2} + b^{2}}}\right )\right )} b^{5}}{\sqrt {-a^{2} + b^{2}} a^{5}} + \frac {{\left (3 \, a^{4} + 4 \, a^{2} b^{2} + 8 \, b^{4}\right )} x}{8 \, a^{5}} + \frac {9 \, a^{3} \tan \left (\frac {1}{2} \, x\right )^{7} + 12 \, a b^{2} \tan \left (\frac {1}{2} \, x\right )^{7} + 24 \, b^{3} \tan \left (\frac {1}{2} \, x\right )^{6} + 33 \, a^{3} \tan \left (\frac {1}{2} \, x\right )^{5} + 12 \, a b^{2} \tan \left (\frac {1}{2} \, x\right )^{5} + 48 \, a^{2} b \tan \left (\frac {1}{2} \, x\right )^{4} + 72 \, b^{3} \tan \left (\frac {1}{2} \, x\right )^{4} - 33 \, a^{3} \tan \left (\frac {1}{2} \, x\right )^{3} - 12 \, a b^{2} \tan \left (\frac {1}{2} \, x\right )^{3} + 64 \, a^{2} b \tan \left (\frac {1}{2} \, x\right )^{2} + 72 \, b^{3} \tan \left (\frac {1}{2} \, x\right )^{2} - 9 \, a^{3} \tan \left (\frac {1}{2} \, x\right ) - 12 \, a b^{2} \tan \left (\frac {1}{2} \, x\right ) + 16 \, a^{2} b + 24 \, b^{3}}{12 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{4} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.38, size = 405, normalized size = 2.81 \[ -\frac {2 b^{5} \arctan \left (\frac {2 \tan \left (\frac {x}{2}\right ) b +2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{a^{5} \sqrt {-a^{2}+b^{2}}}+\frac {3 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{4 a \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}+\frac {\left (\tan ^{7}\left (\frac {x}{2}\right )\right ) b^{2}}{a^{3} \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}+\frac {2 b^{3} \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a^{4} \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}+\frac {\left (\tan ^{5}\left (\frac {x}{2}\right )\right ) b^{2}}{a^{3} \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}+\frac {11 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{4 a \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}+\frac {4 \left (\tan ^{4}\left (\frac {x}{2}\right )\right ) b}{a^{2} \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}+\frac {6 \left (\tan ^{4}\left (\frac {x}{2}\right )\right ) b^{3}}{a^{4} \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}-\frac {\left (\tan ^{3}\left (\frac {x}{2}\right )\right ) b^{2}}{a^{3} \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}-\frac {11 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{4 a \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}+\frac {6 \left (\tan ^{2}\left (\frac {x}{2}\right )\right ) b^{3}}{a^{4} \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}+\frac {16 \left (\tan ^{2}\left (\frac {x}{2}\right )\right ) b}{3 a^{2} \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}-\frac {3 \tan \left (\frac {x}{2}\right )}{4 a \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}-\frac {\tan \left (\frac {x}{2}\right ) b^{2}}{a^{3} \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}+\frac {4 b}{3 a^{2} \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}+\frac {2 b^{3}}{a^{4} \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{4}}+\frac {2 \arctan \left (\tan \left (\frac {x}{2}\right )\right ) b^{4}}{a^{5}}+\frac {3 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{4 a}+\frac {\arctan \left (\tan \left (\frac {x}{2}\right )\right ) b^{2}}{a^{3}} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 1639, normalized size = 11.38 \[ \text {result too large to display} \]
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 \[ \int \frac {\sin ^{4}{\relax (x )}}{a + b \csc {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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