Optimal. Leaf size=507 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a f g^{3/2}}-\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{a \left (-a^2+b^2\right )^{5/4} f g^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a f g^{3/2}}+\frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{a \left (-a^2+b^2\right )^{5/4} f g^{3/2}}+\frac {2}{a f g \sqrt {g \cos (e+f x)}}+\frac {2 b \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{\left (a^2-b^2\right ) f g^2 \sqrt {\cos (e+f x)}}+\frac {b^2 \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{\left (a^2-b^2\right ) \left (b-\sqrt {-a^2+b^2}\right ) f g \sqrt {g \cos (e+f x)}}+\frac {b^2 \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{\left (a^2-b^2\right ) \left (b+\sqrt {-a^2+b^2}\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 b (b-a \sin (e+f x))}{a \left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}} \]
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Rubi [A]
time = 0.88, antiderivative size = 507, normalized size of antiderivative = 1.00, number of steps
used = 21, number of rules used = 16, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.516, Rules used = {2977,
2645, 331, 335, 304, 209, 212, 2775, 2946, 2721, 2719, 2780, 2886, 2884, 211, 214}
\begin {gather*} -\frac {b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{a f g^{3/2} \left (b^2-a^2\right )^{5/4}}+\frac {2 b E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{f g^2 \left (a^2-b^2\right ) \sqrt {\cos (e+f x)}}+\frac {2 b (b-a \sin (e+f x))}{a f g \left (a^2-b^2\right ) \sqrt {g \cos (e+f x)}}+\frac {b^2 \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b-\sqrt {b^2-a^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{f g \left (a^2-b^2\right ) \left (b-\sqrt {b^2-a^2}\right ) \sqrt {g \cos (e+f x)}}+\frac {b^2 \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b+\sqrt {b^2-a^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{f g \left (a^2-b^2\right ) \left (\sqrt {b^2-a^2}+b\right ) \sqrt {g \cos (e+f x)}}+\frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{a f g^{3/2} \left (b^2-a^2\right )^{5/4}}+\frac {\text {ArcTan}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a f g^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a f g^{3/2}}+\frac {2}{a f g \sqrt {g \cos (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 211
Rule 212
Rule 214
Rule 304
Rule 331
Rule 335
Rule 2645
Rule 2719
Rule 2721
Rule 2775
Rule 2780
Rule 2884
Rule 2886
Rule 2946
Rule 2977
Rubi steps
\begin {align*} \int \frac {\csc (e+f x)}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx &=\int \left (\frac {\csc (e+f x)}{a (g \cos (e+f x))^{3/2}}-\frac {b}{a (g \cos (e+f x))^{3/2} (a+b \sin (e+f x))}\right ) \, dx\\ &=\frac {\int \frac {\csc (e+f x)}{(g \cos (e+f x))^{3/2}} \, dx}{a}-\frac {b \int \frac {1}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx}{a}\\ &=\frac {2 b (b-a \sin (e+f x))}{a \left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}+\frac {(2 b) \int \frac {\sqrt {g \cos (e+f x)} \left (\frac {a^2}{2}+\frac {b^2}{2}+\frac {1}{2} a b \sin (e+f x)\right )}{a+b \sin (e+f x)} \, dx}{a \left (a^2-b^2\right ) g^2}-\frac {\text {Subst}\left (\int \frac {1}{x^{3/2} \left (1-\frac {x^2}{g^2}\right )} \, dx,x,g \cos (e+f x)\right )}{a f g}\\ &=\frac {2}{a f g \sqrt {g \cos (e+f x)}}+\frac {2 b (b-a \sin (e+f x))}{a \left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}-\frac {\text {Subst}\left (\int \frac {\sqrt {x}}{1-\frac {x^2}{g^2}} \, dx,x,g \cos (e+f x)\right )}{a f g^3}+\frac {b \int \sqrt {g \cos (e+f x)} \, dx}{\left (a^2-b^2\right ) g^2}+\frac {b^3 \int \frac {\sqrt {g \cos (e+f x)}}{a+b \sin (e+f x)} \, dx}{a \left (a^2-b^2\right ) g^2}\\ &=\frac {2}{a f g \sqrt {g \cos (e+f x)}}+\frac {2 b (b-a \sin (e+f x))}{a \left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}-\frac {2 \text {Subst}\left (\int \frac {x^2}{1-\frac {x^4}{g^2}} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{a f g^3}-\frac {b^2 \int \frac {1}{\sqrt {g \cos (e+f x)} \left (\sqrt {-a^2+b^2}-b \cos (e+f x)\right )} \, dx}{2 \left (a^2-b^2\right ) g}+\frac {b^2 \int \frac {1}{\sqrt {g \cos (e+f x)} \left (\sqrt {-a^2+b^2}+b \cos (e+f x)\right )} \, dx}{2 \left (a^2-b^2\right ) g}+\frac {b^4 \text {Subst}\left (\int \frac {\sqrt {x}}{\left (a^2-b^2\right ) g^2+b^2 x^2} \, dx,x,g \cos (e+f x)\right )}{a \left (a^2-b^2\right ) f g}+\frac {\left (b \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{\left (a^2-b^2\right ) g^2 \sqrt {\cos (e+f x)}}\\ &=\frac {2}{a f g \sqrt {g \cos (e+f x)}}+\frac {2 b \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{\left (a^2-b^2\right ) f g^2 \sqrt {\cos (e+f x)}}+\frac {2 b (b-a \sin (e+f x))}{a \left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}-\frac {\text {Subst}\left (\int \frac {1}{g-x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{a f g}+\frac {\text {Subst}\left (\int \frac {1}{g+x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{a f g}+\frac {\left (2 b^4\right ) \text {Subst}\left (\int \frac {x^2}{\left (a^2-b^2\right ) g^2+b^2 x^4} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{a \left (a^2-b^2\right ) f g}-\frac {\left (b^2 \sqrt {\cos (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)} \left (\sqrt {-a^2+b^2}-b \cos (e+f x)\right )} \, dx}{2 \left (a^2-b^2\right ) g \sqrt {g \cos (e+f x)}}+\frac {\left (b^2 \sqrt {\cos (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)} \left (\sqrt {-a^2+b^2}+b \cos (e+f x)\right )} \, dx}{2 \left (a^2-b^2\right ) g \sqrt {g \cos (e+f x)}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a f g^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a f g^{3/2}}+\frac {2}{a f g \sqrt {g \cos (e+f x)}}+\frac {2 b \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{\left (a^2-b^2\right ) f g^2 \sqrt {\cos (e+f x)}}+\frac {b^2 \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{\left (a^2-b^2\right ) \left (b-\sqrt {-a^2+b^2}\right ) f g \sqrt {g \cos (e+f x)}}+\frac {b^2 \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{\left (a^2-b^2\right ) \left (b+\sqrt {-a^2+b^2}\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 b (b-a \sin (e+f x))}{a \left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}-\frac {b^3 \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} g-b x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{a \left (a^2-b^2\right ) f g}+\frac {b^3 \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} g+b x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{a \left (a^2-b^2\right ) f g}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a f g^{3/2}}-\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{a \left (-a^2+b^2\right )^{5/4} f g^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a f g^{3/2}}+\frac {b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{a \left (-a^2+b^2\right )^{5/4} f g^{3/2}}+\frac {2}{a f g \sqrt {g \cos (e+f x)}}+\frac {2 b \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{\left (a^2-b^2\right ) f g^2 \sqrt {\cos (e+f x)}}+\frac {b^2 \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{\left (a^2-b^2\right ) \left (b-\sqrt {-a^2+b^2}\right ) f g \sqrt {g \cos (e+f x)}}+\frac {b^2 \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{\left (a^2-b^2\right ) \left (b+\sqrt {-a^2+b^2}\right ) f g \sqrt {g \cos (e+f x)}}+\frac {2 b (b-a \sin (e+f x))}{a \left (a^2-b^2\right ) f g \sqrt {g \cos (e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 58.85, size = 1587, normalized size = 3.13 \begin {gather*} -\frac {\cos ^{\frac {3}{2}}(e+f x) \left (\frac {8 a b \left (a+b \sqrt {1-\cos ^2(e+f x)}\right ) \left (\frac {a F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(e+f x)}{3 \left (a^2-b^2\right )}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 \tan ^{-1}\left (1-\frac {(1+i) \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \tan ^{-1}\left (1+\frac {(1+i) \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{-a^2+b^2}}\right )-\log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (e+f x)}+i b \cos (e+f x)\right )+\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (e+f x)}+i b \cos (e+f x)\right )\right )}{\sqrt {b} \sqrt [4]{-a^2+b^2}}\right )}{\sqrt {1-\cos ^2(e+f x)} (b+a \csc (e+f x))}-\frac {\left (-2 a^2+b^2\right ) \left (-1+\cos ^2(e+f x)\right ) \left (a+b \sqrt {1-\cos ^2(e+f x)}\right ) \csc (e+f x) \left (6 \sqrt {2} \sqrt {b} \left (a^2-b^2\right )^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )-6 \sqrt {2} \sqrt {b} \left (a^2-b^2\right )^{3/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )+12 \left (a^2-b^2\right ) \tan ^{-1}\left (\sqrt {\cos (e+f x)}\right )+8 a b F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(e+f x)+6 a^2 \log \left (1-\sqrt {\cos (e+f x)}\right )-6 b^2 \log \left (1-\sqrt {\cos (e+f x)}\right )-6 a^2 \log \left (1+\sqrt {\cos (e+f x)}\right )+6 b^2 \log \left (1+\sqrt {\cos (e+f x)}\right )-3 \sqrt {2} \sqrt {b} \left (a^2-b^2\right )^{3/4} \log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (e+f x)}+b \cos (e+f x)\right )+3 \sqrt {2} \sqrt {b} \left (a^2-b^2\right )^{3/4} \log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (e+f x)}+b \cos (e+f x)\right )\right )}{12 \left (a^3-a b^2\right ) \left (1-\cos ^2(e+f x)\right ) (b+a \csc (e+f x))}-\frac {\sqrt {b} \left (-1+\cos ^2(e+f x)\right ) \left (a+b \sqrt {1-\cos ^2(e+f x)}\right ) \cos (2 (e+f x)) \csc (e+f x) \left (-42 \sqrt {2} \left (a^2-b^2\right )^{3/4} \left (2 a^2-b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )+42 \sqrt {2} \left (a^2-b^2\right )^{3/4} \left (2 a^2-b^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )+84 b^{3/2} \left (a^2-b^2\right ) \tan ^{-1}\left (\sqrt {\cos (e+f x)}\right )-56 a b^{5/2} F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(e+f x)+48 a b^{5/2} F_1\left (\frac {7}{4};\frac {1}{2},1;\frac {11}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right ) \cos ^{\frac {7}{2}}(e+f x)+42 b^{3/2} \left (a^2-b^2\right ) \log \left (1-\sqrt {\cos (e+f x)}\right )+42 b^{3/2} \left (-a^2+b^2\right ) \log \left (1+\sqrt {\cos (e+f x)}\right )+21 \sqrt {2} \left (a^2-b^2\right )^{3/4} \left (2 a^2-b^2\right ) \log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (e+f x)}+b \cos (e+f x)\right )-21 \sqrt {2} \left (a^2-b^2\right )^{3/4} \left (2 a^2-b^2\right ) \log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (e+f x)}+b \cos (e+f x)\right )\right )}{84 \left (a^3-a b^2\right ) \left (1-\cos ^2(e+f x)\right ) \left (-1+2 \cos ^2(e+f x)\right ) (b+a \csc (e+f x))}\right )}{2 (a-b) (a+b) f (g \cos (e+f x))^{3/2}}+\frac {2 \cos (e+f x) (a-b \sin (e+f x))}{\left (a^2-b^2\right ) f (g \cos (e+f x))^{3/2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 28.87, size = 425, normalized size = 0.84
method | result | size |
default | \(\frac {-\left (4 g^{\frac {5}{2}} \ln \left (\frac {2 \sqrt {-g}\, \sqrt {-2 \left (\sin ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) g +g}-2 g}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )}\right )+2 \sqrt {-g}\, \ln \left (\frac {2 \sqrt {g}\, \sqrt {-2 \left (\sin ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) g +g}+4 g \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-2 g}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )-1}\right ) g^{2}+2 \sqrt {-g}\, \ln \left (\frac {2 \sqrt {g}\, \sqrt {-2 \left (\sin ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) g +g}-4 g \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-2 g}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right ) g^{2}\right ) \left (\sin ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 g^{\frac {5}{2}} \ln \left (\frac {2 \sqrt {-g}\, \sqrt {-2 \left (\sin ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) g +g}-2 g}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )}\right )-4 \sqrt {-2 \left (\sin ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) g +g}\, \sqrt {-g}\, g^{\frac {3}{2}}+\sqrt {-g}\, \ln \left (\frac {2 \sqrt {g}\, \sqrt {-2 \left (\sin ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) g +g}+4 g \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-2 g}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )-1}\right ) g^{2}+\sqrt {-g}\, \ln \left (\frac {2 \sqrt {g}\, \sqrt {-2 \left (\sin ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) g +g}-4 g \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-2 g}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right ) g^{2}}{2 g^{\frac {7}{2}} \sqrt {-g}\, a \left (2 \left (\sin ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-1\right ) f}\) | \(425\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\csc {\left (e + f x \right )}}{\left (g \cos {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (a + b \sin {\left (e + f x \right )}\right )}\, dx \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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{\sin \left (e+f\,x\right )\,{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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