Optimal. Leaf size=432 \[ -\frac {2 b \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{3 f g^2 \left (a^2-b^2\right ) \sqrt {g \cos (e+f x)}}+\frac {a^2 b \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b-\sqrt {b^2-a^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{f g^2 \left (a^2-b^2\right ) \left (a^2-b \left (b-\sqrt {b^2-a^2}\right )\right ) \sqrt {g \cos (e+f x)}}+\frac {a^2 b \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b+\sqrt {b^2-a^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{f g^2 \left (a^2-b^2\right ) \left (a^2-b \left (\sqrt {b^2-a^2}+b\right )\right ) \sqrt {g \cos (e+f x)}}+\frac {2 (a-b \sin (e+f x))}{3 f g \left (a^2-b^2\right ) (g \cos (e+f x))^{3/2}}+\frac {a b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{f g^{5/2} \left (b^2-a^2\right )^{7/4}}+\frac {a b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{f g^{5/2} \left (b^2-a^2\right )^{7/4}} \]
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Rubi [A] time = 0.92, antiderivative size = 432, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {2866, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205} \[ \frac {a b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{f g^{5/2} \left (b^2-a^2\right )^{7/4}}+\frac {a b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{f g^{5/2} \left (b^2-a^2\right )^{7/4}}-\frac {2 b \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{3 f g^2 \left (a^2-b^2\right ) \sqrt {g \cos (e+f x)}}+\frac {a^2 b \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b-\sqrt {b^2-a^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{f g^2 \left (a^2-b^2\right ) \left (a^2-b \left (b-\sqrt {b^2-a^2}\right )\right ) \sqrt {g \cos (e+f x)}}+\frac {a^2 b \sqrt {\cos (e+f x)} \Pi \left (\frac {2 b}{b+\sqrt {b^2-a^2}};\left .\frac {1}{2} (e+f x)\right |2\right )}{f g^2 \left (a^2-b^2\right ) \left (a^2-b \left (\sqrt {b^2-a^2}+b\right )\right ) \sqrt {g \cos (e+f x)}}+\frac {2 (a-b \sin (e+f x))}{3 f g \left (a^2-b^2\right ) (g \cos (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 212
Rule 329
Rule 2641
Rule 2642
Rule 2702
Rule 2805
Rule 2807
Rule 2866
Rule 2867
Rubi steps
\begin {align*} \int \frac {\sin (e+f x)}{(g \cos (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx &=\frac {2 (a-b \sin (e+f x))}{3 \left (a^2-b^2\right ) f g (g \cos (e+f x))^{3/2}}-\frac {2 \int \frac {-a b+\frac {1}{2} b^2 \sin (e+f x)}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))} \, dx}{3 \left (a^2-b^2\right ) g^2}\\ &=\frac {2 (a-b \sin (e+f x))}{3 \left (a^2-b^2\right ) f g (g \cos (e+f x))^{3/2}}-\frac {b \int \frac {1}{\sqrt {g \cos (e+f x)}} \, dx}{3 \left (a^2-b^2\right ) g^2}+\frac {(a b) \int \frac {1}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))} \, dx}{\left (a^2-b^2\right ) g^2}\\ &=\frac {2 (a-b \sin (e+f x))}{3 \left (a^2-b^2\right ) f g (g \cos (e+f x))^{3/2}}+\frac {\left (a^2 b\right ) \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 )^{3/2} g^2}+\frac {\left (a^2 b\right ) \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 )^{3/2} g^2}+\frac {\left (a b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (\left (a^2-b^2\right ) g^2+b^2 x^2\right )} \, dx,x,g \cos (e+f x)\right )}{\left (a^2-b^2\right ) f g}-\frac {\left (b \sqrt {\cos (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx}{3 \left (a^2-b^2\right ) g^2 \sqrt {g \cos (e+f x)}}\\ &=-\frac {2 b \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{3 \left (a^2-b^2\right ) f g^2 \sqrt {g \cos (e+f x)}}+\frac {2 (a-b \sin (e+f x))}{3 \left (a^2-b^2\right ) f g (g \cos (e+f x))^{3/2}}+\frac {\left (2 a b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a^2-b^2\right ) g^2+b^2 x^4} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{\left (a^2-b^2\right ) f g}+\frac {\left (a^2 b \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 )^{3/2} g^2 \sqrt {g \cos (e+f x)}}+\frac {\left (a^2 b \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 )^{3/2} g^2 \sqrt {g \cos (e+f x)}}\\ &=-\frac {2 b \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{3 \left (a^2-b^2\right ) f g^2 \sqrt {g \cos (e+f x)}}-\frac {a^2 b \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 )^{3/2} \left (b-\sqrt {-a^2+b^2}\right ) f g^2 \sqrt {g \cos (e+f x)}}+\frac {a^2 b \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 )^{3/2} \left (b+\sqrt {-a^2+b^2}\right ) f g^2 \sqrt {g \cos (e+f x)}}+\frac {2 (a-b \sin (e+f x))}{3 \left (a^2-b^2\right ) f g (g \cos (e+f x))^{3/2}}+\frac {\left (a b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} g-b x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{\left (-a^2+b^2\right )^{3/2} f g^2}+\frac {\left (a b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} g+b x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{\left (-a^2+b^2\right )^{3/2} f g^2}\\ &=\frac {a b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{\left (-a^2+b^2\right )^{7/4} f g^{5/2}}+\frac {a b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{\left (-a^2+b^2\right )^{7/4} f g^{5/2}}-\frac {2 b \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{3 \left (a^2-b^2\right ) f g^2 \sqrt {g \cos (e+f x)}}-\frac {a^2 b \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 )^{3/2} \left (b-\sqrt {-a^2+b^2}\right ) f g^2 \sqrt {g \cos (e+f x)}}+\frac {a^2 b \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 )^{3/2} \left (b+\sqrt {-a^2+b^2}\right ) f g^2 \sqrt {g \cos (e+f x)}}+\frac {2 (a-b \sin (e+f x))}{3 \left (a^2-b^2\right ) f g (g \cos (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [C] time = 23.25, size = 1183, normalized size = 2.74 \[ \frac {b \left (\frac {2 b \left (a+b \sqrt {1-\cos ^2(e+f x)}\right ) \left (\frac {5 b \left (a^2-b^2\right ) \sqrt {\cos (e+f x)} \sqrt {1-\cos ^2(e+f x)} F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{b^2-a^2}\right )}{\left (2 \left (2 F_1\left (\frac {5}{4};-\frac {1}{2},2;\frac {9}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{b^2-a^2}\right ) b^2+\left (a^2-b^2\right ) F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{b^2-a^2}\right )\right ) \cos ^2(e+f x)-5 \left (a^2-b^2\right ) F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{b^2-a^2}\right )\right ) \left (a^2+b^2 \left (\cos ^2(e+f x)-1\right )\right )}+\frac {a \left (-2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{a^2-b^2}}\right )+2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{a^2-b^2}}+1\right )-\log \left (b \cos (e+f x)-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (e+f x)}+\sqrt {a^2-b^2}\right )+\log \left (b \cos (e+f x)+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (e+f x)}+\sqrt {a^2-b^2}\right )\right )}{4 \sqrt {2} \sqrt {b} \left (a^2-b^2\right )^{3/4}}\right ) \sin ^2(e+f x)}{\left (1-\cos ^2(e+f x)\right ) (a+b \sin (e+f x))}-\frac {4 a \left (a+b \sqrt {1-\cos ^2(e+f x)}\right ) \left (\frac {5 a \left (a^2-b^2\right ) F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{b^2-a^2}\right ) \sqrt {\cos (e+f x)}}{\sqrt {1-\cos ^2(e+f x)} \left (5 \left (a^2-b^2\right ) F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{b^2-a^2}\right )-2 \left (2 F_1\left (\frac {5}{4};\frac {1}{2},2;\frac {9}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{b^2-a^2}\right ) b^2+\left (b^2-a^2\right ) F_1\left (\frac {5}{4};\frac {3}{2},1;\frac {9}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{b^2-a^2}\right )\right ) \cos ^2(e+f x)\right ) \left (a^2+b^2 \left (\cos ^2(e+f x)-1\right )\right )}-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \sqrt {b} \left (2 \tan ^{-1}\left (1-\frac {(1+i) \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{b^2-a^2}}\right )-2 \tan ^{-1}\left (\frac {(1+i) \sqrt {b} \sqrt {\cos (e+f x)}}{\sqrt [4]{b^2-a^2}}+1\right )+\log \left (i b \cos (e+f x)-(1+i) \sqrt {b} \sqrt [4]{b^2-a^2} \sqrt {\cos (e+f x)}+\sqrt {b^2-a^2}\right )-\log \left (i b \cos (e+f x)+(1+i) \sqrt {b} \sqrt [4]{b^2-a^2} \sqrt {\cos (e+f x)}+\sqrt {b^2-a^2}\right )\right )}{\left (b^2-a^2\right )^{3/4}}\right ) \sin (e+f x)}{\sqrt {1-\cos ^2(e+f x)} (a+b \sin (e+f x))}\right ) \cos ^{\frac {5}{2}}(e+f x)}{3 (a-b) (a+b) f (g \cos (e+f x))^{5/2}}+\frac {2 (a-b \sin (e+f x)) \cos (e+f x)}{3 \left (a^2-b^2\right ) f (g \cos (e+f x))^{5/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {\sin \left (f x + e\right )}{\left (g \cos \left (f x + e\right )\right )^{\frac {5}{2}} {\left (b \sin \left (f x + e\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 13.25, size = 2322, normalized size = 5.38 \[ \text {Expression too large to display} \]
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 \[ \int \frac {\sin \left (f x + e\right )}{\left (g \cos \left (f x + e\right )\right )^{\frac {5}{2}} {\left (b \sin \left (f x + e\right ) + a\right )}}\,{d x} \]
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 \[ \int \frac {\sin \left (e+f\,x\right )}{{\left (g\,\cos \left (e+f\,x\right )\right )}^{5/2}\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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