Optimal. Leaf size=514 \[ \frac {5 a b (e \cos (c+d x))^{3/2}}{4 d e \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}+\frac {b (e \cos (c+d x))^{3/2}}{2 d e \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}+\frac {\sqrt {e} \left (3 a^2+2 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 \sqrt {b} d \left (b^2-a^2\right )^{9/4}}-\frac {\sqrt {e} \left (3 a^2+2 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 \sqrt {b} d \left (b^2-a^2\right )^{9/4}}+\frac {5 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{4 d \left (a^2-b^2\right )^2 \sqrt {\cos (c+d x)}}+\frac {a e \left (3 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b d \left (a^2-b^2\right )^2 \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}+\frac {a e \left (3 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b d \left (a^2-b^2\right )^2 \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}} \]
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Rubi [A] time = 1.19, antiderivative size = 514, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {2694, 2864, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208} \[ \frac {5 a b (e \cos (c+d x))^{3/2}}{4 d e \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}+\frac {b (e \cos (c+d x))^{3/2}}{2 d e \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}+\frac {\sqrt {e} \left (3 a^2+2 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 \sqrt {b} d \left (b^2-a^2\right )^{9/4}}-\frac {\sqrt {e} \left (3 a^2+2 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 \sqrt {b} d \left (b^2-a^2\right )^{9/4}}+\frac {5 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{4 d \left (a^2-b^2\right )^2 \sqrt {\cos (c+d x)}}+\frac {a e \left (3 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b d \left (a^2-b^2\right )^2 \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}+\frac {a e \left (3 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b d \left (a^2-b^2\right )^2 \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 298
Rule 329
Rule 2639
Rule 2640
Rule 2694
Rule 2701
Rule 2805
Rule 2807
Rule 2864
Rule 2867
Rubi steps
\begin {align*} \int \frac {\sqrt {e \cos (c+d x)}}{(a+b \sin (c+d x))^3} \, dx &=\frac {b (e \cos (c+d x))^{3/2}}{2 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^2}-\frac {\int \frac {\sqrt {e \cos (c+d x)} \left (-2 a+\frac {1}{2} b \sin (c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{2 \left (a^2-b^2\right )}\\ &=\frac {b (e \cos (c+d x))^{3/2}}{2 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^2}+\frac {5 a b (e \cos (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))}+\frac {\int \frac {\sqrt {e \cos (c+d x)} \left (\frac {1}{2} \left (4 a^2+b^2\right )+\frac {5}{4} a b \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2 \left (a^2-b^2\right )^2}\\ &=\frac {b (e \cos (c+d x))^{3/2}}{2 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^2}+\frac {5 a b (e \cos (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))}+\frac {(5 a) \int \sqrt {e \cos (c+d x)} \, dx}{8 \left (a^2-b^2\right )^2}+\frac {\left (3 a^2+2 b^2\right ) \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)} \, dx}{8 \left (a^2-b^2\right )^2}\\ &=\frac {b (e \cos (c+d x))^{3/2}}{2 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^2}+\frac {5 a b (e \cos (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))}-\frac {\left (a \left (3 a^2+2 b^2\right ) e\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{16 b \left (a^2-b^2\right )^2}+\frac {\left (a \left (3 a^2+2 b^2\right ) e\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{16 b \left (a^2-b^2\right )^2}+\frac {\left (b \left (3 a^2+2 b^2\right ) e\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{\left (a^2-b^2\right ) e^2+b^2 x^2} \, dx,x,e \cos (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d}+\frac {\left (5 a \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{8 \left (a^2-b^2\right )^2 \sqrt {\cos (c+d x)}}\\ &=\frac {5 a \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}+\frac {b (e \cos (c+d x))^{3/2}}{2 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^2}+\frac {5 a b (e \cos (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))}+\frac {\left (b \left (3 a^2+2 b^2\right ) e\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a^2-b^2\right ) e^2+b^2 x^4} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{4 \left (a^2-b^2\right )^2 d}-\frac {\left (a \left (3 a^2+2 b^2\right ) e \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \left (\sqrt {-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{16 b \left (a^2-b^2\right )^2 \sqrt {e \cos (c+d x)}}+\frac {\left (a \left (3 a^2+2 b^2\right ) e \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \left (\sqrt {-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{16 b \left (a^2-b^2\right )^2 \sqrt {e \cos (c+d x)}}\\ &=\frac {5 a \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}+\frac {a \left (3 a^2+2 b^2\right ) e \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b \left (a^2-b^2\right )^2 \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {a \left (3 a^2+2 b^2\right ) e \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b \left (a^2-b^2\right )^2 \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {b (e \cos (c+d x))^{3/2}}{2 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^2}+\frac {5 a b (e \cos (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))}-\frac {\left (\left (3 a^2+2 b^2\right ) e\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e-b x^2} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{8 \left (a^2-b^2\right )^2 d}+\frac {\left (\left (3 a^2+2 b^2\right ) e\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e+b x^2} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{8 \left (a^2-b^2\right )^2 d}\\ &=\frac {\left (3 a^2+2 b^2\right ) \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \sqrt {b} \left (-a^2+b^2\right )^{9/4} d}-\frac {\left (3 a^2+2 b^2\right ) \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \sqrt {b} \left (-a^2+b^2\right )^{9/4} d}+\frac {5 a \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}+\frac {a \left (3 a^2+2 b^2\right ) e \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b \left (a^2-b^2\right )^2 \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {a \left (3 a^2+2 b^2\right ) e \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b \left (a^2-b^2\right )^2 \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {b (e \cos (c+d x))^{3/2}}{2 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^2}+\frac {5 a b (e \cos (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [C] time = 24.33, size = 837, normalized size = 1.63 \[ \frac {\sqrt {e \cos (c+d x)} \left (\frac {5 a b \cos (c+d x)}{4 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}+\frac {b \cos (c+d x)}{2 \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\right )}{d}+\frac {\sqrt {e \cos (c+d x)} \left (-\frac {5 a \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (8 F_1\left (\frac {3}{4};-\frac {1}{2},1;\frac {7}{4};\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{b^2-a^2}\right ) \cos ^{\frac {3}{2}}(c+d x) b^{5/2}+3 \sqrt {2} a \left (a^2-b^2\right )^{3/4} \left (2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}+1\right )-\log \left (b \cos (c+d x)-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+\sqrt {a^2-b^2}\right )+\log \left (b \cos (c+d x)+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+\sqrt {a^2-b^2}\right )\right )\right ) \sin ^2(c+d x)}{12 \sqrt {b} \left (b^2-a^2\right ) \left (1-\cos ^2(c+d x)\right ) (a+b \sin (c+d x))}-\frac {2 \left (8 a^2+2 b^2\right ) \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (\frac {a F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{b^2-a^2}\right ) \cos ^{\frac {3}{2}}(c+d x)}{3 \left (a^2-b^2\right )}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 \tan ^{-1}\left (1-\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{b^2-a^2}}\right )-2 \tan ^{-1}\left (\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{b^2-a^2}}+1\right )-\log \left (i b \cos (c+d x)-(1+i) \sqrt {b} \sqrt [4]{b^2-a^2} \sqrt {\cos (c+d x)}+\sqrt {b^2-a^2}\right )+\log \left (i b \cos (c+d x)+(1+i) \sqrt {b} \sqrt [4]{b^2-a^2} \sqrt {\cos (c+d x)}+\sqrt {b^2-a^2}\right )\right )}{\sqrt {b} \sqrt [4]{b^2-a^2}}\right ) \sin (c+d x)}{\sqrt {1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right )}{8 (a-b)^2 (a+b)^2 d \sqrt {\cos (c+d x)}} \]
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 \[ \int \frac {\sqrt {e \cos \left (d x + c\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 30.27, size = 36688, normalized size = 71.38 \[ \text {output too large to display} \]
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 \[ \int \frac {\sqrt {e \cos \left (d x + c\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]
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 \[ \int \frac {\sqrt {e\,\cos \left (c+d\,x\right )}}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3} \,d x \]
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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