Optimal. Leaf size=422 \[ \frac {b (e \cos (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} d \left (b^2-a^2\right )^{5/4}}+\frac {a \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} d \left (b^2-a^2\right )^{5/4}}+\frac {E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)}}+\frac {a^2 e \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{2 b d \left (a^2-b^2\right ) \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}+\frac {a^2 e \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{2 b d \left (a^2-b^2\right ) \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}} \]
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Rubi [A] time = 0.87, antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {2694, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208} \[ \frac {b (e \cos (c+d x))^{3/2}}{d e \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} d \left (b^2-a^2\right )^{5/4}}+\frac {a \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{2 \sqrt {b} d \left (b^2-a^2\right )^{5/4}}+\frac {E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)}}+\frac {a^2 e \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{2 b d \left (a^2-b^2\right ) \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}+\frac {a^2 e \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{2 b d \left (a^2-b^2\right ) \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}} \]
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 2867
Rubi steps
\begin {align*} \int \frac {\sqrt {e \cos (c+d x)}}{(a+b \sin (c+d x))^2} \, dx &=\frac {b (e \cos (c+d x))^{3/2}}{\left (a^2-b^2\right ) d e (a+b \sin (c+d x))}+\frac {\int \frac {\sqrt {e \cos (c+d x)} \left (-a-\frac {1}{2} b \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{-a^2+b^2}\\ &=\frac {b (e \cos (c+d x))^{3/2}}{\left (a^2-b^2\right ) d e (a+b \sin (c+d x))}+\frac {\int \sqrt {e \cos (c+d x)} \, dx}{2 \left (a^2-b^2\right )}+\frac {a \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)} \, dx}{2 \left (a^2-b^2\right )}\\ &=\frac {b (e \cos (c+d x))^{3/2}}{\left (a^2-b^2\right ) d e (a+b \sin (c+d x))}-\frac {\left (a^2 e\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{4 b \left (a^2-b^2\right )}+\frac {\left (a^2 e\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{4 b \left (a^2-b^2\right )}+\frac {(a b e) \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 )}{2 \left (a^2-b^2\right ) d}+\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)} \, dx}{2 \left (a^2-b^2\right ) \sqrt {\cos (c+d x)}}\\ &=\frac {\sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}+\frac {b (e \cos (c+d x))^{3/2}}{\left (a^2-b^2\right ) d e (a+b \sin (c+d x))}+\frac {(a b e) \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 )}{\left (a^2-b^2\right ) d}-\frac {\left (a^2 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}{4 b \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}+\frac {\left (a^2 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}{4 b \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)}}\\ &=\frac {\sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}+\frac {a^2 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 )}{2 b \left (a^2-b^2\right ) \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {a^2 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 )}{2 b \left (a^2-b^2\right ) \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {b (e \cos (c+d x))^{3/2}}{\left (a^2-b^2\right ) d e (a+b \sin (c+d x))}-\frac {(a e) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e-b x^2} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{2 \left (a^2-b^2\right ) d}+\frac {(a e) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e+b x^2} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{2 \left (a^2-b^2\right ) d}\\ &=-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \sqrt {b} \left (-a^2+b^2\right )^{5/4} d}+\frac {a \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \sqrt {b} \left (-a^2+b^2\right )^{5/4} d}+\frac {\sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}+\frac {a^2 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 )}{2 b \left (a^2-b^2\right ) \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {a^2 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 )}{2 b \left (a^2-b^2\right ) \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {b (e \cos (c+d x))^{3/2}}{\left (a^2-b^2\right ) d e (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [C] time = 16.38, size = 787, normalized size = 1.86 \[ -\frac {b \cos (c+d x) \sqrt {e \cos (c+d x)}}{d \left (b^2-a^2\right ) (a+b \sin (c+d x))}+\frac {\sqrt {e \cos (c+d x)} \left (-\frac {\sin ^2(c+d x) \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (8 b^{5/2} \cos ^{\frac {3}{2}}(c+d x) F_1\left (\frac {3}{4};-\frac {1}{2},1;\frac {7}{4};\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{b^2-a^2}\right )+3 \sqrt {2} a \left (a^2-b^2\right )^{3/4} \left (-\log \left (-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+\sqrt {a^2-b^2}+b \cos (c+d x)\right )+\log \left (\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+\sqrt {a^2-b^2}+b \cos (c+d x)\right )+2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}+1\right )\right )\right )}{12 \sqrt {b} \left (b^2-a^2\right ) \left (1-\cos ^2(c+d x)\right ) (a+b \sin (c+d x))}-\frac {4 a \sin (c+d x) \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (\frac {a \cos ^{\frac {3}{2}}(c+d x) F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{b^2-a^2}\right )}{3 \left (a^2-b^2\right )}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (-\log \left (-(1+i) \sqrt {b} \sqrt [4]{b^2-a^2} \sqrt {\cos (c+d x)}+\sqrt {b^2-a^2}+i b \cos (c+d x)\right )+\log \left ((1+i) \sqrt {b} \sqrt [4]{b^2-a^2} \sqrt {\cos (c+d x)}+\sqrt {b^2-a^2}+i b \cos (c+d x)\right )+2 \tan ^{-1}\left (1-\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{b^2-a^2}}\right )-2 \tan ^{-1}\left (1+\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{b^2-a^2}}\right )\right )}{\sqrt {b} \sqrt [4]{b^2-a^2}}\right )}{\sqrt {1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right )}{2 d (a-b) (a+b) \sqrt {\cos (c+d x)}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 113.33, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {e \cos \left (d x + c\right )}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}, x\right ) \]
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 \[ \int \frac {\sqrt {e \cos \left (d x + c\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 9.15, size = 7033, normalized size = 16.67 \[ \text {output too large to display} \]
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 \[ \int \frac {\sqrt {e \cos \left (d x + c\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
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 \[ \int \frac {\sqrt {e\,\cos \left (c+d\,x\right )}}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^2} \,d x \]
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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