Optimal. Leaf size=422 \[ -\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \sqrt {b} \left (-a^2+b^2\right )^{5/4} d}+\frac {a \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \sqrt {b} \left (-a^2+b^2\right )^{5/4} d}+\frac {\sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}+\frac {a^2 e \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{2 b \left (a^2-b^2\right ) \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {a^2 e \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{2 b \left (a^2-b^2\right ) \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {b (e \cos (c+d x))^{3/2}}{\left (a^2-b^2\right ) d e (a+b \sin (c+d x))} \]
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Rubi [A]
time = 0.57, 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 = {2773, 2946,
2721, 2719, 2780, 2886, 2884, 335, 304, 211, 214} \begin {gather*} -\frac {a \sqrt {e} \text {ArcTan}\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 {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} \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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 214
Rule 304
Rule 335
Rule 2719
Rule 2721
Rule 2773
Rule 2780
Rule 2884
Rule 2886
Rule 2946
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) \text {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) \text {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) \text {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) \text {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] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 36.13, size = 787, normalized size = 1.86 \begin {gather*} -\frac {b \cos (c+d x) \sqrt {e \cos (c+d x)}}{\left (-a^2+b^2\right ) d (a+b \sin (c+d x))}+\frac {\sqrt {e \cos (c+d x)} \left (-\frac {4 a \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)}{-a^2+b^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]{-a^2+b^2}}\right )-2 \tan ^{-1}\left (1+\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d 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 (c+d x)}+i b \cos (c+d x)\right )+\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (c+d x)}+i b \cos (c+d x)\right )\right )}{\sqrt {b} \sqrt [4]{-a^2+b^2}}\right ) \sin (c+d x)}{\sqrt {1-\cos ^2(c+d x)} (a+b \sin (c+d x))}-\frac {\left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (8 b^{5/2} 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)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(c+d x)+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 (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+b \cos (c+d x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+b \cos (c+d x)\right )\right )\right ) \sin ^2(c+d x)}{12 \sqrt {b} \left (-a^2+b^2\right ) \left (1-\cos ^2(c+d x)\right ) (a+b \sin (c+d x))}\right )}{2 (a-b) (a+b) d \sqrt {\cos (c+d x)}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 27.31, size = 9239, normalized size = 21.89
method | result | size |
default | \(\text {Expression too large to display}\) | \(9239\) |
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]
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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Sympy [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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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 {\sqrt {e\,\cos \left (c+d\,x\right )}}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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