Optimal. Leaf size=492 \[ -\frac {5 a b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \left (-a^2+b^2\right )^{9/4} d e^{3/2}}+\frac {5 a b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \left (-a^2+b^2\right )^{9/4} d e^{3/2}}-\frac {\left (2 a^2+3 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\left (a^2-b^2\right )^2 d e^2 \sqrt {\cos (c+d x)}}-\frac {5 a^2 b \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{2 \left (a^2-b^2\right )^2 \left (b-\sqrt {-a^2+b^2}\right ) d e \sqrt {e \cos (c+d x)}}-\frac {5 a^2 b \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{2 \left (a^2-b^2\right )^2 \left (b+\sqrt {-a^2+b^2}\right ) d e \sqrt {e \cos (c+d x)}}+\frac {b}{\left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)}{\left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)}} \]
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Rubi [A]
time = 0.81, antiderivative size = 492, 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 = {2773, 2945,
2946, 2721, 2719, 2780, 2886, 2884, 335, 304, 211, 214} \begin {gather*} -\frac {5 a b^{3/2} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{2 d e^{3/2} \left (b^2-a^2\right )^{9/4}}-\frac {\left (2 a^2+3 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d e^2 \left (a^2-b^2\right )^2 \sqrt {\cos (c+d x)}}-\frac {5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)}{d e \left (a^2-b^2\right )^2 \sqrt {e \cos (c+d x)}}+\frac {b}{d e \left (a^2-b^2\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {5 a^2 b \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 d e \left (a^2-b^2\right )^2 \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}-\frac {5 a^2 b \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 d e \left (a^2-b^2\right )^2 \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}}+\frac {5 a b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{2 d e^{3/2} \left (b^2-a^2\right )^{9/4}} \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 2945
Rule 2946
Rubi steps
\begin {align*} \int \frac {1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2} \, dx &=\frac {b}{\left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}+\frac {\int \frac {-a+\frac {3}{2} b \sin (c+d x)}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))} \, dx}{-a^2+b^2}\\ &=\frac {b}{\left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)}{\left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)}}+\frac {2 \int \frac {\sqrt {e \cos (c+d x)} \left (-\frac {1}{2} a \left (a^2+4 b^2\right )-\frac {1}{4} b \left (2 a^2+3 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{\left (a^2-b^2\right )^2 e^2}\\ &=\frac {b}{\left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)}{\left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)}}-\frac {\left (5 a b^2\right ) \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)} \, dx}{2 \left (a^2-b^2\right )^2 e^2}-\frac {\left (2 a^2+3 b^2\right ) \int \sqrt {e \cos (c+d x)} \, dx}{2 \left (a^2-b^2\right )^2 e^2}\\ &=\frac {b}{\left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)}{\left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)}}+\frac {\left (5 a^2 b\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{4 \left (a^2-b^2\right )^2 e}-\frac {\left (5 a^2 b\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{4 \left (a^2-b^2\right )^2 e}-\frac {\left (5 a b^3\right ) \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 )^2 d e}-\frac {\left (\left (2 a^2+3 b^2\right ) \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{2 \left (a^2-b^2\right )^2 e^2 \sqrt {\cos (c+d x)}}\\ &=-\frac {\left (2 a^2+3 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\left (a^2-b^2\right )^2 d e^2 \sqrt {\cos (c+d x)}}+\frac {b}{\left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)}{\left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)}}-\frac {\left (5 a b^3\right ) \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 )^2 d e}+\frac {\left (5 a^2 b \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 \left (a^2-b^2\right )^2 e \sqrt {e \cos (c+d x)}}-\frac {\left (5 a^2 b \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 \left (a^2-b^2\right )^2 e \sqrt {e \cos (c+d x)}}\\ &=-\frac {\left (2 a^2+3 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\left (a^2-b^2\right )^2 d e^2 \sqrt {\cos (c+d x)}}-\frac {5 a^2 b \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 \left (a^2-b^2\right )^2 \left (b-\sqrt {-a^2+b^2}\right ) d e \sqrt {e \cos (c+d x)}}-\frac {5 a^2 b \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 \left (a^2-b^2\right )^2 \left (b+\sqrt {-a^2+b^2}\right ) d e \sqrt {e \cos (c+d x)}}+\frac {b}{\left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)}{\left (a^2-b^2\right )^2 d e \sqrt {e \cos (c+d x)}}+\frac {\left (5 a b^2\right ) \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 )^2 d e}-\frac {\left (5 a b^2\right ) \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 )^2 d e}\\ &=-\frac {5 a b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \left (-a^2+b^2\right )^{9/4} d e^{3/2}}+\frac {5 a b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{2 \left (-a^2+b^2\right )^{9/4} d e^{3/2}}-\frac {\left (2 a^2+3 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\left (a^2-b^2\right )^2 d e^2 \sqrt {\cos (c+d x)}}-\frac {5 a^2 b \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 \left (a^2-b^2\right )^2 \left (b-\sqrt {-a^2+b^2}\right ) d e \sqrt {e \cos (c+d x)}}-\frac {5 a^2 b \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 \left (a^2-b^2\right )^2 \left (b+\sqrt {-a^2+b^2}\right ) d e \sqrt {e \cos (c+d x)}}+\frac {b}{\left (a^2-b^2\right ) d e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))}-\frac {5 a b-\left (2 a^2+3 b^2\right ) \sin (c+d x)}{\left (a^2-b^2\right )^2 d e \sqrt {e \cos (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 = 26.42, size = 777, normalized size = 1.58 \begin {gather*} \frac {\cos ^{\frac {3}{2}}(c+d x) \left (\frac {12 \left (-6 a^2 b+b^3-\left (2 a^2 b+3 b^3\right ) \cos (2 (c+d x))+4 a \left (a^2-b^2\right ) \sin (c+d x)\right )}{\left (a^2-b^2\right )^2 \sqrt {\cos (c+d x)}}-\frac {\sin (c+d x) \left (-\frac {\left (2 a^2+3 b^2\right ) \csc (c+d x) \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 )}{\sqrt {b} \left (-a^2+b^2\right )}-\frac {48 a \left (a^2+4 b^2\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 )}{\sqrt {\sin ^2(c+d x)}}\right ) \left (a+b \sqrt {\sin ^2(c+d x)}\right )}{(a-b)^2 (a+b)^2}\right )}{24 d (e \cos (c+d x))^{3/2} (a+b \sin (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 = 40.78, size = 7528, normalized size = 15.30
method | result | size |
default | \(\text {Expression too large to display}\) | \(7528\) |
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(-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 {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\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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