Optimal. Leaf size=589 \[ \frac {9 \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^{11/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 b^{11/2} \left (-a^2+b^2\right )^{3/4} d}+\frac {9 \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^{11/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 b^{11/2} \left (-a^2+b^2\right )^{3/4} d}+\frac {3 a \left (21 a^2-13 b^2\right ) e^6 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^6 d \sqrt {e \cos (c+d x)}}-\frac {9 a \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6 \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b^6 \left (a^2-b \left (b-\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \cos (c+d x)}}-\frac {9 a \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6 \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {-a^2+b^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{8 b^6 \left (a^2-b \left (b+\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \cos (c+d x)}}-\frac {e (e \cos (c+d x))^{9/2}}{2 b d (a+b \sin (c+d x))^2}-\frac {9 e^3 (e \cos (c+d x))^{5/2} (7 a+2 b \sin (c+d x))}{20 b^3 d (a+b \sin (c+d x))}+\frac {3 e^5 \sqrt {e \cos (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \sin (c+d x)\right )}{4 b^5 d} \]
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Rubi [A]
time = 1.03, antiderivative size = 589, normalized size of antiderivative = 1.00, number
of steps used = 15, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules
used = {2772, 2942, 2944, 2946, 2721, 2720, 2781, 2886, 2884, 335, 218, 214, 211}
\begin {gather*} \frac {3 a e^6 \left (21 a^2-13 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^6 d \sqrt {e \cos (c+d x)}}+\frac {3 e^5 \sqrt {e \cos (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \sin (c+d x)\right )}{4 b^5 d}+\frac {9 e^{11/2} \left (7 a^4-9 a^2 b^2+2 b^4\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 b^{11/2} d \left (b^2-a^2\right )^{3/4}}+\frac {9 e^{11/2} \left (7 a^4-9 a^2 b^2+2 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 b^{11/2} d \left (b^2-a^2\right )^{3/4}}-\frac {9 a e^6 \left (7 a^4-9 a^2 b^2+2 b^4\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^6 d \left (a^2-b \left (b-\sqrt {b^2-a^2}\right )\right ) \sqrt {e \cos (c+d x)}}-\frac {9 a e^6 \left (7 a^4-9 a^2 b^2+2 b^4\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^6 d \left (a^2-b \left (\sqrt {b^2-a^2}+b\right )\right ) \sqrt {e \cos (c+d x)}}-\frac {9 e^3 (e \cos (c+d x))^{5/2} (7 a+2 b \sin (c+d x))}{20 b^3 d (a+b \sin (c+d x))}-\frac {e (e \cos (c+d x))^{9/2}}{2 b d (a+b \sin (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 214
Rule 218
Rule 335
Rule 2720
Rule 2721
Rule 2772
Rule 2781
Rule 2884
Rule 2886
Rule 2942
Rule 2944
Rule 2946
Rubi steps
\begin {align*} \int \frac {(e \cos (c+d x))^{11/2}}{(a+b \sin (c+d x))^3} \, dx &=-\frac {e (e \cos (c+d x))^{9/2}}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (9 e^2\right ) \int \frac {(e \cos (c+d x))^{7/2} \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx}{4 b}\\ &=-\frac {e (e \cos (c+d x))^{9/2}}{2 b d (a+b \sin (c+d x))^2}-\frac {9 e^3 (e \cos (c+d x))^{5/2} (7 a+2 b \sin (c+d x))}{20 b^3 d (a+b \sin (c+d x))}+\frac {\left (9 e^4\right ) \int \frac {(e \cos (c+d x))^{3/2} \left (-b-\frac {7}{2} a \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4 b^3}\\ &=-\frac {e (e \cos (c+d x))^{9/2}}{2 b d (a+b \sin (c+d x))^2}-\frac {9 e^3 (e \cos (c+d x))^{5/2} (7 a+2 b \sin (c+d x))}{20 b^3 d (a+b \sin (c+d x))}+\frac {3 e^5 \sqrt {e \cos (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \sin (c+d x)\right )}{4 b^5 d}+\frac {\left (3 e^6\right ) \int \frac {\frac {1}{2} b \left (7 a^2-3 b^2\right )+\frac {1}{4} a \left (21 a^2-13 b^2\right ) \sin (c+d x)}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))} \, dx}{2 b^5}\\ &=-\frac {e (e \cos (c+d x))^{9/2}}{2 b d (a+b \sin (c+d x))^2}-\frac {9 e^3 (e \cos (c+d x))^{5/2} (7 a+2 b \sin (c+d x))}{20 b^3 d (a+b \sin (c+d x))}+\frac {3 e^5 \sqrt {e \cos (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \sin (c+d x)\right )}{4 b^5 d}+\frac {\left (3 a \left (21 a^2-13 b^2\right ) e^6\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{8 b^6}-\frac {\left (9 \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))} \, dx}{8 b^6}\\ &=-\frac {e (e \cos (c+d x))^{9/2}}{2 b d (a+b \sin (c+d x))^2}-\frac {9 e^3 (e \cos (c+d x))^{5/2} (7 a+2 b \sin (c+d x))}{20 b^3 d (a+b \sin (c+d x))}+\frac {3 e^5 \sqrt {e \cos (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \sin (c+d x)\right )}{4 b^5 d}+\frac {\left (9 a \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6\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^6 \sqrt {-a^2+b^2}}+\frac {\left (9 a \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6\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^6 \sqrt {-a^2+b^2}}-\frac {\left (9 \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^7\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (\left (a^2-b^2\right ) e^2+b^2 x^2\right )} \, dx,x,e \cos (c+d x)\right )}{8 b^5 d}+\frac {\left (3 a \left (21 a^2-13 b^2\right ) e^6 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{8 b^6 \sqrt {e \cos (c+d x)}}\\ &=\frac {3 a \left (21 a^2-13 b^2\right ) e^6 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^6 d \sqrt {e \cos (c+d x)}}-\frac {e (e \cos (c+d x))^{9/2}}{2 b d (a+b \sin (c+d x))^2}-\frac {9 e^3 (e \cos (c+d x))^{5/2} (7 a+2 b \sin (c+d x))}{20 b^3 d (a+b \sin (c+d x))}+\frac {3 e^5 \sqrt {e \cos (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \sin (c+d x)\right )}{4 b^5 d}-\frac {\left (9 \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^7\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-b^2\right ) e^2+b^2 x^4} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{4 b^5 d}+\frac {\left (9 a \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6 \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^6 \sqrt {-a^2+b^2} \sqrt {e \cos (c+d x)}}+\frac {\left (9 a \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6 \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^6 \sqrt {-a^2+b^2} \sqrt {e \cos (c+d x)}}\\ &=\frac {3 a \left (21 a^2-13 b^2\right ) e^6 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^6 d \sqrt {e \cos (c+d x)}}-\frac {9 a \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6 \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^6 \left (a^2-b \left (b-\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \cos (c+d x)}}+\frac {9 a \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6 \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^6 \sqrt {-a^2+b^2} \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {e (e \cos (c+d x))^{9/2}}{2 b d (a+b \sin (c+d x))^2}-\frac {9 e^3 (e \cos (c+d x))^{5/2} (7 a+2 b \sin (c+d x))}{20 b^3 d (a+b \sin (c+d x))}+\frac {3 e^5 \sqrt {e \cos (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \sin (c+d x)\right )}{4 b^5 d}+\frac {\left (9 \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6\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 )}{8 b^5 \sqrt {-a^2+b^2} d}+\frac {\left (9 \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6\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 )}{8 b^5 \sqrt {-a^2+b^2} d}\\ &=\frac {9 \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^{11/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 b^{11/2} \left (-a^2+b^2\right )^{3/4} d}+\frac {9 \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^{11/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 b^{11/2} \left (-a^2+b^2\right )^{3/4} d}+\frac {3 a \left (21 a^2-13 b^2\right ) e^6 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^6 d \sqrt {e \cos (c+d x)}}-\frac {9 a \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6 \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^6 \left (a^2-b \left (b-\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \cos (c+d x)}}+\frac {9 a \left (7 a^4-9 a^2 b^2+2 b^4\right ) e^6 \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^6 \sqrt {-a^2+b^2} \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {e (e \cos (c+d x))^{9/2}}{2 b d (a+b \sin (c+d x))^2}-\frac {9 e^3 (e \cos (c+d x))^{5/2} (7 a+2 b \sin (c+d x))}{20 b^3 d (a+b \sin (c+d x))}+\frac {3 e^5 \sqrt {e \cos (c+d x)} \left (3 \left (7 a^2-2 b^2\right )-7 a b \sin (c+d x)\right )}{4 b^5 d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 54.33, size = 2024, normalized size = 3.44 \begin {gather*} \text {Result too large to show} \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 = 116.91, size = 78449, normalized size = 133.19
method | result | size |
default | \(\text {Expression too large to display}\) | \(78449\) |
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 {{\left (e\,\cos \left (c+d\,x\right )\right )}^{11/2}}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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