Optimal. Leaf size=671 \[ \frac {13 e^7 \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{56 b^7 d}-\frac {39 e^5 (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{280 b^5 d (a+b \sin (c+d x))}+\frac {39 a e^{15/2} \left (11 a^4-17 a^2 b^2+6 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{16 b^{15/2} d \left (b^2-a^2\right )^{3/4}}+\frac {39 a e^{15/2} \left (11 a^4-17 a^2 b^2+6 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{16 b^{15/2} d \left (b^2-a^2\right )^{3/4}}+\frac {13 e^8 \left (231 a^4-203 a^2 b^2+20 b^4\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{56 b^8 d \sqrt {e \cos (c+d x)}}-\frac {39 a^2 e^8 \left (11 a^4-17 a^2 b^2+6 b^4\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{16 b^8 d \left (a^2-b \left (b-\sqrt {b^2-a^2}\right )\right ) \sqrt {e \cos (c+d x)}}-\frac {39 a^2 e^8 \left (11 a^4-17 a^2 b^2+6 b^4\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{16 b^8 d \left (a^2-b \left (\sqrt {b^2-a^2}+b\right )\right ) \sqrt {e \cos (c+d x)}}-\frac {13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3} \]
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Rubi [A] time = 1.83, antiderivative size = 671, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {2693, 2863, 2865, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205} \[ \frac {13 e^7 \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{56 b^7 d}-\frac {39 e^5 (e \cos (c+d x))^{5/2} \left (77 a^2+22 a b \sin (c+d x)-20 b^2\right )}{280 b^5 d (a+b \sin (c+d x))}+\frac {39 a e^{15/2} \left (-17 a^2 b^2+11 a^4+6 b^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{16 b^{15/2} d \left (b^2-a^2\right )^{3/4}}+\frac {39 a e^{15/2} \left (-17 a^2 b^2+11 a^4+6 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{16 b^{15/2} d \left (b^2-a^2\right )^{3/4}}+\frac {13 e^8 \left (-203 a^2 b^2+231 a^4+20 b^4\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{56 b^8 d \sqrt {e \cos (c+d x)}}-\frac {39 a^2 e^8 \left (-17 a^2 b^2+11 a^4+6 b^4\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b-\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{16 b^8 d \left (a^2-b \left (b-\sqrt {b^2-a^2}\right )\right ) \sqrt {e \cos (c+d x)}}-\frac {39 a^2 e^8 \left (-17 a^2 b^2+11 a^4+6 b^4\right ) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{b+\sqrt {b^2-a^2}};\left .\frac {1}{2} (c+d x)\right |2\right )}{16 b^8 d \left (a^2-b \left (\sqrt {b^2-a^2}+b\right )\right ) \sqrt {e \cos (c+d x)}}-\frac {13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3} \]
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 212
Rule 329
Rule 2641
Rule 2642
Rule 2693
Rule 2702
Rule 2805
Rule 2807
Rule 2863
Rule 2865
Rule 2867
Rubi steps
\begin {align*} \int \frac {(e \cos (c+d x))^{15/2}}{(a+b \sin (c+d x))^4} \, dx &=-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}-\frac {\left (13 e^2\right ) \int \frac {(e \cos (c+d x))^{11/2} \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx}{6 b}\\ &=-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}-\frac {13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}+\frac {\left (39 e^4\right ) \int \frac {(e \cos (c+d x))^{7/2} \left (-2 b-\frac {11}{2} a \sin (c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{28 b^3}\\ &=-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}-\frac {13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}-\frac {39 e^5 (e \cos (c+d x))^{5/2} \left (77 a^2-20 b^2+22 a b \sin (c+d x)\right )}{280 b^5 d (a+b \sin (c+d x))}-\frac {\left (39 e^6\right ) \int \frac {(e \cos (c+d x))^{3/2} \left (\frac {11 a b}{2}+\frac {1}{4} \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{28 b^5}\\ &=-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}-\frac {13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}-\frac {39 e^5 (e \cos (c+d x))^{5/2} \left (77 a^2-20 b^2+22 a b \sin (c+d x)\right )}{280 b^5 d (a+b \sin (c+d x))}+\frac {13 e^7 \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{56 b^7 d}-\frac {\left (13 e^8\right ) \int \frac {-\frac {1}{4} a b \left (77 a^2-53 b^2\right )-\frac {1}{8} \left (231 a^4-203 a^2 b^2+20 b^4\right ) \sin (c+d x)}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))} \, dx}{14 b^7}\\ &=-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}-\frac {13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}-\frac {39 e^5 (e \cos (c+d x))^{5/2} \left (77 a^2-20 b^2+22 a b \sin (c+d x)\right )}{280 b^5 d (a+b \sin (c+d x))}+\frac {13 e^7 \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{56 b^7 d}-\frac {\left (39 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))} \, dx}{16 b^8}+\frac {\left (13 \left (231 a^4-203 a^2 b^2+20 b^4\right ) e^8\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{112 b^8}\\ &=-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}-\frac {13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}-\frac {39 e^5 (e \cos (c+d x))^{5/2} \left (77 a^2-20 b^2+22 a b \sin (c+d x)\right )}{280 b^5 d (a+b \sin (c+d x))}+\frac {13 e^7 \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{56 b^7 d}+\frac {\left (39 a^2 \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{32 b^8 \sqrt {-a^2+b^2}}+\frac {\left (39 a^2 \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{32 b^8 \sqrt {-a^2+b^2}}-\frac {\left (39 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^9\right ) \operatorname {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 )}{16 b^7 d}+\frac {\left (13 \left (231 a^4-203 a^2 b^2+20 b^4\right ) e^8 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{112 b^8 \sqrt {e \cos (c+d x)}}\\ &=\frac {13 \left (231 a^4-203 a^2 b^2+20 b^4\right ) e^8 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{56 b^8 d \sqrt {e \cos (c+d x)}}-\frac {e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}-\frac {13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}-\frac {39 e^5 (e \cos (c+d x))^{5/2} \left (77 a^2-20 b^2+22 a b \sin (c+d x)\right )}{280 b^5 d (a+b \sin (c+d x))}+\frac {13 e^7 \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{56 b^7 d}-\frac {\left (39 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^9\right ) \operatorname {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 )}{8 b^7 d}+\frac {\left (39 a^2 \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8 \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}{32 b^8 \sqrt {-a^2+b^2} \sqrt {e \cos (c+d x)}}+\frac {\left (39 a^2 \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8 \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}{32 b^8 \sqrt {-a^2+b^2} \sqrt {e \cos (c+d x)}}\\ &=\frac {13 \left (231 a^4-203 a^2 b^2+20 b^4\right ) e^8 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{56 b^8 d \sqrt {e \cos (c+d x)}}-\frac {39 a^2 \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8 \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 )}{16 b^8 \sqrt {-a^2+b^2} \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {39 a^2 \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8 \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 )}{16 b^8 \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))^{13/2}}{3 b d (a+b \sin (c+d x))^3}-\frac {13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}-\frac {39 e^5 (e \cos (c+d x))^{5/2} \left (77 a^2-20 b^2+22 a b \sin (c+d x)\right )}{280 b^5 d (a+b \sin (c+d x))}+\frac {13 e^7 \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{56 b^7 d}+\frac {\left (39 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8\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 )}{16 b^7 \sqrt {-a^2+b^2} d}+\frac {\left (39 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8\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 )}{16 b^7 \sqrt {-a^2+b^2} d}\\ &=\frac {39 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^{15/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{16 b^{15/2} \left (-a^2+b^2\right )^{3/4} d}+\frac {39 a \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^{15/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{16 b^{15/2} \left (-a^2+b^2\right )^{3/4} d}+\frac {13 \left (231 a^4-203 a^2 b^2+20 b^4\right ) e^8 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{56 b^8 d \sqrt {e \cos (c+d x)}}-\frac {39 a^2 \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8 \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 )}{16 b^8 \sqrt {-a^2+b^2} \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {39 a^2 \left (11 a^4-17 a^2 b^2+6 b^4\right ) e^8 \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 )}{16 b^8 \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))^{13/2}}{3 b d (a+b \sin (c+d x))^3}-\frac {13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}-\frac {39 e^5 (e \cos (c+d x))^{5/2} \left (77 a^2-20 b^2+22 a b \sin (c+d x)\right )}{280 b^5 d (a+b \sin (c+d x))}+\frac {13 e^7 \sqrt {e \cos (c+d x)} \left (21 a \left (11 a^2-6 b^2\right )-b \left (77 a^2-20 b^2\right ) \sin (c+d x)\right )}{56 b^7 d}\\ \end {align*}
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Mathematica [C] time = 27.81, size = 2102, normalized size = 3.13 \[ \text {Result too large to show} \]
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(-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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maple [C] time = 139.89, size = 300244, normalized size = 447.46 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{15/2}}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^4} \,d x \]
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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