Optimal. Leaf size=282 \[ -\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{15 d \left (a^2-b^2\right )^3 \sqrt {a+b \cos (c+d x)}}-\frac {16 a b \sin (c+d x)}{15 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^{3/2}}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}-\frac {16 a \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 d \left (a^2-b^2\right )^2 \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (23 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 d \left (a^2-b^2\right )^3 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}} \]
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Rubi [A] time = 0.36, antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2664, 2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{15 d \left (a^2-b^2\right )^3 \sqrt {a+b \cos (c+d x)}}-\frac {16 a b \sin (c+d x)}{15 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^{3/2}}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}-\frac {16 a \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 d \left (a^2-b^2\right )^2 \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (23 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 d \left (a^2-b^2\right )^3 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2664
Rule 2752
Rule 2754
Rubi steps
\begin {align*} \int \frac {1}{(a+b \cos (c+d x))^{7/2}} \, dx &=-\frac {2 b \sin (c+d x)}{5 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{5/2}}-\frac {2 \int \frac {-\frac {5 a}{2}+\frac {3}{2} b \cos (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx}{5 \left (a^2-b^2\right )}\\ &=-\frac {2 b \sin (c+d x)}{5 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{5/2}}-\frac {16 a b \sin (c+d x)}{15 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^{3/2}}+\frac {4 \int \frac {\frac {3}{4} \left (5 a^2+3 b^2\right )-2 a b \cos (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx}{15 \left (a^2-b^2\right )^2}\\ &=-\frac {2 b \sin (c+d x)}{5 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{5/2}}-\frac {16 a b \sin (c+d x)}{15 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^{3/2}}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{15 \left (a^2-b^2\right )^3 d \sqrt {a+b \cos (c+d x)}}-\frac {8 \int \frac {-\frac {1}{8} a \left (15 a^2+17 b^2\right )-\frac {1}{8} b \left (23 a^2+9 b^2\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{15 \left (a^2-b^2\right )^3}\\ &=-\frac {2 b \sin (c+d x)}{5 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{5/2}}-\frac {16 a b \sin (c+d x)}{15 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^{3/2}}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{15 \left (a^2-b^2\right )^3 d \sqrt {a+b \cos (c+d x)}}-\frac {(8 a) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{15 \left (a^2-b^2\right )^2}+\frac {\left (23 a^2+9 b^2\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{15 \left (a^2-b^2\right )^3}\\ &=-\frac {2 b \sin (c+d x)}{5 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{5/2}}-\frac {16 a b \sin (c+d x)}{15 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^{3/2}}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{15 \left (a^2-b^2\right )^3 d \sqrt {a+b \cos (c+d x)}}+\frac {\left (\left (23 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{15 \left (a^2-b^2\right )^3 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (8 a \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{15 \left (a^2-b^2\right )^2 \sqrt {a+b \cos (c+d x)}}\\ &=\frac {2 \left (23 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 \left (a^2-b^2\right )^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {16 a \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {2 b \sin (c+d x)}{5 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{5/2}}-\frac {16 a b \sin (c+d x)}{15 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^{3/2}}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{15 \left (a^2-b^2\right )^3 d \sqrt {a+b \cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 1.42, size = 189, normalized size = 0.67 \[ \frac {2 \left (\frac {\left (\frac {a+b \cos (c+d x)}{a+b}\right )^{5/2} \left (\left (23 a^2+9 b^2\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+8 a (b-a) F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )\right )}{(a-b)^3}+\frac {b \sin (c+d x) \left (34 a^4+b^2 \left (23 a^2+9 b^2\right ) \cos ^2(c+d x)+2 a b \left (27 a^2+5 b^2\right ) \cos (c+d x)-5 a^2 b^2+3 b^4\right )}{\left (b^2-a^2\right )^3}\right )}{15 d (a+b \cos (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \cos \left (d x + c\right ) + a}}{b^{4} \cos \left (d x + c\right )^{4} + 4 \, a b^{3} \cos \left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \cos \left (d x + c\right )^{2} + 4 \, a^{3} b \cos \left (d x + c\right ) + a^{4}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.36, size = 616, normalized size = 2.18 \[ -\frac {\sqrt {-\left (-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (\frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +\left (a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{10 b^{2} \left (a -b \right ) \left (a +b \right ) \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {a -b}{2 b}\right )^{3}}+\frac {8 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +\left (a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{15 b \left (a -b \right )^{2} \left (a +b \right )^{2} \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {a -b}{2 b}\right )^{2}}+\frac {4 b \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (23 a^{2}+9 b^{2}\right )}{15 \left (a -b \right )^{3} \left (a +b \right )^{3} \sqrt {-\left (-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}+\frac {2 \left (15 a^{2}-8 a b +9 b^{2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a -b}{a -b}}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right )}{\left (15 a^{5}+15 a^{4} b -30 a^{3} b^{2}-30 a^{2} b^{3}+15 a \,b^{4}+15 b^{5}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +\left (a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}-\frac {2 \left (23 a^{2}+9 b^{2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a -b}{a -b}}\, \left (\EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right )-\EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right )\right )}{15 \left (a -b \right )^{2} \left (a +b \right )^{3} \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +\left (a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}\right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b}\, d} \]
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 \[ \int \frac {1}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \]
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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