Optimal. Leaf size=155 \[ \frac {F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{2 a^3 d}-\frac {9 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {\sin (c+d x)}{2 d \sqrt {\cos (c+d x)} \left (a^3 \sec (c+d x)+a^3\right )}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}+\frac {2 \sin (c+d x)}{5 a d \sqrt {\cos (c+d x)} (a \sec (c+d x)+a)^2} \]
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Rubi [A] time = 0.38, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {4264, 3817, 4019, 4020, 3787, 3771, 2639, 2641} \[ \frac {F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{2 a^3 d}-\frac {9 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {\sin (c+d x)}{2 d \sqrt {\cos (c+d x)} \left (a^3 \sec (c+d x)+a^3\right )}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3}+\frac {2 \sin (c+d x)}{5 a d \sqrt {\cos (c+d x)} (a \sec (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 3771
Rule 3787
Rule 3817
Rule 4019
Rule 4020
Rule 4264
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\cos (c+d x)} (a+a \sec (c+d x))^3} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\sec (c+d x)}}{(a+a \sec (c+d x))^3} \, dx\\ &=-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\sec (c+d x)} \left (-\frac {9 a}{2}+\frac {3}{2} a \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}+\frac {2 \sin (c+d x)}{5 a d \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^2}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {3 a^2-\frac {9}{2} a^2 \sec (c+d x)}{\sqrt {\sec (c+d x)} (a+a \sec (c+d x))} \, dx}{15 a^4}\\ &=-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}+\frac {2 \sin (c+d x)}{5 a d \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^2}+\frac {\sin (c+d x)}{2 d \sqrt {\cos (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {27 a^3}{4}-\frac {15}{4} a^3 \sec (c+d x)}{\sqrt {\sec (c+d x)}} \, dx}{15 a^6}\\ &=-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}+\frac {2 \sin (c+d x)}{5 a d \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^2}+\frac {\sin (c+d x)}{2 d \sqrt {\cos (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \, dx}{4 a^3}-\frac {\left (9 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{20 a^3}\\ &=-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}+\frac {2 \sin (c+d x)}{5 a d \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^2}+\frac {\sin (c+d x)}{2 d \sqrt {\cos (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}+\frac {\int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{4 a^3}-\frac {9 \int \sqrt {\cos (c+d x)} \, dx}{20 a^3}\\ &=-\frac {9 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{2 a^3 d}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}+\frac {2 \sin (c+d x)}{5 a d \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^2}+\frac {\sin (c+d x)}{2 d \sqrt {\cos (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}\\ \end {align*}
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Mathematica [C] time = 6.26, size = 721, normalized size = 4.65 \[ -\frac {2 \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^3(c+d x) \sqrt {1-\sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt {\sin (c) \left (-\sqrt {\cot ^2(c)+1}\right ) \sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt {\sin \left (d x-\tan ^{-1}(\cot (c))\right )+1} \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right )}{d \sqrt {\cot ^2(c)+1} (a \sec (c+d x)+a)^3}-\frac {9 i \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^3(c+d x) \left (\frac {2 e^{2 i d x} \sqrt {e^{-i d x} \left (2 i \sin (c) \left (-1+e^{2 i d x}\right )+2 \cos (c) \left (1+e^{2 i d x}\right )\right )} \sqrt {i \sin (2 c) e^{2 i d x}+\cos (2 c) e^{2 i d x}+1} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right )}{3 i d \cos (c) \left (1+e^{2 i d x}\right )-3 d \sin (c) \left (-1+e^{2 i d x}\right )}-\frac {2 \sqrt {e^{-i d x} \left (2 i \sin (c) \left (-1+e^{2 i d x}\right )+2 \cos (c) \left (1+e^{2 i d x}\right )\right )} \sqrt {i \sin (2 c) e^{2 i d x}+\cos (2 c) e^{2 i d x}+1} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right )}{d \sin (c) \left (-1+e^{2 i d x}\right )-i d \cos (c) \left (1+e^{2 i d x}\right )}\right )}{10 (a \sec (c+d x)+a)^3}+\frac {\cos ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\frac {2 \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \sec ^5\left (\frac {c}{2}+\frac {d x}{2}\right )}{5 d}+\frac {2 \tan \left (\frac {c}{2}\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{5 d}-\frac {12 \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \sec ^3\left (\frac {c}{2}+\frac {d x}{2}\right )}{5 d}-\frac {12 \tan \left (\frac {c}{2}\right ) \sec ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{5 d}+\frac {36 \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right )}{5 d}+\frac {36 \csc (c)}{5 d}\right )}{\cos ^{\frac {5}{2}}(c+d x) (a \sec (c+d x)+a)^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\cos \left (d x + c\right )}}{a^{3} \cos \left (d x + c\right ) \sec \left (d x + c\right )^{3} + 3 \, a^{3} \cos \left (d x + c\right ) \sec \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) \sec \left (d x + c\right ) + a^{3} \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {1}{{\left (a \sec \left (d x + c\right ) + a\right )}^{3} \sqrt {\cos \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 3.61, size = 270, normalized size = 1.74 \[ -\frac {\sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (36 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+18 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-66 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+38 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-9 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right )}{20 a^{3} \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
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 = -0.01 \[ \int \frac {1}{\sqrt {\cos \left (c+d\,x\right )}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{\sqrt {\cos {\left (c + d x \right )}} \sec ^{3}{\left (c + d x \right )} + 3 \sqrt {\cos {\left (c + d x \right )}} \sec ^{2}{\left (c + d x \right )} + 3 \sqrt {\cos {\left (c + d x \right )}} \sec {\left (c + d x \right )} + \sqrt {\cos {\left (c + d x \right )}}}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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