Optimal. Leaf size=247 \[ -\frac {63 \sin (c+d x)}{10 d \sec ^{\frac {3}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}+\frac {77 \sin (c+d x)}{10 a^3 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {21 \sin (c+d x)}{2 a^3 d \sqrt {\sec (c+d x)}}-\frac {21 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{2 a^3 d}+\frac {231 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac {4 \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac {\sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3} \]
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Rubi [A] time = 0.37, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3817, 4020, 3787, 3769, 3771, 2639, 2641} \[ -\frac {63 \sin (c+d x)}{10 d \sec ^{\frac {3}{2}}(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}+\frac {77 \sin (c+d x)}{10 a^3 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {21 \sin (c+d x)}{2 a^3 d \sqrt {\sec (c+d x)}}-\frac {21 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{2 a^3 d}+\frac {231 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac {4 \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac {\sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 3769
Rule 3771
Rule 3787
Rule 3817
Rule 4020
Rubi steps
\begin {align*} \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx &=-\frac {\sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {\int \frac {-\frac {15 a}{2}+\frac {9}{2} a \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac {\sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {4 \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {\int \frac {-\frac {105 a^2}{2}+42 a^2 \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))} \, dx}{15 a^4}\\ &=-\frac {\sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {4 \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {63 \sin (c+d x)}{10 d \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\int \frac {-\frac {1155 a^3}{4}+\frac {945}{4} a^3 \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{15 a^6}\\ &=-\frac {\sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {4 \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {63 \sin (c+d x)}{10 d \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}-\frac {63 \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{4 a^3}+\frac {77 \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)} \, dx}{4 a^3}\\ &=\frac {77 \sin (c+d x)}{10 a^3 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {21 \sin (c+d x)}{2 a^3 d \sqrt {\sec (c+d x)}}-\frac {\sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {4 \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {63 \sin (c+d x)}{10 d \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}-\frac {21 \int \sqrt {\sec (c+d x)} \, dx}{4 a^3}+\frac {231 \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{20 a^3}\\ &=\frac {77 \sin (c+d x)}{10 a^3 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {21 \sin (c+d x)}{2 a^3 d \sqrt {\sec (c+d x)}}-\frac {\sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {4 \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {63 \sin (c+d x)}{10 d \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}-\frac {\left (21 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{4 a^3}+\frac {\left (231 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{20 a^3}\\ &=\frac {231 \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{10 a^3 d}-\frac {21 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{2 a^3 d}+\frac {77 \sin (c+d x)}{10 a^3 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {21 \sin (c+d x)}{2 a^3 d \sqrt {\sec (c+d x)}}-\frac {\sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^3}-\frac {4 \sin (c+d x)}{5 a d \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2}-\frac {63 \sin (c+d x)}{10 d \sec ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \sec (c+d x)\right )}\\ \end {align*}
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Mathematica [C] time = 2.86, size = 297, normalized size = 1.20 \[ -\frac {e^{-i d x} \cos \left (\frac {1}{2} (c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left (77 i e^{-\frac {3}{2} i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \left (1+e^{i (c+d x)}\right )^5 \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )+125 \sin \left (\frac {1}{2} (c+d x)\right )+359 \sin \left (\frac {3}{2} (c+d x)\right )+350 \sin \left (\frac {5}{2} (c+d x)\right )+138 \sin \left (\frac {7}{2} (c+d x)\right )+5 \sin \left (\frac {9}{2} (c+d x)\right )-\sin \left (\frac {11}{2} (c+d x)\right )-3465 i \cos \left (\frac {1}{2} (c+d x)\right )-2541 i \cos \left (\frac {3}{2} (c+d x)\right )-1155 i \cos \left (\frac {5}{2} (c+d x)\right )-231 i \cos \left (\frac {7}{2} (c+d x)\right )+3360 \cos ^5\left (\frac {1}{2} (c+d x)\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{40 a^3 d (\sec (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\sec \left (d x + c\right )}}{a^{3} \sec \left (d x + c\right )^{6} + 3 \, a^{3} \sec \left (d x + c\right )^{5} + 3 \, a^{3} \sec \left (d x + c\right )^{4} + a^{3} \sec \left (d x + c\right )^{3}}, x\right ) \]
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 \[ \int \frac {1}{{\left (a \sec \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 4.24, size = 296, normalized size = 1.20 \[ -\frac {\sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (64 \left (\cos ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-288 \left (\cos ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-76 \left (\cos ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-210 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-462 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+530 \left (\cos ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-248 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+19 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}{20 a^{3} \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a \sec \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
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 \[ \frac {\int \frac {1}{\sec ^{\frac {11}{2}}{\left (c + d x \right )} + 3 \sec ^{\frac {9}{2}}{\left (c + d x \right )} + 3 \sec ^{\frac {7}{2}}{\left (c + d x \right )} + \sec ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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