Optimal. Leaf size=264 \[ \frac {(39 A-20 B) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{4 a^{5/2} d}-\frac {(219 A-115 B) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{16 \sqrt {2} a^{5/2} d}-\frac {7 (9 A-5 B) \sin (c+d x)}{16 a^2 d \sqrt {a \sec (c+d x)+a}}+\frac {(31 A-15 B) \sin (c+d x) \cos (c+d x)}{16 a^2 d \sqrt {a \sec (c+d x)+a}}-\frac {(19 A-11 B) \sin (c+d x) \cos (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac {(A-B) \sin (c+d x) \cos (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}} \]
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Rubi [A] time = 0.79, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4020, 4022, 3920, 3774, 203, 3795} \[ -\frac {7 (9 A-5 B) \sin (c+d x)}{16 a^2 d \sqrt {a \sec (c+d x)+a}}+\frac {(39 A-20 B) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{4 a^{5/2} d}-\frac {(219 A-115 B) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {(31 A-15 B) \sin (c+d x) \cos (c+d x)}{16 a^2 d \sqrt {a \sec (c+d x)+a}}-\frac {(19 A-11 B) \sin (c+d x) \cos (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac {(A-B) \sin (c+d x) \cos (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 3774
Rule 3795
Rule 3920
Rule 4020
Rule 4022
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx &=-\frac {(A-B) \cos (c+d x) \sin (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}+\frac {\int \frac {\cos ^2(c+d x) \left (2 a (3 A-B)-\frac {7}{2} a (A-B) \sec (c+d x)\right )}{(a+a \sec (c+d x))^{3/2}} \, dx}{4 a^2}\\ &=-\frac {(A-B) \cos (c+d x) \sin (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}-\frac {(19 A-11 B) \cos (c+d x) \sin (c+d x)}{16 a d (a+a \sec (c+d x))^{3/2}}+\frac {\int \frac {\cos ^2(c+d x) \left (a^2 (31 A-15 B)-\frac {5}{4} a^2 (19 A-11 B) \sec (c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx}{8 a^4}\\ &=-\frac {(A-B) \cos (c+d x) \sin (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}-\frac {(19 A-11 B) \cos (c+d x) \sin (c+d x)}{16 a d (a+a \sec (c+d x))^{3/2}}+\frac {(31 A-15 B) \cos (c+d x) \sin (c+d x)}{16 a^2 d \sqrt {a+a \sec (c+d x)}}+\frac {\int \frac {\cos (c+d x) \left (-7 a^3 (9 A-5 B)+\frac {3}{2} a^3 (31 A-15 B) \sec (c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx}{16 a^5}\\ &=-\frac {(A-B) \cos (c+d x) \sin (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}-\frac {(19 A-11 B) \cos (c+d x) \sin (c+d x)}{16 a d (a+a \sec (c+d x))^{3/2}}-\frac {7 (9 A-5 B) \sin (c+d x)}{16 a^2 d \sqrt {a+a \sec (c+d x)}}+\frac {(31 A-15 B) \cos (c+d x) \sin (c+d x)}{16 a^2 d \sqrt {a+a \sec (c+d x)}}+\frac {\int \frac {2 a^4 (39 A-20 B)-\frac {7}{2} a^4 (9 A-5 B) \sec (c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx}{16 a^6}\\ &=-\frac {(A-B) \cos (c+d x) \sin (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}-\frac {(19 A-11 B) \cos (c+d x) \sin (c+d x)}{16 a d (a+a \sec (c+d x))^{3/2}}-\frac {7 (9 A-5 B) \sin (c+d x)}{16 a^2 d \sqrt {a+a \sec (c+d x)}}+\frac {(31 A-15 B) \cos (c+d x) \sin (c+d x)}{16 a^2 d \sqrt {a+a \sec (c+d x)}}-\frac {(219 A-115 B) \int \frac {\sec (c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx}{32 a^2}+\frac {(39 A-20 B) \int \sqrt {a+a \sec (c+d x)} \, dx}{8 a^3}\\ &=-\frac {(A-B) \cos (c+d x) \sin (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}-\frac {(19 A-11 B) \cos (c+d x) \sin (c+d x)}{16 a d (a+a \sec (c+d x))^{3/2}}-\frac {7 (9 A-5 B) \sin (c+d x)}{16 a^2 d \sqrt {a+a \sec (c+d x)}}+\frac {(31 A-15 B) \cos (c+d x) \sin (c+d x)}{16 a^2 d \sqrt {a+a \sec (c+d x)}}+\frac {(219 A-115 B) \operatorname {Subst}\left (\int \frac {1}{2 a+x^2} \, dx,x,-\frac {a \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{16 a^2 d}-\frac {(39 A-20 B) \operatorname {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,-\frac {a \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{4 a^2 d}\\ &=\frac {(39 A-20 B) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{4 a^{5/2} d}-\frac {(219 A-115 B) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}-\frac {(A-B) \cos (c+d x) \sin (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}-\frac {(19 A-11 B) \cos (c+d x) \sin (c+d x)}{16 a d (a+a \sec (c+d x))^{3/2}}-\frac {7 (9 A-5 B) \sin (c+d x)}{16 a^2 d \sqrt {a+a \sec (c+d x)}}+\frac {(31 A-15 B) \cos (c+d x) \sin (c+d x)}{16 a^2 d \sqrt {a+a \sec (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 6.17, size = 512, normalized size = 1.94 \[ -\frac {A (\sec (c+d x)+1)^{5/2} \left (\frac {760 \tan (c+d x) \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};1-\sec (c+d x)\right )}{d \sqrt {\sec (c+d x)+1}}+\frac {152 \sin (c+d x) \cos (c+d x)}{d (\sec (c+d x)+1)^{3/2}}-\frac {219 \tan (c+d x) \left (2 \cos ^2(c+d x) \sqrt {1-\sec (c+d x)}-\cos (c+d x) \sqrt {1-\sec (c+d x)}+7 \tanh ^{-1}\left (\sqrt {1-\sec (c+d x)}\right )-4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-\sec (c+d x)}}{\sqrt {2}}\right )\right )}{d \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1}}\right )}{128 (a (\sec (c+d x)+1))^{5/2}}-\frac {A \sin (c+d x) \cos (c+d x)}{4 d (a (\sec (c+d x)+1))^{5/2}}-\frac {B \sin (c+d x)}{4 d (a (\sec (c+d x)+1))^{5/2}}-\frac {5 B (\sec (c+d x)+1)^{5/2} \left (\frac {6 \sin (c+d x)}{d (\sec (c+d x)+1)^{3/2}}+\frac {9 \tan (c+d x) \left (\cos (c+d x)+\frac {\tanh ^{-1}\left (\sqrt {1-\sec (c+d x)}\right )}{\sqrt {1-\sec (c+d x)}}\right )}{d \sqrt {\sec (c+d x)+1}}+\frac {23 \tan (c+d x) \left (-\cos (c+d x) \sqrt {1-\sec (c+d x)}+\tanh ^{-1}\left (\sqrt {1-\sec (c+d x)}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-\sec (c+d x)}}{\sqrt {2}}\right )\right )}{d \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1}}\right )}{32 (a (\sec (c+d x)+1))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 10.81, size = 776, normalized size = 2.94 \[ \left [\frac {\sqrt {2} {\left ({\left (219 \, A - 115 \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (219 \, A - 115 \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (219 \, A - 115 \, B\right )} \cos \left (d x + c\right ) + 219 \, A - 115 \, B\right )} \sqrt {-a} \log \left (\frac {2 \, \sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 3 \, a \cos \left (d x + c\right )^{2} + 2 \, a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + 8 \, {\left ({\left (39 \, A - 20 \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (39 \, A - 20 \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (39 \, A - 20 \, B\right )} \cos \left (d x + c\right ) + 39 \, A - 20 \, B\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 4 \, {\left (8 \, A \cos \left (d x + c\right )^{4} - 4 \, {\left (5 \, A - 4 \, B\right )} \cos \left (d x + c\right )^{3} - 5 \, {\left (19 \, A - 11 \, B\right )} \cos \left (d x + c\right )^{2} - 7 \, {\left (9 \, A - 5 \, B\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{64 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}}, \frac {\sqrt {2} {\left ({\left (219 \, A - 115 \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (219 \, A - 115 \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (219 \, A - 115 \, B\right )} \cos \left (d x + c\right ) + 219 \, A - 115 \, B\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) - 8 \, {\left ({\left (39 \, A - 20 \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (39 \, A - 20 \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (39 \, A - 20 \, B\right )} \cos \left (d x + c\right ) + 39 \, A - 20 \, B\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) + 2 \, {\left (8 \, A \cos \left (d x + c\right )^{4} - 4 \, {\left (5 \, A - 4 \, B\right )} \cos \left (d x + c\right )^{3} - 5 \, {\left (19 \, A - 11 \, B\right )} \cos \left (d x + c\right )^{2} - 7 \, {\left (9 \, A - 5 \, B\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{32 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.67, size = 720, normalized size = 2.73 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.90, size = 1427, normalized size = 5.41 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (B \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{2}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^2\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\left (A + B \sec {\left (c + d x \right )}\right ) \cos ^{2}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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