Optimal. Leaf size=221 \[ \frac {(19 A-12 B) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{4 a^{3/2} d}-\frac {(13 A-9 B) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(7 A-6 B) \sin (c+d x)}{4 a d \sqrt {a \sec (c+d x)+a}}+\frac {(2 A-B) \sin (c+d x) \cos (c+d x)}{2 a d \sqrt {a \sec (c+d x)+a}}-\frac {(A-B) \sin (c+d x) \cos (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.58, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {(19 A-12 B) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{4 a^{3/2} d}-\frac {(13 A-9 B) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(7 A-6 B) \sin (c+d x)}{4 a d \sqrt {a \sec (c+d x)+a}}+\frac {(2 A-B) \sin (c+d x) \cos (c+d x)}{2 a d \sqrt {a \sec (c+d x)+a}}-\frac {(A-B) \sin (c+d x) \cos (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}} \]
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))^{3/2}} \, dx &=-\frac {(A-B) \cos (c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {\int \frac {\cos ^2(c+d x) \left (2 a (2 A-B)-\frac {5}{2} a (A-B) \sec (c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx}{2 a^2}\\ &=-\frac {(A-B) \cos (c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {(2 A-B) \cos (c+d x) \sin (c+d x)}{2 a d \sqrt {a+a \sec (c+d x)}}+\frac {\int \frac {\cos (c+d x) \left (-a^2 (7 A-6 B)+3 a^2 (2 A-B) \sec (c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx}{4 a^3}\\ &=-\frac {(A-B) \cos (c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}-\frac {(7 A-6 B) \sin (c+d x)}{4 a d \sqrt {a+a \sec (c+d x)}}+\frac {(2 A-B) \cos (c+d x) \sin (c+d x)}{2 a d \sqrt {a+a \sec (c+d x)}}+\frac {\int \frac {\frac {1}{2} a^3 (19 A-12 B)-\frac {1}{2} a^3 (7 A-6 B) \sec (c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx}{4 a^4}\\ &=-\frac {(A-B) \cos (c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}-\frac {(7 A-6 B) \sin (c+d x)}{4 a d \sqrt {a+a \sec (c+d x)}}+\frac {(2 A-B) \cos (c+d x) \sin (c+d x)}{2 a d \sqrt {a+a \sec (c+d x)}}+\frac {(19 A-12 B) \int \sqrt {a+a \sec (c+d x)} \, dx}{8 a^2}-\frac {(13 A-9 B) \int \frac {\sec (c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx}{4 a}\\ &=-\frac {(A-B) \cos (c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}-\frac {(7 A-6 B) \sin (c+d x)}{4 a d \sqrt {a+a \sec (c+d x)}}+\frac {(2 A-B) \cos (c+d x) \sin (c+d x)}{2 a d \sqrt {a+a \sec (c+d x)}}-\frac {(19 A-12 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 d}+\frac {(13 A-9 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 )}{2 a d}\\ &=\frac {(19 A-12 B) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{4 a^{3/2} d}-\frac {(13 A-9 B) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(A-B) \cos (c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}-\frac {(7 A-6 B) \sin (c+d x)}{4 a d \sqrt {a+a \sec (c+d x)}}+\frac {(2 A-B) \cos (c+d x) \sin (c+d x)}{2 a d \sqrt {a+a \sec (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 2.44, size = 395, normalized size = 1.79 \[ \frac {\sec (c+d x) \left ((91 A-48 B) (\sin (c+d x)+\tan (c+d x)) \tanh ^{-1}\left (\sqrt {1-\sec (c+d x)}\right )-40 A \sqrt {1-\sec (c+d x)} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};1-\sec (c+d x)\right ) (\sin (c+d x)+\tan (c+d x))-13 A \sin (c+d x) \sqrt {1-\sec (c+d x)}+\frac {13}{2} A \sin (2 (c+d x)) \sqrt {1-\sec (c+d x)}+18 A \sin (c+d x) \cos ^2(c+d x) \sqrt {1-\sec (c+d x)}-52 \sqrt {2} A \sin (c+d x) \tanh ^{-1}\left (\frac {\sqrt {1-\sec (c+d x)}}{\sqrt {2}}\right )-52 \sqrt {2} A \tan (c+d x) \tanh ^{-1}\left (\frac {\sqrt {1-\sec (c+d x)}}{\sqrt {2}}\right )+24 B \sin (c+d x) \sqrt {1-\sec (c+d x)}+8 B \sin (2 (c+d x)) \sqrt {1-\sec (c+d x)}+36 \sqrt {2} B \sin (c+d x) \tanh ^{-1}\left (\frac {\sqrt {1-\sec (c+d x)}}{\sqrt {2}}\right )+36 \sqrt {2} B \tan (c+d x) \tanh ^{-1}\left (\frac {\sqrt {1-\sec (c+d x)}}{\sqrt {2}}\right )\right )}{16 d \sqrt {1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 5.84, size = 644, normalized size = 2.91 \[ \left [\frac {\sqrt {2} {\left ({\left (13 \, A - 9 \, B\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (13 \, A - 9 \, B\right )} \cos \left (d x + c\right ) + 13 \, A - 9 \, 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 ) + {\left ({\left (19 \, A - 12 \, B\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (19 \, A - 12 \, B\right )} \cos \left (d x + c\right ) + 19 \, A - 12 \, 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 ) + 2 \, {\left (2 \, A \cos \left (d x + c\right )^{3} - {\left (3 \, A - 4 \, B\right )} \cos \left (d x + c\right )^{2} - {\left (7 \, A - 6 \, 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 )}{8 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}}, \frac {\sqrt {2} {\left ({\left (13 \, A - 9 \, B\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (13 \, A - 9 \, B\right )} \cos \left (d x + c\right ) + 13 \, A - 9 \, 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 ) - {\left ({\left (19 \, A - 12 \, B\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (19 \, A - 12 \, B\right )} \cos \left (d x + c\right ) + 19 \, A - 12 \, 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 ) + {\left (2 \, A \cos \left (d x + c\right )^{3} - {\left (3 \, A - 4 \, B\right )} \cos \left (d x + c\right )^{2} - {\left (7 \, A - 6 \, 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 )}{4 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.18, size = 673, normalized size = 3.05 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.80, size = 1075, normalized size = 4.86 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (B \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{2}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^2\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (A + B \sec {\left (c + d x \right )}\right ) \cos ^{2}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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