Optimal. Leaf size=562 \[ -\frac {7 b \sin (c+d x)}{4 a^2 d (a+b \sec (c+d x))^{3/2}}-\frac {\sqrt {a+b} \left (4 a^2+35 b^2\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac {a+b}{a};\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{4 a^5 d}-\frac {b^2 \left (33 a^4-170 a^2 b^2+105 b^4\right ) \tan (c+d x)}{12 a^4 d \left (a^2-b^2\right )^2 \sqrt {a+b \sec (c+d x)}}-\frac {\left (33 a^4-170 a^2 b^2+105 b^4\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{12 a^4 d (a-b) (a+b)^{3/2}}-\frac {b^2 \left (27 a^2-35 b^2\right ) \tan (c+d x)}{12 a^3 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}+\frac {(a+3 b) \left (6 a^3-45 a^2 b+35 b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{12 a^4 d (a-b) (a+b)^{3/2}}+\frac {\sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}} \]
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Rubi [A] time = 1.17, antiderivative size = 562, 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 = {3846, 4104, 4060, 4058, 3921, 3784, 3832, 4004} \[ -\frac {b^2 \left (-170 a^2 b^2+33 a^4+105 b^4\right ) \tan (c+d x)}{12 a^4 d \left (a^2-b^2\right )^2 \sqrt {a+b \sec (c+d x)}}-\frac {b^2 \left (27 a^2-35 b^2\right ) \tan (c+d x)}{12 a^3 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}+\frac {(a+3 b) \left (-45 a^2 b+6 a^3+35 b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{12 a^4 d (a-b) (a+b)^{3/2}}-\frac {\left (-170 a^2 b^2+33 a^4+105 b^4\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{12 a^4 d (a-b) (a+b)^{3/2}}-\frac {\sqrt {a+b} \left (4 a^2+35 b^2\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac {a+b}{a};\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{4 a^5 d}-\frac {7 b \sin (c+d x)}{4 a^2 d (a+b \sec (c+d x))^{3/2}}+\frac {\sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3784
Rule 3832
Rule 3846
Rule 3921
Rule 4004
Rule 4058
Rule 4060
Rule 4104
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx &=\frac {\cos (c+d x) \sin (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}+\frac {\int \frac {\cos (c+d x) \left (-\frac {7 b}{2}+a \sec (c+d x)+\frac {5}{2} b \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{5/2}} \, dx}{2 a}\\ &=-\frac {7 b \sin (c+d x)}{4 a^2 d (a+b \sec (c+d x))^{3/2}}+\frac {\cos (c+d x) \sin (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {\int \frac {\frac {1}{4} \left (-4 a^2-35 b^2\right )-\frac {5}{2} a b \sec (c+d x)+\frac {21}{4} b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx}{2 a^2}\\ &=-\frac {7 b \sin (c+d x)}{4 a^2 d (a+b \sec (c+d x))^{3/2}}+\frac {\cos (c+d x) \sin (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {b^2 \left (27 a^2-35 b^2\right ) \tan (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}+\frac {\int \frac {\frac {3}{8} \left (a^2-b^2\right ) \left (4 a^2+35 b^2\right )+\frac {3}{4} a b \left (3 a^2-7 b^2\right ) \sec (c+d x)-\frac {1}{8} b^2 \left (27 a^2-35 b^2\right ) \sec ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx}{3 a^3 \left (a^2-b^2\right )}\\ &=-\frac {7 b \sin (c+d x)}{4 a^2 d (a+b \sec (c+d x))^{3/2}}+\frac {\cos (c+d x) \sin (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {b^2 \left (27 a^2-35 b^2\right ) \tan (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac {b^2 \left (33 a^4-170 a^2 b^2+105 b^4\right ) \tan (c+d x)}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}}-\frac {2 \int \frac {-\frac {3}{16} \left (a^2-b^2\right )^2 \left (4 a^2+35 b^2\right )-\frac {1}{8} a b \left (3 a^4-54 a^2 b^2+35 b^4\right ) \sec (c+d x)-\frac {1}{16} b^2 \left (33 a^4-170 a^2 b^2+105 b^4\right ) \sec ^2(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{3 a^4 \left (a^2-b^2\right )^2}\\ &=-\frac {7 b \sin (c+d x)}{4 a^2 d (a+b \sec (c+d x))^{3/2}}+\frac {\cos (c+d x) \sin (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {b^2 \left (27 a^2-35 b^2\right ) \tan (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac {b^2 \left (33 a^4-170 a^2 b^2+105 b^4\right ) \tan (c+d x)}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}}-\frac {2 \int \frac {-\frac {3}{16} \left (a^2-b^2\right )^2 \left (4 a^2+35 b^2\right )+\left (-\frac {1}{8} a b \left (3 a^4-54 a^2 b^2+35 b^4\right )+\frac {1}{16} b^2 \left (33 a^4-170 a^2 b^2+105 b^4\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{3 a^4 \left (a^2-b^2\right )^2}+\frac {\left (b^2 \left (33 a^4-170 a^2 b^2+105 b^4\right )\right ) \int \frac {\sec (c+d x) (1+\sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx}{24 a^4 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (33 a^4-170 a^2 b^2+105 b^4\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{12 a^4 (a-b) (a+b)^{3/2} d}-\frac {7 b \sin (c+d x)}{4 a^2 d (a+b \sec (c+d x))^{3/2}}+\frac {\cos (c+d x) \sin (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {b^2 \left (27 a^2-35 b^2\right ) \tan (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac {b^2 \left (33 a^4-170 a^2 b^2+105 b^4\right ) \tan (c+d x)}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}}+\frac {\left (4 a^2+35 b^2\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}} \, dx}{8 a^4}+\frac {\left (b (a+3 b) \left (6 a^3-45 a^2 b+35 b^3\right )\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{24 a^4 (a-b) (a+b)^2}\\ &=-\frac {\left (33 a^4-170 a^2 b^2+105 b^4\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{12 a^4 (a-b) (a+b)^{3/2} d}+\frac {(a+3 b) \left (6 a^3-45 a^2 b+35 b^3\right ) \cot (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{12 a^4 (a-b) (a+b)^{3/2} d}-\frac {\sqrt {a+b} \left (4 a^2+35 b^2\right ) \cot (c+d x) \Pi \left (\frac {a+b}{a};\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{4 a^5 d}-\frac {7 b \sin (c+d x)}{4 a^2 d (a+b \sec (c+d x))^{3/2}}+\frac {\cos (c+d x) \sin (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}-\frac {b^2 \left (27 a^2-35 b^2\right ) \tan (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac {b^2 \left (33 a^4-170 a^2 b^2+105 b^4\right ) \tan (c+d x)}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 16.38, size = 2285, normalized size = 4.07 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.29, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sec \left (d x + c\right ) + a} \cos \left (d x + c\right )^{2}}{b^{3} \sec \left (d x + c\right )^{3} + 3 \, a b^{2} \sec \left (d x + c\right )^{2} + 3 \, a^{2} b \sec \left (d x + c\right ) + a^{3}}, x\right ) \]
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 \[ \int \frac {\cos \left (d x + c\right )^{2}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.84, size = 5638, normalized size = 10.03 \[ \text {output too large to display} \]
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 \[ \int \frac {\cos \left (d x + c\right )^{2}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^2}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\cos ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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