Optimal. Leaf size=280 \[ \frac {203 A \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{8 a^{5/2} d}-\frac {287 A \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{8 \sqrt {2} a^{5/2} d}+\frac {21 A \sin (c+d x)}{2 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {77 A \sin (c+d x) \cos ^2(c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {119 A \sin (c+d x) \cos (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}-\frac {19 A \sin (c+d x) \cos ^2(c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}} \]
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Rubi [A] time = 0.91, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4020, 4022, 3920, 3774, 203, 3795} \[ \frac {21 A \sin (c+d x)}{2 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {203 A \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{8 a^{5/2} d}-\frac {287 A \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{8 \sqrt {2} a^{5/2} d}+\frac {77 A \sin (c+d x) \cos ^2(c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {119 A \sin (c+d x) \cos (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}-\frac {19 A \sin (c+d x) \cos ^2(c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}-\frac {A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}} \]
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 ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx &=-\frac {A \cos ^2(c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}+\frac {\int \frac {\cos ^3(c+d x) (10 a A+9 a A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx}{4 a^2}\\ &=-\frac {A \cos ^2(c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {19 A \cos ^2(c+d x) \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {\int \frac {\cos ^3(c+d x) \left (77 a^2 A+\frac {133}{2} a^2 A \sec (c+d x)\right )}{\sqrt {a-a \sec (c+d x)}} \, dx}{8 a^4}\\ &=-\frac {A \cos ^2(c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {19 A \cos ^2(c+d x) \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {77 A \cos ^2(c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}-\frac {\int \frac {\cos ^2(c+d x) \left (-238 a^3 A-\frac {385}{2} a^3 A \sec (c+d x)\right )}{\sqrt {a-a \sec (c+d x)}} \, dx}{24 a^5}\\ &=-\frac {A \cos ^2(c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {19 A \cos ^2(c+d x) \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {119 A \cos (c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {77 A \cos ^2(c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {\int \frac {\cos (c+d x) \left (504 a^4 A+357 a^4 A \sec (c+d x)\right )}{\sqrt {a-a \sec (c+d x)}} \, dx}{48 a^6}\\ &=-\frac {A \cos ^2(c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {19 A \cos ^2(c+d x) \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {21 A \sin (c+d x)}{2 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {119 A \cos (c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {77 A \cos ^2(c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}-\frac {\int \frac {-609 a^5 A-252 a^5 A \sec (c+d x)}{\sqrt {a-a \sec (c+d x)}} \, dx}{48 a^7}\\ &=-\frac {A \cos ^2(c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {19 A \cos ^2(c+d x) \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {21 A \sin (c+d x)}{2 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {119 A \cos (c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {77 A \cos ^2(c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {(203 A) \int \sqrt {a-a \sec (c+d x)} \, dx}{16 a^3}+\frac {(287 A) \int \frac {\sec (c+d x)}{\sqrt {a-a \sec (c+d x)}} \, dx}{16 a^2}\\ &=-\frac {A \cos ^2(c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {19 A \cos ^2(c+d x) \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {21 A \sin (c+d x)}{2 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {119 A \cos (c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {77 A \cos ^2(c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {(203 A) \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 )}{8 a^2 d}-\frac {(287 A) \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 )}{8 a^2 d}\\ &=\frac {203 A \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a-a \sec (c+d x)}}\right )}{8 a^{5/2} d}-\frac {287 A \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a-a \sec (c+d x)}}\right )}{8 \sqrt {2} a^{5/2} d}-\frac {A \cos ^2(c+d x) \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}-\frac {19 A \cos ^2(c+d x) \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}+\frac {21 A \sin (c+d x)}{2 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {119 A \cos (c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}+\frac {77 A \cos ^2(c+d x) \sin (c+d x)}{24 a^2 d \sqrt {a-a \sec (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 6.84, size = 514, normalized size = 1.84 \[ A \left (\frac {\sin ^5\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^3(c+d x) \left (\frac {7 \sin \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )}{6 d}-\frac {92 \sin \left (\frac {3 c}{2}\right ) \sin \left (\frac {3 d x}{2}\right )}{3 d}-\frac {7 \sin \left (\frac {5 c}{2}\right ) \sin \left (\frac {5 d x}{2}\right )}{2 d}-\frac {\sin \left (\frac {7 c}{2}\right ) \sin \left (\frac {7 d x}{2}\right )}{3 d}-\frac {7 \cos \left (\frac {c}{2}\right ) \cos \left (\frac {d x}{2}\right )}{6 d}+\frac {92 \cos \left (\frac {3 c}{2}\right ) \cos \left (\frac {3 d x}{2}\right )}{3 d}+\frac {7 \cos \left (\frac {5 c}{2}\right ) \cos \left (\frac {5 d x}{2}\right )}{2 d}+\frac {\cos \left (\frac {7 c}{2}\right ) \cos \left (\frac {7 d x}{2}\right )}{3 d}-\frac {\cot \left (\frac {c}{2}\right ) \csc ^3\left (\frac {c}{2}+\frac {d x}{2}\right )}{d}+\frac {39 \cot \left (\frac {c}{2}\right ) \csc \left (\frac {c}{2}+\frac {d x}{2}\right )}{2 d}+\frac {\csc \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \csc ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {39 \csc \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \csc ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{2 d}\right )}{(a-a \sec (c+d x))^{5/2}}+\frac {7 e^{-\frac {1}{2} i (c+d x)} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \sin ^5\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^{\frac {5}{2}}(c+d x) \left (29 \sinh ^{-1}\left (e^{i (c+d x)}\right )-41 \sqrt {2} \tanh ^{-1}\left (\frac {1+e^{i (c+d x)}}{\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}}\right )+29 \tanh ^{-1}\left (\sqrt {1+e^{2 i (c+d x)}}\right )\right )}{2 \sqrt {2} d (a-a \sec (c+d x))^{5/2}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 656, normalized size = 2.34 \[ \left [-\frac {861 \, \sqrt {2} {\left (A \cos \left (d x + c\right )^{2} - 2 \, A \cos \left (d x + c\right ) + A\right )} \sqrt {-a} \log \left (\frac {2 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} + {\left (3 \, a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )}{{\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + 1218 \, {\left (A \cos \left (d x + c\right )^{2} - 2 \, A \cos \left (d x + c\right ) + A\right )} \sqrt {-a} \log \left (\frac {2 \, {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} - {\left (2 \, a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )}{\sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + 4 \, {\left (8 \, A \cos \left (d x + c\right )^{6} + 30 \, A \cos \left (d x + c\right )^{5} + 113 \, A \cos \left (d x + c\right )^{4} - 294 \, A \cos \left (d x + c\right )^{3} - 133 \, A \cos \left (d x + c\right )^{2} + 252 \, A \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}}{96 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )} \sin \left (d x + c\right )}, \frac {861 \, \sqrt {2} {\left (A \cos \left (d x + c\right )^{2} - 2 \, A \cos \left (d x + c\right ) + A\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 ) \sin \left (d x + c\right ) - 1218 \, {\left (A \cos \left (d x + c\right )^{2} - 2 \, A \cos \left (d x + c\right ) + A\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 ) \sin \left (d x + c\right ) - 2 \, {\left (8 \, A \cos \left (d x + c\right )^{6} + 30 \, A \cos \left (d x + c\right )^{5} + 113 \, A \cos \left (d x + c\right )^{4} - 294 \, A \cos \left (d x + c\right )^{3} - 133 \, A \cos \left (d x + c\right )^{2} + 252 \, A \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}}{48 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )} \sin \left (d x + c\right )}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.94, size = 346, normalized size = 1.24 \[ -\frac {\frac {861 \, \sqrt {2} A \arctan \left (\frac {\sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right ) \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {1218 \, A \arctan \left (\frac {\sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}}{2 \, \sqrt {a}}\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right ) \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {2 \, \sqrt {2} {\left (129 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{\frac {5}{2}} A + 560 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{\frac {3}{2}} A a + 636 \, \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a} A a^{2}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{3} a^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right ) \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {3 \, \sqrt {2} {\left (33 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{\frac {3}{2}} A + 31 \, \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a} A a\right )}}{a^{4} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right ) \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.07, size = 1964, normalized size = 7.01 \[ \text {Expression too large to display} \]
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 \[ \int \frac {{\left (A \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{{\left (-a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^3\,\left (A+\frac {A}{\cos \left (c+d\,x\right )}\right )}{{\left (a-\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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