Optimal. Leaf size=126 \[ \frac {(3 A+5 B) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {(3 A+5 B) \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}+\frac {(A-B) \tan (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}} \]
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Rubi [A] time = 0.16, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {4000, 3796, 3795, 203} \[ \frac {(3 A+5 B) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {(3 A+5 B) \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}+\frac {(A-B) \tan (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 3795
Rule 3796
Rule 4000
Rubi steps
\begin {align*} \int \frac {\sec (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx &=\frac {(A-B) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}+\frac {(3 A+5 B) \int \frac {\sec (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx}{8 a}\\ &=\frac {(A-B) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}+\frac {(3 A+5 B) \tan (c+d x)}{16 a d (a+a \sec (c+d x))^{3/2}}+\frac {(3 A+5 B) \int \frac {\sec (c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx}{32 a^2}\\ &=\frac {(A-B) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}+\frac {(3 A+5 B) \tan (c+d x)}{16 a d (a+a \sec (c+d x))^{3/2}}-\frac {(3 A+5 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 {(3 A+5 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) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}+\frac {(3 A+5 B) \tan (c+d x)}{16 a d (a+a \sec (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [C] time = 1.59, size = 206, normalized size = 1.63 \[ \frac {64 A \sin \left (\frac {1}{2} (c+d x)\right ) \cos ^5\left (\frac {1}{2} (c+d x)\right ) \sqrt {1-\sec (c+d x)} \sec (c+d x) \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};\frac {1}{2} (1-\sec (c+d x))\right )+B (10 \sin (c+d x)+\sin (2 (c+d x))) \sqrt {1-\sec (c+d x)}+40 \sqrt {2} B \sin \left (\frac {1}{2} (c+d x)\right ) \cos ^5\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \tanh ^{-1}\left (\frac {\sqrt {1-\sec (c+d x)}}{\sqrt {2}}\right )}{32 a^2 d (\cos (c+d x)+1)^2 \sqrt {1-\sec (c+d x)} \sqrt {a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 475, normalized size = 3.77 \[ \left [-\frac {\sqrt {2} {\left ({\left (3 \, A + 5 \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (3 \, A + 5 \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (3 \, A + 5 \, B\right )} \cos \left (d x + c\right ) + 3 \, A + 5 \, B\right )} \sqrt {-a} \log \left (\frac {2 \, \sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 3 \, a \cos \left (d x + c\right )^{2} + 2 \, a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) - 4 \, {\left ({\left (7 \, A + B\right )} \cos \left (d x + c\right )^{2} + {\left (3 \, A + 5 \, B\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{64 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}}, -\frac {\sqrt {2} {\left ({\left (3 \, A + 5 \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (3 \, A + 5 \, B\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (3 \, A + 5 \, B\right )} \cos \left (d x + c\right ) + 3 \, A + 5 \, B\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) - 2 \, {\left ({\left (7 \, A + B\right )} \cos \left (d x + c\right )^{2} + {\left (3 \, A + 5 \, B\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{32 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 7.47, size = 191, normalized size = 1.52 \[ \frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} {\left (\frac {2 \, \sqrt {2} {\left (A a^{5} - B a^{5}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a^{8} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} - \frac {\sqrt {2} {\left (5 \, A a^{5} + 3 \, B a^{5}\right )}}{a^{8} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {\sqrt {2} {\left (3 \, A + 5 \, B\right )} \log \left ({\left | -\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} \right |}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.49, size = 594, normalized size = 4.71 \[ \frac {\sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (3 A \sin \left (d x +c \right ) \ln \left (-\frac {-\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+\cos \left (d x +c \right )-1}{\sin \left (d x +c \right )}\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right )+5 B \sin \left (d x +c \right ) \ln \left (-\frac {-\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+\cos \left (d x +c \right )-1}{\sin \left (d x +c \right )}\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right )+6 A \cos \left (d x +c \right ) \sin \left (d x +c \right ) \ln \left (-\frac {-\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+\cos \left (d x +c \right )-1}{\sin \left (d x +c \right )}\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+10 B \cos \left (d x +c \right ) \sin \left (d x +c \right ) \ln \left (-\frac {-\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+\cos \left (d x +c \right )-1}{\sin \left (d x +c \right )}\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+3 A \ln \left (-\frac {-\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+\cos \left (d x +c \right )-1}{\sin \left (d x +c \right )}\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )-14 A \left (\cos ^{3}\left (d x +c \right )\right )+5 B \ln \left (-\frac {-\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+\cos \left (d x +c \right )-1}{\sin \left (d x +c \right )}\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )-2 B \left (\cos ^{3}\left (d x +c \right )\right )+8 A \left (\cos ^{2}\left (d x +c \right )\right )-8 B \left (\cos ^{2}\left (d x +c \right )\right )+6 A \cos \left (d x +c \right )+10 B \cos \left (d x +c \right )\right )}{32 d \left (1+\cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right ) a^{3}} \]
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 \[ \int \frac {{\left (B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}}{\cos \left (c+d\,x\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\left (A + B \sec {\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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