Optimal. Leaf size=158 \[ -\frac {2 (b B-a C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 d}+\frac {2 C F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 b d}+\frac {2 \left (A b^2-a (b B-a C)\right ) \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 (a+b) d}+\frac {2 C \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (b B-a C) \sin (c+d x)}{b^2 d \sqrt {\cos (c+d x)}} \]
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Rubi [A]
time = 0.62, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {4197, 3134,
3138, 2719, 3081, 2720, 2884} \begin {gather*} \frac {2 \left (A b^2-a (b B-a C)\right ) \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 d (a+b)}-\frac {2 (b B-a C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 d}+\frac {2 (b B-a C) \sin (c+d x)}{b^2 d \sqrt {\cos (c+d x)}}+\frac {2 C F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 b d}+\frac {2 C \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2720
Rule 2884
Rule 3081
Rule 3134
Rule 3138
Rule 4197
Rubi steps
\begin {align*} \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx &=\int \frac {C+B \cos (c+d x)+A \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (b+a \cos (c+d x))} \, dx\\ &=\frac {2 C \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \int \frac {\frac {3}{2} (b B-a C)+\frac {1}{2} b (3 A+C) \cos (c+d x)+\frac {1}{2} a C \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))} \, dx}{3 b}\\ &=\frac {2 C \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (b B-a C) \sin (c+d x)}{b^2 d \sqrt {\cos (c+d x)}}+\frac {4 \int \frac {\frac {1}{4} \left (b^2 (3 A+C)-3 a (b B-a C)\right )-\frac {1}{4} b (3 b B-4 a C) \cos (c+d x)-\frac {3}{4} a (b B-a C) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{3 b^2}\\ &=\frac {2 C \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (b B-a C) \sin (c+d x)}{b^2 d \sqrt {\cos (c+d x)}}-\frac {4 \int \frac {-\frac {1}{4} a \left (b^2 (3 A+C)-3 a (b B-a C)\right )-\frac {1}{4} a^2 b C \cos (c+d x)}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{3 a b^2}-\frac {(b B-a C) \int \sqrt {\cos (c+d x)} \, dx}{b^2}\\ &=-\frac {2 (b B-a C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 d}+\frac {2 C \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (b B-a C) \sin (c+d x)}{b^2 d \sqrt {\cos (c+d x)}}+\frac {C \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 b}+\left (A-\frac {a (b B-a C)}{b^2}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))} \, dx\\ &=-\frac {2 (b B-a C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 d}+\frac {2 C F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 b d}+\frac {2 \left (A-\frac {a (b B-a C)}{b^2}\right ) \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{(a+b) d}+\frac {2 C \sin (c+d x)}{3 b d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (b B-a C) \sin (c+d x)}{b^2 d \sqrt {\cos (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 12.54, size = 266, normalized size = 1.68 \begin {gather*} \frac {\frac {2 b \left (6 A b^2-9 a b B+9 a^2 C+2 b^2 C\right ) \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a+b}+\frac {b \left (-6 b^2 B+8 a b C\right ) \left (2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-\frac {2 b \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a+b}\right )}{a}+\frac {4 b^2 C \sin (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}+\frac {12 b (b B-a C) \sin (c+d x)}{\sqrt {\cos (c+d x)}}+\frac {6 (-b B+a C) \left (-2 a b E\left (\left .\text {ArcSin}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 b (a+b) F\left (\left .\text {ArcSin}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+\left (a^2-2 b^2\right ) \Pi \left (-\frac {a}{b};\left .\text {ArcSin}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )\right ) \sin (c+d x)}{a \sqrt {\sin ^2(c+d x)}}}{6 b^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 445, normalized size = 2.82
method | result | size |
default | \(-\frac {\sqrt {-\left (-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-\frac {2 \left (A \,b^{2}-a b B +a^{2} C \right ) a \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticPi \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 a}{a -b}, \sqrt {2}\right )}{b^{2} \left (a^{2}-a b \right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}+\frac {2 \left (b B -a C \right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{b^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}+\frac {2 C \left (-\frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}{6 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{2}\right )^{2}}+\frac {\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{3 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}\right )}{b}\right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\) | \(445\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right ) \cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\cos \left (c+d\,x\right )}^{3/2}\,\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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