Optimal. Leaf size=217 \[ \frac {2 a^2 (A b-a B) \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{b^3 d (a+b)}+\frac {2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^3 d}-\frac {2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x)}{5 b^3 d \sqrt {\cos (c+d x)}}+\frac {2 (A b-a B) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 b^2 d}+\frac {2 (A b-a B) \sin (c+d x)}{3 b^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 B \sin (c+d x)}{5 b d \cos ^{\frac {5}{2}}(c+d x)} \]
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Rubi [A] time = 1.19, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {2954, 3000, 3055, 3059, 2639, 3002, 2641, 2805} \[ \frac {2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^3 d}+\frac {2 a^2 (A b-a B) \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{b^3 d (a+b)}-\frac {2 \left (-5 a^2 B+5 a A b-3 b^2 B\right ) \sin (c+d x)}{5 b^3 d \sqrt {\cos (c+d x)}}+\frac {2 (A b-a B) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 b^2 d}+\frac {2 (A b-a B) \sin (c+d x)}{3 b^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 B \sin (c+d x)}{5 b d \cos ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2805
Rule 2954
Rule 3000
Rule 3002
Rule 3055
Rule 3059
Rubi steps
\begin {align*} \int \frac {A+B \sec (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))} \, dx &=\int \frac {B+A \cos (c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (b+a \cos (c+d x))} \, dx\\ &=\frac {2 B \sin (c+d x)}{5 b d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \int \frac {\frac {5}{2} (A b-a B)+\frac {3}{2} b B \cos (c+d x)+\frac {3}{2} a B \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (b+a \cos (c+d x))} \, dx}{5 b}\\ &=\frac {2 B \sin (c+d x)}{5 b d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 (A b-a B) \sin (c+d x)}{3 b^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {4 \int \frac {-\frac {3}{4} \left (5 a A b-5 a^2 B-3 b^2 B\right )+\frac {1}{4} b (5 A b+4 a B) \cos (c+d x)+\frac {5}{4} a (A b-a B) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))} \, dx}{15 b^2}\\ &=\frac {2 B \sin (c+d x)}{5 b d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 (A b-a B) \sin (c+d x)}{3 b^2 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (5 a A b-5 a^2 B-3 b^2 B\right ) \sin (c+d x)}{5 b^3 d \sqrt {\cos (c+d x)}}+\frac {8 \int \frac {\frac {5}{8} \left (3 a^2+b^2\right ) (A b-a B)+\frac {1}{8} b \left (20 a A b-20 a^2 B-9 b^2 B\right ) \cos (c+d x)+\frac {3}{8} a \left (5 a A b-5 a^2 B-3 b^2 B\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{15 b^3}\\ &=\frac {2 B \sin (c+d x)}{5 b d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 (A b-a B) \sin (c+d x)}{3 b^2 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (5 a A b-5 a^2 B-3 b^2 B\right ) \sin (c+d x)}{5 b^3 d \sqrt {\cos (c+d x)}}-\frac {8 \int \frac {-\frac {5}{8} a \left (3 a^2+b^2\right ) (A b-a B)-\frac {5}{8} a^2 b (A b-a B) \cos (c+d x)}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{15 a b^3}+\frac {\left (5 a A b-5 a^2 B-3 b^2 B\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 b^3}\\ &=\frac {2 \left (5 a A b-5 a^2 B-3 b^2 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^3 d}+\frac {2 B \sin (c+d x)}{5 b d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 (A b-a B) \sin (c+d x)}{3 b^2 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (5 a A b-5 a^2 B-3 b^2 B\right ) \sin (c+d x)}{5 b^3 d \sqrt {\cos (c+d x)}}+\frac {\left (a^2 (A b-a B)\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{b^3}+\frac {(A b-a B) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 b^2}\\ &=\frac {2 \left (5 a A b-5 a^2 B-3 b^2 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^3 d}+\frac {2 (A b-a B) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 b^2 d}+\frac {2 a^2 (A b-a B) \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{b^3 (a+b) d}+\frac {2 B \sin (c+d x)}{5 b d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 (A b-a B) \sin (c+d x)}{3 b^2 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (5 a A b-5 a^2 B-3 b^2 B\right ) \sin (c+d x)}{5 b^3 d \sqrt {\cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 4.86, size = 326, normalized size = 1.50 \[ \frac {\frac {6 b \left (5 a^2 B-5 a A b+3 b^2 B\right ) \sin (c+d x)}{\sqrt {\cos (c+d x)}}-\frac {b^2 \left (20 a^2 B-20 a A b+9 b^2 B\right ) \left (2 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-\frac {2 b \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a+b}\right )}{a}-\frac {3 \left (5 a^2 B-5 a A b+3 b^2 B\right ) \sin (c+d x) \left (\left (a^2-2 b^2\right ) \Pi \left (-\frac {a}{b};\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 b (a+b) F\left (\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )-2 a b E\left (\left .\sin ^{-1}\left (\sqrt {\cos (c+d x)}\right )\right |-1\right )\right )}{a \sqrt {\sin ^2(c+d x)}}+\frac {b \left (-45 a^3 B+45 a^2 A b-19 a b^2 B+10 A b^3\right ) \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a+b}+\frac {10 b^2 (A b-a B) \sin (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}+\frac {6 b^3 B \sin (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)}}{15 b^4 d} \]
Antiderivative was successfully verified.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 15.60, size = 785, normalized size = 3.62 \[ \text {result too large to display} \]
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 \[ \int \frac {B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}}{{\cos \left (c+d\,x\right )}^{7/2}\,\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )} \,d x \]
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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