Optimal. Leaf size=217 \[ \frac {2 \left (5 a A b-5 a^2 B-3 b^2 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^3 d}+\frac {2 (A b-a B) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 b^2 d}+\frac {2 a^2 (A b-a B) \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{b^3 (a+b) d}+\frac {2 B \sin (c+d x)}{5 b d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 (A b-a B) \sin (c+d x)}{3 b^2 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (5 a A b-5 a^2 B-3 b^2 B\right ) \sin (c+d x)}{5 b^3 d \sqrt {\cos (c+d x)}} \]
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Rubi [A]
time = 0.76, 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 = {3033, 3079,
3134, 3138, 2719, 3081, 2720, 2884} \begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2720
Rule 2884
Rule 3033
Rule 3079
Rule 3081
Rule 3134
Rule 3138
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 = 14.60, size = 326, normalized size = 1.50 \begin {gather*} \frac {\frac {b \left (45 a^2 A b+10 A b^3-45 a^3 B-19 a b^2 B\right ) \Pi \left (\frac {2 a}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{a+b}-\frac {b^2 \left (-20 a A b+20 a^2 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 {6 b^3 B \sin (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)}+\frac {10 b^2 (A b-a B) \sin (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}+\frac {6 b \left (-5 a A b+5 a^2 B+3 b^2 B\right ) \sin (c+d x)}{\sqrt {\cos (c+d x)}}-\frac {3 \left (-5 a A b+5 a^2 B+3 b^2 B\right ) \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)}}}{15 b^4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(757\) vs.
\(2(279)=558\).
time = 8.67, size = 758, normalized size = 3.49
method | result | size |
default | \(\text {Expression too large to display}\) | \(758\) |
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 \begin {gather*} \text {Failed to integrate} \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(-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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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 = -0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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