Optimal. Leaf size=333 \[ -\frac {2 a (a-b) \sqrt {a+b} \left (3 a^2+29 b^2\right ) \cot (c+d x) E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{21 b^2 d}-\frac {2 (a-b) \sqrt {a+b} \left (3 a^2-24 a b+5 b^2\right ) \cot (c+d x) F\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{21 b d}+\frac {2 \left (3 a^2+5 b^2\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{21 d}+\frac {2 a (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{7 d}+\frac {2 (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{7 d} \]
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Rubi [A]
time = 0.39, antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3920, 4087,
4090, 3917, 4089} \begin {gather*} -\frac {2 (a-b) \sqrt {a+b} \left (3 a^2-24 a b+5 b^2\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} F\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{21 b d}-\frac {2 a (a-b) \sqrt {a+b} \left (3 a^2+29 b^2\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{21 b^2 d}+\frac {2 \left (3 a^2+5 b^2\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{21 d}+\frac {2 \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d}+\frac {2 a \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3917
Rule 3920
Rule 4087
Rule 4089
Rule 4090
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \, dx &=\frac {2 (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac {5}{7} \int \sec (c+d x) (b+a \sec (c+d x)) (a+b \sec (c+d x))^{3/2} \, dx\\ &=\frac {2 a (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{7 d}+\frac {2 (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac {2}{7} \int \sec (c+d x) \sqrt {a+b \sec (c+d x)} \left (4 a b+\frac {1}{2} \left (3 a^2+5 b^2\right ) \sec (c+d x)\right ) \, dx\\ &=\frac {2 \left (3 a^2+5 b^2\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{21 d}+\frac {2 a (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{7 d}+\frac {2 (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac {4}{21} \int \frac {\sec (c+d x) \left (\frac {1}{4} b \left (27 a^2+5 b^2\right )+\frac {1}{4} a \left (3 a^2+29 b^2\right ) \sec (c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx\\ &=\frac {2 \left (3 a^2+5 b^2\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{21 d}+\frac {2 a (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{7 d}+\frac {2 (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}-\frac {1}{21} \left ((a-b) \left (3 a^2-24 a b+5 b^2\right )\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx+\frac {1}{21} \left (a \left (3 a^2+29 b^2\right )\right ) \int \frac {\sec (c+d x) (1+\sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx\\ &=-\frac {2 a (a-b) \sqrt {a+b} \left (3 a^2+29 b^2\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{21 b^2 d}-\frac {2 (a-b) \sqrt {a+b} \left (3 a^2-24 a b+5 b^2\right ) \cot (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{21 b d}+\frac {2 \left (3 a^2+5 b^2\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{21 d}+\frac {2 a (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{7 d}+\frac {2 (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}\\ \end {align*}
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Mathematica [A]
time = 13.93, size = 474, normalized size = 1.42 \begin {gather*} -\frac {2 \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left (2 a \left (3 a^3+3 a^2 b+29 a b^2+29 b^3\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\frac {b+a \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} E\left (\text {ArcSin}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right )-2 b \left (3 a^3+27 a^2 b+29 a b^2+5 b^3\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\frac {b+a \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} F\left (\text {ArcSin}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right )+a \left (3 a^2+29 b^2\right ) \cos (c+d x) (b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{21 b d (b+a \cos (c+d x))^3 \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} \sec ^{\frac {5}{2}}(c+d x)}+\frac {\cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left (\frac {2 a \left (3 a^2+29 b^2\right ) \sin (c+d x)}{21 b}+\frac {2}{21} \sec (c+d x) \left (9 a^2 \sin (c+d x)+5 b^2 \sin (c+d x)\right )+\frac {6}{7} a b \sec (c+d x) \tan (c+d x)+\frac {2}{7} b^2 \sec ^2(c+d x) \tan (c+d x)\right )}{d (b+a \cos (c+d x))^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1851\) vs.
\(2(299)=598\).
time = 0.35, size = 1852, normalized size = 5.56
method | result | size |
default | \(\text {Expression too large to display}\) | \(1852\) |
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]
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \sec {\left (c + d x \right )}\right )^{\frac {5}{2}} \sec ^{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 = -0.00 \begin {gather*} \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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