Optimal. Leaf size=549 \[ \frac {(a-b) \sqrt {a+b} \left (54 a b B+3 b^2 (11 A-16 C)+8 a^2 (2 A+3 C)\right ) \cot (c+d x) E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 b d}+\frac {\sqrt {a+b} \left (3 b^2 (11 A+16 (B-C))+4 a^2 (4 A+3 B+6 C)+2 a b (13 A+27 B+72 C)\right ) \cot (c+d x) F\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 d}-\frac {\sqrt {a+b} \left (5 A b^3+8 a^3 B+30 a b^2 B+20 a^2 b (A+2 C)\right ) \cot (c+d x) \Pi \left (\frac {a+b}{a};\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{8 a d}+\frac {\left (15 A b^2+42 a b B+8 a^2 (2 A+3 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {(5 A b+6 a B) \cos (c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d} \]
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Rubi [A]
time = 0.86, antiderivative size = 549, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {4179, 4143,
4006, 3869, 3917, 4089} \begin {gather*} \frac {\sqrt {a+b} \cot (c+d x) \left (4 a^2 (4 A+3 B+6 C)+2 a b (13 A+27 B+72 C)+3 b^2 (11 A+16 (B-C))\right ) \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 )}{24 d}+\frac {(a-b) \sqrt {a+b} \cot (c+d x) \left (8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right ) \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 )}{24 b d}+\frac {\sin (c+d x) \left (8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{24 d}-\frac {\sqrt {a+b} \cot (c+d x) \left (8 a^3 B+20 a^2 b (A+2 C)+30 a b^2 B+5 A b^3\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac {a+b}{a};\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{8 a d}+\frac {(6 a B+5 A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{12 d}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3869
Rule 3917
Rule 4006
Rule 4089
Rule 4143
Rule 4179
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac {1}{3} \int \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left (\frac {1}{2} (5 A b+6 a B)+(2 a A+3 b B+3 a C) \sec (c+d x)-\frac {1}{2} b (A-6 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {(5 A b+6 a B) \cos (c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac {1}{6} \int \cos (c+d x) \sqrt {a+b \sec (c+d x)} \left (\frac {1}{4} \left (15 A b^2+42 a b B+4 a^2 (4 A+6 C)\right )+\frac {1}{2} \left (6 a^2 B+12 b^2 B+a b (11 A+24 C)\right ) \sec (c+d x)-\frac {3}{4} b (3 A b+2 a B-8 b C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {\left (15 A b^2+42 a b B+8 a^2 (2 A+3 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {(5 A b+6 a B) \cos (c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac {1}{6} \int \frac {\frac {3}{8} \left (5 A b^3+8 a^3 B+30 a b^2 B+20 a^2 b (A+2 C)\right )+\frac {1}{4} b \left (6 a^2 B+24 b^2 B+a b (13 A+72 C)\right ) \sec (c+d x)-\frac {1}{8} b \left (54 a b B+3 b^2 (11 A-16 C)+8 a^2 (2 A+3 C)\right ) \sec ^2(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx\\ &=\frac {\left (15 A b^2+42 a b B+8 a^2 (2 A+3 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {(5 A b+6 a B) \cos (c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac {1}{6} \int \frac {\frac {3}{8} \left (5 A b^3+8 a^3 B+30 a b^2 B+20 a^2 b (A+2 C)\right )+\left (\frac {1}{8} b \left (54 a b B+3 b^2 (11 A-16 C)+8 a^2 (2 A+3 C)\right )+\frac {1}{4} b \left (6 a^2 B+24 b^2 B+a b (13 A+72 C)\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx-\frac {1}{48} \left (b \left (54 a b B+3 b^2 (11 A-16 C)+8 a^2 (2 A+3 C)\right )\right ) \int \frac {\sec (c+d x) (1+\sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx\\ &=\frac {(a-b) \sqrt {a+b} \left (54 a b B+3 b^2 (11 A-16 C)+8 a^2 (2 A+3 C)\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}}}{24 b d}+\frac {\left (15 A b^2+42 a b B+8 a^2 (2 A+3 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {(5 A b+6 a B) \cos (c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}+\frac {1}{16} \left (5 A b^3+8 a^3 B+30 a b^2 B+20 a^2 b (A+2 C)\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}} \, dx+\frac {1}{48} \left (b \left (3 b^2 (11 A+16 (B-C))+4 a^2 (4 A+3 B+6 C)+2 a b (13 A+27 B+72 C)\right )\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx\\ &=\frac {(a-b) \sqrt {a+b} \left (54 a b B+3 b^2 (11 A-16 C)+8 a^2 (2 A+3 C)\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}}}{24 b d}+\frac {\sqrt {a+b} \left (3 b^2 (11 A+16 (B-C))+4 a^2 (4 A+3 B+6 C)+2 a b (13 A+27 B+72 C)\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}}}{24 d}-\frac {\sqrt {a+b} \left (5 A b^3+8 a^3 B+30 a b^2 B+20 a^2 b (A+2 C)\right ) \cot (c+d x) \Pi \left (\frac {a+b}{a};\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}}}{8 a d}+\frac {\left (15 A b^2+42 a b B+8 a^2 (2 A+3 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {(5 A b+6 a B) \cos (c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(5347\) vs. \(2(549)=1098\).
time = 25.70, size = 5347, normalized size = 9.74 \begin {gather*} \text {Result too large to show} \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. \(5112\) vs.
\(2(503)=1006\).
time = 0.51, size = 5113, normalized size = 9.31
method | result | size |
default | \(\text {Expression too large to display}\) | \(5113\) |
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(-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 {\cos \left (c+d\,x\right )}^3\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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