Optimal. Leaf size=507 \[ \frac {(a-b) \sqrt {a+b} \left (3 b^2 (11 A-16 C)+8 a^2 (2 A+3 C)\right ) \cot (c+d x) E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 b d}+\frac {\sqrt {a+b} \left (16 a^2 A+26 a A b+33 A b^2+24 a^2 C+144 a b C-48 b^2 C\right ) \cot (c+d x) F\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 d}-\frac {5 b \sqrt {a+b} \left (A b^2+4 a^2 (A+2 C)\right ) \cot (c+d x) \Pi \left (\frac {a+b}{a};\text {ArcSin}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{8 a d}+\frac {\left (15 A b^2+8 a^2 (2 A+3 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {5 A b \cos (c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d} \]
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Rubi [A]
time = 0.83, antiderivative size = 507, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4180, 4179,
4143, 4006, 3869, 3917, 4089} \begin {gather*} \frac {\sqrt {a+b} \left (16 a^2 A+24 a^2 C+26 a A b+144 a b C+33 A b^2-48 b^2 C\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 )}{24 d}+\frac {(a-b) \sqrt {a+b} \left (8 a^2 (2 A+3 C)+3 b^2 (11 A-16 C)\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 )}{24 b d}-\frac {5 b \sqrt {a+b} \left (4 a^2 (A+2 C)+A 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}} \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 {\left (8 a^2 (2 A+3 C)+15 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{24 d}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{3 d}+\frac {5 A b \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{12 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3869
Rule 3917
Rule 4006
Rule 4089
Rule 4143
Rule 4179
Rule 4180
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+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 {5 A b}{2}+a (2 A+3 C) \sec (c+d x)-\frac {1}{2} b (A-6 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {5 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+4 a^2 (4 A+6 C)\right )+\frac {1}{2} a b (11 A+24 C) \sec (c+d x)-\frac {3}{4} b^2 (3 A-8 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {\left (15 A b^2+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 \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 {15}{8} b \left (A b^2+4 a^2 (A+2 C)\right )+\frac {1}{4} a b^2 (13 A+72 C) \sec (c+d x)-\frac {1}{8} b \left (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+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 \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 {15}{8} b \left (A b^2+4 a^2 (A+2 C)\right )+\left (\frac {1}{4} a b^2 (13 A+72 C)+\frac {1}{8} b \left (3 b^2 (11 A-16 C)+8 a^2 (2 A+3 C)\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx-\frac {1}{48} \left (b \left (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 (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+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 \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}{48} \left (b \left (16 a^2 A+26 a A b+33 A b^2+24 a^2 C+144 a b C-48 b^2 C\right )\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx+\frac {1}{16} \left (5 b \left (A b^2+4 a^2 (A+2 C)\right )\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}} \, dx\\ &=\frac {(a-b) \sqrt {a+b} \left (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 (16 a^2 A+26 a A b+33 A b^2+24 a^2 C+144 a b C-48 b^2 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 {5 b \sqrt {a+b} \left (A b^2+4 a^2 (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+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 \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. \(1501\) vs. \(2(507)=1014\).
time = 19.57, size = 1501, normalized size = 2.96 \begin {gather*} \frac {\cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \left (\frac {1}{6} \left (a^2 A+24 b^2 C\right ) \sin (c+d x)+\frac {13}{12} a A b \sin (2 (c+d x))+\frac {1}{6} a^2 A \sin (3 (c+d x))\right )}{d (b+a \cos (c+d x))^2 (A+2 C+A \cos (2 c+2 d x))}+\frac {(a+b \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \sqrt {\frac {1}{1-\tan ^2\left (\frac {1}{2} (c+d x)\right )}} \left (16 a^3 A \tan \left (\frac {1}{2} (c+d x)\right )+16 a^2 A b \tan \left (\frac {1}{2} (c+d x)\right )+33 a A b^2 \tan \left (\frac {1}{2} (c+d x)\right )+33 A b^3 \tan \left (\frac {1}{2} (c+d x)\right )+24 a^3 C \tan \left (\frac {1}{2} (c+d x)\right )+24 a^2 b C \tan \left (\frac {1}{2} (c+d x)\right )-48 a b^2 C \tan \left (\frac {1}{2} (c+d x)\right )-48 b^3 C \tan \left (\frac {1}{2} (c+d x)\right )-32 a^3 A \tan ^3\left (\frac {1}{2} (c+d x)\right )-66 a A b^2 \tan ^3\left (\frac {1}{2} (c+d x)\right )-48 a^3 C \tan ^3\left (\frac {1}{2} (c+d x)\right )+96 a b^2 C \tan ^3\left (\frac {1}{2} (c+d x)\right )+16 a^3 A \tan ^5\left (\frac {1}{2} (c+d x)\right )-16 a^2 A b \tan ^5\left (\frac {1}{2} (c+d x)\right )+33 a A b^2 \tan ^5\left (\frac {1}{2} (c+d x)\right )-33 A b^3 \tan ^5\left (\frac {1}{2} (c+d x)\right )+24 a^3 C \tan ^5\left (\frac {1}{2} (c+d x)\right )-24 a^2 b C \tan ^5\left (\frac {1}{2} (c+d x)\right )-48 a b^2 C \tan ^5\left (\frac {1}{2} (c+d x)\right )+48 b^3 C \tan ^5\left (\frac {1}{2} (c+d x)\right )+120 a^2 A b \Pi \left (-1;\text {ArcSin}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+30 A b^3 \Pi \left (-1;\text {ArcSin}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+240 a^2 b C \Pi \left (-1;\text {ArcSin}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+120 a^2 A b \Pi \left (-1;\text {ArcSin}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+30 A b^3 \Pi \left (-1;\text {ArcSin}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+240 a^2 b C \Pi \left (-1;\text {ArcSin}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+(a+b) \left (3 b^2 (11 A-16 C)+8 a^2 (2 A+3 C)\right ) E\left (\text {ArcSin}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}-2 b \left (24 b^2 (A-C)-a b (13 A+72 C)+a^2 (38 A+72 C)\right ) F\left (\text {ArcSin}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (c+d x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}\right )}{12 d (b+a \cos (c+d x))^{5/2} (A+2 C+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x) \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^{3/2} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+b \tan ^2\left (\frac {1}{2} (c+d x)\right )}{1+\tan ^2\left (\frac {1}{2} (c+d x)\right )}}} \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. \(3511\) vs.
\(2(462)=924\).
time = 0.42, size = 3512, normalized size = 6.93
method | result | size |
default | \(\text {Expression too large to display}\) | \(3512\) |
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 {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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