Optimal. Leaf size=1122 \[ \frac {2 (a-b) \sqrt {a+b} \left (2 a b c d \left (35 c^4-8 c^2 d^2+5 d^4\right )-a^2 d^2 \left (58 c^4-41 c^2 d^2+15 d^4\right )-b^2 \left (15 c^6+19 c^4 d^2-2 c^2 d^4\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left (\text {ArcSin}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{15 c^3 (c-d)^3 (c+d)^{5/2} (b c-a d)^2 f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 \sqrt {a+b} \left (b^2 c^3 \left (15 c^3+10 c^2 d+9 c d^2-2 d^3\right )-2 a b c^2 \left (15 c^4+20 c^3 d-4 c^2 d^2-4 c d^3+5 d^4\right )+a^2 d \left (60 c^5-2 c^4 d-66 c^3 d^2+25 c^2 d^3+30 c d^4-15 d^5\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) F\left (\text {ArcSin}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{15 c^4 (c-d)^3 (c+d)^{5/2} (b c-a d) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 a \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left (\frac {(a+b) c}{a (c+d)};\text {ArcSin}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{c^4 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 d^2 (b+a \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{5 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2 \sqrt {c+d \sec (e+f x)}}-\frac {2 d \left (10 b c^3-13 a c^2 d-2 b c d^2+5 a d^3\right ) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{15 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}} \]
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Rubi [A]
time = 2.16, antiderivative size = 1122, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {4027, 3127,
3126, 3132, 2890, 3077, 2897, 3075} \begin {gather*} \frac {2 (b+a \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \sin (e+f x) d^2}{5 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2 \sqrt {c+d \sec (e+f x)}}-\frac {2 \left (10 b c^3-13 a d c^2-2 b d^2 c+5 a d^3\right ) \sqrt {a+b \sec (e+f x)} \sin (e+f x) d}{15 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}}+\frac {2 (a-b) \sqrt {a+b} \left (-\left (\left (15 c^6+19 d^2 c^4-2 d^4 c^2\right ) b^2\right )+2 a c d \left (35 c^4-8 d^2 c^2+5 d^4\right ) b-a^2 d^2 \left (58 c^4-41 d^2 c^2+15 d^4\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left (\text {ArcSin}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{15 c^3 (c-d)^3 (c+d)^{5/2} (b c-a d)^2 f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 \sqrt {a+b} \left (b^2 \left (15 c^3+10 d c^2+9 d^2 c-2 d^3\right ) c^3-2 a b \left (15 c^4+20 d c^3-4 d^2 c^2-4 d^3 c+5 d^4\right ) c^2+a^2 d \left (60 c^5-2 d c^4-66 d^2 c^3+25 d^3 c^2+30 d^4 c-15 d^5\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) F\left (\text {ArcSin}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{15 c^4 (c-d)^3 (c+d)^{5/2} (b c-a d) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 a \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left (\frac {(a+b) c}{a (c+d)};\text {ArcSin}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{c^4 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2890
Rule 2897
Rule 3075
Rule 3077
Rule 3126
Rule 3127
Rule 3132
Rule 4027
Rubi steps
\begin {align*} \int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{7/2}} \, dx &=\frac {\left (\sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {\cos ^2(e+f x) (b+a \cos (e+f x))^{3/2}}{(d+c \cos (e+f x))^{7/2}} \, dx}{\sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\\ &=\frac {2 d^2 (b+a \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{5 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2 \sqrt {c+d \sec (e+f x)}}+\frac {\left (2 \sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {\sqrt {b+a \cos (e+f x)} \left (-\frac {1}{2} d (5 b c-3 a d)+\frac {1}{2} \left (5 b c^2-5 a c d-2 b d^2\right ) \cos (e+f x)+\frac {5}{2} a \left (c^2-d^2\right ) \cos ^2(e+f x)\right )}{(d+c \cos (e+f x))^{5/2}} \, dx}{5 c \left (c^2-d^2\right ) \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\\ &=\frac {2 d^2 (b+a \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{5 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2 \sqrt {c+d \sec (e+f x)}}-\frac {2 d \left (10 b c^3-13 a c^2 d-2 b c d^2+5 a d^3\right ) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{15 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}}+\frac {\left (4 \sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {\frac {1}{4} \left (a^2 d^2 \left (13 c^2-5 d^2\right )-8 a b c d \left (5 c^2-d^2\right )+3 b^2 \left (5 c^4+3 c^2 d^2\right )\right )-\frac {1}{2} \left (b^2 c d \left (5 c^2-d^2\right )+3 a^2 \left (5 c^3 d-c d^3\right )-a b \left (15 c^4-4 c^2 d^2+5 d^4\right )\right ) \cos (e+f x)+\frac {15}{4} a^2 \left (c^2-d^2\right )^2 \cos ^2(e+f x)}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}} \, dx}{15 c^2 \left (c^2-d^2\right )^2 \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\\ &=\frac {2 d^2 (b+a \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{5 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2 \sqrt {c+d \sec (e+f x)}}-\frac {2 d \left (10 b c^3-13 a c^2 d-2 b c d^2+5 a d^3\right ) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{15 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}}+\frac {\left (a^2 \sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {\sqrt {d+c \cos (e+f x)}}{\sqrt {b+a \cos (e+f x)}} \, dx}{c^4 \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {\left (4 \sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {-\frac {15}{4} a^2 d^2 \left (c^2-d^2\right )^2+\frac {1}{4} c^2 \left (a^2 d^2 \left (13 c^2-5 d^2\right )-8 a b c d \left (5 c^2-d^2\right )+3 b^2 \left (5 c^4+3 c^2 d^2\right )\right )+c \left (-\frac {15}{2} a^2 d \left (c^2-d^2\right )^2+\frac {1}{2} c \left (-b^2 c d \left (5 c^2-d^2\right )-3 a^2 \left (5 c^3 d-c d^3\right )+a b \left (15 c^4-4 c^2 d^2+5 d^4\right )\right )\right ) \cos (e+f x)}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}} \, dx}{15 c^4 \left (c^2-d^2\right )^2 \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\\ &=-\frac {2 a \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left (\frac {(a+b) c}{a (c+d)};\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{c^4 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 d^2 (b+a \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{5 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2 \sqrt {c+d \sec (e+f x)}}-\frac {2 d \left (10 b c^3-13 a c^2 d-2 b c d^2+5 a d^3\right ) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{15 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}}-\frac {\left (\left (2 a b c d \left (35 c^4-8 c^2 d^2+5 d^4\right )-a^2 d^2 \left (58 c^4-41 c^2 d^2+15 d^4\right )-b^2 \left (15 c^6+19 c^4 d^2-2 c^2 d^4\right )\right ) \sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {1+\cos (e+f x)}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}} \, dx}{15 c^3 (c-d) \left (c^2-d^2\right )^2 \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {\left (\left (b^2 c^3 \left (15 c^3+10 c^2 d+9 c d^2-2 d^3\right )-2 a b c^2 \left (15 c^4+20 c^3 d-4 c^2 d^2-4 c d^3+5 d^4\right )+a^2 d \left (60 c^5-2 c^4 d-66 c^3 d^2+25 c^2 d^3+30 c d^4-15 d^5\right )\right ) \sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)}\right ) \int \frac {1}{\sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}} \, dx}{15 c^4 (c-d) \left (c^2-d^2\right )^2 \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\\ &=\frac {2 (a-b) \sqrt {a+b} \left (2 a b c d \left (35 c^4-8 c^2 d^2+5 d^4\right )-a^2 d^2 \left (58 c^4-41 c^2 d^2+15 d^4\right )-b^2 \left (15 c^6+19 c^4 d^2-2 c^2 d^4\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{15 c^3 (c-d)^3 (c+d)^{5/2} (b c-a d)^2 f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 \sqrt {a+b} \left (b^2 c^3 \left (15 c^3+10 c^2 d+9 c d^2-2 d^3\right )-2 a b c^2 \left (15 c^4+20 c^3 d-4 c^2 d^2-4 c d^3+5 d^4\right )+a^2 d \left (60 c^5-2 c^4 d-66 c^3 d^2+25 c^2 d^3+30 c d^4-15 d^5\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{15 c^4 (c-d)^3 (c+d)^{5/2} (b c-a d) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 a \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \Pi \left (\frac {(a+b) c}{a (c+d)};\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{c^4 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 d^2 (b+a \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{5 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2 \sqrt {c+d \sec (e+f x)}}-\frac {2 d \left (10 b c^3-13 a c^2 d-2 b c d^2+5 a d^3\right ) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{15 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(2385\) vs. \(2(1122)=2244\).
time = 7.52, size = 2385, normalized size = 2.13 \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. \(39417\) vs.
\(2(1037)=2074\).
time = 3.48, size = 39418, normalized size = 35.13
method | result | size |
default | \(\text {Expression too large to display}\) | \(39418\) |
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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