Optimal. Leaf size=1150 \[ -\frac {2 \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)};\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)} a^2}{c^4 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 (b c-a d) \left (5 b c^3-13 a d c^2+3 b d^2 c+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 {2 d (b c-a d) (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 (a-b) \sqrt {a+b} \left (\left (15 d^5-41 c^2 d^3+58 c^4 d\right ) a^2-b c \left (35 c^4+34 d^2 c^2-5 d^4\right ) a+b^2 c^2 d \left (29 c^2+3 d^2\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 (\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) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 \sqrt {a+b} \left (b^3 \left (5 c^2+24 d c+3 d^2\right ) c^4-a b^2 \left (35 c^3+42 d c^2+21 d^2 c-2 d^3\right ) c^3+a^2 b \left (45 c^4+48 d c^3+d^2 c^2-8 d^3 c+10 d^4\right ) c^2-a^3 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 (\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)}} \]
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Rubi [A] time = 3.44, antiderivative size = 1150, 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 = {3942, 2989, 3047, 3053, 2811, 2998, 2818, 2996} \[ -\frac {2 \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)};\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)} a^2}{c^4 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 (b c-a d) \left (5 b c^3-13 a d c^2+3 b d^2 c+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 {2 d (b c-a d) (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 (a-b) \sqrt {a+b} \left (\left (15 d^5-41 c^2 d^3+58 c^4 d\right ) a^2-b c \left (35 c^4+34 d^2 c^2-5 d^4\right ) a+b^2 c^2 d \left (29 c^2+3 d^2\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 (\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) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 \sqrt {a+b} \left (b^3 \left (5 c^2+24 d c+3 d^2\right ) c^4-a b^2 \left (35 c^3+42 d c^2+21 d^2 c-2 d^3\right ) c^3+a^2 b \left (45 c^4+48 d c^3+d^2 c^2-8 d^3 c+10 d^4\right ) c^2-a^3 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 (\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)}} \]
Antiderivative was successfully verified.
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Rule 2811
Rule 2818
Rule 2989
Rule 2996
Rule 2998
Rule 3047
Rule 3053
Rule 3942
Rubi steps
\begin {align*} \int \frac {(a+b \sec (e+f x))^{5/2}}{(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 (e+f x) (b+a \cos (e+f x))^{5/2}}{(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 (b c-a d) (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} (5 b c-3 a d) (b c-a d)-\frac {1}{2} \left (5 a^2 c d+3 b^2 c d-2 a b \left (5 c^2-d^2\right )\right ) \cos (e+f x)+\frac {5}{2} a^2 \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 (b c-a d) (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 (b c-a d) \left (5 b c^3-13 a c^2 d+3 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} (b c-a d) \left (35 a b c^3-13 a^2 c^2 d-24 b^2 c^2 d-3 a b c d^2+5 a^2 d^3\right )-\frac {1}{4} \left (2 a b^2 c d \left (21 c^2-d^2\right )-b^3 c^2 \left (5 c^2+3 d^2\right )+6 a^3 \left (5 c^3 d-c d^3\right )-a^2 b \left (45 c^4+c^2 d^2+10 d^4\right )\right ) \cos (e+f x)+\frac {15}{4} a^3 \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 (b c-a d) (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 (b c-a d) \left (5 b c^3-13 a c^2 d+3 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^3 \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^3 d^2 \left (c^2-d^2\right )^2+\frac {1}{4} c^2 (b c-a d) \left (35 a b c^3-13 a^2 c^2 d-24 b^2 c^2 d-3 a b c d^2+5 a^2 d^3\right )+c \left (-\frac {15}{2} a^3 d \left (c^2-d^2\right )^2+\frac {1}{4} c \left (-2 a b^2 c d \left (21 c^2-d^2\right )+b^3 c^2 \left (5 c^2+3 d^2\right )-6 a^3 \left (5 c^3 d-c d^3\right )+a^2 b \left (45 c^4+c^2 d^2+10 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^2 \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 (b c-a d) (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 (b c-a d) \left (5 b c^3-13 a c^2 d+3 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 (b^3 c^4 \left (5 c^2+24 c d+3 d^2\right )-a b^2 c^3 \left (35 c^3+42 c^2 d+21 c d^2-2 d^3\right )+a^2 b c^2 \left (45 c^4+48 c^3 d+c^2 d^2-8 c d^3+10 d^4\right )-a^3 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 {\left (4 \left (c \left (-\frac {15}{4} a^3 d^2 \left (c^2-d^2\right )^2+\frac {1}{4} c^2 (b c-a d) \left (35 a b c^3-13 a^2 c^2 d-24 b^2 c^2 d-3 a b c d^2+5 a^2 d^3\right )\right )-c d \left (-\frac {15}{2} a^3 d \left (c^2-d^2\right )^2+\frac {1}{4} c \left (-2 a b^2 c d \left (21 c^2-d^2\right )+b^3 c^2 \left (5 c^2+3 d^2\right )-6 a^3 \left (5 c^3 d-c d^3\right )+a^2 b \left (45 c^4+c^2 d^2+10 d^4\right )\right )\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^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 (b^2 c^2 d \left (29 c^2+3 d^2\right )-a b c \left (35 c^4+34 c^2 d^2-5 d^4\right )+a^2 \left (58 c^4 d-41 c^2 d^3+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) 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) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 \sqrt {a+b} \left (b^3 c^4 \left (5 c^2+24 c d+3 d^2\right )-a b^2 c^3 \left (35 c^3+42 c^2 d+21 c d^2-2 d^3\right )+a^2 b c^2 \left (45 c^4+48 c^3 d+c^2 d^2-8 c d^3+10 d^4\right )-a^3 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^2 \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 (b c-a d) (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 (b c-a d) \left (5 b c^3-13 a c^2 d+3 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] time = 7.32, size = 2344, normalized size = 2.04 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.98, size = 32283, normalized size = 28.07 \[ \text {output too large to display} \]
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 \[ \int \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ \text {Hanged} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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