Optimal. Leaf size=802 \[ -\frac {\tan (e+f x)}{2 a (c-d)^3 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} c^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\sqrt {2} (c-4 d) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} (c-d)^4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{2 \sqrt {2} \sqrt {a} (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {3 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{4 \sqrt {a} c (c-d)^2 (c+d)^{5/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {(3 c-d) d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{\sqrt {a} c^2 (c-d)^3 (c+d)^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {2 d^{5/2} \left (6 c^2-4 c d+d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{\sqrt {a} c^3 (c-d)^4 \sqrt {c+d} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {d^3 \tan (e+f x)}{2 a c (c-d)^2 (c+d) f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^2}-\frac {(3 c-d) d^3 \tan (e+f x)}{a c^2 (c-d)^3 (c+d) f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))}-\frac {3 d^3 \tan (e+f x)}{4 a c \left (c^2-d^2\right )^2 f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \]
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Rubi [A]
time = 0.57, antiderivative size = 802, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4025, 186, 65,
212, 44, 214} \begin {gather*} -\frac {(3 c-d) \tan (e+f x) d^3}{a c^2 (c-d)^3 (c+d) f \sqrt {\sec (e+f x) a+a} (c+d \sec (e+f x))}-\frac {3 \tan (e+f x) d^3}{4 a c \left (c^2-d^2\right )^2 f \sqrt {\sec (e+f x) a+a} (c+d \sec (e+f x))}-\frac {\tan (e+f x) d^3}{2 a c (c-d)^2 (c+d) f \sqrt {\sec (e+f x) a+a} (c+d \sec (e+f x))^2}-\frac {2 \left (6 c^2-4 d c+d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x) d^{5/2}}{\sqrt {a} c^3 (c-d)^4 \sqrt {c+d} f \sqrt {a-a \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}-\frac {(3 c-d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x) d^{5/2}}{\sqrt {a} c^2 (c-d)^3 (c+d)^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x) d^{5/2}}{4 \sqrt {a} c (c-d)^2 (c+d)^{5/2} f \sqrt {a-a \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} c^3 f \sqrt {a-a \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{2 \sqrt {2} \sqrt {a} (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}-\frac {\sqrt {2} (c-4 d) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} (c-d)^4 f \sqrt {a-a \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}-\frac {\tan (e+f x)}{2 a (c-d)^3 f (\sec (e+f x)+1) \sqrt {\sec (e+f x) a+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 186
Rule 212
Rule 214
Rule 4025
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3} \, dx &=-\frac {\left (a^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a-a x} (a+a x)^2 (c+d x)^3} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {\left (a^2 \tan (e+f x)\right ) \text {Subst}\left (\int \left (\frac {1}{a^2 c^3 x \sqrt {a-a x}}-\frac {1}{a^2 (c-d)^3 (1+x)^2 \sqrt {a-a x}}+\frac {-c+4 d}{a^2 (c-d)^4 (1+x) \sqrt {a-a x}}-\frac {d^3}{a^2 c (c-d)^2 \sqrt {a-a x} (c+d x)^3}-\frac {(3 c-d) d^3}{a^2 c^2 (c-d)^3 \sqrt {a-a x} (c+d x)^2}-\frac {d^3 \left (6 c^2-4 c d+d^2\right )}{a^2 c^3 (c-d)^4 \sqrt {a-a x} (c+d x)}\right ) \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {\tan (e+f x) \text {Subst}\left (\int \frac {1}{x \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{c^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {((c-4 d) \tan (e+f x)) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{(c-d)^4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {1}{(1+x)^2 \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{(c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left (d^3 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} (c+d x)^3} \, dx,x,\sec (e+f x)\right )}{c (c-d)^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left ((3 c-d) d^3 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} (c+d x)^2} \, dx,x,\sec (e+f x)\right )}{c^2 (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left (d^3 \left (6 c^2-4 c d+d^2\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{c^3 (c-d)^4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {\tan (e+f x)}{2 a (c-d)^3 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}-\frac {d^3 \tan (e+f x)}{2 a c (c-d)^2 (c+d) f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^2}-\frac {(3 c-d) d^3 \tan (e+f x)}{a c^2 (c-d)^3 (c+d) f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))}+\frac {(2 \tan (e+f x)) \text {Subst}\left (\int \frac {1}{1-\frac {x^2}{a}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{a c^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {(2 (c-4 d) \tan (e+f x)) \text {Subst}\left (\int \frac {1}{2-\frac {x^2}{a}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{a (c-d)^4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{4 (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left (3 d^3 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} (c+d x)^2} \, dx,x,\sec (e+f x)\right )}{4 c (c-d)^2 (c+d) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left ((3 c-d) d^3 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{2 c^2 (c-d)^3 (c+d) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\left (2 d^3 \left (6 c^2-4 c d+d^2\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{c+d-\frac {d x^2}{a}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{a c^3 (c-d)^4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {\tan (e+f x)}{2 a (c-d)^3 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} c^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\sqrt {2} (c-4 d) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} (c-d)^4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {2 d^{5/2} \left (6 c^2-4 c d+d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{\sqrt {a} c^3 (c-d)^4 \sqrt {c+d} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {d^3 \tan (e+f x)}{2 a c (c-d)^2 (c+d) f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^2}-\frac {(3 c-d) d^3 \tan (e+f x)}{a c^2 (c-d)^3 (c+d) f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))}-\frac {3 d^3 \tan (e+f x)}{4 a c \left (c^2-d^2\right )^2 f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))}-\frac {\tan (e+f x) \text {Subst}\left (\int \frac {1}{2-\frac {x^2}{a}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{2 a (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left (3 d^3 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{8 c (c-d)^2 (c+d)^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\left ((3 c-d) d^3 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{c+d-\frac {d x^2}{a}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{a c^2 (c-d)^3 (c+d) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {\tan (e+f x)}{2 a (c-d)^3 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} c^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\sqrt {2} (c-4 d) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} (c-d)^4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{2 \sqrt {2} \sqrt {a} (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {(3 c-d) d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{\sqrt {a} c^2 (c-d)^3 (c+d)^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {2 d^{5/2} \left (6 c^2-4 c d+d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{\sqrt {a} c^3 (c-d)^4 \sqrt {c+d} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {d^3 \tan (e+f x)}{2 a c (c-d)^2 (c+d) f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^2}-\frac {(3 c-d) d^3 \tan (e+f x)}{a c^2 (c-d)^3 (c+d) f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))}-\frac {3 d^3 \tan (e+f x)}{4 a c \left (c^2-d^2\right )^2 f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))}-\frac {\left (3 d^3 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{c+d-\frac {d x^2}{a}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{4 a c (c-d)^2 (c+d)^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {\tan (e+f x)}{2 a (c-d)^3 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} c^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\sqrt {2} (c-4 d) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} (c-d)^4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{2 \sqrt {2} \sqrt {a} (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {3 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{4 \sqrt {a} c (c-d)^2 (c+d)^{5/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {(3 c-d) d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{\sqrt {a} c^2 (c-d)^3 (c+d)^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {2 d^{5/2} \left (6 c^2-4 c d+d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{\sqrt {a} c^3 (c-d)^4 \sqrt {c+d} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {d^3 \tan (e+f x)}{2 a c (c-d)^2 (c+d) f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^2}-\frac {(3 c-d) d^3 \tan (e+f x)}{a c^2 (c-d)^3 (c+d) f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))}-\frac {3 d^3 \tan (e+f x)}{4 a c \left (c^2-d^2\right )^2 f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 40.80, size = 776222, normalized size = 967.86 \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. \(480552\) vs.
\(2(694)=1388\).
time = 15.05, size = 480553, normalized size = 599.19
method | result | size |
default | \(\text {Expression too large to display}\) | \(480553\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \left (c + d \sec {\left (e + f x \right )}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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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