Optimal. Leaf size=756 \[ \frac{2 d^{7/2} (4 c-d) \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right )}{a^{3/2} c^2 f (c-d)^4 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} \left (c^2-4 c d+6 d^2\right ) \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right )}{a^{3/2} f (c-d)^4 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right )}{a^{3/2} c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{d^4 \tan (e+f x)}{a^2 c f (c-d)^3 (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}+\frac{d^{7/2} \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right )}{a^{3/2} c f (c-d)^3 (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{3 \tan (e+f x)}{16 a^2 f (c-d)^2 (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-3 d) \tan (e+f x)}{2 a^2 f (c-d)^3 (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x)}{4 a^2 f (c-d)^2 (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a}}-\frac{3 \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right )}{16 \sqrt{2} a^{3/2} f (c-d)^2 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{(c-3 d) \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right )}{2 \sqrt{2} a^{3/2} f (c-d)^3 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}} \]
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Rubi [A] time = 0.633782, antiderivative size = 756, normalized size of antiderivative = 1., 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 = {3940, 180, 63, 206, 51, 208} \[ \frac{2 d^{7/2} (4 c-d) \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right )}{a^{3/2} c^2 f (c-d)^4 \sqrt{c+d} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{\sqrt{2} \left (c^2-4 c d+6 d^2\right ) \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right )}{a^{3/2} f (c-d)^4 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{2 \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right )}{a^{3/2} c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{d^4 \tan (e+f x)}{a^2 c f (c-d)^3 (c+d) \sqrt{a \sec (e+f x)+a} (c+d \sec (e+f x))}+\frac{d^{7/2} \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right )}{a^{3/2} c f (c-d)^3 (c+d)^{3/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{3 \tan (e+f x)}{16 a^2 f (c-d)^2 (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{(c-3 d) \tan (e+f x)}{2 a^2 f (c-d)^3 (\sec (e+f x)+1) \sqrt{a \sec (e+f x)+a}}-\frac{\tan (e+f x)}{4 a^2 f (c-d)^2 (\sec (e+f x)+1)^2 \sqrt{a \sec (e+f x)+a}}-\frac{3 \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right )}{16 \sqrt{2} a^{3/2} f (c-d)^2 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{(c-3 d) \tan (e+f x) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right )}{2 \sqrt{2} a^{3/2} f (c-d)^3 \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 3940
Rule 180
Rule 63
Rule 206
Rule 51
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))^2} \, dx &=-\frac{\left (a^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a-a x} (a+a x)^3 (c+d x)^2} \, 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 ) \operatorname{Subst}\left (\int \left (\frac{1}{a^3 c^2 x \sqrt{a-a x}}-\frac{1}{a^3 (c-d)^2 (1+x)^3 \sqrt{a-a x}}+\frac{-c+3 d}{a^3 (c-d)^3 (1+x)^2 \sqrt{a-a x}}+\frac{-c^2+4 c d-6 d^2}{a^3 (c-d)^4 (1+x) \sqrt{a-a x}}+\frac{d^4}{a^3 c (c-d)^3 \sqrt{a-a x} (c+d x)^2}+\frac{(4 c-d) d^4}{a^3 c^2 (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) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a-a x}} \, dx,x,\sec (e+f x)\right )}{a c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{((c-3 d) \tan (e+f x)) \operatorname{Subst}\left (\int \frac{1}{(1+x)^2 \sqrt{a-a x}} \, dx,x,\sec (e+f x)\right )}{a (c-d)^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{\tan (e+f x) \operatorname{Subst}\left (\int \frac{1}{(1+x)^3 \sqrt{a-a x}} \, dx,x,\sec (e+f x)\right )}{a (c-d)^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{\left (d^4 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-a x} (c+d x)^2} \, dx,x,\sec (e+f x)\right )}{a c (c-d)^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{\left ((4 c-d) d^4 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{a c^2 (c-d)^4 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{\left (\left (c^2-4 c d+6 d^2\right ) \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{(1+x) \sqrt{a-a x}} \, dx,x,\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)}{4 a^2 (c-d)^2 f (1+\sec (e+f x))^2 \sqrt{a+a \sec (e+f x)}}-\frac{(c-3 d) \tan (e+f x)}{2 a^2 (c-d)^3 f (1+\sec (e+f x)) \sqrt{a+a \sec (e+f x)}}+\frac{d^4 \tan (e+f x)}{a^2 c (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)) \operatorname{Subst}\left (\int \frac{1}{1-\frac{x^2}{a}} \, dx,x,\sqrt{a-a \sec (e+f x)}\right )}{a^2 c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{((c-3 d) \tan (e+f x)) \operatorname{Subst}\left (\int \frac{1}{(1+x) \sqrt{a-a x}} \, dx,x,\sec (e+f x)\right )}{4 a (c-d)^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{(3 \tan (e+f x)) \operatorname{Subst}\left (\int \frac{1}{(1+x)^2 \sqrt{a-a x}} \, dx,x,\sec (e+f x)\right )}{8 a (c-d)^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{\left (2 (4 c-d) d^4 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{c+d-\frac{d x^2}{a}} \, dx,x,\sqrt{a-a \sec (e+f x)}\right )}{a^2 c^2 (c-d)^4 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{\left (d^4 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{2 a c (c-d)^3 (c+d) f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{\left (2 \left (c^2-4 c d+6 d^2\right ) \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{2-\frac{x^2}{a}} \, dx,x,\sqrt{a-a \sec (e+f x)}\right )}{a^2 (c-d)^4 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=-\frac{\tan (e+f x)}{4 a^2 (c-d)^2 f (1+\sec (e+f x))^2 \sqrt{a+a \sec (e+f x)}}-\frac{(c-3 d) \tan (e+f x)}{2 a^2 (c-d)^3 f (1+\sec (e+f x)) \sqrt{a+a \sec (e+f x)}}-\frac{3 \tan (e+f x)}{16 a^2 (c-d)^2 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)}{a^{3/2} c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{\sqrt{2} \left (c^2-4 c d+6 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right ) \tan (e+f x)}{a^{3/2} (c-d)^4 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{2 (4 c-d) d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right ) \tan (e+f x)}{a^{3/2} c^2 (c-d)^4 \sqrt{c+d} f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{d^4 \tan (e+f x)}{a^2 c (c-d)^3 (c+d) f \sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))}-\frac{((c-3 d) \tan (e+f x)) \operatorname{Subst}\left (\int \frac{1}{2-\frac{x^2}{a}} \, dx,x,\sqrt{a-a \sec (e+f x)}\right )}{2 a^2 (c-d)^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{(3 \tan (e+f x)) \operatorname{Subst}\left (\int \frac{1}{(1+x) \sqrt{a-a x}} \, dx,x,\sec (e+f x)\right )}{32 a (c-d)^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{\left (d^4 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{c+d-\frac{d x^2}{a}} \, dx,x,\sqrt{a-a \sec (e+f x)}\right )}{a^2 c (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)}{4 a^2 (c-d)^2 f (1+\sec (e+f x))^2 \sqrt{a+a \sec (e+f x)}}-\frac{(c-3 d) \tan (e+f x)}{2 a^2 (c-d)^3 f (1+\sec (e+f x)) \sqrt{a+a \sec (e+f x)}}-\frac{3 \tan (e+f x)}{16 a^2 (c-d)^2 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)}{a^{3/2} c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{(c-3 d) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right ) \tan (e+f x)}{2 \sqrt{2} a^{3/2} (c-d)^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{\sqrt{2} \left (c^2-4 c d+6 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right ) \tan (e+f x)}{a^{3/2} (c-d)^4 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right ) \tan (e+f x)}{a^{3/2} c (c-d)^3 (c+d)^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{2 (4 c-d) d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right ) \tan (e+f x)}{a^{3/2} c^2 (c-d)^4 \sqrt{c+d} f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{d^4 \tan (e+f x)}{a^2 c (c-d)^3 (c+d) f \sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))}-\frac{(3 \tan (e+f x)) \operatorname{Subst}\left (\int \frac{1}{2-\frac{x^2}{a}} \, dx,x,\sqrt{a-a \sec (e+f x)}\right )}{16 a^2 (c-d)^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=-\frac{\tan (e+f x)}{4 a^2 (c-d)^2 f (1+\sec (e+f x))^2 \sqrt{a+a \sec (e+f x)}}-\frac{(c-3 d) \tan (e+f x)}{2 a^2 (c-d)^3 f (1+\sec (e+f x)) \sqrt{a+a \sec (e+f x)}}-\frac{3 \tan (e+f x)}{16 a^2 (c-d)^2 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)}{a^{3/2} c^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{(c-3 d) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right ) \tan (e+f x)}{2 \sqrt{2} a^{3/2} (c-d)^3 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right ) \tan (e+f x)}{16 \sqrt{2} a^{3/2} (c-d)^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}-\frac{\sqrt{2} \left (c^2-4 c d+6 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right ) \tan (e+f x)}{a^{3/2} (c-d)^4 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right ) \tan (e+f x)}{a^{3/2} c (c-d)^3 (c+d)^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{2 (4 c-d) d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right ) \tan (e+f x)}{a^{3/2} c^2 (c-d)^4 \sqrt{c+d} f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{d^4 \tan (e+f x)}{a^2 c (c-d)^3 (c+d) f \sqrt{a+a \sec (e+f x)} (c+d \sec (e+f x))}\\ \end{align*}
Mathematica [C] time = 39.9961, size = 686248, normalized size = 907.74 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 5.676, size = 197500, normalized size = 261.2 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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