Optimal. Leaf size=802 \[ -\frac{(3 c-d) \tan (e+f x) d^3}{a c^2 (c-d)^3 (c+d) f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))}-\frac{3 \tan (e+f x) d^3}{4 a c \left (c^2-d^2\right )^2 f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))}-\frac{\tan (e+f x) d^3}{2 a c (c-d)^2 (c+d) f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))^2}-\frac{2 \left (6 c^2-4 d c+d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right ) \tan (e+f x) d^{5/2}}{\sqrt{a} c^3 (c-d)^4 \sqrt{c+d} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{(3 c-d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right ) \tan (e+f x) d^{5/2}}{\sqrt{a} c^2 (c-d)^3 (c+d)^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right ) \tan (e+f x) d^{5/2}}{4 \sqrt{a} c (c-d)^2 (c+d)^{5/2} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right ) \tan (e+f x)}{\sqrt{a} c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right ) \tan (e+f x)}{2 \sqrt{2} \sqrt{a} (c-d)^3 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{\sqrt{2} (c-4 d) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right ) \tan (e+f x)}{\sqrt{a} (c-d)^4 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{\tan (e+f x)}{2 a (c-d)^3 f (\sec (e+f x)+1) \sqrt{\sec (e+f x) a+a}} \]
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Rubi [A] time = 0.796566, antiderivative size = 802, 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} \[ -\frac{(3 c-d) \tan (e+f x) d^3}{a c^2 (c-d)^3 (c+d) f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))}-\frac{3 \tan (e+f x) d^3}{4 a c \left (c^2-d^2\right )^2 f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))}-\frac{\tan (e+f x) d^3}{2 a c (c-d)^2 (c+d) f \sqrt{\sec (e+f x) a+a} (c+d \sec (e+f x))^2}-\frac{2 \left (6 c^2-4 d c+d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right ) \tan (e+f x) d^{5/2}}{\sqrt{a} c^3 (c-d)^4 \sqrt{c+d} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{(3 c-d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right ) \tan (e+f x) d^{5/2}}{\sqrt{a} c^2 (c-d)^3 (c+d)^{3/2} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a-a \sec (e+f x)}}{\sqrt{a} \sqrt{c+d}}\right ) \tan (e+f x) d^{5/2}}{4 \sqrt{a} c (c-d)^2 (c+d)^{5/2} f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a}}\right ) \tan (e+f x)}{\sqrt{a} c^3 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right ) \tan (e+f x)}{2 \sqrt{2} \sqrt{a} (c-d)^3 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{\sqrt{2} (c-4 d) \tanh ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{2} \sqrt{a}}\right ) \tan (e+f x)}{\sqrt{a} (c-d)^4 f \sqrt{a-a \sec (e+f x)} \sqrt{\sec (e+f x) a+a}}-\frac{\tan (e+f x)}{2 a (c-d)^3 f (\sec (e+f x)+1) \sqrt{\sec (e+f x) a+a}} \]
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))^{3/2} (c+d \sec (e+f x))^3} \, dx &=-\frac{\left (a^2 \tan (e+f x)\right ) \operatorname{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 ) \operatorname{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) \operatorname{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)) \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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)) \operatorname{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)) \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [C] time = 40.8732, size = 774154, normalized size = 965.28 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 28.445, size = 480553, normalized size = 599.2 \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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