Optimal. Leaf size=559 \[ \frac{\left (21 a^2 A b^2+a^3 b (3 A-2 C)+6 a^4 C-5 a A b^3-15 A b^4\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{3 a^3 b d (a-b) (a+b)^{3/2}}-\frac{b \left (26 a^2 A b^2+a^4 (-(3 A-8 C))-15 A b^4\right ) \tan (c+d x)}{3 a^3 d \left (a^2-b^2\right )^2 \sqrt{a+b \sec (c+d x)}}-\frac{b \left (5 A b^2-a^2 (3 A-2 C)\right ) \tan (c+d x)}{3 a^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}-\frac{\left (26 a^2 A b^2+a^4 (-(3 A-8 C))-15 A b^4\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{3 a^3 b d (a-b) (a+b)^{3/2}}+\frac{5 A b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{a^4 d}+\frac{A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}} \]
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Rubi [A] time = 1.19001, antiderivative size = 559, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {4105, 4060, 4058, 3921, 3784, 3832, 4004} \[ -\frac{b \left (26 a^2 A b^2+a^4 (-(3 A-8 C))-15 A b^4\right ) \tan (c+d x)}{3 a^3 d \left (a^2-b^2\right )^2 \sqrt{a+b \sec (c+d x)}}-\frac{b \left (5 A b^2-a^2 (3 A-2 C)\right ) \tan (c+d x)}{3 a^2 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}+\frac{\left (21 a^2 A b^2+a^3 b (3 A-2 C)+6 a^4 C-5 a A b^3-15 A b^4\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{3 a^3 b d (a-b) (a+b)^{3/2}}-\frac{\left (26 a^2 A b^2+a^4 (-(3 A-8 C))-15 A b^4\right ) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{3 a^3 b d (a-b) (a+b)^{3/2}}+\frac{5 A b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{a^4 d}+\frac{A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4105
Rule 4060
Rule 4058
Rule 3921
Rule 3784
Rule 3832
Rule 4004
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{5/2}} \, dx &=\frac{A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac{\int \frac{\frac{5 A b}{2}-a C \sec (c+d x)-\frac{3}{2} A b \sec ^2(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx}{a}\\ &=\frac{A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac{b \left (5 A b^2-a^2 (3 A-2 C)\right ) \tan (c+d x)}{3 a^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}+\frac{2 \int \frac{-\frac{15}{4} A b \left (a^2-b^2\right )+\frac{3}{2} a \left (A b^2+a^2 C\right ) \sec (c+d x)-\frac{1}{4} b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sec ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx}{3 a^2 \left (a^2-b^2\right )}\\ &=\frac{A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac{b \left (5 A b^2-a^2 (3 A-2 C)\right ) \tan (c+d x)}{3 a^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac{b \left (26 a^2 A b^2-15 A b^4-a^4 (3 A-8 C)\right ) \tan (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt{a+b \sec (c+d x)}}-\frac{4 \int \frac{\frac{15}{8} A b \left (a^2-b^2\right )^2+\frac{1}{4} a \left (5 A b^4-3 a^4 C-a^2 b^2 (9 A+C)\right ) \sec (c+d x)-\frac{1}{8} b \left (26 a^2 A b^2-15 A b^4-a^4 (3 A-8 C)\right ) \sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{3 a^3 \left (a^2-b^2\right )^2}\\ &=\frac{A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac{b \left (5 A b^2-a^2 (3 A-2 C)\right ) \tan (c+d x)}{3 a^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac{b \left (26 a^2 A b^2-15 A b^4-a^4 (3 A-8 C)\right ) \tan (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt{a+b \sec (c+d x)}}-\frac{4 \int \frac{\frac{15}{8} A b \left (a^2-b^2\right )^2+\left (\frac{1}{8} b \left (26 a^2 A b^2-15 A b^4-a^4 (3 A-8 C)\right )+\frac{1}{4} a \left (5 A b^4-3 a^4 C-a^2 b^2 (9 A+C)\right )\right ) \sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{3 a^3 \left (a^2-b^2\right )^2}+\frac{\left (b \left (26 a^2 A b^2-15 A b^4-a^4 (3 A-8 C)\right )\right ) \int \frac{\sec (c+d x) (1+\sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx}{6 a^3 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (26 a^2 A b^2-15 A b^4-a^4 (3 A-8 C)\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{3 a^3 (a-b) b (a+b)^{3/2} d}+\frac{A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac{b \left (5 A b^2-a^2 (3 A-2 C)\right ) \tan (c+d x)}{3 a^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac{b \left (26 a^2 A b^2-15 A b^4-a^4 (3 A-8 C)\right ) \tan (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt{a+b \sec (c+d x)}}-\frac{(5 A b) \int \frac{1}{\sqrt{a+b \sec (c+d x)}} \, dx}{2 a^3}+\frac{\left (21 a^2 A b^2-5 a A b^3-15 A b^4+a^3 b (3 A-2 C)+6 a^4 C\right ) \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx}{6 a^3 (a-b) (a+b)^2}\\ &=-\frac{\left (26 a^2 A b^2-15 A b^4-a^4 (3 A-8 C)\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{3 a^3 (a-b) b (a+b)^{3/2} d}+\frac{\left (21 a^2 A b^2-5 a A b^3-15 A b^4+a^3 b (3 A-2 C)+6 a^4 C\right ) \cot (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{3 a^3 (a-b) b (a+b)^{3/2} d}+\frac{5 A b \sqrt{a+b} \cot (c+d x) \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{a^4 d}+\frac{A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}-\frac{b \left (5 A b^2-a^2 (3 A-2 C)\right ) \tan (c+d x)}{3 a^2 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac{b \left (26 a^2 A b^2-15 A b^4-a^4 (3 A-8 C)\right ) \tan (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt{a+b \sec (c+d x)}}\\ \end{align*}
Mathematica [B] time = 20.4938, size = 1714, normalized size = 3.07 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.683, size = 6418, normalized size = 11.5 \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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C \cos \left (d x + c\right ) \sec \left (d x + c\right )^{2} + A \cos \left (d x + c\right )\right )} \sqrt{b \sec \left (d x + c\right ) + a}}{b^{3} \sec \left (d x + c\right )^{3} + 3 \, a b^{2} \sec \left (d x + c\right )^{2} + 3 \, a^{2} b \sec \left (d x + c\right ) + a^{3}}, x\right ) \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \sec \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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