Optimal. Leaf size=158 \[ \frac{2 C \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 b d}+\frac{2 \left (A b^2-a (b B-a C)\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d (a+b)}-\frac{2 (b B-a C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d}+\frac{2 (b B-a C) \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x)}{3 b d \cos ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 0.934632, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {4112, 3055, 3059, 2639, 3002, 2641, 2805} \[ \frac{2 \left (A b^2-a (b B-a C)\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d (a+b)}-\frac{2 (b B-a C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d}+\frac{2 (b B-a C) \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 C F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 b d}+\frac{2 C \sin (c+d x)}{3 b d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4112
Rule 3055
Rule 3059
Rule 2639
Rule 3002
Rule 2641
Rule 2805
Rubi steps
\begin{align*} \int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx &=\int \frac{C+B \cos (c+d x)+A \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (b+a \cos (c+d x))} \, dx\\ &=\frac{2 C \sin (c+d x)}{3 b d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \int \frac{\frac{3}{2} (b B-a C)+\frac{1}{2} b (3 A+C) \cos (c+d x)+\frac{1}{2} a C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x))} \, dx}{3 b}\\ &=\frac{2 C \sin (c+d x)}{3 b d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (b B-a C) \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)}}+\frac{4 \int \frac{\frac{1}{4} \left (b^2 (3 A+C)-3 a (b B-a C)\right )-\frac{1}{4} b (3 b B-4 a C) \cos (c+d x)-\frac{3}{4} a (b B-a C) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{3 b^2}\\ &=\frac{2 C \sin (c+d x)}{3 b d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (b B-a C) \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)}}-\frac{4 \int \frac{-\frac{1}{4} a \left (b^2 (3 A+C)-3 a (b B-a C)\right )-\frac{1}{4} a^2 b C \cos (c+d x)}{\sqrt{\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{3 a b^2}-\frac{(b B-a C) \int \sqrt{\cos (c+d x)} \, dx}{b^2}\\ &=-\frac{2 (b B-a C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d}+\frac{2 C \sin (c+d x)}{3 b d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (b B-a C) \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)}}+\frac{C \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 b}+\left (A-\frac{a (b B-a C)}{b^2}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} (b+a \cos (c+d x))} \, dx\\ &=-\frac{2 (b B-a C) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d}+\frac{2 C F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 b d}+\frac{2 \left (A-\frac{a (b B-a C)}{b^2}\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{(a+b) d}+\frac{2 C \sin (c+d x)}{3 b d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (b B-a C) \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 2.48184, size = 269, normalized size = 1.7 \[ \frac{-\frac{6 (b B-a C) \sin (c+d x) \left (2 b (a+b) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right ),-1\right )-\left (a^2-2 b^2\right ) \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )-2 a b E\left (\left .\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )\right )}{a \sqrt{\sin ^2(c+d x)}}+\frac{b \left (8 a b C-6 b^2 B\right ) \left (2 \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )-\frac{2 b \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a+b}\right )}{a}+\frac{2 b \left (9 a^2 C-9 a b B+6 A b^2+2 b^2 C\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a+b}+\frac{12 b (b B-a C) \sin (c+d x)}{\sqrt{\cos (c+d x)}}+\frac{4 b^2 C \sin (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)}}{6 b^3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 7.112, size = 472, normalized size = 3. \begin{align*} \text{result 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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