Optimal. Leaf size=307 \[ -\frac{\text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (A b^2-a (b B-a C)\right )}{a b d \left (a^2-b^2\right )}-\frac{E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^2 C-a b B+A b^2-2 b^2 C\right )}{b^2 d \left (a^2-b^2\right )}+\frac{\left (a^2 b^2 (A+5 C)+a^3 b B-3 a^4 C-3 a b^3 B+A b^4\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a b^2 d (a+b) \left (a^2-b^2\right )}+\frac{\sin (c+d x) \left (3 a^2 C-a b B+A b^2-2 b^2 C\right )}{b^2 d \left (a^2-b^2\right ) \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{b d \left (a^2-b^2\right ) \sqrt{\cos (c+d x)} (a \cos (c+d x)+b)} \]
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Rubi [A] time = 1.1022, antiderivative size = 307, 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} \[ -\frac{F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (A b^2-a (b B-a C)\right )}{a b d \left (a^2-b^2\right )}-\frac{E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^2 C-a b B+A b^2-2 b^2 C\right )}{b^2 d \left (a^2-b^2\right )}+\frac{\left (a^2 b^2 (A+5 C)+a^3 b B-3 a^4 C-3 a b^3 B+A b^4\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a b^2 d (a+b) \left (a^2-b^2\right )}+\frac{\sin (c+d x) \left (3 a^2 C-a b B+A b^2-2 b^2 C\right )}{b^2 d \left (a^2-b^2\right ) \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{b d \left (a^2-b^2\right ) \sqrt{\cos (c+d x)} (a \cos (c+d x)+b)} \]
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))^2} \, dx &=\int \frac{C+B \cos (c+d x)+A \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x))^2} \, dx\\ &=-\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt{\cos (c+d x)} (b+a \cos (c+d x))}-\frac{\int \frac{\frac{1}{2} \left (-A b^2+a b B-3 a^2 C+2 b^2 C\right )+b (b B-a (A+C)) \cos (c+d x)+\frac{1}{2} \left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (b+a \cos (c+d x))} \, dx}{b \left (a^2-b^2\right )}\\ &=\frac{\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d \sqrt{\cos (c+d x)}}-\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt{\cos (c+d x)} (b+a \cos (c+d x))}-\frac{2 \int \frac{\frac{1}{4} \left (-a^2 b B+2 b^3 B+3 a^3 C-a b^2 (A+4 C)\right )+\frac{1}{2} b \left (A b^2-a b B+2 a^2 C-b^2 C\right ) \cos (c+d x)+\frac{1}{4} a \left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{b^2 \left (a^2-b^2\right )}\\ &=\frac{\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d \sqrt{\cos (c+d x)}}-\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt{\cos (c+d x)} (b+a \cos (c+d x))}+\frac{2 \int \frac{\frac{1}{4} a \left (a^2 b B-2 b^3 B-3 a^3 C+a b^2 (A+4 C)\right )-\frac{1}{4} a b \left (A b^2-a (b B-a C)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{a b^2 \left (a^2-b^2\right )}-\frac{\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \int \sqrt{\cos (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )}\\ &=-\frac{\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 \left (a^2-b^2\right ) d}+\frac{\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d \sqrt{\cos (c+d x)}}-\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt{\cos (c+d x)} (b+a \cos (c+d x))}+\frac{\left (A b^4+a^3 b B-3 a b^3 B-3 a^4 C+a^2 b^2 (A+5 C)\right ) \int \frac{1}{\sqrt{\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{2 a b^2 \left (a^2-b^2\right )}-\frac{\left (A b^2-a (b B-a C)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{2 a b \left (a^2-b^2\right )}\\ &=-\frac{\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 \left (a^2-b^2\right ) d}-\frac{\left (A b^2-a (b B-a C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a b \left (a^2-b^2\right ) d}+\frac{\left (A b^4+a^3 b B-3 a b^3 B-3 a^4 C+a^2 b^2 (A+5 C)\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a (a-b) b^2 (a+b)^2 d}+\frac{\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d \sqrt{\cos (c+d x)}}-\frac{\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt{\cos (c+d x)} (b+a \cos (c+d x))}\\ \end{align*}
Mathematica [A] time = 5.02124, size = 340, normalized size = 1.11 \[ \frac{4 \sqrt{\cos (c+d x)} \left (\frac{a \sin (c+d x) \left (a (a C-b B)+A b^2\right )}{\left (a^2-b^2\right ) (a \cos (c+d x)+b)}+2 C \tan (c+d x)\right )-\frac{\frac{4 b \left (2 a^2 C-a b B+A b^2-b^2 C\right ) \left (2 \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )-\frac{2 b \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a+b}\right )}{a}+\frac{2 \sin (c+d x) \left (3 a^2 C-a b B+A b^2-2 b^2 C\right ) \left (2 b (a+b) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right ),-1\right )-\left (a^2-2 b^2\right ) \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )-2 a b E\left (\left .\sin ^{-1}\left (\sqrt{\cos (c+d x)}\right )\right |-1\right )\right )}{a b \sqrt{\sin ^2(c+d x)}}+\frac{2 \left (-3 a^2 b B+9 a^3 C-a b^2 (A+10 C)+4 b^3 B\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a+b}}{(a-b) (a+b)}}{4 b^2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 9.099, size = 897, normalized size = 2.9 \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 )}^{2} \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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