Optimal. Leaf size=342 \[ \frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \left (7 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)-7 a^3 b B-21 a b^3 B+21 A b^4\right )}{21 a^5 d}+\frac{2 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{21 a^3 d \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^2 b (3 A+5 C)-3 a^3 B-5 a b^2 B+5 A b^3\right )}{5 a^4 d}-\frac{2 b^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (A b^2-a (b B-a C)\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^5 d (a+b)}-\frac{2 (A b-a B) \sin (c+d x)}{5 a^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)} \]
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Rubi [A] time = 1.22208, antiderivative size = 342, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.186, Rules used = {4104, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ \frac{2 \sin (c+d x) \left (a^2 (5 A+7 C)-7 a b B+7 A b^2\right )}{21 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (7 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)-7 a^3 b B-21 a b^3 B+21 A b^4\right )}{21 a^5 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (a^2 b (3 A+5 C)-3 a^3 B-5 a b^2 B+5 A b^3\right )}{5 a^4 d}-\frac{2 b^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (A b^2-a (b B-a C)\right ) \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^5 d (a+b)}-\frac{2 (A b-a B) \sin (c+d x)}{5 a^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4104
Rule 4106
Rule 3849
Rule 2805
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))} \, dx &=\frac{2 A \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{2 \int \frac{\frac{7}{2} (A b-a B)-\frac{1}{2} a (5 A+7 C) \sec (c+d x)-\frac{5}{2} A b \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))} \, dx}{7 a}\\ &=\frac{2 A \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{2 (A b-a B) \sin (c+d x)}{5 a^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 \int \frac{\frac{5}{4} \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right )+\frac{1}{4} a (4 A b+21 a B) \sec (c+d x)-\frac{21}{4} b (A b-a B) \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx}{35 a^2}\\ &=\frac{2 A \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{2 (A b-a B) \sin (c+d x)}{5 a^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \sqrt{\sec (c+d x)}}-\frac{8 \int \frac{\frac{21}{8} \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right )+\frac{1}{8} a \left (28 A b^2-28 a b B-5 a^2 (5 A+7 C)\right ) \sec (c+d x)-\frac{5}{8} b \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{105 a^3}\\ &=\frac{2 A \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{2 (A b-a B) \sin (c+d x)}{5 a^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \sqrt{\sec (c+d x)}}-\frac{8 \int \frac{\frac{21}{8} a \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right )-\left (\frac{21}{8} b \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right )-\frac{1}{8} a^2 \left (28 A b^2-28 a b B-5 a^2 (5 A+7 C)\right )\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{105 a^5}-\frac{\left (b^3 \left (A b^2-a (b B-a C)\right )\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{a^5}\\ &=\frac{2 A \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{2 (A b-a B) \sin (c+d x)}{5 a^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \sqrt{\sec (c+d x)}}-\frac{\left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{5 a^4}+\frac{\left (21 A b^4-7 a^3 b B-21 a b^3 B+7 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \int \sqrt{\sec (c+d x)} \, dx}{21 a^5}-\frac{\left (b^3 \left (A b^2-a (b B-a C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{a^5}\\ &=-\frac{2 b^3 \left (A b^2-a (b B-a C)\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{a^5 (a+b) d}+\frac{2 A \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{2 (A b-a B) \sin (c+d x)}{5 a^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \sqrt{\sec (c+d x)}}-\frac{\left (\left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 a^4}+\frac{\left (\left (21 A b^4-7 a^3 b B-21 a b^3 B+7 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 a^5}\\ &=-\frac{2 \left (5 A b^3-3 a^3 B-5 a b^2 B+a^2 b (3 A+5 C)\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{5 a^4 d}+\frac{2 \left (21 A b^4-7 a^3 b B-21 a b^3 B+7 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{21 a^5 d}-\frac{2 b^3 \left (A b^2-a (b B-a C)\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{a^5 (a+b) d}+\frac{2 A \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}-\frac{2 (A b-a B) \sin (c+d x)}{5 a^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (7 A b^2-7 a b B+a^2 (5 A+7 C)\right ) \sin (c+d x)}{21 a^3 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [F] time = 68.7677, size = 0, normalized size = 0. \[ \int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))} \, dx \]
Verification is Not applicable to the result.
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Maple [B] time = 7.123, size = 1095, normalized size = 3.2 \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 )} \sec \left (d x + c\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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