Optimal. Leaf size=78 \[ \frac {2 b^2 (A+3 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \sec (c+d x)}}{3 d}+\frac {2 A b^2 \tan (c+d x) \sqrt {b \sec (c+d x)}}{3 d} \]
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Rubi [A] time = 0.10, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3238, 4046, 3771, 2641} \[ \frac {2 b^2 (A+3 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \sec (c+d x)}}{3 d}+\frac {2 A b^2 \tan (c+d x) \sqrt {b \sec (c+d x)}}{3 d} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3238
Rule 3771
Rule 4046
Rubi steps
\begin {align*} \int \left (A+C \cos ^2(c+d x)\right ) (b \sec (c+d x))^{5/2} \, dx &=b^2 \int \sqrt {b \sec (c+d x)} \left (C+A \sec ^2(c+d x)\right ) \, dx\\ &=\frac {2 A b^2 \sqrt {b \sec (c+d x)} \tan (c+d x)}{3 d}+\frac {1}{3} \left (b^2 (A+3 C)\right ) \int \sqrt {b \sec (c+d x)} \, dx\\ &=\frac {2 A b^2 \sqrt {b \sec (c+d x)} \tan (c+d x)}{3 d}+\frac {1}{3} \left (b^2 (A+3 C) \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 b^2 (A+3 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \sec (c+d x)}}{3 d}+\frac {2 A b^2 \sqrt {b \sec (c+d x)} \tan (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 58, normalized size = 0.74 \[ \frac {2 b^2 \sqrt {b \sec (c+d x)} \left ((A+3 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+A \tan (c+d x)\right )}{3 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b^{2} \cos \left (d x + c\right )^{2} + A b^{2}\right )} \sqrt {b \sec \left (d x + c\right )} \sec \left (d x + c\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.32, size = 199, normalized size = 2.55 \[ -\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (i A \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+3 i C \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )-A \cos \left (d x +c \right )+A \right ) \cos \left (d x +c \right ) \left (1+\cos \left (d x +c \right )\right )^{2} \left (\frac {b}{\cos \left (d x +c \right )}\right )^{\frac {5}{2}}}{3 d \sin \left (d x +c \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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