Optimal. Leaf size=112 \[ \frac {2 (5 A+7 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \sec (c+d x)}}{21 b^4 d}+\frac {2 (5 A+7 C) \sin (c+d x)}{21 b^3 d \sqrt {b \sec (c+d x)}}+\frac {2 A \tan (c+d x)}{7 d (b \sec (c+d x))^{7/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 112, 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 = {4045, 3769, 3771, 2641} \[ \frac {2 (5 A+7 C) \sin (c+d x)}{21 b^3 d \sqrt {b \sec (c+d x)}}+\frac {2 (5 A+7 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \sec (c+d x)}}{21 b^4 d}+\frac {2 A \tan (c+d x)}{7 d (b \sec (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3769
Rule 3771
Rule 4045
Rubi steps
\begin {align*} \int \frac {A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{7/2}} \, dx &=\frac {2 A \tan (c+d x)}{7 d (b \sec (c+d x))^{7/2}}+\frac {(5 A+7 C) \int \frac {1}{(b \sec (c+d x))^{3/2}} \, dx}{7 b^2}\\ &=\frac {2 (5 A+7 C) \sin (c+d x)}{21 b^3 d \sqrt {b \sec (c+d x)}}+\frac {2 A \tan (c+d x)}{7 d (b \sec (c+d x))^{7/2}}+\frac {(5 A+7 C) \int \sqrt {b \sec (c+d x)} \, dx}{21 b^4}\\ &=\frac {2 (5 A+7 C) \sin (c+d x)}{21 b^3 d \sqrt {b \sec (c+d x)}}+\frac {2 A \tan (c+d x)}{7 d (b \sec (c+d x))^{7/2}}+\frac {\left ((5 A+7 C) \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 b^4}\\ &=\frac {2 (5 A+7 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \sec (c+d x)}}{21 b^4 d}+\frac {2 (5 A+7 C) \sin (c+d x)}{21 b^3 d \sqrt {b \sec (c+d x)}}+\frac {2 A \tan (c+d x)}{7 d (b \sec (c+d x))^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.75, size = 79, normalized size = 0.71 \[ \frac {2 \sin (c+d x) (3 A \cos (2 (c+d x))+13 A+14 C)+\frac {4 (5 A+7 C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\sqrt {\cos (c+d x)}}}{42 b^3 d \sqrt {b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \sqrt {b \sec \left (d x + c\right )}}{b^{4} \sec \left (d x + c\right )^{4}}, x\right ) \]
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 \[ \int \frac {C \sec \left (d x + c\right )^{2} + A}{\left (b \sec \left (d x + c\right )\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.76, size = 241, normalized size = 2.15 \[ -\frac {2 \left (1+\cos \left (d x +c \right )\right )^{2} \left (-1+\cos \left (d x +c \right )\right ) \left (5 i A \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+7 i C \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )-3 A \left (\cos ^{4}\left (d x +c \right )\right )+3 A \left (\cos ^{3}\left (d x +c \right )\right )-5 A \left (\cos ^{2}\left (d x +c \right )\right )-7 C \left (\cos ^{2}\left (d x +c \right )\right )+5 A \cos \left (d x +c \right )+7 C \cos \left (d x +c \right )\right )}{21 d \cos \left (d x +c \right )^{4} \left (\frac {b}{\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \sin \left (d x +c \right )^{3}} \]
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 \[ \int \frac {C \sec \left (d x + c\right )^{2} + A}{\left (b \sec \left (d x + c\right )\right )^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {A+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{7/2}} \,d x \]
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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