Optimal. Leaf size=147 \[ \frac {2 C \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \sec (c+d x)}}{3 b^4 d}+\frac {6 B E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^3 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 C \sin (c+d x)}{3 b^3 d \sqrt {b \sec (c+d x)}}+\frac {2 B \sin (c+d x)}{5 b^2 d (b \sec (c+d x))^{3/2}} \]
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Rubi [A] time = 0.13, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {4047, 3769, 3771, 2639, 12, 16, 2641} \[ \frac {2 B \sin (c+d x)}{5 b^2 d (b \sec (c+d x))^{3/2}}+\frac {6 B E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^3 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 C \sin (c+d x)}{3 b^3 d \sqrt {b \sec (c+d x)}}+\frac {2 C \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \sec (c+d x)}}{3 b^4 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 16
Rule 2639
Rule 2641
Rule 3769
Rule 3771
Rule 4047
Rubi steps
\begin {align*} \int \frac {B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{7/2}} \, dx &=\frac {B \int \frac {1}{(b \sec (c+d x))^{5/2}} \, dx}{b}+\int \frac {C \sec ^2(c+d x)}{(b \sec (c+d x))^{7/2}} \, dx\\ &=\frac {2 B \sin (c+d x)}{5 b^2 d (b \sec (c+d x))^{3/2}}+\frac {(3 B) \int \frac {1}{\sqrt {b \sec (c+d x)}} \, dx}{5 b^3}+C \int \frac {\sec ^2(c+d x)}{(b \sec (c+d x))^{7/2}} \, dx\\ &=\frac {2 B \sin (c+d x)}{5 b^2 d (b \sec (c+d x))^{3/2}}+\frac {C \int \frac {1}{(b \sec (c+d x))^{3/2}} \, dx}{b^2}+\frac {(3 B) \int \sqrt {\cos (c+d x)} \, dx}{5 b^3 \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}\\ &=\frac {6 B E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^3 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 B \sin (c+d x)}{5 b^2 d (b \sec (c+d x))^{3/2}}+\frac {2 C \sin (c+d x)}{3 b^3 d \sqrt {b \sec (c+d x)}}+\frac {C \int \sqrt {b \sec (c+d x)} \, dx}{3 b^4}\\ &=\frac {6 B E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^3 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 B \sin (c+d x)}{5 b^2 d (b \sec (c+d x))^{3/2}}+\frac {2 C \sin (c+d x)}{3 b^3 d \sqrt {b \sec (c+d x)}}+\frac {\left (C \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 b^4}\\ &=\frac {6 B E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^3 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 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 b^4 d}+\frac {2 B \sin (c+d x)}{5 b^2 d (b \sec (c+d x))^{3/2}}+\frac {2 C \sin (c+d x)}{3 b^3 d \sqrt {b \sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.58, size = 91, normalized size = 0.62 \[ \frac {2 \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)} \left (\sin (c+d x) \sqrt {\cos (c+d x)} (3 B \cos (c+d x)+5 C)+9 B E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+5 C F\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{15 b^4 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \sec \left (d x + c\right ) + B\right )} \sqrt {b \sec \left (d x + c\right )}}{b^{4} \sec \left (d x + c\right )^{3}}, x\right ) \]
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 \[ \int \frac {C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )}{\left (b \sec \left (d x + c\right )\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.16, size = 482, normalized size = 3.28 \[ -\frac {2 \left (9 i B \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 )}}\, \sin \left (d x +c \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \cos \left (d x +c \right )-9 i B \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 ) \sin \left (d x +c \right )-5 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 )+9 i B \sin \left (d x +c \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\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 )}}-9 i B \sin \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 ) \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 )}}-5 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 B \left (\cos ^{4}\left (d x +c \right )\right )+5 C \left (\cos ^{3}\left (d x +c \right )\right )+6 B \left (\cos ^{2}\left (d x +c \right )\right )-9 B \cos \left (d x +c \right )-5 C \cos \left (d x +c \right )\right )}{15 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 )} \]
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 \[ \int \frac {C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )}{\left (b \sec \left (d x + c\right )\right )^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{7/2}} \,d x \]
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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