Optimal. Leaf size=142 \[ -\frac {2 C (b \cos (c+d x))^n \sin (c+d x)}{d (3-2 n) \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 (A (3-2 n)+C (5-2 n)) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (-5+2 n);\frac {1}{4} (-1+2 n);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (3-2 n) (5-2 n) \cos ^{\frac {5}{2}}(c+d x) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.08, antiderivative size = 132, normalized size of antiderivative = 0.93, number of steps
used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {20, 3093, 2722}
\begin {gather*} \frac {2 \left (\frac {A}{5-2 n}+\frac {C}{3-2 n}\right ) \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2 n-5);\frac {1}{4} (2 n-1);\cos ^2(c+d x)\right )}{d \sqrt {\sin ^2(c+d x)} \cos ^{\frac {5}{2}}(c+d x)}-\frac {2 C \sin (c+d x) (b \cos (c+d x))^n}{d (3-2 n) \cos ^{\frac {5}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 20
Rule 2722
Rule 3093
Rubi steps
\begin {align*} \int \frac {(b \cos (c+d x))^n \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx &=\left (\cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{-\frac {7}{2}+n}(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx\\ &=-\frac {2 C (b \cos (c+d x))^n \sin (c+d x)}{d (3-2 n) \cos ^{\frac {5}{2}}(c+d x)}+\frac {\left (\left (C \left (-\frac {5}{2}+n\right )+A \left (-\frac {3}{2}+n\right )\right ) \cos ^{-n}(c+d x) (b \cos (c+d x))^n\right ) \int \cos ^{-\frac {7}{2}+n}(c+d x) \, dx}{-\frac {3}{2}+n}\\ &=-\frac {2 C (b \cos (c+d x))^n \sin (c+d x)}{d (3-2 n) \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 (A (3-2 n)+C (5-2 n)) (b \cos (c+d x))^n \, _2F_1\left (\frac {1}{2},\frac {1}{4} (-5+2 n);\frac {1}{4} (-1+2 n);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (3-2 n) (5-2 n) \cos ^{\frac {5}{2}}(c+d x) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 140, normalized size = 0.99 \begin {gather*} -\frac {2 (b \cos (c+d x))^n \csc (c+d x) \left (A (-1+2 n) \, _2F_1\left (\frac {1}{2},\frac {1}{4} (-5+2 n);\frac {1}{4} (-1+2 n);\cos ^2(c+d x)\right )+C (-5+2 n) \cos ^2(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{4} (-1+2 n);\frac {1}{4} (3+2 n);\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (-5+2 n) (-1+2 n) \cos ^{\frac {5}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.21, size = 0, normalized size = 0.00 \[\int \frac {\left (b \cos \left (d x +c \right )\right )^{n} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right )}{\cos \left (d x +c \right )^{\frac {7}{2}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^n}{{\cos \left (c+d\,x\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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