Optimal. Leaf size=139 \[ \frac {b^2 B (b \cos (c+d x))^{-2+n} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-2+n);\frac {n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d (2-n) \sqrt {\sin ^2(c+d x)}}+\frac {b C (b \cos (c+d x))^{-1+n} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-1+n);\frac {1+n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1-n) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.12, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {16, 3089, 2827,
2722} \begin {gather*} \frac {b^2 B \sin (c+d x) (b \cos (c+d x))^{n-2} \, _2F_1\left (\frac {1}{2},\frac {n-2}{2};\frac {n}{2};\cos ^2(c+d x)\right )}{d (2-n) \sqrt {\sin ^2(c+d x)}}+\frac {b C \sin (c+d x) (b \cos (c+d x))^{n-1} \, _2F_1\left (\frac {1}{2},\frac {n-1}{2};\frac {n+1}{2};\cos ^2(c+d x)\right )}{d (1-n) \sqrt {\sin ^2(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 2722
Rule 2827
Rule 3089
Rubi steps
\begin {align*} \int (b \cos (c+d x))^n \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx &=b^4 \int (b \cos (c+d x))^{-4+n} \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=b^3 \int (b \cos (c+d x))^{-3+n} (B+C \cos (c+d x)) \, dx\\ &=\left (b^3 B\right ) \int (b \cos (c+d x))^{-3+n} \, dx+\left (b^2 C\right ) \int (b \cos (c+d x))^{-2+n} \, dx\\ &=\frac {b^2 B (b \cos (c+d x))^{-2+n} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-2+n);\frac {n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d (2-n) \sqrt {\sin ^2(c+d x)}}+\frac {b C (b \cos (c+d x))^{-1+n} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-1+n);\frac {1+n}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1-n) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 118, normalized size = 0.85 \begin {gather*} -\frac {(b \cos (c+d x))^n \csc (c+d x) \left (B (-1+n) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-2+n);\frac {n}{2};\cos ^2(c+d x)\right )+C (-2+n) \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-1+n);\frac {1+n}{2};\cos ^2(c+d x)\right )\right ) \sec ^2(c+d x) \sqrt {\sin ^2(c+d x)}}{d (-2+n) (-1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.26, size = 0, normalized size = 0.00 \[\int \left (b \cos \left (d x +c \right )\right )^{n} \left (B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right ) \left (\sec ^{4}\left (d x +c \right )\right )\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 (b\,\cos \left (c+d\,x\right )\right )}^n\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right )}{{\cos \left (c+d\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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