Optimal. Leaf size=189 \[ \frac {(C (2+n)+A (3+n)) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-1-n);\frac {1-n}{2};\cos ^2(c+d x)\right ) (b \sec (c+d x))^{1+n} \sin (c+d x)}{b d (1+n) (3+n) \sqrt {\sin ^2(c+d x)}}+\frac {B \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-2-n);-\frac {n}{2};\cos ^2(c+d x)\right ) (b \sec (c+d x))^{2+n} \sin (c+d x)}{b^2 d (2+n) \sqrt {\sin ^2(c+d x)}}+\frac {C (b \sec (c+d x))^{2+n} \tan (c+d x)}{b^2 d (3+n)} \]
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Rubi [A]
time = 0.14, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {16, 4132, 3857,
2722, 4131} \begin {gather*} \frac {(A (n+3)+C (n+2)) \sin (c+d x) (b \sec (c+d x))^{n+1} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-n-1);\frac {1-n}{2};\cos ^2(c+d x)\right )}{b d (n+1) (n+3) \sqrt {\sin ^2(c+d x)}}+\frac {B \sin (c+d x) (b \sec (c+d x))^{n+2} \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-n-2);-\frac {n}{2};\cos ^2(c+d x)\right )}{b^2 d (n+2) \sqrt {\sin ^2(c+d x)}}+\frac {C \tan (c+d x) (b \sec (c+d x))^{n+2}}{b^2 d (n+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 2722
Rule 3857
Rule 4131
Rule 4132
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (b \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {\int (b \sec (c+d x))^{2+n} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx}{b^2}\\ &=\frac {\int (b \sec (c+d x))^{2+n} \left (A+C \sec ^2(c+d x)\right ) \, dx}{b^2}+\frac {B \int (b \sec (c+d x))^{3+n} \, dx}{b^3}\\ &=\frac {C (b \sec (c+d x))^{2+n} \tan (c+d x)}{b^2 d (3+n)}+\frac {\left (A+\frac {C (2+n)}{3+n}\right ) \int (b \sec (c+d x))^{2+n} \, dx}{b^2}+\frac {\left (B \left (\frac {\cos (c+d x)}{b}\right )^n (b \sec (c+d x))^n\right ) \int \left (\frac {\cos (c+d x)}{b}\right )^{-3-n} \, dx}{b^3}\\ &=\frac {C (b \sec (c+d x))^{2+n} \tan (c+d x)}{b^2 d (3+n)}+\frac {B \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-2-n);-\frac {n}{2};\cos ^2(c+d x)\right ) \sec (c+d x) (b \sec (c+d x))^n \tan (c+d x)}{d (2+n) \sqrt {\sin ^2(c+d x)}}+\frac {\left (\left (A+\frac {C (2+n)}{3+n}\right ) \left (\frac {\cos (c+d x)}{b}\right )^n (b \sec (c+d x))^n\right ) \int \left (\frac {\cos (c+d x)}{b}\right )^{-2-n} \, dx}{b^2}\\ &=\frac {\left (A+\frac {C (2+n)}{3+n}\right ) \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-1-n);\frac {1-n}{2};\cos ^2(c+d x)\right ) (b \sec (c+d x))^{1+n} \sin (c+d x)}{b d (1+n) \sqrt {\sin ^2(c+d x)}}+\frac {C (b \sec (c+d x))^{2+n} \tan (c+d x)}{b^2 d (3+n)}+\frac {B \, _2F_1\left (\frac {1}{2},\frac {1}{2} (-2-n);-\frac {n}{2};\cos ^2(c+d x)\right ) \sec (c+d x) (b \sec (c+d x))^n \tan (c+d x)}{d (2+n) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.88, size = 296, normalized size = 1.57 \begin {gather*} -\frac {i 2^{3+n} e^{2 i (c+d x)} \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^n \left (\frac {2 B e^{i (c+d x)} \left (1+e^{2 i (c+d x)}\right ) \, _2F_1\left (1,\frac {1}{2} (-1-n);\frac {5+n}{2};-e^{2 i (c+d x)}\right )}{3+n}+\frac {4 C e^{2 i (c+d x)} \, _2F_1\left (1,-1-\frac {n}{2};\frac {6+n}{2};-e^{2 i (c+d x)}\right )}{4+n}+\frac {A \left (1+e^{2 i (c+d x)}\right )^2 \, _2F_1\left (1,-\frac {n}{2};\frac {4+n}{2};-e^{2 i (c+d x)}\right )}{2+n}\right ) \sec ^{-2-n}(c+d x) (b \sec (c+d x))^n \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{d \left (1+e^{2 i (c+d x)}\right )^3 (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x)))} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.26, size = 0, normalized size = 0.00 \[\int \left (\sec ^{2}\left (d x +c \right )\right ) \left (b \sec \left (d x +c \right )\right )^{n} \left (A +B \sec \left (d x +c \right )+C \left (\sec ^{2}\left (d x +c \right )\right )\right )\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b \sec {\left (c + d x \right )}\right )^{n} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}\, dx \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 (\frac {b}{\cos \left (c+d\,x\right )}\right )}^n\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\cos \left (c+d\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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