Optimal. Leaf size=544 \[ -\frac{\sin (c+d x) \left (6 a^2 A b^2 m (m+3)+a^4 A \left (m^2+4 m+3\right )+4 a^3 b B m (m+3)+4 a b^3 B m (m+2)+A b^4 m (m+2)\right ) \sec ^{m-1}(c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1-m}{2},\frac{3-m}{2},\cos ^2(c+d x)\right )}{d (1-m) (m+1) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{\sin (c+d x) \left (4 a^3 A b \left (m^2+6 m+8\right )+6 a^2 b^2 B \left (m^2+5 m+4\right )+a^4 B \left (m^2+6 m+8\right )+4 a A b^3 \left (m^2+5 m+4\right )+b^4 B \left (m^2+4 m+3\right )\right ) \sec ^m(c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{m}{2},\frac{2-m}{2},\cos ^2(c+d x)\right )}{d m (m+2) (m+4) \sqrt{\sin ^2(c+d x)}}+\frac{b \sin (c+d x) \left (a^2 A b \left (5 m^2+37 m+68\right )+2 a^3 B \left (m^2+8 m+19\right )+4 a b^2 B \left (m^2+6 m+8\right )+A b^3 \left (m^2+6 m+8\right )\right ) \sec ^{m+1}(c+d x)}{d (m+1) (m+3) (m+4)}+\frac{b^2 \sin (c+d x) \left (a^2 B \left (m^2+9 m+26\right )+2 a A b (m+4)^2+b^2 B (m+3)^2\right ) \sec ^{m+2}(c+d x)}{d (m+2) (m+3) (m+4)}+\frac{b \sin (c+d x) (a B (m+7)+A b (m+4)) \sec ^{m+1}(c+d x) (a+b \sec (c+d x))^2}{d (m+3) (m+4)}+\frac{b B \sin (c+d x) \sec ^{m+1}(c+d x) (a+b \sec (c+d x))^3}{d (m+4)} \]
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Rubi [A] time = 1.63038, antiderivative size = 544, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {4026, 4096, 4076, 4047, 3772, 2643, 4046} \[ -\frac{\sin (c+d x) \left (6 a^2 A b^2 m (m+3)+a^4 A \left (m^2+4 m+3\right )+4 a^3 b B m (m+3)+4 a b^3 B m (m+2)+A b^4 m (m+2)\right ) \sec ^{m-1}(c+d x) \, _2F_1\left (\frac{1}{2},\frac{1-m}{2};\frac{3-m}{2};\cos ^2(c+d x)\right )}{d (1-m) (m+1) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{\sin (c+d x) \left (4 a^3 A b \left (m^2+6 m+8\right )+6 a^2 b^2 B \left (m^2+5 m+4\right )+a^4 B \left (m^2+6 m+8\right )+4 a A b^3 \left (m^2+5 m+4\right )+b^4 B \left (m^2+4 m+3\right )\right ) \sec ^m(c+d x) \, _2F_1\left (\frac{1}{2},-\frac{m}{2};\frac{2-m}{2};\cos ^2(c+d x)\right )}{d m (m+2) (m+4) \sqrt{\sin ^2(c+d x)}}+\frac{b \sin (c+d x) \left (a^2 A b \left (5 m^2+37 m+68\right )+2 a^3 B \left (m^2+8 m+19\right )+4 a b^2 B \left (m^2+6 m+8\right )+A b^3 \left (m^2+6 m+8\right )\right ) \sec ^{m+1}(c+d x)}{d (m+1) (m+3) (m+4)}+\frac{b^2 \sin (c+d x) \left (a^2 B \left (m^2+9 m+26\right )+2 a A b (m+4)^2+b^2 B (m+3)^2\right ) \sec ^{m+2}(c+d x)}{d (m+2) (m+3) (m+4)}+\frac{b \sin (c+d x) (a B (m+7)+A b (m+4)) \sec ^{m+1}(c+d x) (a+b \sec (c+d x))^2}{d (m+3) (m+4)}+\frac{b B \sin (c+d x) \sec ^{m+1}(c+d x) (a+b \sec (c+d x))^3}{d (m+4)} \]
Antiderivative was successfully verified.
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Rule 4026
Rule 4096
Rule 4076
Rule 4047
Rule 3772
Rule 2643
Rule 4046
Rubi steps
\begin{align*} \int \sec ^m(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx &=\frac{b B \sec ^{1+m}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{d (4+m)}+\frac{\int \sec ^m(c+d x) (a+b \sec (c+d x))^2 \left (a (b B m+a A (4+m))+\left (b^2 B (3+m)+a (2 A b+a B) (4+m)\right ) \sec (c+d x)+b (A b (4+m)+a B (7+m)) \sec ^2(c+d x)\right ) \, dx}{4+m}\\ &=\frac{b (A b (4+m)+a B (7+m)) \sec ^{1+m}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{d (3+m) (4+m)}+\frac{b B \sec ^{1+m}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{d (4+m)}+\frac{\int \sec ^m(c+d x) (a+b \sec (c+d x)) \left (a \left (A b^2 m (4+m)+2 a b B m (5+m)+a^2 A \left (12+7 m+m^2\right )\right )+\left (b^2 (2+m) (A b (4+m)+a B (7+m))+a (3+m) \left (3 a A b (4+m)+a^2 B (4+m)+b^2 B (3+2 m)\right )\right ) \sec (c+d x)+b \left (b^2 B (3+m)^2+2 a A b (4+m)^2+a^2 B \left (26+9 m+m^2\right )\right ) \sec ^2(c+d x)\right ) \, dx}{12+7 m+m^2}\\ &=\frac{b^2 \left (b^2 B (3+m)^2+2 a A b (4+m)^2+a^2 B \left (26+9 m+m^2\right )\right ) \sec ^{2+m}(c+d x) \sin (c+d x)}{d (2+m) \left (12+7 m+m^2\right )}+\frac{b (A b (4+m)+a B (7+m)) \sec ^{1+m}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{d (3+m) (4+m)}+\frac{b B \sec ^{1+m}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{d (4+m)}+\frac{\int \sec ^m(c+d x) \left (a^2 (2+m) \left (A b^2 m (4+m)+2 a b B m (5+m)+a^2 A \left (12+7 m+m^2\right )\right )+(3+m) \left (b^4 B \left (3+4 m+m^2\right )+4 a A b^3 \left (4+5 m+m^2\right )+6 a^2 b^2 B \left (4+5 m+m^2\right )+4 a^3 A b \left (8+6 m+m^2\right )+a^4 B \left (8+6 m+m^2\right )\right ) \sec (c+d x)+b (2+m) \left (A b^3 \left (8+6 m+m^2\right )+4 a b^2 B \left (8+6 m+m^2\right )+2 a^3 B \left (19+8 m+m^2\right )+a^2 A b \left (68+37 m+5 m^2\right )\right ) \sec ^2(c+d x)\right ) \, dx}{24+26 m+9 m^2+m^3}\\ &=\frac{b^2 \left (b^2 B (3+m)^2+2 a A b (4+m)^2+a^2 B \left (26+9 m+m^2\right )\right ) \sec ^{2+m}(c+d x) \sin (c+d x)}{d (2+m) \left (12+7 m+m^2\right )}+\frac{b (A b (4+m)+a B (7+m)) \sec ^{1+m}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{d (3+m) (4+m)}+\frac{b B \sec ^{1+m}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{d (4+m)}+\frac{\int \sec ^m(c+d x) \left (a^2 (2+m) \left (A b^2 m (4+m)+2 a b B m (5+m)+a^2 A \left (12+7 m+m^2\right )\right )+b (2+m) \left (A b^3 \left (8+6 m+m^2\right )+4 a b^2 B \left (8+6 m+m^2\right )+2 a^3 B \left (19+8 m+m^2\right )+a^2 A b \left (68+37 m+5 m^2\right )\right ) \sec ^2(c+d x)\right ) \, dx}{24+26 m+9 m^2+m^3}+\frac{\left (b^4 B \left (3+4 m+m^2\right )+4 a A b^3 \left (4+5 m+m^2\right )+6 a^2 b^2 B \left (4+5 m+m^2\right )+4 a^3 A b \left (8+6 m+m^2\right )+a^4 B \left (8+6 m+m^2\right )\right ) \int \sec ^{1+m}(c+d x) \, dx}{8+6 m+m^2}\\ &=\frac{b \left (A b^3 \left (8+6 m+m^2\right )+4 a b^2 B \left (8+6 m+m^2\right )+2 a^3 B \left (19+8 m+m^2\right )+a^2 A b \left (68+37 m+5 m^2\right )\right ) \sec ^{1+m}(c+d x) \sin (c+d x)}{d (1+m) \left (12+7 m+m^2\right )}+\frac{b^2 \left (b^2 B (3+m)^2+2 a A b (4+m)^2+a^2 B \left (26+9 m+m^2\right )\right ) \sec ^{2+m}(c+d x) \sin (c+d x)}{d (2+m) \left (12+7 m+m^2\right )}+\frac{b (A b (4+m)+a B (7+m)) \sec ^{1+m}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{d (3+m) (4+m)}+\frac{b B \sec ^{1+m}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{d (4+m)}+\frac{\left (A b^4 m (2+m)+4 a b^3 B m (2+m)+6 a^2 A b^2 m (3+m)+4 a^3 b B m (3+m)+a^4 A \left (3+4 m+m^2\right )\right ) \int \sec ^m(c+d x) \, dx}{(1+m) (3+m)}+\frac{\left (\left (b^4 B \left (3+4 m+m^2\right )+4 a A b^3 \left (4+5 m+m^2\right )+6 a^2 b^2 B \left (4+5 m+m^2\right )+4 a^3 A b \left (8+6 m+m^2\right )+a^4 B \left (8+6 m+m^2\right )\right ) \cos ^m(c+d x) \sec ^m(c+d x)\right ) \int \cos ^{-1-m}(c+d x) \, dx}{8+6 m+m^2}\\ &=\frac{b \left (A b^3 \left (8+6 m+m^2\right )+4 a b^2 B \left (8+6 m+m^2\right )+2 a^3 B \left (19+8 m+m^2\right )+a^2 A b \left (68+37 m+5 m^2\right )\right ) \sec ^{1+m}(c+d x) \sin (c+d x)}{d (1+m) \left (12+7 m+m^2\right )}+\frac{b^2 \left (b^2 B (3+m)^2+2 a A b (4+m)^2+a^2 B \left (26+9 m+m^2\right )\right ) \sec ^{2+m}(c+d x) \sin (c+d x)}{d (2+m) \left (12+7 m+m^2\right )}+\frac{b (A b (4+m)+a B (7+m)) \sec ^{1+m}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{d (3+m) (4+m)}+\frac{b B \sec ^{1+m}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{d (4+m)}+\frac{\left (b^4 B \left (3+4 m+m^2\right )+4 a A b^3 \left (4+5 m+m^2\right )+6 a^2 b^2 B \left (4+5 m+m^2\right )+4 a^3 A b \left (8+6 m+m^2\right )+a^4 B \left (8+6 m+m^2\right )\right ) \, _2F_1\left (\frac{1}{2},-\frac{m}{2};\frac{2-m}{2};\cos ^2(c+d x)\right ) \sec ^m(c+d x) \sin (c+d x)}{d m \left (8+6 m+m^2\right ) \sqrt{\sin ^2(c+d x)}}+\frac{\left (\left (A b^4 m (2+m)+4 a b^3 B m (2+m)+6 a^2 A b^2 m (3+m)+4 a^3 b B m (3+m)+a^4 A \left (3+4 m+m^2\right )\right ) \cos ^m(c+d x) \sec ^m(c+d x)\right ) \int \cos ^{-m}(c+d x) \, dx}{(1+m) (3+m)}\\ &=\frac{b \left (A b^3 \left (8+6 m+m^2\right )+4 a b^2 B \left (8+6 m+m^2\right )+2 a^3 B \left (19+8 m+m^2\right )+a^2 A b \left (68+37 m+5 m^2\right )\right ) \sec ^{1+m}(c+d x) \sin (c+d x)}{d (1+m) \left (12+7 m+m^2\right )}+\frac{b^2 \left (b^2 B (3+m)^2+2 a A b (4+m)^2+a^2 B \left (26+9 m+m^2\right )\right ) \sec ^{2+m}(c+d x) \sin (c+d x)}{d (2+m) \left (12+7 m+m^2\right )}+\frac{b (A b (4+m)+a B (7+m)) \sec ^{1+m}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{d (3+m) (4+m)}+\frac{b B \sec ^{1+m}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{d (4+m)}-\frac{\left (A b^4 m (2+m)+4 a b^3 B m (2+m)+6 a^2 A b^2 m (3+m)+4 a^3 b B m (3+m)+a^4 A \left (3+4 m+m^2\right )\right ) \, _2F_1\left (\frac{1}{2},\frac{1-m}{2};\frac{3-m}{2};\cos ^2(c+d x)\right ) \sec ^{-1+m}(c+d x) \sin (c+d x)}{d (1-m) (1+m) (3+m) \sqrt{\sin ^2(c+d x)}}+\frac{\left (b^4 B \left (3+4 m+m^2\right )+4 a A b^3 \left (4+5 m+m^2\right )+6 a^2 b^2 B \left (4+5 m+m^2\right )+4 a^3 A b \left (8+6 m+m^2\right )+a^4 B \left (8+6 m+m^2\right )\right ) \, _2F_1\left (\frac{1}{2},-\frac{m}{2};\frac{2-m}{2};\cos ^2(c+d x)\right ) \sec ^m(c+d x) \sin (c+d x)}{d m \left (8+6 m+m^2\right ) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 4.42286, size = 365, normalized size = 0.67 \[ \frac{\sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sec ^{m-1}(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \left (\frac{a^3 (a B+4 A b) \cos ^4(c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+1}{2},\frac{m+3}{2},\sec ^2(c+d x)\right )}{m+1}+b \left (\frac{2 a^2 (2 a B+3 A b) \cos ^3(c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},\sec ^2(c+d x)\right )}{m+2}+b \left (\frac{2 a (3 a B+2 A b) \cos ^2(c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+3}{2},\frac{m+5}{2},\sec ^2(c+d x)\right )}{m+3}+b \left (\frac{(4 a B+A b) \cos (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+4}{2},\frac{m+6}{2},\sec ^2(c+d x)\right )}{m+4}+\frac{b B \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+5}{2},\frac{m+7}{2},\sec ^2(c+d x)\right )}{m+5}\right )\right )\right )+\frac{a^4 A \cos ^5(c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m}{2},\frac{m+2}{2},\sec ^2(c+d x)\right )}{m}\right )}{d (a \cos (c+d x)+b)^4 (A \cos (c+d x)+B)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.804, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( dx+c \right ) \right ) ^{m} \left ( a+b\sec \left ( dx+c \right ) \right ) ^{4} \left ( A+B\sec \left ( dx+c \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B b^{4} \sec \left (d x + c\right )^{5} + A a^{4} +{\left (4 \, B a b^{3} + A b^{4}\right )} \sec \left (d x + c\right )^{4} + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \sec \left (d x + c\right )^{3} + 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} \sec \left (d x + c\right )^{2} +{\left (B a^{4} + 4 \, A a^{3} b\right )} \sec \left (d x + c\right )\right )} \sec \left (d x + c\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (A + B \sec{\left (c + d x \right )}\right ) \left (a + b \sec{\left (c + d x \right )}\right )^{4} \sec ^{m}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \sec \left (d x + c\right ) + A\right )}{\left (b \sec \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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