Optimal. Leaf size=322 \[ -\frac {8 c^2 \left (B \left (21-8 m-4 m^2\right )-C \left (19-8 m+4 m^2\right )-A \left (35+24 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{f (5+2 m) (7+2 m) \left (3+8 m+4 m^2\right ) \sqrt {c-c \sin (e+f x)}}-\frac {2 c \left (B \left (21-8 m-4 m^2\right )-C \left (19-8 m+4 m^2\right )-A \left (35+24 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)}}{f (3+2 m) (5+2 m) (7+2 m)}-\frac {2 (7 B+2 C+2 B m+4 C m) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m) (7+2 m)}+\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{c f (7+2 m)} \]
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Rubi [A]
time = 0.46, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3118, 3052,
2819, 2817} \begin {gather*} -\frac {8 c^2 \left (-A \left (4 m^2+24 m+35\right )+B \left (-4 m^2-8 m+21\right )-C \left (4 m^2-8 m+19\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) \left (4 m^2+8 m+3\right ) \sqrt {c-c \sin (e+f x)}}-\frac {2 c \left (-A \left (4 m^2+24 m+35\right )+B \left (-4 m^2-8 m+21\right )-C \left (4 m^2-8 m+19\right )\right ) \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+3) (2 m+5) (2 m+7)}-\frac {2 (2 B m+7 B+4 C m+2 C) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{c f (2 m+7)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2817
Rule 2819
Rule 3052
Rule 3118
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right ) \, dx &=\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{c f (7+2 m)}-\frac {2 \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \left (-\frac {1}{2} a c (C (5-2 m)+A (7+2 m))-\frac {1}{2} a c (7 B+2 C+2 B m+4 C m) \sin (e+f x)\right ) \, dx}{a c (7+2 m)}\\ &=-\frac {2 (7 B+2 C+2 B m+4 C m) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m) (7+2 m)}+\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{c f (7+2 m)}-\frac {\left (B \left (21-8 m-4 m^2\right )-C \left (19-8 m+4 m^2\right )-A \left (35+24 m+4 m^2\right )\right ) \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \, dx}{(5+2 m) (7+2 m)}\\ &=-\frac {2 c \left (B \left (21-8 m-4 m^2\right )-C \left (19-8 m+4 m^2\right )-A \left (35+24 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)}}{f (3+2 m) (5+2 m) (7+2 m)}-\frac {2 (7 B+2 C+2 B m+4 C m) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m) (7+2 m)}+\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{c f (7+2 m)}-\frac {\left (4 c \left (B \left (21-8 m-4 m^2\right )-C \left (19-8 m+4 m^2\right )-A \left (35+24 m+4 m^2\right )\right )\right ) \int (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)} \, dx}{(3+2 m) (5+2 m) (7+2 m)}\\ &=-\frac {8 c^2 \left (B \left (21-8 m-4 m^2\right )-C \left (19-8 m+4 m^2\right )-A \left (35+24 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{f (1+2 m) (3+2 m) (5+2 m) (7+2 m) \sqrt {c-c \sin (e+f x)}}-\frac {2 c \left (B \left (21-8 m-4 m^2\right )-C \left (19-8 m+4 m^2\right )-A \left (35+24 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)}}{f (3+2 m) (5+2 m) (7+2 m)}-\frac {2 (7 B+2 C+2 B m+4 C m) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m) (7+2 m)}+\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{c f (7+2 m)}\\ \end {align*}
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Mathematica [A]
time = 3.43, size = 306, normalized size = 0.95 \begin {gather*} \frac {c \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a (1+\sin (e+f x)))^m \sqrt {c-c \sin (e+f x)} \left (700 A-546 B+494 C+760 A m-380 B m+284 C m+272 A m^2-120 B m^2+136 C m^2+32 A m^3-16 B m^3+16 C m^3+2 \left (3+8 m+4 m^2\right ) (B (7+2 m)-C (13+2 m)) \cos (2 (e+f x))-(1+2 m) \left (4 A \left (35+24 m+4 m^2\right )-4 B \left (63+32 m+4 m^2\right )+C \left (253+80 m+12 m^2\right )\right ) \sin (e+f x)+15 C \sin (3 (e+f x))+46 C m \sin (3 (e+f x))+36 C m^2 \sin (3 (e+f x))+8 C m^3 \sin (3 (e+f x))\right )}{2 f (1+2 m) (3+2 m) (5+2 m) (7+2 m) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.74, size = 0, normalized size = 0.00 \[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{\frac {3}{2}} \left (A +B \sin \left (f x +e \right )+C \left (\sin ^{2}\left (f x +e \right )\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1004 vs.
\(2 (314) = 628\).
time = 0.60, size = 1004, normalized size = 3.12 \begin {gather*} -\frac {2 \, {\left (\frac {{\left (a^{m} c^{\frac {3}{2}} {\left (2 \, m + 5\right )} - \frac {a^{m} c^{\frac {3}{2}} {\left (2 \, m - 3\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {a^{m} c^{\frac {3}{2}} {\left (2 \, m - 3\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{m} c^{\frac {3}{2}} {\left (2 \, m + 5\right )} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )} A e^{\left (2 \, m \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right ) - m \log \left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )\right )}}{{\left (4 \, m^{2} + 8 \, m + 3\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (a^{m} c^{\frac {3}{2}} {\left (2 \, m + 9\right )} - \frac {2 \, {\left (2 \, m^{2} + 9 \, m\right )} a^{m} c^{\frac {3}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {{\left (4 \, m^{2} + 15\right )} a^{m} c^{\frac {3}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {{\left (4 \, m^{2} + 15\right )} a^{m} c^{\frac {3}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {2 \, {\left (2 \, m^{2} + 9 \, m\right )} a^{m} c^{\frac {3}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{m} c^{\frac {3}{2}} {\left (2 \, m + 9\right )} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )} B e^{\left (2 \, m \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right ) - m \log \left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )\right )}}{{\left (8 \, m^{3} + 36 \, m^{2} + 46 \, m + \frac {{\left (8 \, m^{3} + 36 \, m^{2} + 46 \, m + 15\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 15\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {3}{2}}} + \frac {4 \, {\left (2 \, a^{m} c^{\frac {3}{2}} {\left (2 \, m + 13\right )} - \frac {4 \, {\left (2 \, m^{2} + 13 \, m\right )} a^{m} c^{\frac {3}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {{\left (8 \, m^{3} + 60 \, m^{2} + 66 \, m + 91\right )} a^{m} c^{\frac {3}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {{\left (8 \, m^{3} + 20 \, m^{2} + 82 \, m - 35\right )} a^{m} c^{\frac {3}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {{\left (8 \, m^{3} + 20 \, m^{2} + 82 \, m - 35\right )} a^{m} c^{\frac {3}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {{\left (8 \, m^{3} + 60 \, m^{2} + 66 \, m + 91\right )} a^{m} c^{\frac {3}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {4 \, {\left (2 \, m^{2} + 13 \, m\right )} a^{m} c^{\frac {3}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {2 \, a^{m} c^{\frac {3}{2}} {\left (2 \, m + 13\right )} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}\right )} C e^{\left (2 \, m \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right ) - m \log \left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )\right )}}{{\left (16 \, m^{4} + 128 \, m^{3} + 344 \, m^{2} + 352 \, m + \frac {2 \, {\left (16 \, m^{4} + 128 \, m^{3} + 344 \, m^{2} + 352 \, m + 105\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {{\left (16 \, m^{4} + 128 \, m^{3} + 344 \, m^{2} + 352 \, m + 105\right )} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 105\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {3}{2}}}\right )}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 575, normalized size = 1.79 \begin {gather*} -\frac {2 \, {\left ({\left (8 \, C c m^{3} + 36 \, C c m^{2} + 46 \, C c m + 15 \, C c\right )} \cos \left (f x + e\right )^{4} - 16 \, {\left (A + B + C\right )} c m^{2} - {\left (8 \, {\left (B - C\right )} c m^{3} + 4 \, {\left (11 \, B - 17 \, C\right )} c m^{2} + 2 \, {\left (31 \, B - 55 \, C\right )} c m + 3 \, {\left (7 \, B - 13 \, C\right )} c\right )} \cos \left (f x + e\right )^{3} - 32 \, {\left (3 \, A + B - C\right )} c m - {\left (8 \, {\left (A + C\right )} c m^{3} + 4 \, {\left (13 \, A - 6 \, B + 5 \, C\right )} c m^{2} + 2 \, {\left (47 \, A - 48 \, B + 47 \, C\right )} c m + {\left (35 \, A - 42 \, B + 43 \, C\right )} c\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left (35 \, A - 21 \, B + 19 \, C\right )} c - {\left (8 \, {\left (A - B + C\right )} c m^{3} + 4 \, {\left (17 \, A - 13 \, B + 17 \, C\right )} c m^{2} + 2 \, {\left (95 \, A - 63 \, B + 63 \, C\right )} c m + {\left (175 \, A - 147 \, B + 143 \, C\right )} c\right )} \cos \left (f x + e\right ) - {\left (16 \, {\left (A + B + C\right )} c m^{2} + {\left (8 \, C c m^{3} + 36 \, C c m^{2} + 46 \, C c m + 15 \, C c\right )} \cos \left (f x + e\right )^{3} + 32 \, {\left (3 \, A + B - C\right )} c m + {\left (8 \, B c m^{3} + 4 \, {\left (11 \, B - 8 \, C\right )} c m^{2} + 2 \, {\left (31 \, B - 32 \, C\right )} c m + 3 \, {\left (7 \, B - 8 \, C\right )} c\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left (35 \, A - 21 \, B + 19 \, C\right )} c - {\left (8 \, {\left (A - B + C\right )} c m^{3} + 4 \, {\left (13 \, A - 17 \, B + 13 \, C\right )} c m^{2} + 2 \, {\left (47 \, A - 79 \, B + 79 \, C\right )} c m + {\left (35 \, A - 63 \, B + 67 \, C\right )} c\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{16 \, f m^{4} + 128 \, f m^{3} + 344 \, f m^{2} + 352 \, f m + {\left (16 \, f m^{4} + 128 \, f m^{3} + 344 \, f m^{2} + 352 \, f m + 105 \, f\right )} \cos \left (f x + e\right ) - {\left (16 \, f m^{4} + 128 \, f m^{3} + 344 \, f m^{2} + 352 \, f m + 105 \, f\right )} \sin \left (f x + e\right ) + 105 \, f} \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 [B]
time = 23.16, size = 790, normalized size = 2.45 \begin {gather*} \frac {\sqrt {c-c\,\sin \left (e+f\,x\right )}\,\left (\frac {C\,c\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (m^3\,8{}\mathrm {i}+m^2\,36{}\mathrm {i}+m\,46{}\mathrm {i}+15{}\mathrm {i}\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}+\frac {c\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (1260\,A-840\,B+735\,C+1144\,A\,m-128\,B\,m-18\,C\,m+336\,A\,m^2+32\,A\,m^3+32\,B\,m^2+100\,C\,m^2+8\,C\,m^3\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}-\frac {c\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (A\,1260{}\mathrm {i}-B\,840{}\mathrm {i}+C\,735{}\mathrm {i}+A\,m\,1144{}\mathrm {i}-B\,m\,128{}\mathrm {i}-C\,m\,18{}\mathrm {i}+A\,m^2\,336{}\mathrm {i}+A\,m^3\,32{}\mathrm {i}+B\,m^2\,32{}\mathrm {i}+C\,m^2\,100{}\mathrm {i}+C\,m^3\,8{}\mathrm {i}\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}+\frac {c\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\left (2\,m+1\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (140\,A-210\,B+175\,C+96\,A\,m-88\,B\,m+16\,C\,m+16\,A\,m^2-8\,B\,m^2+4\,C\,m^2\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}-\frac {c\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (2\,m+1\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (A\,140{}\mathrm {i}-B\,210{}\mathrm {i}+C\,175{}\mathrm {i}+A\,m\,96{}\mathrm {i}-B\,m\,88{}\mathrm {i}+C\,m\,16{}\mathrm {i}+A\,m^2\,16{}\mathrm {i}-B\,m^2\,8{}\mathrm {i}+C\,m^2\,4{}\mathrm {i}\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}+\frac {c\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (4\,m^2+8\,m+3\right )\,\left (14\,B-21\,C+4\,B\,m-2\,C\,m\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}-\frac {c\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (4\,m^2+8\,m+3\right )\,\left (B\,14{}\mathrm {i}-C\,21{}\mathrm {i}+B\,m\,4{}\mathrm {i}-C\,m\,2{}\mathrm {i}\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}-\frac {C\,c\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (8\,m^3+36\,m^2+46\,m+15\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}\right )}{{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}-\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\left (m^4\,16{}\mathrm {i}+m^3\,128{}\mathrm {i}+m^2\,344{}\mathrm {i}+m\,352{}\mathrm {i}+105{}\mathrm {i}\right )}{16\,m^4+128\,m^3+344\,m^2+352\,m+105}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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