Optimal. Leaf size=229 \[ \frac {a^3 \cos (c+d x) \sin ^{m+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(c+d x)\right )}{d (m+1) \sqrt {\cos ^2(c+d x)}}+\frac {3 a^2 b \sin ^{m+2}(c+d x) \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};\sin ^2(c+d x)\right )}{d (m+2)}+\frac {3 a b^2 \sqrt {\cos ^2(c+d x)} \sec (c+d x) \sin ^{m+3}(c+d x) \, _2F_1\left (\frac {3}{2},\frac {m+3}{2};\frac {m+5}{2};\sin ^2(c+d x)\right )}{d (m+3)}+\frac {b^3 \sin ^{m+4}(c+d x) \, _2F_1\left (2,\frac {m+4}{2};\frac {m+6}{2};\sin ^2(c+d x)\right )}{d (m+4)} \]
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Rubi [A] time = 0.45, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {4401, 2643, 2564, 364, 2577} \[ \frac {3 a^2 b \sin ^{m+2}(c+d x) \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};\sin ^2(c+d x)\right )}{d (m+2)}+\frac {a^3 \cos (c+d x) \sin ^{m+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(c+d x)\right )}{d (m+1) \sqrt {\cos ^2(c+d x)}}+\frac {3 a b^2 \sqrt {\cos ^2(c+d x)} \sec (c+d x) \sin ^{m+3}(c+d x) \, _2F_1\left (\frac {3}{2},\frac {m+3}{2};\frac {m+5}{2};\sin ^2(c+d x)\right )}{d (m+3)}+\frac {b^3 \sin ^{m+4}(c+d x) \, _2F_1\left (2,\frac {m+4}{2};\frac {m+6}{2};\sin ^2(c+d x)\right )}{d (m+4)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 2564
Rule 2577
Rule 2643
Rule 4401
Rubi steps
\begin {align*} \int \sin ^m(c+d x) (a+b \tan (c+d x))^3 \, dx &=\int \left (a^3 \sin ^m(c+d x)+3 a^2 b \sec (c+d x) \sin ^{1+m}(c+d x)+3 a b^2 \sec ^2(c+d x) \sin ^{2+m}(c+d x)+b^3 \sec ^3(c+d x) \sin ^{3+m}(c+d x)\right ) \, dx\\ &=a^3 \int \sin ^m(c+d x) \, dx+\left (3 a^2 b\right ) \int \sec (c+d x) \sin ^{1+m}(c+d x) \, dx+\left (3 a b^2\right ) \int \sec ^2(c+d x) \sin ^{2+m}(c+d x) \, dx+b^3 \int \sec ^3(c+d x) \sin ^{3+m}(c+d x) \, dx\\ &=\frac {a^3 \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};\sin ^2(c+d x)\right ) \sin ^{1+m}(c+d x)}{d (1+m) \sqrt {\cos ^2(c+d x)}}+\frac {3 a b^2 \sqrt {\cos ^2(c+d x)} \, _2F_1\left (\frac {3}{2},\frac {3+m}{2};\frac {5+m}{2};\sin ^2(c+d x)\right ) \sec (c+d x) \sin ^{3+m}(c+d x)}{d (3+m)}+\frac {\left (3 a^2 b\right ) \operatorname {Subst}\left (\int \frac {x^{1+m}}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}+\frac {b^3 \operatorname {Subst}\left (\int \frac {x^{3+m}}{\left (1-x^2\right )^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {a^3 \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};\sin ^2(c+d x)\right ) \sin ^{1+m}(c+d x)}{d (1+m) \sqrt {\cos ^2(c+d x)}}+\frac {3 a^2 b \, _2F_1\left (1,\frac {2+m}{2};\frac {4+m}{2};\sin ^2(c+d x)\right ) \sin ^{2+m}(c+d x)}{d (2+m)}+\frac {3 a b^2 \sqrt {\cos ^2(c+d x)} \, _2F_1\left (\frac {3}{2},\frac {3+m}{2};\frac {5+m}{2};\sin ^2(c+d x)\right ) \sec (c+d x) \sin ^{3+m}(c+d x)}{d (3+m)}+\frac {b^3 \, _2F_1\left (2,\frac {4+m}{2};\frac {6+m}{2};\sin ^2(c+d x)\right ) \sin ^{4+m}(c+d x)}{d (4+m)}\\ \end {align*}
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Mathematica [A] time = 2.56, size = 205, normalized size = 0.90 \[ \frac {\sin ^{m+1}(c+d x) \left (\frac {a^3 \sqrt {\cos ^2(c+d x)} \sec (c+d x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(c+d x)\right )}{m+1}+b \sin (c+d x) \left (\frac {3 a^2 \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};\sin ^2(c+d x)\right )}{m+2}+b \left (\frac {3 a \sqrt {\cos ^2(c+d x)} \tan (c+d x) \, _2F_1\left (\frac {3}{2},\frac {m+3}{2};\frac {m+5}{2};\sin ^2(c+d x)\right )}{m+3}+\frac {b \sin ^2(c+d x) \, _2F_1\left (2,\frac {m+4}{2};\frac {m+6}{2};\sin ^2(c+d x)\right )}{m+4}\right )\right )\right )}{d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{3} \tan \left (d x + c\right )^{3} + 3 \, a b^{2} \tan \left (d x + c\right )^{2} + 3 \, a^{2} b \tan \left (d x + c\right ) + a^{3}\right )} \sin \left (d x + c\right )^{m}, x\right ) \]
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 \[ \int {\left (b \tan \left (d x + c\right ) + a\right )}^{3} \sin \left (d x + c\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.20, size = 0, normalized size = 0.00 \[ \int \left (\sin ^{m}\left (d x +c \right )\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}\, dx \]
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 \[ \int {\left (b \tan \left (d x + c\right ) + a\right )}^{3} \sin \left (d x + c\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\sin \left (c+d\,x\right )}^m\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3 \,d x \]
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 \[ \int \left (a + b \tan {\left (c + d x \right )}\right )^{3} \sin ^{m}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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