Integrand size = 21, antiderivative size = 229 \[ \int \sin ^m(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {a^3 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(c+d x)\right ) \sin ^{1+m}(c+d x)}{d (1+m) \sqrt {\cos ^2(c+d x)}}+\frac {3 a^2 b \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},\sin ^2(c+d x)\right ) \sin ^{2+m}(c+d x)}{d (2+m)}+\frac {3 a b^2 \sqrt {\cos ^2(c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {3+m}{2},\frac {5+m}{2},\sin ^2(c+d x)\right ) \sec (c+d x) \sin ^{3+m}(c+d x)}{d (3+m)}+\frac {b^3 \operatorname {Hypergeometric2F1}\left (2,\frac {4+m}{2},\frac {6+m}{2},\sin ^2(c+d x)\right ) \sin ^{4+m}(c+d x)}{d (4+m)} \]
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Time = 0.56 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {4486, 2722, 2644, 371, 2657} \[ \int \sin ^m(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {a^3 \cos (c+d x) \sin ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(c+d x)\right )}{d (m+1) \sqrt {\cos ^2(c+d x)}}+\frac {3 a^2 b \sin ^{m+2}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},\sin ^2(c+d x)\right )}{d (m+2)}+\frac {3 a b^2 \sqrt {\cos ^2(c+d x)} \sec (c+d x) \sin ^{m+3}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {m+3}{2},\frac {m+5}{2},\sin ^2(c+d x)\right )}{d (m+3)}+\frac {b^3 \sin ^{m+4}(c+d x) \operatorname {Hypergeometric2F1}\left (2,\frac {m+4}{2},\frac {m+6}{2},\sin ^2(c+d x)\right )}{d (m+4)} \]
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Rule 371
Rule 2644
Rule 2657
Rule 2722
Rule 4486
Rubi steps \begin{align*} \text {integral}& = \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) \operatorname {Hypergeometric2F1}\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)} \operatorname {Hypergeometric2F1}\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 ) \text {Subst}\left (\int \frac {x^{1+m}}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}+\frac {b^3 \text {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) \operatorname {Hypergeometric2F1}\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 \operatorname {Hypergeometric2F1}\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)} \operatorname {Hypergeometric2F1}\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 \operatorname {Hypergeometric2F1}\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*}
Time = 2.76 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.90 \[ \int \sin ^m(c+d x) (a+b \tan (c+d x))^3 \, dx=\frac {\sin ^{1+m}(c+d x) \left (\frac {a^3 \sqrt {\cos ^2(c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(c+d x)\right ) \sec (c+d x)}{1+m}+b \sin (c+d x) \left (\frac {3 a^2 \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},\sin ^2(c+d x)\right )}{2+m}+b \left (\frac {b \operatorname {Hypergeometric2F1}\left (2,\frac {4+m}{2},\frac {6+m}{2},\sin ^2(c+d x)\right ) \sin ^2(c+d x)}{4+m}+\frac {3 a \sqrt {\cos ^2(c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {3+m}{2},\frac {5+m}{2},\sin ^2(c+d x)\right ) \tan (c+d x)}{3+m}\right )\right )\right )}{d} \]
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\[\int \left (\sin ^{m}\left (d x +c \right )\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}d x\]
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\[ \int \sin ^m(c+d x) (a+b \tan (c+d x))^3 \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{3} \sin \left (d x + c\right )^{m} \,d x } \]
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\[ \int \sin ^m(c+d x) (a+b \tan (c+d x))^3 \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{3} \sin ^{m}{\left (c + d x \right )}\, dx \]
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\[ \int \sin ^m(c+d x) (a+b \tan (c+d x))^3 \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{3} \sin \left (d x + c\right )^{m} \,d x } \]
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\[ \int \sin ^m(c+d x) (a+b \tan (c+d x))^3 \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{3} \sin \left (d x + c\right )^{m} \,d x } \]
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Timed out. \[ \int \sin ^m(c+d x) (a+b \tan (c+d x))^3 \, dx=\int {\sin \left (c+d\,x\right )}^m\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3 \,d x \]
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