3.11.0 ∫ cos5(e + fx)(a + a sin(e + fx))m(A + B sin(e + fx)) dx [1021]

Integrand size = 31, antiderivative size = 123 \[ \int \cos ^5(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\frac {4 (A-B) (a+a \sin (e+f x))^{3+m}}{a^3 f (3+m)}-\frac {4 (A-2 B) (a+a \sin (e+f x))^{4+m}}{a^4 f (4+m)}+\frac {(A-5 B) (a+a \sin (e+f x))^{5+m}}{a^5 f (5+m)}+\frac {B (a+a \sin (e+f x))^{6+m}}{a^6 f (6+m)} \]

Rubi [A] (veriﬁed)

Time = 0.10 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2915, 78} \[ \int \cos ^5(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\frac {B (a \sin (e+f x)+a)^{m+6}}{a^6 f (m+6)}+\frac {(A-5 B) (a \sin (e+f x)+a)^{m+5}}{a^5 f (m+5)}-\frac {4 (A-2 B) (a \sin (e+f x)+a)^{m+4}}{a^4 f (m+4)}+\frac {4 (A-B) (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)} \]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a-x)^2 (a+x)^{2+m} \left (A+\frac {B x}{a}\right ) \, dx,x,a \sin (e+f x)\right )}{a^5 f} \\ & = \frac {\text {Subst}\left (\int \left (4 a^2 (A-B) (a+x)^{2+m}-4 a (A-2 B) (a+x)^{3+m}+(A-5 B) (a+x)^{4+m}+\frac {B (a+x)^{5+m}}{a}\right ) \, dx,x,a \sin (e+f x)\right )}{a^5 f} \\ & = \frac {4 (A-B) (a+a \sin (e+f x))^{3+m}}{a^3 f (3+m)}-\frac {4 (A-2 B) (a+a \sin (e+f x))^{4+m}}{a^4 f (4+m)}+\frac {(A-5 B) (a+a \sin (e+f x))^{5+m}}{a^5 f (5+m)}+\frac {B (a+a \sin (e+f x))^{6+m}}{a^6 f (6+m)} \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 0.26 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.84 \[ \int \cos ^5(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\frac {(a (1+\sin (e+f x)))^{3+m} \left (\frac {4 a^3 (A-B)}{3+m}-\frac {4 a^3 (A-2 B) (1+\sin (e+f x))}{4+m}+\frac {a^3 (A-5 B) (1+\sin (e+f x))^2}{5+m}+\frac {B (a+a \sin (e+f x))^3}{6+m}\right )}{a^6 f} \]

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(254\) vs. \(2(123)=246\).

Time = 6.23 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.07


method	result	size

parallelrisch	\(\frac {\left (\left (\left (4 A +\frac {B}{4}\right ) m^{3}+\left (68 A -B \right ) m^{2}+\left (264 A -\frac {705 B}{4}\right ) m -225 B \right ) \cos \left (2 f x +2 e \right )+\left (\left (A -\frac {B}{2}\right ) m^{2}+\left (6 A -\frac {27 B}{2}\right ) m -30 B \right ) \left (3+m \right ) \cos \left (4 f x +4 e \right )+\frac {3 \left (m^{2}+\frac {53}{3} m +\frac {100}{3}\right ) \left (\left (A +B \right ) m +6 A \right ) \sin \left (3 f x +3 e \right )}{2}+\frac {\left (4+m \right ) \left (\left (A +B \right ) m +6 A \right ) \left (3+m \right ) \sin \left (5 f x +5 e \right )}{2}-\frac {B \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \cos \left (6 f x +6 e \right )}{4}+\left (m^{2}+23 m +300\right ) \left (\left (A +B \right ) m +6 A \right ) \sin \left (f x +e \right )+\left (\frac {B}{2}+3 A \right ) m^{3}+\left (11 B +59 A \right ) m^{2}+\left (502 A +\frac {277 B}{2}\right ) m +1536 A -150 B \right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{m}}{8 \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (6+m \right ) f}\)	\(255\)
derivativedivides	\(\frac {\left (A \,m^{3}+17 A \,m^{2}-B \,m^{2}+98 A m -15 B m +192 A -60 B \right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{4}+18 m^{3}+119 m^{2}+342 m +360\right )}+\frac {B \left (\sin ^{6}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (6+m \right )}+\frac {\left (A m +B m +6 A \right ) \left (\sin ^{5}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{2}+11 m +30\right )}+\frac {\left (A \,m^{2}-2 B \,m^{2}+6 A m -27 B m -60 B \right ) \left (\sin ^{4}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{3}+15 m^{2}+74 m +120\right )}+\frac {\left (A \,m^{3}+B \,m^{3}+21 A \,m^{2}+15 B \,m^{2}+150 A m +60 B m +360 A \right ) \sin \left (f x +e \right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{4}+18 m^{3}+119 m^{2}+342 m +360\right )}-\frac {\left (2 A \,m^{3}-B \,m^{3}+26 A \,m^{2}-22 B \,m^{2}+84 A m -141 B m -180 B \right ) \left (\sin ^{2}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{4}+18 m^{3}+119 m^{2}+342 m +360\right )}-\frac {2 \left (A \,m^{3}+B \,m^{3}+17 A \,m^{2}+11 B \,m^{2}+86 A m +20 B m +120 A \right ) \left (\sin ^{3}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{4}+18 m^{3}+119 m^{2}+342 m +360\right )}\)	\(456\)
default	\(\frac {\left (A \,m^{3}+17 A \,m^{2}-B \,m^{2}+98 A m -15 B m +192 A -60 B \right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{4}+18 m^{3}+119 m^{2}+342 m +360\right )}+\frac {B \left (\sin ^{6}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (6+m \right )}+\frac {\left (A m +B m +6 A \right ) \left (\sin ^{5}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{2}+11 m +30\right )}+\frac {\left (A \,m^{2}-2 B \,m^{2}+6 A m -27 B m -60 B \right ) \left (\sin ^{4}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{3}+15 m^{2}+74 m +120\right )}+\frac {\left (A \,m^{3}+B \,m^{3}+21 A \,m^{2}+15 B \,m^{2}+150 A m +60 B m +360 A \right ) \sin \left (f x +e \right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{4}+18 m^{3}+119 m^{2}+342 m +360\right )}-\frac {\left (2 A \,m^{3}-B \,m^{3}+26 A \,m^{2}-22 B \,m^{2}+84 A m -141 B m -180 B \right ) \left (\sin ^{2}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{4}+18 m^{3}+119 m^{2}+342 m +360\right )}-\frac {2 \left (A \,m^{3}+B \,m^{3}+17 A \,m^{2}+11 B \,m^{2}+86 A m +20 B m +120 A \right ) \left (\sin ^{3}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a \left (1+\sin \left (f x +e \right )\right )\right )}}{f \left (m^{4}+18 m^{3}+119 m^{2}+342 m +360\right )}\)	\(456\)

Fricas [A] (veriﬁcation not implemented)

Time = 0.32 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.80 \[ \int \cos ^5(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=-\frac {{\left ({\left (B m^{3} + 12 \, B m^{2} + 47 \, B m + 60 \, B\right )} \cos \left (f x + e\right )^{6} - {\left ({\left (A + B\right )} m^{3} + 3 \, {\left (3 \, A + B\right )} m^{2} + 18 \, A m\right )} \cos \left (f x + e\right )^{4} - 8 \, {\left ({\left (A + B\right )} m^{2} + 6 \, A m\right )} \cos \left (f x + e\right )^{2} - 32 \, {\left (A + B\right )} m - {\left ({\left ({\left (A + B\right )} m^{3} + {\left (13 \, A + 7 \, B\right )} m^{2} + 6 \, {\left (9 \, A + 2 \, B\right )} m + 72 \, A\right )} \cos \left (f x + e\right )^{4} + 8 \, {\left ({\left (A + B\right )} m^{2} + 2 \, {\left (4 \, A + B\right )} m + 12 \, A\right )} \cos \left (f x + e\right )^{2} + 32 \, {\left (A + B\right )} m + 192 \, A\right )} \sin \left (f x + e\right ) - 192 \, A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{f m^{4} + 18 \, f m^{3} + 119 \, f m^{2} + 342 \, f m + 360 \, f} \]

Sympy [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 22522 vs. \(2 (107) = 214\).

Time = 95.27 (sec) , antiderivative size = 22522, normalized size of antiderivative = 183.11 \[ \int \cos ^5(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\text {Too large to display} \]

Maxima [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 643 vs. \(2 (123) = 246\).

Time = 0.23 (sec) , antiderivative size = 643, normalized size of antiderivative = 5.23 \[ \int \cos ^5(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\frac {\frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} a^{m} \sin \left (f x + e\right )^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a^{m} \sin \left (f x + e\right )^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (f x + e\right )^{3} + 12 \, {\left (m^{2} + m\right )} a^{m} \sin \left (f x + e\right )^{2} - 24 \, a^{m} m \sin \left (f x + e\right ) + 24 \, a^{m}\right )} A {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} - \frac {2 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} a^{m} \sin \left (f x + e\right )^{3} + {\left (m^{2} + m\right )} a^{m} \sin \left (f x + e\right )^{2} - 2 \, a^{m} m \sin \left (f x + e\right ) + 2 \, a^{m}\right )} A {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {{\left ({\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} a^{m} \sin \left (f x + e\right )^{6} + {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} a^{m} \sin \left (f x + e\right )^{5} - 5 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a^{m} \sin \left (f x + e\right )^{4} + 20 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (f x + e\right )^{3} - 60 \, {\left (m^{2} + m\right )} a^{m} \sin \left (f x + e\right )^{2} + 120 \, a^{m} m \sin \left (f x + e\right ) - 120 \, a^{m}\right )} B {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} - \frac {2 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} a^{m} \sin \left (f x + e\right )^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{m} \sin \left (f x + e\right )^{3} - 3 \, {\left (m^{2} + m\right )} a^{m} \sin \left (f x + e\right )^{2} + 6 \, a^{m} m \sin \left (f x + e\right ) - 6 \, a^{m}\right )} B {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} + \frac {{\left (a^{m} {\left (m + 1\right )} \sin \left (f x + e\right )^{2} + a^{m} m \sin \left (f x + e\right ) - a^{m}\right )} B {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{2} + 3 \, m + 2} + \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m + 1} A}{a {\left (m + 1\right )}}}{f} \]

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 811 vs. \(2 (123) = 246\).

Time = 0.34 (sec) , antiderivative size = 811, normalized size of antiderivative = 6.59 \[ \int \cos ^5(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\text {Too large to display} \]

Mupad [B] (veriﬁcation not implemented)

Time = 17.29 (sec) , antiderivative size = 517, normalized size of antiderivative = 4.20 \[ \int \cos ^5(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=-{\mathrm {e}}^{-e\,6{}\mathrm {i}-f\,x\,6{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (-\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (12288\,A-1200\,B+4016\,A\,m+1108\,B\,m+472\,A\,m^2+24\,A\,m^3+88\,B\,m^2+4\,B\,m^3\right )}{64\,f\,\left (m^4+18\,m^3+119\,m^2+342\,m+360\right )}-\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (2\,e+2\,f\,x\right )\,\left (1056\,A\,m-900\,B-705\,B\,m+272\,A\,m^2+16\,A\,m^3-4\,B\,m^2+B\,m^3\right )}{32\,f\,\left (m^4+18\,m^3+119\,m^2+342\,m+360\right )}+\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (4\,e+4\,f\,x\right )\,\left (m+3\right )\,\left (60\,B-12\,A\,m+27\,B\,m-2\,A\,m^2+B\,m^2\right )}{16\,f\,\left (m^4+18\,m^3+119\,m^2+342\,m+360\right )}+\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,\left (A\,6{}\mathrm {i}+A\,m\,1{}\mathrm {i}+B\,m\,1{}\mathrm {i}\right )\,\left (m^2+23\,m+300\right )\,1{}\mathrm {i}}{8\,f\,\left (m^4+18\,m^3+119\,m^2+342\,m+360\right )}+\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (5\,e+5\,f\,x\right )\,\left (A\,6{}\mathrm {i}+A\,m\,1{}\mathrm {i}+B\,m\,1{}\mathrm {i}\right )\,\left (m^2+7\,m+12\right )\,1{}\mathrm {i}}{16\,f\,\left (m^4+18\,m^3+119\,m^2+342\,m+360\right )}+\frac {B\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (6\,e+6\,f\,x\right )\,\left (m^3+12\,m^2+47\,m+60\right )}{32\,f\,\left (m^4+18\,m^3+119\,m^2+342\,m+360\right )}+\frac {{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (3\,e+3\,f\,x\right )\,\left (A\,6{}\mathrm {i}+A\,m\,1{}\mathrm {i}+B\,m\,1{}\mathrm {i}\right )\,\left (3\,m^2+53\,m+100\right )\,1{}\mathrm {i}}{16\,f\,\left (m^4+18\,m^3+119\,m^2+342\,m+360\right )}\right ) \]

\(\int \cos ^5(e+f x) (a+a \sin (e+f x))^m (A+B \sin (e+f x)) \, dx\) [1021]

Optimal result