Optimal. Leaf size=205 \[ \frac {8 d^4 \sin ^3(a+b x)}{81 b^5}+\frac {160 d^4 \sin (a+b x)}{27 b^5}-\frac {160 d^3 (c+d x) \cos (a+b x)}{27 b^4}-\frac {8 d^3 (c+d x) \sin ^2(a+b x) \cos (a+b x)}{27 b^4}-\frac {4 d^2 (c+d x)^2 \sin ^3(a+b x)}{9 b^3}-\frac {8 d^2 (c+d x)^2 \sin (a+b x)}{3 b^3}+\frac {8 d (c+d x)^3 \cos (a+b x)}{9 b^2}+\frac {4 d (c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{9 b^2}+\frac {(c+d x)^4 \sin ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.20, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {4404, 3311, 3296, 2637, 3310} \[ -\frac {4 d^2 (c+d x)^2 \sin ^3(a+b x)}{9 b^3}-\frac {8 d^2 (c+d x)^2 \sin (a+b x)}{3 b^3}-\frac {160 d^3 (c+d x) \cos (a+b x)}{27 b^4}-\frac {8 d^3 (c+d x) \sin ^2(a+b x) \cos (a+b x)}{27 b^4}+\frac {8 d (c+d x)^3 \cos (a+b x)}{9 b^2}+\frac {4 d (c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{9 b^2}+\frac {8 d^4 \sin ^3(a+b x)}{81 b^5}+\frac {160 d^4 \sin (a+b x)}{27 b^5}+\frac {(c+d x)^4 \sin ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 3310
Rule 3311
Rule 4404
Rubi steps
\begin {align*} \int (c+d x)^4 \cos (a+b x) \sin ^2(a+b x) \, dx &=\frac {(c+d x)^4 \sin ^3(a+b x)}{3 b}-\frac {(4 d) \int (c+d x)^3 \sin ^3(a+b x) \, dx}{3 b}\\ &=\frac {4 d (c+d x)^3 \cos (a+b x) \sin ^2(a+b x)}{9 b^2}-\frac {4 d^2 (c+d x)^2 \sin ^3(a+b x)}{9 b^3}+\frac {(c+d x)^4 \sin ^3(a+b x)}{3 b}-\frac {(8 d) \int (c+d x)^3 \sin (a+b x) \, dx}{9 b}+\frac {\left (8 d^3\right ) \int (c+d x) \sin ^3(a+b x) \, dx}{9 b^3}\\ &=\frac {8 d (c+d x)^3 \cos (a+b x)}{9 b^2}-\frac {8 d^3 (c+d x) \cos (a+b x) \sin ^2(a+b x)}{27 b^4}+\frac {4 d (c+d x)^3 \cos (a+b x) \sin ^2(a+b x)}{9 b^2}+\frac {8 d^4 \sin ^3(a+b x)}{81 b^5}-\frac {4 d^2 (c+d x)^2 \sin ^3(a+b x)}{9 b^3}+\frac {(c+d x)^4 \sin ^3(a+b x)}{3 b}-\frac {\left (8 d^2\right ) \int (c+d x)^2 \cos (a+b x) \, dx}{3 b^2}+\frac {\left (16 d^3\right ) \int (c+d x) \sin (a+b x) \, dx}{27 b^3}\\ &=-\frac {16 d^3 (c+d x) \cos (a+b x)}{27 b^4}+\frac {8 d (c+d x)^3 \cos (a+b x)}{9 b^2}-\frac {8 d^2 (c+d x)^2 \sin (a+b x)}{3 b^3}-\frac {8 d^3 (c+d x) \cos (a+b x) \sin ^2(a+b x)}{27 b^4}+\frac {4 d (c+d x)^3 \cos (a+b x) \sin ^2(a+b x)}{9 b^2}+\frac {8 d^4 \sin ^3(a+b x)}{81 b^5}-\frac {4 d^2 (c+d x)^2 \sin ^3(a+b x)}{9 b^3}+\frac {(c+d x)^4 \sin ^3(a+b x)}{3 b}+\frac {\left (16 d^3\right ) \int (c+d x) \sin (a+b x) \, dx}{3 b^3}+\frac {\left (16 d^4\right ) \int \cos (a+b x) \, dx}{27 b^4}\\ &=-\frac {160 d^3 (c+d x) \cos (a+b x)}{27 b^4}+\frac {8 d (c+d x)^3 \cos (a+b x)}{9 b^2}+\frac {16 d^4 \sin (a+b x)}{27 b^5}-\frac {8 d^2 (c+d x)^2 \sin (a+b x)}{3 b^3}-\frac {8 d^3 (c+d x) \cos (a+b x) \sin ^2(a+b x)}{27 b^4}+\frac {4 d (c+d x)^3 \cos (a+b x) \sin ^2(a+b x)}{9 b^2}+\frac {8 d^4 \sin ^3(a+b x)}{81 b^5}-\frac {4 d^2 (c+d x)^2 \sin ^3(a+b x)}{9 b^3}+\frac {(c+d x)^4 \sin ^3(a+b x)}{3 b}+\frac {\left (16 d^4\right ) \int \cos (a+b x) \, dx}{3 b^4}\\ &=-\frac {160 d^3 (c+d x) \cos (a+b x)}{27 b^4}+\frac {8 d (c+d x)^3 \cos (a+b x)}{9 b^2}+\frac {160 d^4 \sin (a+b x)}{27 b^5}-\frac {8 d^2 (c+d x)^2 \sin (a+b x)}{3 b^3}-\frac {8 d^3 (c+d x) \cos (a+b x) \sin ^2(a+b x)}{27 b^4}+\frac {4 d (c+d x)^3 \cos (a+b x) \sin ^2(a+b x)}{9 b^2}+\frac {8 d^4 \sin ^3(a+b x)}{81 b^5}-\frac {4 d^2 (c+d x)^2 \sin ^3(a+b x)}{9 b^3}+\frac {(c+d x)^4 \sin ^3(a+b x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 1.45, size = 385, normalized size = 1.88 \[ \frac {81 b^4 c^4 \sin (a+b x)-27 b^4 c^4 \sin (3 (a+b x))+324 b^4 c^3 d x \sin (a+b x)-108 b^4 c^3 d x \sin (3 (a+b x))+486 b^4 c^2 d^2 x^2 \sin (a+b x)-162 b^4 c^2 d^2 x^2 \sin (3 (a+b x))+324 b^4 c d^3 x^3 \sin (a+b x)-108 b^4 c d^3 x^3 \sin (3 (a+b x))+81 b^4 d^4 x^4 \sin (a+b x)-27 b^4 d^4 x^4 \sin (3 (a+b x))-972 b^2 c^2 d^2 \sin (a+b x)+36 b^2 c^2 d^2 \sin (3 (a+b x))-1944 b^2 c d^3 x \sin (a+b x)+72 b^2 c d^3 x \sin (3 (a+b x))+324 b d (c+d x) \cos (a+b x) \left (b^2 (c+d x)^2-6 d^2\right )-12 b d (c+d x) \cos (3 (a+b x)) \left (3 b^2 (c+d x)^2-2 d^2\right )-972 b^2 d^4 x^2 \sin (a+b x)+36 b^2 d^4 x^2 \sin (3 (a+b x))+1944 d^4 \sin (a+b x)-8 d^4 \sin (3 (a+b x))}{324 b^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 352, normalized size = 1.72 \[ -\frac {12 \, {\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d - 2 \, b c d^{3} + {\left (9 \, b^{3} c^{2} d^{2} - 2 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{3} - 36 \, {\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d - 14 \, b c d^{3} + {\left (9 \, b^{3} c^{2} d^{2} - 14 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right ) - {\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 27 \, b^{4} c^{4} - 252 \, b^{2} c^{2} d^{2} + 488 \, d^{4} + 18 \, {\left (9 \, b^{4} c^{2} d^{2} - 14 \, b^{2} d^{4}\right )} x^{2} - {\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 27 \, b^{4} c^{4} - 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4} + 18 \, {\left (9 \, b^{4} c^{2} d^{2} - 2 \, b^{2} d^{4}\right )} x^{2} + 36 \, {\left (3 \, b^{4} c^{3} d - 2 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{2} + 36 \, {\left (3 \, b^{4} c^{3} d - 14 \, b^{2} c d^{3}\right )} x\right )} \sin \left (b x + a\right )}{81 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 350, normalized size = 1.71 \[ -\frac {{\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 9 \, b^{3} c^{2} d^{2} x + 3 \, b^{3} c^{3} d - 2 \, b d^{4} x - 2 \, b c d^{3}\right )} \cos \left (3 \, b x + 3 \, a\right )}{27 \, b^{5}} + \frac {{\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{2} d^{2} x + b^{3} c^{3} d - 6 \, b d^{4} x - 6 \, b c d^{3}\right )} \cos \left (b x + a\right )}{b^{5}} - \frac {{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 162 \, b^{4} c^{2} d^{2} x^{2} + 108 \, b^{4} c^{3} d x + 27 \, b^{4} c^{4} - 36 \, b^{2} d^{4} x^{2} - 72 \, b^{2} c d^{3} x - 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4}\right )} \sin \left (3 \, b x + 3 \, a\right )}{324 \, b^{5}} + \frac {{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{4} c^{3} d x + b^{4} c^{4} - 12 \, b^{2} d^{4} x^{2} - 24 \, b^{2} c d^{3} x - 12 \, b^{2} c^{2} d^{2} + 24 \, d^{4}\right )} \sin \left (b x + a\right )}{4 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 835, normalized size = 4.07 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 880, normalized size = 4.29 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.41, size = 448, normalized size = 2.19 \[ \frac {{\sin \left (a+b\,x\right )}^3\,\left (27\,b^4\,c^4-252\,b^2\,c^2\,d^2+488\,d^4\right )}{81\,b^5}-\frac {8\,{\cos \left (a+b\,x\right )}^3\,\left (20\,c\,d^3-3\,b^2\,c^3\,d\right )}{27\,b^4}+\frac {8\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )\,\left (20\,d^4-9\,b^2\,c^2\,d^2\right )}{27\,b^5}-\frac {4\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2\,\left (14\,c\,d^3-3\,b^2\,c^3\,d\right )}{9\,b^4}+\frac {8\,d^4\,x^3\,{\cos \left (a+b\,x\right )}^3}{9\,b^2}-\frac {8\,x\,{\cos \left (a+b\,x\right )}^3\,\left (20\,d^4-9\,b^2\,c^2\,d^2\right )}{27\,b^4}+\frac {d^4\,x^4\,{\sin \left (a+b\,x\right )}^3}{3\,b}-\frac {4\,x\,{\sin \left (a+b\,x\right )}^3\,\left (14\,c\,d^3-3\,b^2\,c^3\,d\right )}{9\,b^3}-\frac {2\,x^2\,{\sin \left (a+b\,x\right )}^3\,\left (14\,d^4-9\,b^2\,c^2\,d^2\right )}{9\,b^3}+\frac {8\,c\,d^3\,x^2\,{\cos \left (a+b\,x\right )}^3}{3\,b^2}+\frac {4\,d^4\,x^3\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2}{3\,b^2}-\frac {8\,d^4\,x^2\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{3\,b^3}+\frac {4\,c\,d^3\,x^3\,{\sin \left (a+b\,x\right )}^3}{3\,b}-\frac {4\,x\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2\,\left (14\,d^4-9\,b^2\,c^2\,d^2\right )}{9\,b^4}+\frac {4\,c\,d^3\,x^2\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2}{b^2}-\frac {16\,c\,d^3\,x\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{3\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.48, size = 646, normalized size = 3.15 \[ \begin {cases} \frac {c^{4} \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac {4 c^{3} d x \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac {2 c^{2} d^{2} x^{2} \sin ^{3}{\left (a + b x \right )}}{b} + \frac {4 c d^{3} x^{3} \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac {d^{4} x^{4} \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac {4 c^{3} d \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{3 b^{2}} + \frac {8 c^{3} d \cos ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {4 c^{2} d^{2} x \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b^{2}} + \frac {8 c^{2} d^{2} x \cos ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {4 c d^{3} x^{2} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b^{2}} + \frac {8 c d^{3} x^{2} \cos ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {4 d^{4} x^{3} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{3 b^{2}} + \frac {8 d^{4} x^{3} \cos ^{3}{\left (a + b x \right )}}{9 b^{2}} - \frac {28 c^{2} d^{2} \sin ^{3}{\left (a + b x \right )}}{9 b^{3}} - \frac {8 c^{2} d^{2} \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b^{3}} - \frac {56 c d^{3} x \sin ^{3}{\left (a + b x \right )}}{9 b^{3}} - \frac {16 c d^{3} x \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b^{3}} - \frac {28 d^{4} x^{2} \sin ^{3}{\left (a + b x \right )}}{9 b^{3}} - \frac {8 d^{4} x^{2} \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b^{3}} - \frac {56 c d^{3} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{9 b^{4}} - \frac {160 c d^{3} \cos ^{3}{\left (a + b x \right )}}{27 b^{4}} - \frac {56 d^{4} x \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{9 b^{4}} - \frac {160 d^{4} x \cos ^{3}{\left (a + b x \right )}}{27 b^{4}} + \frac {488 d^{4} \sin ^{3}{\left (a + b x \right )}}{81 b^{5}} + \frac {160 d^{4} \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{27 b^{5}} & \text {for}\: b \neq 0 \\\left (c^{4} x + 2 c^{3} d x^{2} + 2 c^{2} d^{2} x^{3} + c d^{3} x^{4} + \frac {d^{4} x^{5}}{5}\right ) \sin ^{2}{\relax (a )} \cos {\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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