Optimal. Leaf size=351 \[ \frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \sin \left (4 a-\frac {4 b c}{d}\right ) C\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{512 b^{5/2}}-\frac {3 \sqrt {\pi } d^{3/2} \sin \left (2 a-\frac {2 b c}{d}\right ) C\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{64 b^{5/2}}+\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \cos \left (4 a-\frac {4 b c}{d}\right ) S\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{512 b^{5/2}}-\frac {3 \sqrt {\pi } d^{3/2} \cos \left (2 a-\frac {2 b c}{d}\right ) S\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{64 b^{5/2}}+\frac {3 d \sqrt {c+d x} \sin (2 a+2 b x)}{32 b^2}-\frac {3 d \sqrt {c+d x} \sin (4 a+4 b x)}{256 b^2}-\frac {(c+d x)^{3/2} \cos (2 a+2 b x)}{8 b}+\frac {(c+d x)^{3/2} \cos (4 a+4 b x)}{32 b} \]
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Rubi [A] time = 0.57, antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {4406, 3296, 3306, 3305, 3351, 3304, 3352} \[ \frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \sin \left (4 a-\frac {4 b c}{d}\right ) \text {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{512 b^{5/2}}-\frac {3 \sqrt {\pi } d^{3/2} \sin \left (2 a-\frac {2 b c}{d}\right ) \text {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {\pi } \sqrt {d}}\right )}{64 b^{5/2}}+\frac {3 \sqrt {\frac {\pi }{2}} d^{3/2} \cos \left (4 a-\frac {4 b c}{d}\right ) S\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{512 b^{5/2}}-\frac {3 \sqrt {\pi } d^{3/2} \cos \left (2 a-\frac {2 b c}{d}\right ) S\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{64 b^{5/2}}+\frac {3 d \sqrt {c+d x} \sin (2 a+2 b x)}{32 b^2}-\frac {3 d \sqrt {c+d x} \sin (4 a+4 b x)}{256 b^2}-\frac {(c+d x)^{3/2} \cos (2 a+2 b x)}{8 b}+\frac {(c+d x)^{3/2} \cos (4 a+4 b x)}{32 b} \]
Antiderivative was successfully verified.
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Rule 3296
Rule 3304
Rule 3305
Rule 3306
Rule 3351
Rule 3352
Rule 4406
Rubi steps
\begin {align*} \int (c+d x)^{3/2} \cos (a+b x) \sin ^3(a+b x) \, dx &=\int \left (\frac {1}{4} (c+d x)^{3/2} \sin (2 a+2 b x)-\frac {1}{8} (c+d x)^{3/2} \sin (4 a+4 b x)\right ) \, dx\\ &=-\left (\frac {1}{8} \int (c+d x)^{3/2} \sin (4 a+4 b x) \, dx\right )+\frac {1}{4} \int (c+d x)^{3/2} \sin (2 a+2 b x) \, dx\\ &=-\frac {(c+d x)^{3/2} \cos (2 a+2 b x)}{8 b}+\frac {(c+d x)^{3/2} \cos (4 a+4 b x)}{32 b}-\frac {(3 d) \int \sqrt {c+d x} \cos (4 a+4 b x) \, dx}{64 b}+\frac {(3 d) \int \sqrt {c+d x} \cos (2 a+2 b x) \, dx}{16 b}\\ &=-\frac {(c+d x)^{3/2} \cos (2 a+2 b x)}{8 b}+\frac {(c+d x)^{3/2} \cos (4 a+4 b x)}{32 b}+\frac {3 d \sqrt {c+d x} \sin (2 a+2 b x)}{32 b^2}-\frac {3 d \sqrt {c+d x} \sin (4 a+4 b x)}{256 b^2}+\frac {\left (3 d^2\right ) \int \frac {\sin (4 a+4 b x)}{\sqrt {c+d x}} \, dx}{512 b^2}-\frac {\left (3 d^2\right ) \int \frac {\sin (2 a+2 b x)}{\sqrt {c+d x}} \, dx}{64 b^2}\\ &=-\frac {(c+d x)^{3/2} \cos (2 a+2 b x)}{8 b}+\frac {(c+d x)^{3/2} \cos (4 a+4 b x)}{32 b}+\frac {3 d \sqrt {c+d x} \sin (2 a+2 b x)}{32 b^2}-\frac {3 d \sqrt {c+d x} \sin (4 a+4 b x)}{256 b^2}+\frac {\left (3 d^2 \cos \left (4 a-\frac {4 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {4 b c}{d}+4 b x\right )}{\sqrt {c+d x}} \, dx}{512 b^2}-\frac {\left (3 d^2 \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{64 b^2}+\frac {\left (3 d^2 \sin \left (4 a-\frac {4 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {4 b c}{d}+4 b x\right )}{\sqrt {c+d x}} \, dx}{512 b^2}-\frac {\left (3 d^2 \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{\sqrt {c+d x}} \, dx}{64 b^2}\\ &=-\frac {(c+d x)^{3/2} \cos (2 a+2 b x)}{8 b}+\frac {(c+d x)^{3/2} \cos (4 a+4 b x)}{32 b}+\frac {3 d \sqrt {c+d x} \sin (2 a+2 b x)}{32 b^2}-\frac {3 d \sqrt {c+d x} \sin (4 a+4 b x)}{256 b^2}+\frac {\left (3 d \cos \left (4 a-\frac {4 b c}{d}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {4 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{256 b^2}-\frac {\left (3 d \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{32 b^2}+\frac {\left (3 d \sin \left (4 a-\frac {4 b c}{d}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {4 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{256 b^2}-\frac {\left (3 d \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {2 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{32 b^2}\\ &=-\frac {(c+d x)^{3/2} \cos (2 a+2 b x)}{8 b}+\frac {(c+d x)^{3/2} \cos (4 a+4 b x)}{32 b}+\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (4 a-\frac {4 b c}{d}\right ) S\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{512 b^{5/2}}-\frac {3 d^{3/2} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) S\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{64 b^{5/2}}+\frac {3 d^{3/2} \sqrt {\frac {\pi }{2}} C\left (\frac {2 \sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (4 a-\frac {4 b c}{d}\right )}{512 b^{5/2}}-\frac {3 d^{3/2} \sqrt {\pi } C\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{64 b^{5/2}}+\frac {3 d \sqrt {c+d x} \sin (2 a+2 b x)}{32 b^2}-\frac {3 d \sqrt {c+d x} \sin (4 a+4 b x)}{256 b^2}\\ \end {align*}
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Mathematica [A] time = 2.37, size = 393, normalized size = 1.12 \[ \frac {3 \sqrt {2 \pi } d \sin \left (4 a-\frac {4 b c}{d}\right ) C\left (2 \sqrt {\frac {b}{d}} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}\right )-48 \sqrt {\pi } d \sin \left (2 a-\frac {2 b c}{d}\right ) C\left (\frac {2 \sqrt {\frac {b}{d}} \sqrt {c+d x}}{\sqrt {\pi }}\right )+3 \sqrt {2 \pi } d \cos \left (4 a-\frac {4 b c}{d}\right ) S\left (2 \sqrt {\frac {b}{d}} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}\right )-48 \sqrt {\pi } d \cos \left (2 a-\frac {2 b c}{d}\right ) S\left (\frac {2 \sqrt {\frac {b}{d}} \sqrt {c+d x}}{\sqrt {\pi }}\right )+96 d \sqrt {\frac {b}{d}} \sqrt {c+d x} \sin (2 (a+b x))-12 d \sqrt {\frac {b}{d}} \sqrt {c+d x} \sin (4 (a+b x))-128 b d x \sqrt {\frac {b}{d}} \sqrt {c+d x} \cos (2 (a+b x))-128 b c \sqrt {\frac {b}{d}} \sqrt {c+d x} \cos (2 (a+b x))+32 b d x \sqrt {\frac {b}{d}} \sqrt {c+d x} \cos (4 (a+b x))+32 b c \sqrt {\frac {b}{d}} \sqrt {c+d x} \cos (4 (a+b x))}{1024 b^2 \sqrt {\frac {b}{d}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 316, normalized size = 0.90 \[ \frac {3 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (2 \, \sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 3 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (2 \, \sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) - 48 \, \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 48 \, \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 16 \, {\left (16 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{4} + 10 \, b^{2} d x + 10 \, b^{2} c - 32 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{2} - 3 \, {\left (2 \, b d \cos \left (b x + a\right )^{3} - 5 \, b d \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{1024 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 6.77, size = 1503, normalized size = 4.28 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 376, normalized size = 1.07 \[ \frac {-\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {2 \left (d x +c \right ) b}{d}+\frac {2 d a -2 c b}{d}\right )}{8 b}+\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {2 \left (d x +c \right ) b}{d}+\frac {2 d a -2 c b}{d}\right )}{4 b}-\frac {d \sqrt {\pi }\, \left (\cos \left (\frac {2 d a -2 c b}{d}\right ) \mathrm {S}\left (\frac {2 \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {2 d a -2 c b}{d}\right ) \FresnelC \left (\frac {2 \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{8 b \sqrt {\frac {b}{d}}}\right )}{8 b}+\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {4 \left (d x +c \right ) b}{d}+\frac {4 d a -4 c b}{d}\right )}{32 b}-\frac {3 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {4 \left (d x +c \right ) b}{d}+\frac {4 d a -4 c b}{d}\right )}{8 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {4 d a -4 c b}{d}\right ) \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {4 d a -4 c b}{d}\right ) \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{32 b \sqrt {\frac {b}{d}}}\right )}{32 b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.50, size = 503, normalized size = 1.43 \[ \frac {{\left (\frac {256 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \cos \left (\frac {4 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right )}{d} - \frac {1024 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} \cos \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right )}{d} - 96 \, \sqrt {d x + c} b^{2} \sin \left (\frac {4 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) + 768 \, \sqrt {d x + c} b^{2} \sin \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right ) - {\left (\left (48 i + 48\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - \left (48 i - 48\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {2 i \, b}{d}}\right ) - {\left (-\left (6 i + 6\right ) \, \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) + \left (6 i - 6\right ) \, \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (2 \, \sqrt {d x + c} \sqrt {\frac {i \, b}{d}}\right ) - {\left (\left (6 i - 6\right ) \, \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) - \left (6 i + 6\right ) \, \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (2 \, \sqrt {d x + c} \sqrt {-\frac {i \, b}{d}}\right ) - {\left (-\left (48 i - 48\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + \left (48 i + 48\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b d \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {2 i \, b}{d}}\right )\right )} d}{8192 \, b^{4}} \]
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 \[ \int \cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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