Optimal. Leaf size=299 \[ -\frac {\sqrt {\frac {\pi }{3}} \sqrt {d} \cos \left (6 a-\frac {6 b c}{d}\right ) C\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{384 b^{3/2}}+\frac {3 \sqrt {\pi } \sqrt {d} \cos \left (2 a-\frac {2 b c}{d}\right ) C\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{128 b^{3/2}}+\frac {\sqrt {\frac {\pi }{3}} \sqrt {d} \sin \left (6 a-\frac {6 b c}{d}\right ) S\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{384 b^{3/2}}-\frac {3 \sqrt {\pi } \sqrt {d} \sin \left (2 a-\frac {2 b c}{d}\right ) S\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{128 b^{3/2}}-\frac {3 \sqrt {c+d x} \cos (2 a+2 b x)}{64 b}+\frac {\sqrt {c+d x} \cos (6 a+6 b x)}{192 b} \]
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Rubi [A] time = 0.46, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {4406, 3296, 3306, 3305, 3351, 3304, 3352} \[ -\frac {\sqrt {\frac {\pi }{3}} \sqrt {d} \cos \left (6 a-\frac {6 b c}{d}\right ) \text {FresnelC}\left (\frac {2 \sqrt {\frac {3}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{384 b^{3/2}}+\frac {3 \sqrt {\pi } \sqrt {d} \cos \left (2 a-\frac {2 b c}{d}\right ) \text {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {\pi } \sqrt {d}}\right )}{128 b^{3/2}}+\frac {\sqrt {\frac {\pi }{3}} \sqrt {d} \sin \left (6 a-\frac {6 b c}{d}\right ) S\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{384 b^{3/2}}-\frac {3 \sqrt {\pi } \sqrt {d} \sin \left (2 a-\frac {2 b c}{d}\right ) S\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{128 b^{3/2}}-\frac {3 \sqrt {c+d x} \cos (2 a+2 b x)}{64 b}+\frac {\sqrt {c+d x} \cos (6 a+6 b x)}{192 b} \]
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 \sqrt {c+d x} \cos ^3(a+b x) \sin ^3(a+b x) \, dx &=\int \left (\frac {3}{32} \sqrt {c+d x} \sin (2 a+2 b x)-\frac {1}{32} \sqrt {c+d x} \sin (6 a+6 b x)\right ) \, dx\\ &=-\left (\frac {1}{32} \int \sqrt {c+d x} \sin (6 a+6 b x) \, dx\right )+\frac {3}{32} \int \sqrt {c+d x} \sin (2 a+2 b x) \, dx\\ &=-\frac {3 \sqrt {c+d x} \cos (2 a+2 b x)}{64 b}+\frac {\sqrt {c+d x} \cos (6 a+6 b x)}{192 b}-\frac {d \int \frac {\cos (6 a+6 b x)}{\sqrt {c+d x}} \, dx}{384 b}+\frac {(3 d) \int \frac {\cos (2 a+2 b x)}{\sqrt {c+d x}} \, dx}{128 b}\\ &=-\frac {3 \sqrt {c+d x} \cos (2 a+2 b x)}{64 b}+\frac {\sqrt {c+d x} \cos (6 a+6 b x)}{192 b}-\frac {\left (d \cos \left (6 a-\frac {6 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {6 b c}{d}+6 b x\right )}{\sqrt {c+d x}} \, dx}{384 b}+\frac {\left (3 d \cos \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}{128 b}+\frac {\left (d \sin \left (6 a-\frac {6 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {6 b c}{d}+6 b x\right )}{\sqrt {c+d x}} \, dx}{384 b}-\frac {\left (3 d \sin \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}{128 b}\\ &=-\frac {3 \sqrt {c+d x} \cos (2 a+2 b x)}{64 b}+\frac {\sqrt {c+d x} \cos (6 a+6 b x)}{192 b}-\frac {\cos \left (6 a-\frac {6 b c}{d}\right ) \operatorname {Subst}\left (\int \cos \left (\frac {6 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{192 b}+\frac {\left (3 \cos \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 )}{64 b}+\frac {\sin \left (6 a-\frac {6 b c}{d}\right ) \operatorname {Subst}\left (\int \sin \left (\frac {6 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{192 b}-\frac {\left (3 \sin \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 )}{64 b}\\ &=-\frac {3 \sqrt {c+d x} \cos (2 a+2 b x)}{64 b}+\frac {\sqrt {c+d x} \cos (6 a+6 b x)}{192 b}-\frac {\sqrt {d} \sqrt {\frac {\pi }{3}} \cos \left (6 a-\frac {6 b c}{d}\right ) C\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{384 b^{3/2}}+\frac {3 \sqrt {d} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) C\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{128 b^{3/2}}+\frac {\sqrt {d} \sqrt {\frac {\pi }{3}} S\left (\frac {2 \sqrt {b} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (6 a-\frac {6 b c}{d}\right )}{384 b^{3/2}}-\frac {3 \sqrt {d} \sqrt {\pi } S\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{128 b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 1.25, size = 264, normalized size = 0.88 \[ \frac {-\sqrt {3 \pi } \cos \left (6 a-\frac {6 b c}{d}\right ) C\left (2 \sqrt {\frac {b}{d}} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}\right )+27 \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) C\left (\frac {2 \sqrt {\frac {b}{d}} \sqrt {c+d x}}{\sqrt {\pi }}\right )+\sqrt {3 \pi } \sin \left (6 a-\frac {6 b c}{d}\right ) S\left (2 \sqrt {\frac {b}{d}} \sqrt {\frac {3}{\pi }} \sqrt {c+d x}\right )-27 \sqrt {\pi } \sin \left (2 a-\frac {2 b c}{d}\right ) S\left (\frac {2 \sqrt {\frac {b}{d}} \sqrt {c+d x}}{\sqrt {\pi }}\right )-54 \sqrt {\frac {b}{d}} \sqrt {c+d x} \cos (2 (a+b x))+6 \sqrt {\frac {b}{d}} \sqrt {c+d x} \cos (6 (a+b x))}{1152 b \sqrt {\frac {b}{d}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 242, normalized size = 0.81 \[ -\frac {\sqrt {3} \pi d \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (2 \, \sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - \sqrt {3} \pi d \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (2 \, \sqrt {3} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) - 27 \, \pi d \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + 27 \, \pi d \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - 48 \, {\left (4 \, b \cos \left (b x + a\right )^{6} - 6 \, b \cos \left (b x + a\right )^{4} + b\right )} \sqrt {d x + c}}{1152 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 7.83, size = 818, normalized size = 2.74 \[ -\frac {\frac {i \, \sqrt {3} \sqrt {\pi } {\left (12 \, b c + i \, d\right )} d \operatorname {erf}\left (-\frac {\sqrt {3} \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (\frac {6 i \, b c - 6 i \, a d}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} - \frac {i \, \sqrt {3} \sqrt {\pi } {\left (12 \, b c - i \, d\right )} d \operatorname {erf}\left (-\frac {\sqrt {3} \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (\frac {-6 i \, b c + 6 i \, a d}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} + 12 \, {\left (-\frac {i \, \sqrt {3} \sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {3} \sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (\frac {6 i \, b c - 6 i \, a d}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}} + \frac {i \, \sqrt {3} \sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {3} \sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (\frac {-6 i \, b c + 6 i \, a d}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}} + \frac {9 i \, \sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (\frac {2 i \, b c - 2 i \, a d}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}} - \frac {9 i \, \sqrt {\pi } d \operatorname {erf}\left (-\frac {\sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (\frac {-2 i \, b c + 2 i \, a d}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}\right )} c - \frac {9 i \, \sqrt {\pi } {\left (12 \, b c + 3 i \, d\right )} d \operatorname {erf}\left (-\frac {\sqrt {b d} \sqrt {d x + c} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (\frac {2 i \, b c - 2 i \, a d}{d}\right )}}{\sqrt {b d} {\left (\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} + \frac {9 i \, \sqrt {\pi } {\left (12 \, b c - 3 i \, d\right )} d \operatorname {erf}\left (-\frac {\sqrt {b d} \sqrt {d x + c} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (\frac {-2 i \, b c + 2 i \, a d}{d}\right )}}{\sqrt {b d} {\left (-\frac {i \, b d}{\sqrt {b^{2} d^{2}}} + 1\right )} b} - \frac {6 \, \sqrt {d x + c} d e^{\left (\frac {6 i \, {\left (d x + c\right )} b - 6 i \, b c + 6 i \, a d}{d}\right )}}{b} + \frac {54 \, \sqrt {d x + c} d e^{\left (\frac {2 i \, {\left (d x + c\right )} b - 2 i \, b c + 2 i \, a d}{d}\right )}}{b} + \frac {54 \, \sqrt {d x + c} d e^{\left (\frac {-2 i \, {\left (d x + c\right )} b + 2 i \, b c - 2 i \, a d}{d}\right )}}{b} - \frac {6 \, \sqrt {d x + c} d e^{\left (\frac {-6 i \, {\left (d x + c\right )} b + 6 i \, b c - 6 i \, a d}{d}\right )}}{b}}{2304 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 293, normalized size = 0.98 \[ \frac {-\frac {3 d \sqrt {d x +c}\, \cos \left (\frac {2 \left (d x +c \right ) b}{d}+\frac {2 d a -2 c b}{d}\right )}{64 b}+\frac {3 d \sqrt {\pi }\, \left (\cos \left (\frac {2 d a -2 c b}{d}\right ) \FresnelC \left (\frac {2 \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {2 d a -2 c b}{d}\right ) \mathrm {S}\left (\frac {2 \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{128 b \sqrt {\frac {b}{d}}}+\frac {d \sqrt {d x +c}\, \cos \left (\frac {6 \left (d x +c \right ) b}{d}+\frac {6 d a -6 c b}{d}\right )}{192 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {6}\, \left (\cos \left (\frac {6 d a -6 c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {6}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {6 d a -6 c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {6}\, \sqrt {d x +c}\, b}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{2304 b \sqrt {\frac {b}{d}}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.52, size = 435, normalized size = 1.45 \[ \frac {{\left (\frac {96 \, \sqrt {d x + c} b^{2} \cos \left (\frac {6 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right )}{d} - \frac {864 \, \sqrt {d x + c} b^{2} \cos \left (\frac {2 \, {\left ({\left (d x + c\right )} b - b c + a d\right )}}{d}\right )}{d} + {\left (\left (2 i - 2\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) + \left (2 i + 2\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {\frac {6 i \, b}{d}}\right ) + {\left (-\left (54 i - 54\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - \left (54 i + 54\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \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 (54 i + 54\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + \left (54 i - 54\right ) \cdot 4^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \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 (2 i + 2\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) - \left (2 i - 2\right ) \cdot 36^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } b \left (\frac {b^{2}}{d^{2}}\right )^{\frac {1}{4}} \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right )\right )} \operatorname {erf}\left (\sqrt {d x + c} \sqrt {-\frac {6 i \, b}{d}}\right )\right )} d}{18432 \, b^{3}} \]
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 \[ \int {\cos \left (a+b\,x\right )}^3\,{\sin \left (a+b\,x\right )}^3\,\sqrt {c+d\,x} \,d x \]
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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